Masaryk University
Faculty of Arts
Subatomic Quantification
Doctoral Thesis
Marcin Wągiel
Brno, Spring 2018
Masaryk University
Faculty of Arts
Department of Linguistics and Baltic Languages
Field of study: General Linguistics
Subatomic Quantification
Doctoral Thesis
Marcin Wągiel
Brno, Spring 2018
Declaration
Hereby I declare that this paper is my original authorial work, which
I have worked out on my own. All sources, references, and literature
used or excerpted during elaboration of this work are properly cited
and listed in complete reference to the due source.
Marcin Wągiel
i
Acknowledgement
This thesis is a result of a collaborative effort and I would have never
finished it without the inspiration, support and advice of many people
I met in the last several years. First of all, I would like to sincerely
thank Mojmír Dočekal for introducing me to how fascinating linguistics
can be and raising my initial interest in the subject of pluralities
and quantification. I would have never become a semanticist if I have
not met him. Second, I am deeply grateful to Viola Schmitt for so many
things that it is hard to list them. She convinced me that my idea of
accounting for subatomic quantification is actually interesting and
constantly kept me motivated during the whole process of writing.
Her constructive criticism and inspiring comments were of incalculable
value and this thesis would have never come into existence if it
weren’t for her support and advice. Third, I am very thankful to Nina
Haslinger who proofread long parts of this thesis and with whom I
had many thought-provoking discussions concerning the data, theoretical
implications, and formalism. Without those conversations this
thesis most definitely would not be what it is. Also, I would like to
thank Scott Grimm who inspired me to radically change my way of
thinking about nominal semantics.
The process of clarifying what this thesis should be about and
how to achieve its aim was long. During that time I stayed at several
academic locations each of which had a considerable impact on my
linguistic development. Therefore, I would like to thank the following
people to whom I owe for what I know now (always in the alphabetical
order). Many thanks to my friends and colleagues at the Masaryk
University in Brno, especially Pavel Caha, Hana Strachoňová, and
Markéta Ziková for their support and making Brno a stimulating place
to do linguistics. I am also grateful to Ondřej Šefčík for not making the
administrative part of the whole enterprise to administrative. Next,
I am very grateful to the people with whom I interacted during my
stay at the Humboldt University in Berlin, namely Manfred Bierwisch,
Aleksandra Gogłoza, Robert Hammel, Denisa Lenertová, Manfred
Krifka, Roland Meyer, Uli Sauerland, Stephanie Solt, Luka Szucsich,
Kazuko Yatsushiro, Karolina Zuchewicz, and especially Radek Šimík.
iii
Due to those discussions I have clarified many vague ideas and learned
a great deal of semantics. During two stays at the University of Vienna
I met a number of people with whom discussing linguistics was so
much more than I have ever experienced. Specifically I would like to
thank Muriel Assmann, Daniel Büring, Enrico Flor, Izabela Jordanoska,
Chiara Masnovo, Martin Prinzhorn, Maximilian Prüller, Magdalena
Roszkowski, Jakob Steixner, and Ede Zimmermann for helping me
discover so many new things and realize what I want to say and how.
Finally, for a long time parallel to my Ph.D. program I was working at
the Palacký University in Olomouc. During that period, I have learned
much due to interactions with Jakub Bortlík, Petra Charvátová, Joseph
Emonds, Anna Kozánková, Jeffrey Parrott, and Lída Veselovská. I am
also grateful for support to Ivana Dobrotová, Michał Hanczakowski,
and Jan Jeništa with whom I worked at that time. I would also like to
sincerely thank Jakub Dotlačil, Berit Gehrke, and Yasu Sudo whom I
met several times at different occasions during my PhD program. Their
comments and inspiring insights as well as interesting discussions
were always of great help.
At different conferences, workshops, and other events I met people
whose questions and comments were of extreme importance for
developing my understanding of the phenomena described in this
dissertation as well as realizing problems I have not even noticed
before. Sometimes it was just a question or a remark, sometimes
it was a longer discussion or even several conversations. In each
case, however, it stimulated me to clarify what I actually want to say.
Other people helped me a lot as informants providing examples from
their native languages. Therefore, I would like to thank David Adger,
Curt Anderson, Boban Arsenijević, Petr Biskup, Joanna Błaszczak,
Jonathan Bobaljik, Natalie Boll-Avetisyan, Olga Borik, Ana Bosnić,
Roslyn Burns, Lisa Bylinina, Bożena Cetnarowska, Lucas Champollion,
Guglielmo Cinque, Miloje Despić, Dominika Dziubała-Szrejbrowska,
Regine Eckardt, Kurt Erbach, Katalin É. Kiss, Hana Filip, Salvatore Florio,
Katherine Fraser, Ljudmila Geist, Dariusz Głuch, Julie Goncharov,
Justyna Grudzińska, Veronika Hegedűs, Gillen Martinez de la Hidalga,
Daniel Hole, Krzysztof Hwaszcz, Tetiana Kamyshanova, Jiří Kašpar,
Hanna Kędzierska, Dorota Klimek-Jankowska, Predrag Kovačević,
Jakub Kozakoszczak, Ivona Kučerová, Fred Landman, Daniel Lassiter,
Tomas Lentz, Chang Liu, Aitor Lizardi Ituarte, Sebastian Löbner,
iv
Marijana Marelj, Radvan Markus, Lanko Marušič, Ora Matushansky,
Louise McNally, Erlinde Meertens, Katarzyna Miechowicz-Mathiasen,
Petra Mišmaš, Maša Močnik, Marcin Morzycki, Olav Mueller-Reichau,
Andrew Nevins, David Nicolas, Rick Nouwen, Itziar Orbegozo Arrizabalaga,
Agnieszka Patejuk, Roumyana Pancheva, Asya Pereltsvaig,
Lilla Pintér, Hagen Pitsch, Adam Przepiórkowski, Wiktor Pskit, Zorica
Puškar, Agata Renans, Susan Rothstein, Marta Ruda, Paweł Rutkowski,
Piotr Stalmaszczyk, Balázs Surányi, Branimir Stanković, Peter Sutton,
Iveta Šafratová, Barbara Tomaszewicz, Jan Winkowski, Jacek Witkoś,
Jan Wiślicki, Kata Wohlmuth, Henk Zeevat, Sarah Zobel, Yulia Zinova,
and Eytan Zweig. All errors are of course my own. Furthermore, I am
grateful to my informants without whom.
I gratefully acknowledge that the research was supported by a
Czech Science Foundation (GAČR) grant to the Department of Linguistics
and Baltic Languages at the Masaryk University in Brno
(GA17-16111S) as well as an Aktion Österreich-Tschechien scholarship
awarded to Marcin Wągiel and financed by the Austrian Federal
Ministry of Science, Research and Economy (ICM-2016-05748).
Finally, many thanks for my family and friends for their support
and not asking too often whether I’m already done. In particular,
I am grateful to my mum and dad for teaching me how to think
independently, to Piotr and Zosia, Herok, Bacha, Byrtek, and especially
the Gwizdońs. Last, but not least, I am deeply thankful to Izabela for
her love and unconditional support as well as tons of delicious plum
cake that kept me going in the last weeks. In the last several years, we
have achieved together so much more than this thesis.
v
Contents
1 Introduction 1
1.1 What’s this all about . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Intuitive notions of a whole and part . . . . . . . 3
1.1.2 The claims in a nutshell . . . . . . . . . . . . . . 7
1.2 General assumptions . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Cognitive view of meaning . . . . . . . . . . . . 10
1.2.2 Ontology . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Notation . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 The things to come . . . . . . . . . . . . . . . . . . . . . . 14
2 Partitives and part-whole structures 17
2.1 Partitives . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The analogy . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Entity and set partitives across languages . . . . 25
2.2.2 Partitivity and countability . . . . . . . . . . . . 35
2.3 Italian irregular plurals . . . . . . . . . . . . . . . . . . . . 46
2.3.1 Aggregate meaning . . . . . . . . . . . . . . . . . 51
2.3.2 Interaction with cardinals in partitives . . . . . . 54
2.4 Parts vs. subsets . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Ambiguity or indeterminacy . . . . . . . . . . . . . . . . . 70
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 Exploring topological sensitivity 77
3.1 Continuous and discontinuous parts . . . . . . . . . . . . . 77
3.1.1 Polish half -words . . . . . . . . . . . . . . . . . . 79
3.1.2 Inherent vagueness . . . . . . . . . . . . . . . . . 88
3.2 More topology-sensitive partitive words . . . . . . . . . . . 89
3.2.1 Quarter-words . . . . . . . . . . . . . . . . . . . . 90
3.2.2 Piece-words . . . . . . . . . . . . . . . . . . . . . 94
3.3 Mass, parts, quantities, and pieces . . . . . . . . . . . . . . 99
3.4 Cross-linguistic parallels . . . . . . . . . . . . . . . . . . . 105
3.5 Whole-adjectives . . . . . . . . . . . . . . . . . . . . . . . 115
3.5.1 German ganz and plurals . . . . . . . . . . . . . 119
vii
3.5.2 Integrity vs. maximality . . . . . . . . . . . . . . 121
3.5.3 Universal quantification over parts and wholes . 124
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4 Multipliers 131
4.1 Significance of multipliers . . . . . . . . . . . . . . . . . . 132
4.1.1 Morphological complexity of Slavic multipliers 133
4.1.2 Subatomic quantifiers . . . . . . . . . . . . . . . 136
4.1.3 Cross-linguistic similarities . . . . . . . . . . . . 141
4.1.4 Multipliers as degree modifiers . . . . . . . . . . 144
4.2 Less obvious cases . . . . . . . . . . . . . . . . . . . . . . . 145
4.2.1 Essential parts . . . . . . . . . . . . . . . . . . . . 146
4.2.2 Portions . . . . . . . . . . . . . . . . . . . . . . . 149
4.2.3 Events . . . . . . . . . . . . . . . . . . . . . . . . 152
4.2.4 Social roles . . . . . . . . . . . . . . . . . . . . . . 155
4.3 Multipliers vs. cardinals . . . . . . . . . . . . . . . . . . . 156
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5 Conceptual background 163
5.1 Psychological evidence . . . . . . . . . . . . . . . . . . . . 164
5.1.1 Object/substance distinction . . . . . . . . . . . 164
5.1.2 Whole object assumption . . . . . . . . . . . . . 168
5.1.3 Part-whole perception . . . . . . . . . . . . . . . 169
5.1.4 Number sense . . . . . . . . . . . . . . . . . . . . 171
5.2 The three claims . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.1 Relevance of topological notions with respect to
part-whole structures . . . . . . . . . . . . . . . 176
5.2.2 General counting principles . . . . . . . . . . . . 179
5.2.3 Subatomic quantification . . . . . . . . . . . . . 188
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6 Theory of parts and wholes 195
6.1 Mereology . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.1.1 Mereology vs. set theory . . . . . . . . . . . . . . 201
6.1.2 Limits of mereology . . . . . . . . . . . . . . . . 203
6.2 Mereotopology . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.2.1 Integrated wholes . . . . . . . . . . . . . . . . . . 209
6.2.2 Other types of connection . . . . . . . . . . . . . 214
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6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7 Mereotopological account for subatomic quantification 219
7.1 Doing without atoms . . . . . . . . . . . . . . . . . . . . . 219
7.2 Common nouns . . . . . . . . . . . . . . . . . . . . . . . . 225
7.2.1 Count nouns . . . . . . . . . . . . . . . . . . . . 225
7.2.2 Pluralization . . . . . . . . . . . . . . . . . . . . . 228
7.2.3 Consequences . . . . . . . . . . . . . . . . . . . . 231
7.3 Measure functions . . . . . . . . . . . . . . . . . . . . . . 234
7.3.1 Counting operation . . . . . . . . . . . . . . . . . 235
7.3.2 Contextual conditioning . . . . . . . . . . . . . . 237
7.3.3 Counting essential parts . . . . . . . . . . . . . . 239
7.4 Numerical expressions . . . . . . . . . . . . . . . . . . . . 241
7.4.1 Numeral roots . . . . . . . . . . . . . . . . . . . . 242
7.4.2 Cardinals . . . . . . . . . . . . . . . . . . . . . . 243
7.4.3 Multipliers . . . . . . . . . . . . . . . . . . . . . . 245
7.4.4 Cross-linguistic support . . . . . . . . . . . . . . 247
7.5 Partitives . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
7.5.1 Partitivity . . . . . . . . . . . . . . . . . . . . . . 250
7.5.2 Maximization . . . . . . . . . . . . . . . . . . . . 252
7.5.3 Choice functions . . . . . . . . . . . . . . . . . . 254
7.5.4 Set partitive constraint . . . . . . . . . . . . . . . 255
7.5.5 Partitions . . . . . . . . . . . . . . . . . . . . . . . 257
7.5.6 Individuation . . . . . . . . . . . . . . . . . . . . 259
7.5.7 Partitive words . . . . . . . . . . . . . . . . . . . 261
7.6 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . 264
7.6.1 Explicit entity partitives . . . . . . . . . . . . . . 265
7.6.2 Explicit set partitives . . . . . . . . . . . . . . . . 266
7.6.3 Count explicit partitives . . . . . . . . . . . . . . 268
7.6.4 Topology-neutral proportional partitives . . . . 271
7.6.5 Topology-sensitive proportional partitives . . . 274
7.6.6 Multiplier phrases . . . . . . . . . . . . . . . . . 277
7.7 Possible extensions . . . . . . . . . . . . . . . . . . . . . . 280
7.7.1 Italian partitives with irregular plurals . . . . . 281
7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
8 Conclusion 285
8.1 Overall summary . . . . . . . . . . . . . . . . . . . . . . . 286
ix
8.2 Open questions and future research . . . . . . . . . . . . . . 293
x
List of Tables
2.1 Properties of partitive words 42
2.2 Italian nouns with irregular plurals exclusively (based
on Acquaviva 2008, pp. 126–127) 48
2.3 Italian nouns with both regular and irregular plural
counterparts 48
2.4 Semantic interpretation of Italian plurals 49
2.5 Properties of Italian parte ‘part’ 57
3.1 Distribution of Polish half -words 83
3.2 Denotations of Polish half -words 87
3.3 Distribution of Polish quarter-words 92
3.4 Denotations of Polish quarter-words 94
3.5 Distribution of Polish piece-words 98
3.6 Denotations of Polish piece-words 98
3.7 Distribution of Polish whole adjectives 118
3.8 Interpretations of Polish whole-adjectives 124
3.9 Quantificational properties of Polish cały ‘whole’ 128
4.1 Examples of structured parthood (Simons 1987, p.
10) 147
4.2 Properties of cardinals and multipliers 159
6.1 Correspondences between mereology and set theory
(Champollion and Krifka 2016; adapted) 202
7.1 Attested relative orderings of numerals and classifiers
(Greenberg 1972) 248
7.2 Attributive and counting numerals across languages
(Hurford 2001) 249
7.3 Multipliers across languages 249
8.1 Properties of partitive words 287
8.2 Properties of Italian parte ‘part’ 287
8.3 Denotations of Polish half -words 289
8.4 Interpretations of Polish whole-adjectives 289
8.5 Properties of cardinals and multipliers 291
xi
List of Figures
2.1 Sections of city walls 56
3.1 Continuous half 83
3.2 Discontinuous half 83
3.3 Flag AB 106
3.4 Flag ABA 106
4.1 Pschent 139
4.2 Deshret 139
4.3 Hedjet 139
4.4 Parts of a double murder event in a knife-stabbing
scenario 154
4.5 Parts of a double murder event in a bomb scenario 154
5.1 Part-whole perception (Elkind et al. 1964) 170
5.2 Relevance of integrity in counting (Dehaene 1997, p. 60;
adapted from Shipley and Shepperson 1990) 173
5.3 Counting 183
5.4 Illegal counting 183
5.5 Measuring and integrity 185
5.6 Counting and integrity 185
5.7 Measuring in units 186
5.8 Counting objects 186
5.9 Overlapping parts 189
5.10 Counting of parts 191
5.11 Illegal counting of parts 192
6.1 Object with a solitary proper part 198
6.2 Objects sharing all parts 198
6.3 Semi-lattice 201
6.4 Connectedness and transitivity 206
6.5 Internal part 207
6.6 Internal overlap 207
6.7 Tangential overlap 207
6.8 Interior 209
6.9 Exterior 209
6.10 Closure 209
6.11 Wholes vs. sums 210
xiii
6.12 External connection 210
6.13 Strong connection 211
6.14 Strongly self-connected part 212
6.15 Parthood and connectedness (based on Grimm 2012b, p.
136) 214
6.16 By-connected and mediately connected 216
xiv
1 Introduction
Since the early days of formal semantics (starting with Montague
1973) issues concerning quantification, pluralities, and countability
have been of central importance for the study of meaning in natural
language. A substantial body of research has been dedicated to these
topics which led to a number of influential theories (e.g., Barwise
and Cooper 1981, Scha 1981, Link 1983, Hoeksema 1983, Gillon 1987,
Krifka 1989, Schwarzschild 1996, Landman 2000, Winter 2001). Within
this tradition since at least Scha (1981) until, e.g., Champollion (2017),
the prevailing view is that denotations of count nouns are atomic or,
in other words, involve atoms, i.e., entities that have no proper parts
in a mereological sense. Though at first blush this seems unintuitive
since we know very well from our common experience that often
things do have parts, there seemed to be good reasons to assume that
compositional semantics ignores this fact and treats referents of count
noun as indivisible units. The achievements of that strand of research
are indisputable. At the same time, I believe that the attachment to the
notion of atomicity originates from a limited scope of investigation
and in the face of novel linguistic evidence to be presented needs to
revised. The motivation behind this thesis is to explore what could be
gained for semantic theory if we adopted a different perspective.
1.1 What’s this all about
This study is about parts of singular concrete things such as apples,
walls, and crowns and how we quantify over them. It is also about
parts of pluralities of things such as collections of apples, walls, and
crowns and why we do not quantify over them. It is about various
types of partitives, e.g., structures such as half of the apple, multiplier
phrases, e.g., expressions like double crown, and other natural language
expressions, e.g., adjective phrases such as whole apple. What all those
constructions have in common is that they involve what I will call
throughout this study subatomic quantification. Subatomic quantification
is quantification over parts of things that constitute building
blocks of denotations of nominal expressions such as apple, wall, and
crown. Though the use of the word ‘subatomic’ in the title of this thesis
1
1. Introduction
suggests that I will also adopt the concept of an atom, in fact I will
argue against approaches to nominal semantics based on the notion
of atomicity. I decided for the use of that term since I believe that
it intuitively evokes what I intend to focus on. Ironically, however, I
believe the evidence concerning subatomic quantification forces us to
reconsider what building blocks of denotations of count nouns actually
are. The great source of inspiration for this study was the work of
Grimm (2012a,b) who proposes a novel view on how to capture the
semantic relevance of the mass/count distinction without reference
to atoms.
The empirical aim of this thesis is to provide novel linguistic evidence
showing that natural language semantics is sensitive to subatomic
part-whole structures and to topological relations holding
between elements within such structures. Furthermore, I intend to
demonstrate the linguistic relevance of subatomic quantification as
well as the fact that quantification over parts of a whole is subject to
identical restrictions as quantification over wholes. The evidence to be
examined indicates that some quantificational operations including
counting presuppose particular topological relations.
From a theoretical point of view, this thesis is intended to provide
a novel argument for adopting a theory of wholes called mereotopology,
i.e., a system in which standard mereology based on parthood
is extended with the primitive relation of connectedness as well as
nuanced derived topological notions (Casati and Varzi 1999).
Some of the key questions concerning natural language semantics
I want to address in this study are the following.
1. Are there different types of parts of entities similar to different
types of wholes?
2. How do we count parts of an object?
3. Why cannot we count parts of a plurality?
At first blush, the answers at which I arrive might seem somewhat
surprising.
1. Yes, there are continuous and discontinuous parts. The first
resemble referents of singular count nouns, whereas the latter
are more like pluralities.
2
1. Introduction
2. We count parts of an object in the very same way how we
count whole objects. The mechanism is unified, thus identical
restrictions apply.
3. For the same reason we cannot count pluralities. Only objects
with certain topological characteristics can be assigned a number
in numerical quantification.
The most important result of this study is a proposal of an account
for subatomic quantification. I will argue not only that it correctly
predicts why only some parts are countable, but also that extending it
to the level of wholes will enable us to seek for more advantageous
explanations for known problems. For instance, it could explain why
a natural language expression such as two apples refers to a plurality of
two apples rather than to two pluralities of apples. Or why object mass
nouns such as furniture are not countable despite involving reference
to discrete entities. There are also a number of other questions I will
attempt to tackle, many of which will remain without a definite answer.
However, I believe that both the evidence presented here and the
proposed account provide a novel exciting perspective to think about
partitivity and countability.
Note that though I will sometimes mention them in passing, this
study is not about abstract terms (see, e.g., Asher 1993, Tovena 2001,
Nicolas 2002, Moltmann 2013), event nominals (see, e.g., Grimshaw
1990, 2011, Bierwisch 1990, Borer 2005b), and collective nouns (see,
e.g., Landman 1989a,b, Barker 1992, Schwarzschild 1996, Pearson 2011,
de Vries 2015), nor about parts of their referents. Despite the fact that in
principle I believe that the approach developed here can be extended
to these expressions, for the most part I will not consider part-whole
structures of abstract entities. Before I introduce the main claims of this
thesis in a bit more detail and then delve into intricacies of linguistic
evidence, let us first consider my view on what it intuitively means to
be a whole and part.
1.1.1 Intuitive notions of a whole and part
The starting point of this journey concerns an ontological intuition
dating back at least to Pre-Socratics that entities are often made up of
smaller entities which are related to each other in a particular manner
3
1. Introduction
(see Varzi 2016 for a historical overview). In other words, when thinking
about what is, humans often assume that objects are generally
configurations of parts. This ontological intuition seems to stem from a
cognitive fact that human beings often conceive entities as being made
up of smaller entities related to each other in certain ways. In fact, psychological
evidence demonstrates that at least from early childhood
humans are able to perceive entities simultaneously in a twofold way
(e.g., Elkind et al. 1964, Kimchi 1993, Boisvert et al. 1999). On the one
hand, we can discriminate parts from a whole making them more
salient than the entire object, whereas on the other hand we have an
ability to integrate the parts in such a way that the perception of a
complete whole emerges in our cognition. In other words, humans
are able to see entities as collections of parts and as integrated wholes
simultaneously.1
At the same time, from the very early considerations on the partwhole
relation in Plato’s Parmenides and Theaetetus much concern regards
unity. What interests Plato is how to differentiate between a true
unity and an arbitrary sum of parts. The crucial property of the first is
that it is provided by structure which distinguishes it from the latter.
Without going into details of Plato’s ontological views (but see, e.g.,
Priest 2014), it seems that the key intuition behind his deliberations is
that the crucial component of what it means to be a whole is given by
the manner of arrangement of parts. In other words, a whole is not
simply reducible to the sum of its parts (see also Casati and Varzi 1999),
e.g., a glass is not an arbitrary sum of shards, but rather a particular
configuration of shards. It happens that I also share this intuition.
When considering how objects are conceptualized, I believe it is
instructive to make use of categories such as unity, boundedness, and
integrity. The first is about capturing an individual as a complete
whole in itself, i.e., distinguishing between discrete objects as opposed
to fragmentary portions of a continuum of a larger entity. The second
category concerns boundaries. Objects are well-defined in space in
contrast to scattered fragments of entities or a diffused continuum of
matter. Finally, integrity characterizes a mutual bond between entities
perceived as being parts of a certain whole. Though this bond can
be conceptualized in many different ways, one significant mode of
1. I will return to the discussion of findings of cognitive psychology in Section 5.1.
4
1. Introduction
integrity is viewed in terms of topology. Specifically, objects involve
elements that are connected to each other, i.e., stick together, and thus
move along the same trajectories as one unit. The vital question that
inspired this study is to what extent these categories are relevant for
the architecture of natural language semantics? In the following chapters,
I will argue that there are good reasons to believe that it is indeed
relevant. Specifically, I will provide linguistic evidence not only that
the meaning of nominal expressions involves information concerning
integrity but also that grammar is sensitive to such a notion.
Crucially, I argue that the same way of thinking should apply when
talking about parts of wholes. When describing parthood with respect
to entities extended in space or time, employing the topological notion
of contiguity might allow us to capture some non-trivial facts about
part-whole structures. In particular, some parts constitute individuated
continuous strings of matter within a whole, whereas other parts
are just arbitrary sums of portions of substances. To demonstrate the
difference let us discuss the following example (cf. Acquaviva 2008,
pp. 90–93). Consider two parts of a table, its leg and a splinter. Under
ordinary circumstances, we would definitely agree that both these
entities are parts of the table; however, each of them in a somewhat different
sense. A splinter is just some arbitrary portion of matter making
up the table, an element among numerous similar fragments of the
whole. Before mentioning it, we probably had no or at least very little
expectations with respect to its appearance. We might even question
whether it exists as a splinter before being detached from the whole
since normally tables are not perceived as collections of splinters. On
the other hand, a leg is easily identifiable among a definite number of
similar elements. We perceive it as having a clearly specified function
and expect certain traits of its appearance. Furthermore, there is also a
sense in which two separate disconnected splinters or two disjoint legs
are part of the table since they constitute some fragment of its material
make-up. Importantly, however, this type of parthood resembles the
relationship between a splinter and the whole rather than the case of
a leg.
In the philosophical literature, this distinction has been attributed
to different modes of partitioning an object. For instance, Krecz (1986)
proposes different terms for specific subdivisions of a whole, i.e., parts,
as opposed to arbitrary subdivision, i.e., pieces. The first would corre-
5
1. Introduction
spond, e.g., to a leg of a table, whereas the latter would be used with
respect, e.g., a splinter. Similar, Markosian (1998) draws an analogous
distinction between what are called metaphysical parts, i.e., parts that
can be considered as objects in their own right, and conceptual parts,
i.e., equivalents of arbitrary parts. Finally, Jennings (2010) proposes to
differentiate between parts viewed as arbitrary bits of an object one
could cut off and functional parts defined as parts that are considered
essential for an object since they fulfill a particular function. Crucially,
the distinction is conceptually valid but it is also reflected in grammar.
For instance, English distinguishes between the two different flavors
of parthood discussed above syntactically as can be witnessed in (1.1)
(see also Champollion and Krifka 2016). The use of the bare mass
noun part in the sentence in (1.1a) indicates arbitrary partitioning of
the table. On the other hand, the full DP a part in (1.1b) corresponds
to a specific division in parts. Notice also that only the latter use is
countable.
(1.1) English (Acquaviva 2008, p. 90; adapted)
a. A splinter is part of the table.
b. A leg is a part of the table.
It has been acknowledged that the distinction discussed above seems
to have some further linguistic implications (Champollion and Krifka
2016). For instance, consider the minimal pairs in (1.2) and (1.3). The
contrast suggests that transitivity does not hold for arbitrary parts.2
In other words, the fact that the thumb is a part of a hand does not
guarantee that it would be linguistically treated as a part of what that
hand is part of, i.e., an arm.
(1.2) English (Champollion and Krifka 2016)
a. the thumb of this hand
b. a hand without thumb
(1.3) English (Champollion and Krifka 2016)
a. #the thumb of this arm
b. #an arm without thumb
2. Champollion and Krifka (2016) use the terms structured and unstructured
parthood in order to refer arbitrary and specific subdivisions, respectively.
6
1. Introduction
Despite these facts the role of the distinction between arbitrary and
specific parts in natural language semantics is usually either totally
neglected or reduced to lexical relations such as meronymy, i.e., relations
between meronyms and holonyms, e.g. wing and bird, and
hyponymy, i.e., the relation between hyponyms and hyperonyms, e.g.,
swallow and bird. Although it has been recognized that meronymy and
hyponymy have some general properties which establish taxonomies
and structure large parts of the lexicon in natural language (Cruse
1986), compositional approaches to meaning did not assume that such
categories play any role in the proposed models. One of the aims of
this thesis is to provide linguistic evidence for the relevance of the two
aspects of what it means to be part of something and to propose how
first steps towards accommodating a less naive view on parthood into
semantic theory could look like.
1.1.2 The claims in a nutshell
In this thesis, I will argue that a proper treatment of countability in
natural language should take into account the interaction between
partitivity, topology, and quantification. In other words, I believe that
exploring subatomic quantification can reveal some non-trivial phenomena
that were otherwise obscure, and thus contribute to our understanding
of the interaction between nominal semantics and the
meaning of numerical expressions in general. The central claims of
this thesis are the following.
First, natural language is sensitive to the fact that entities denoted
by nominal expressions have parts as well as to topological relations
holding between them. The relevance of part-whole structures is typically
acknowledged with respect to pluralities. However, my claim
is stronger. I postulate that natural language is also sensitive to subatomic
part-whole structures. This is in discordance with a mainstream
view that utilizes the notion of atomicity in order to capture semantic
properties of count nouns. Such an approach postulates that atoms,
i.e., building blocks of denotations of expressions like apple and crown,
are entities that from a linguistic perspective are treated as having
no proper parts. Consequently, countability can be accounted for in
terms of quantification over atoms. Nevertheless, the linguistic evidence
I will present in the following chapters will demonstrate that
7
1. Introduction
in fact there are a number of natural language expressions involving
quantification over such subatomic parts. Crucially, based on the distribution
of partitives I will argue for a unified parthood relation for
both singularities and pluralities. Furthermore, I will postulate that
the difference between the two is a result of different topological relations
encoded in the corresponding part-whole structures. The claim
is that prototypical referents of count singulars are conceptualized as
integrated wholes, whereas regular plurals require entities in their denotations
to comprise such cohesive objects as their parts but impose
no topological constraints on a spatial configuration of those parts. As
we will see from the data concerning topologically sensitive partitive
expressions, the distinction between integrated wholes and scattered
entities also applies at the subatomic level. Specifically, the novel evidence
shows that the contrast between continuous and discontinuous
parts is relevant for interpretation of natural language expressions.
Hence, the notion of topological integrity should be accommodated
into semantic theory.
The second claim concerns counting. In particular, I will argue that
countability is not some syntactic feature nor is it a meaning postulate
on a certain lexical item, but rather it follows from what I refer to as the
general counting principles. In particular, the principle of non-overlap
ensures that entities one quantifies over are disjoint, i.e., they do not
share a part. In other words, things can be counted once and once
only. This rule excludes a possibility of counting entities involving
multiple overlapping parts as denoted by mass nouns and pluralia
tantum. Furthermore, the principle of maximality guarantees that
when counted entities are associated with numbers in their entirety,
i.e., no part is left out. This is especially relevant with respect to homogeneous
entities referred to by nouns such as twig and fence. Given a
particular counting situation, what counts as one always needs to be
the maximal entity no matter how its part-whole structure is defined
in that situation. Finally, the principle of integrity requires that what
can be counted needs to be conceptualized as an entity that comes
in one piece. This restriction rules out arbitrary sums of entities as
well as discontinuous portions of substances. Consequently, count-
8
1. Introduction
ing pluralities is not allowed.3 All things considered, only predicates
denoting entities that satisfy the general counting principles can be
modified by cardinals.
The third and final claim extends the general mechanism behind
counting to the subatomic level. More specifically, I will postulate that
the general counting principles constitute a universal set of constraints
regulating what is fit for being counted and what is not irrespective
whether it is a whole or part. In other words, I will argue that it follows
from the principles of non-overlap, maximality, and integrity that
certain parts are countable whereas other are not. The evidence from
partitive constructions involving cardinal numerals demonstrates that
it is not allowed to count discontinuous parts of entities. Moreover,
the claim is further corroborated by the fact that natural language
developed expressions dedicated to subatomic quantification.
Linguistic evidence for the three claims introduced above comes
from a number of natural language expressions. Specifically, I will
examine different types of partitives, whole-adjectives as well as multipliers
such as English double. Some parts of this thesis will also
investigate the data considering plurals and cardinal numerals. In
order to provide a more general picture, the evidence will be examined
from a cross-linguistic perspective. The languages which will
be discussed most include Polish, German, Italian, and English but
throughout the text I will also address examples from (in the alphabetical
order) Basque, Bosnian-Croatian-Serbian (BCS), Czech, Dutch,
Finnish, French, Hebrew, Hungarian, Irish, Japanese, Lithuanian, Maltese,
Mandarin, Mi’gmaq, Portuguese, Russian, and Yucuna.
1.2 General assumptions
There are several issues concerning conceptual grounds that require a
short commentary. In this section, I will present my view on natural
language semantics and discuss the ontology assumed for the purpose
of developing an account for subatomic quantification.
3. Notice that group nouns are more than pluralities. In fact, I assume that they
denote abstract singular entities with whom pluralities of members can be associated.
I will come back to this issue in Section 2.4.
9
1. Introduction
1.2.1 Cognitive view of meaning
The view on theory of meaning I adopt here is on a par with the position
expressed by Krifka (1998) and shared, e.g., by Partee 2018 and
Grimm (2012b). Though model-theoretic semantics in the tradition
of Montague has commonly been assumed to necessarily endorse
some form of semantic externalism (see, e.g., Putnam 1975, Davidson
1987), i.e., a view on which expressions of natural language designate
objects in the world, and as such is incompatible with approaches to
meaning that seek to develop cognitive models of the world (assuming
that meanings are in the head after all), such a sharp distinction does
not seem plausible. In fact, as pointed out by Krifka it is possible to
combine the use of the methods and techniques developed within
the model-theoretic tradition with an assumption that natural language
expressions are interpreted by conceptual structures that in
turn are associated with external entities by some pragmatic mechanism
responsible for how we use language. Therefore, I assume that
model-theoretic representations discussed in this study are compatible
with a cognitive view of meaning and can be seen as tools one can
use to try to grasp human mental reality.
In other words, in the system developed here I make no metaphysical
claims, nor do I embrace any form of semantic externalism. Instead,
I would like to see the postulated notions and structures as attempts
to capture the way we conceptualize certain aspects of external reality.
That is to say, the theory of subatomic quantification to follow is assumed
to characterize some properties of how human beings perceive
things in the world, not to describe them how they are. This means
that both primitive objects and more complex conceptual forms are
assumed to ‘be in the head’. Of course, it is not unlikely that properties
of such semantic objects may often correspond to the properties of
mind/language-external entities. Consequently, the first could be used
to talk about the latter assuming matching by some extra-linguistic
mechanism. In any case, I do not presume that meanings are out there
in the world. Rather, I will often emphasize the significance for natural
language semantics of how entities are conceptualized. In the next
section, I will describe the minimal assumptions concerning primitive
objects grouped together in the domains of the model.
10
1. Introduction
1.2.2 Ontology
In standard Montague semantics, the domain of discourse is simply a
collection of disjoint non-empty sets of entities from which denotations
of basic expressions as well as more complex constituents are built.
It is usually assumed that the domain of entities is supplemented
with the set of truth values as well as sets of events, possible worlds,
degrees and others. Since the focus of this study is very specific, I will
limit the number of distinct types of primitive objects almost to the
minimum. In doing so, I will try to be faithful to the position famously
formulated by Link (1983) that the guide in ontological considerations
should be natural language itself.
I will use the term entity to talk about anything that can be referred
to by proper names such as Noam Chomsky and Nim Chimpsky and
definite descriptions such as the author of “Syntactic Structures” and
that chimpanzee as well as denoted by common nouns such as apple and
juice. I assume that entities can be either well-defined discrete objects,
pluralities thereof, or shapeless amorphous substances. Though there
are significant differences between them, they all fall into one domain.
In general, I will refer to a thing in the denotation of a count noun as
individual or object. Sometimes, I will use the term scattered entity
to indicate referents of mass terms and arbitrary sum to talk about
denotations of plurals. The words ‘entity’ and ‘thing’ are assumed to
be general terms covering both individuals/objects, portions of matter
as well as pluralities.
I do not assume that all nouns designate entities. Some classes
of nominals such as nominalizations, measure words as well as role
nouns arguably make reference to eventualities, degrees, and roles, respectively.
For instance, murder denotes a set of murdering events (e.g.,
Grimshaw 1990, 2011), liter refers to an interval or a set of degrees on a
scale of volume (e.g., von Stechow 1984, Heim 1985), whereas president
designates a social role, i.e., a function or capacity independent of
the individuals that bear it (Sowa 1984, Steimann 2000). Such social
constructs can be associated with individuals by a special shifting
mechanism (Zobel 2017). Though in this study I will restrict my focus
to entities, or specifically concrete entities, I believe that at least some
of the ideas introduced here can serve as an inspiration for developing
11
1. Introduction
a new way of thinking of partialness with respect to somewhat more
abstract things such as eventualities, degrees, and roles.
Since kinds, or more generally concepts (see Krifka 1995, MuellerReichau
2006), do not play any role in this study, I will refrain from
the discussion concerning their ontological status as well as their
role in nominal semantics. Nevertheless, in principle the approach
developed here is compatible with accounts modeling kinds in the
spirit of Carlson (1977, 1980) and Krifka (1995) where they are treated
as entities in their own right that unlike object-level things are not
spatio-temporally bounded.
A considerable amount of attention in this study will be dedicated
to numerical expressions including cardinal numerals, fractions, and
multipliers. I assume that all those lexical items involve some sort
of reference to mathematical entities. I will use the term number to
refer to abstract entities definite descriptions such as the number two
designate. I adopt here an intuitionist perspective on the epistemology
of mathematics (see, e.g., Kitcher 1984). According to this view,
mathematical objects are constructions of the human mind and do not
exist in the external world. Notice, however, that numbers in this sense
do not necessary coincide with objects defined by Peano’s axioms.
For instance, I do not assume that artificial integers are part of the
ontology described here. Rather, numbers are what we refer to when
we use our everyday language and what might coincide with human
number sense.
1.3 Conventions
Both glossing and notational conventions used in this thesis are more
or less standard in academic linguistics. Nonetheless, for the sake of
clarity I provide a complete list of what to expect in examples and
formulae.
1.3.1 Examples
Since in many parts of this thesis I discuss phrases and sentences from
various languages some of which are not very well-known, in order to
avoid confusion I will always indicate the language of an example, even
12
1. Introduction
if that language is English.4 Furthermore, I will provide information
on the bibliographical source or a name of a native speaker or native
speakers with whom I have consulted a particular example. In the
case of evidence from Polish, all examples and judgments are my own
though I have often confronted them with other native speakers who
confirmed my intuitions. Most of the data come from introspection
with some exceptions found in corpora. I use the symbol # to indicate
both semantic infelicity, i.e., awkwardness in terms of meaning, and
constructions that lack a particular interpretation that is crucial from
the perspective of a discussed phenomenon. For the sake of clarity,
in the latter case I will always provide relevant readings and mark
the one that is non-existent with the symbol #. Unless explicitly stated
otherwise, I will always consider uncoerced meanings of nominal
expressions, i.e., I will ignore the mass-count as well as count-mass
shifts. Occasionally, I will use the symbol * in the examples to indicate
ungrammaticality.
Concerning glossing, I will introduce the complete grammatical
information only when relevant morpho-syntactic issues are discussed
or its lack might cause confusion. Otherwise, for the sake of simplicity
I limit it to the minimum. In the examples, I will use the following
abbreviations. Case: nom – nominative, gen – genitive, dat – dative, acc
– accusative, ins – instrumental, loc – locative; gender: m – masculine, f
– feminine, n – neuter; number: sg – singular, pl – plural, coll – Italian
irregular plural; others: adj – adjectival form, cop – copula, distr –
distributivity marker, int – interrogative marker, lnk – linker, imp –
imperative, poss – possessive, refl – reflexive marker.
1.3.2 Notation
In the following text, I will use a relatively standard notation.5 For completeness,
I specify the following. The symbols x, y, z, and w are used
to represent entity variables, whereas n and P are dedicated for number
and predicate variables, respectively. The small letters a, b, c, and d
stand for entity constants, whereas Arabic numbers, e.g., 1, 2, and 3, for
integers. On the other hand, particular properties are transcribed with
4. Though most of the evidence will be introduced in Chapter 2, 3, and 4, some
crucial data will be also provided in Chapter 5 and 7.
5. Formal expressions will appear mostly in Chapter 6 and 7.
13
1. Introduction
small capitals, e.g., apple stands for the property of being an apple.
The primitive semantic types of entities, numbers, and truth values
are represented by the symbols e, n, and t, respectively. Only complex
types are given in angle brackets, e.g., e, t as opposed to e. Furthermore,
the scope of both the λ operator and quantifiers ∃ and ∀ is
indicated by square brackets, e.g., λx[apple(x)] and ∃x[apple(x)]. For
convenience, in complex formulae I use small and big brackets in order
to designate scope of particular operators. Finally, presuppositions
are introduced always following a dot immediately after the relevant
λ operator, e.g., in the term λx. fruit(x) λP[*P(x) ∧ #(P)(x) = 3] the
sequence fruit(x) is a presupposition.
1.4 The things to come
A substantial body of research has been dedicated to quantification in
natural language. The thesis is structured as follows. The next three
chapters will provide linguistic evidence on the relevance of subatomic
quantification mainly from a cross-linguistic perspective. I will
examine a broad range of constructions involving different types of
partitives, whole-adjectives, and multipliers such as English double. In
Chapter 2, I will discuss certain patterns observed in partitive constructions
cross-linguistically that suggest a unified meaning of partitive
words such as part and half as well as the importance of the topological
notion of integrity in subatomic quantification. The main focus will be
dedicated to the interaction between numerals, partitive words, and
nominals involving count singulars, regular plurals as well as Italian
irregular plurals. Chapter 3 will provide further evidence for the significance
of integrity in quantification over parts. In particular, I will
present novel data concerning topology-sensitive partitive words that
require the whole to be an integrated object or yield only integrated
parts of the whole. Though the evidence will come mainly from Polish,
parallels with other languages will also be drawn including English,
German, and Mandarin. In Chapter 4, I will explore the meaning of
multipliers, i.e., a neglected class of numerical expressions that is
specialized to count parts of entities denoted by the modified noun.
The discussed data will come from Slavic where morphological evidence
supports semantic complexity of those expressions. The main
14
1. Introduction
purpose of Chapter 5 is to provide the three claims that constitute
the conceptual background for the analysis. For that purpose, I will
relate the previously introduced linguistic evidence with psychological
findings concerning the role integrity and part-whole structures
play in human cognition and especially in counting. In Chapter 6, I
will introduce mereotopology, i.e., a theory of wholes extending the
mereological framework based on the notion of parthood with the
topological concept of connectedness. Mereotopology allows us to
capture the difference between distinct types of things such as integrated
wholes, arbitrary sums of individuals, and scattered entities
corresponding to substances and will prove advantageous in modeling
quantification over parts. Chapter 7 will be dedicated to spelling
out a formal analysis of selected issues in subatomic quantification.
The account will be based on the mereotopological framework with
the intuitive notion of integrity playing the main role in the semantic
representation of countable parts. Finally, Chapter 8 will conclude the
thesis and suggest how further research could extend the approach
in order to tackle some open questions. For convenience, the most
relevant contribution of a given part is summarized at the end of each
chapter.
15
2 Partitives and part-whole structures
In this chapter, I will examine cross-linguistic evidence for the relevance
of subatomic quantification in natural language. In particular,
I will focus on the distribution and semantic properties of distinct
types of partitives some of which have not received much attention in
the linguistic literature oriented on the study of meaning. Specifically,
I will investigate constructions involving proportional expressions
such as part and half in order to reveal some non-trivial facts about
partitivity. I will provide novel data indicating that singulars and plurals
share one unified parthood relation. Furthermore, I will show
that distinct part-whole structures result from the fact that certain
entities are conceptualized as cohesive individuals whereas others are
not. Crucially, the distinction turns out to be relevant from the perspective
of countability. In particular, partitive constructions indicate
that cardinal numerals select only for predicates denoting integrated
entities.1
2.1 Partitives
As already mentioned in the introduction, the part-whole relation is
an important notion in human cognition and natural language. For instance,
large parts of the lexicon are structured by means of meronymy,
i.e., the relation that holds between expressions denoting entities conceptualized
as whole objects (holonyms) and expressions denoting
parts of such wholes (meronyms), e.g., the relationship between cat
and tail. However, the relevance of partitivity in natural language is not
restricted to lexical associations but it is also reflected in grammar. As
noticed by Hoeksema (1996), its importance for the syntax-semantics
1. I would like to sincerely thank my informants with whom I have consulted
the structures and meanings of particular constructions. Specifically, many thanks
to Muriel Assmann, Nina Haslinger, Aitor Lizardi Ituarte, Tetiana Kamyshanova,
Radvan Markus, Erlinde Meertens, Lilla Pintér, Martin Prinzhorn, Viola Schmitt,
Hana Strachoňová, Yasutada Sudo, Balázs Surányi, and Guy Tabachnick. Finally, I
am deeply grateful to Enrico Flor and Chiara Masnovo for their judgments and the
discussion of the Italian data. Their introspective reports and insightful comments
were of invaluable help for the relevant part of this chapter.
17
2. Partitives and part-whole structures
interface is corroborated by the fact that the link between a whole and
its parts can support definite descriptions, as witnessed in (2.1).
(2.1) English (Hoeksema 1996, p. 1)
I still have a jar of gooseberry jam, but I cannot get the lid off.
There are many ways how the part-whole relation can be expressed
syntactically and languages differ in what means they employ in order
to indicate it. Sometimes even though a particular construction lacks
a formal exponent expressing partitivity, the relation between the
elements is well understood. For example, English compounds lack
morphological means to express partitivity but often the part-whole
relation between the referents of the first stem and the referents of the
second stem is semantically transparent and can be readily retrieved
(Hoeksema 1996).
(2.2) English (Hoeksema 1996, p. 1)
a. chicken-feet
b. church-door
c. mountain-top
d. bike-steer
In this study, however, I will restrict my focus to the structure that
expresses the part-whole relation formally, i.e. the so-called partitive
construction. The partitive is an expression of natural language that
indicates partialness. In English nominal partitives commonly take
the form as in (2.3).2
(2.3) Partitive structure (Martí i Girbau 2010)
a. [DP Det [PP of [DP Det NP]]]
b. [three [of [my friends]]]
2. In fact, there are at least several proposals concerning the syntax of partitives.
For instance Ionin et al. (2006) argue for different structures for distinct types of
partitives, see (i). However, for the most part I will ignore all the syntactic intricacies
here.
(i) English (Ionin et al. 2006)
a. [NP half [PP of [DP these [NP eight apples]]]]
b. [NP two [NP ∅/apple [PP of [DP these [NP eight apples]]]]]
18
2. Partitives and part-whole structures
The upstairs determiner is a quantifier, whereas the preposition of
links it with the larger whole denoted by its complement, i.e., the
embedded definite or specific DP, from which it is partitioned. A
similar frame is found in many other Germanic languages as well as
in Romance, as demonstrated in (2.4) and (2.5). On the other hand, in
languages such as Estonian and Finnish, a special partitive case can
be used instead of the prepositional element, see (2.6) (de Hoop 2003),
whereas, e.g., German and many Slavic languages can express the
partitive with the genitive (the so-called partitive genitive), see (2.7)
(see, e.g., Hoeing 1997 for German and Valkova 1999, Rutkowski 2007
for Slavic).
(2.4) Dutch
drie
three
van
of
mijn
my
vrienden
friends
(2.5) French
trois
three
de
of
mes
my
amis
friends
(2.6) Finnish (de Hoop 2003)
Anne
Anne
joi
drank
maitoa.
milkpart
‘Anne drank (some) milk.’
(2.7) Russian
Oleg
Oleg
vypil
drank
polovinu
half
moloka.
milkgen
‘Oleg drank half the milk.’
In terms of meaning, the partitive picks out a part from a whole. The
part is associated with the upstairs quantifier, whereas the whole is
denoted by the embedded DP and can be either a singular entity or
a plurality (e.g., Jackendoff 1977, Selkirk 1977, Ladusaw 1982). Thus,
partitives can denote parts of individuals, see (2.8a), as well as subsets
of larger sets of individuals, see (2.8b), and portions of substances, see
(2.8c).
19
2. Partitives and part-whole structures
(2.8) English
a. most of the apple
b. most of the apples
c. most of the juice
An important observation concerning partitive constructions is that
not every kind of DP can be embedded within the structure. A restriction
called the Partitive Constraint excludes certain expressions from
this position (e.g., Jackendoff 1977, Selkirk 1977, Barwise and Cooper
1981, Ladusaw 1982). In particular, its early formulations posited that
only definites can appear in the downstairs DP.3 Later approaches
redefined the restriction in semantic terms postulating that the embedded
DP needs to denote an entity, i.e., it needs to either definite or
specific (de Hoop 1997).
Before I begin a detailed discussion of the relationship between partitivity
and countability it will be useful to make several terminological
remarks. First, I will use the terms part-words and half-words to refer
to expressions such as English part and half, respectively. Despite significant
syntactic differences between those lexical items suggesting
more fine-grained categorial distinctions, they both can appear in the
upstairs Det position in partitives like (2.8a) on a par with quantifiers
such as most and all. From a cross-linguistic perspective different types
of half -words often fall into different syntactic categories. For instance,
German Hälfte ‘half’ appears to have nominal properties, whereas
halb ‘half’ is a standard adjective. As demonstrated in (2.9a)–(2.9c),
while Hälfte has a gender value and either assigns the genitive case
or combines with the prepositional von-phrase assigning the dative
(Durrell 1996; see also Hoeing 1997, Asbury 2007 for the discussion of
the structures), halb agrees with the modified nouns. However, since
this study is primarily concerned with semantic aspects of partitivity,
I take the terms introduced above to refer to expressions explicitly
designating a part of a whole. Therefore, despite the different syntactic
structures expressions such as Hälfte and halb occur in, I will
treat them as semantically related linguistic objects. The general term
partitive words will be used to cover part-words and half -words as
3. See Abbott (1996) for a discussion and a proposal of an analysis not involving
the Partitive Constraint.
20
2. Partitives and part-whole structures
well as different types of proportional quantifiers that can be found in
partitives.4
(2.9) German
a. eine
anom.sg.f
Hälfte
half1.nom.f
des
thegen.sg.m
Apfels
applegen.m
‘a half of the apple’
b. eine
anom.sg.f
Hälfte
half1.nom.f
vom
of-thedat.sg.m
Apfel
appledat.m
‘a half of the apple’
c. ein
anom.sg.m
halber
half2.nom.m
Apfel
applenom.m
‘a half of the apple’
Another issue concerns the grammatical number of the downstairs
DP in the partitive construction. There is a tradition in the literature
on partitivity to distinguish between partitive expressions referring
to a portion of a singular entity and those referring to a subset of a
set of individuals. The first are often called mass partitives and the
latter group partitives (e.g., Hoeksema 1996, Abbott 1996). However,
despite this tradition I will follow de Hoop (1997, 2003) and refer
to phrases such as (2.8a) as entity partitives and to phrases such as
(2.8b) as set partitives.5 Typically, entity partitives involve a singular
term in the embedded DP, whereas set partitives comprise an upstairs
quantifier c-commanding a plural DP. I will restrict the use of the
term mass partitives to constructions such as (2.8c) where the upstairs
determiner combines with the of -phrase taking a mass term as its
complement. The main reason is that I found the terms mass and
group partitives confusing in that they suggest a correspondence with
the mass/count distinction, contrary to fact (see also Martí i Girbau
2010).
Furthermore, I will refer to constructions involving an overt partword
such as those in (2.10a) as explicit partitives. On the other hand,
4. The term more or less corresponds to what Quirk et al. (1985, p. 249) call general
partitive nouns excluding measure terms.
5. Yet another term is used by Korat (2016) to refer to partitives involving regular
count singular nouns as well as collective nouns, namely singular quantified terms.
However, since I do not focus on group nouns in this study, I will ignore the potential
distinction.
21
2. Partitives and part-whole structures
I will call phrases containing proportional quantifiers such as most
or half, see (2.10b), proportional partitives, whereas those involving
fractions, see (2.10c), will be referred to as fraction partitives.
(2.10) English
a. part of the apple
b. half of the apple
c. two thirds of the apple
Finally, I will use the terms count explicit partitives and count proportional
partitives to refer to phrases such as (2.11a) and (2.11b),
respectively, where the partitive word is modified by the cardinal numeral.
These should not be confused with count partitives, the term
sometimes used for phrases such as (2.11c) (e.g., Ionin et al. 2006).6
(2.11) English
a. two parts of the apple
b. two halves of the apple
c. two of the apples
The next sections of this chapter will be dedicated to the interaction
between cardinal numerals and partitives involving different types of
expressions both in the upstairs DP head as well as in the embedded
DP.
2.2 The analogy
At least from Link (1983) on it is commonly assumed that it is necessary
to distinguish between two distinct yet corresponding part-whole
structures employed in natural language, namely one corresponding
to the material constitution of objects and one capturing collections of
distinct objects. Hence, the material part relation provides ordering for
structures representing the relationship between portions of matter
and individuals made out of that matter, whereas the individual part
relation concerns particular objects and sums thereof. The distinction
is motivated by the fact that things and stuff they consist of are
apparently differentiated in natural language. The famous example
6. Quirk et al. (1985, p. 249) call them plural partitives.
22
2. Partitives and part-whole structures
concerns two golden rings. If they were forged recently out of some
old Egyptian gold, then it is definitely true to state that the rings are
new although the gold is old.
However, there are some data that might be interpreted as suggesting
that at least in some expressions natural language semantics does
not distinguish between the two distinct part relations postulated by
Link (1983) and instead employs a unified part-whole structure for different
types of nominal expressions. Moltmann (1997, 1998) observes
an analogy between partitives involving singular and plural terms.
For instance, in German the same expression Teil ‘part’ can be used
both in entity and set partitive constructions in order to quantify over
parts of singular individuals and subsets of groups of individuals, see
(2.12).7 In particular, when used in the entity partitive, Teil quantifies
over material parts of singular objects, i.e., functional parts or portions
of a substance making up the whole individual. For instance, (2.12a)
denotes a set of parts of the relevant apple. It could be used to refer,
e.g., to the stem end of the apple in question or its blossom end as well
as its pulp or pips. On the other hand, when Teil takes the plural DP
as its complement, it quantifies over whole individuals, e.g., (2.12b)
makes reference to a part of a particular collection, i.e., a subset of the
apples rather than to a set of arbitrary parts of singular apples. For
instance, given a situation in which there are three relevant apples a1,
a2, and a3 (2.12b) could not be used to refer to a set involving the stem
end of a1 and a pip of a2. Instead, it would be true of a set of some of
the apples in question, e.g., a set comprising a1 and a2.
7. In fact, Moltmann (1997, 1998) provides English examples to argue for her point,
see (i). However, since the reported felicity judgments for English contrast with
Moltmann’s claim (see Schwarzschild 1996), I take Moltmann’s data not to be reliable
and use uncontroversial German examples instead.
(i) English (Moltmann 1997, p. 11)
a. all of / some of / part of the book
b. all of / some of / part of the wine
c. all of / some of / part of the books
23
2. Partitives and part-whole structures
(2.12) German (Nina Haslinger, p.c.)
a. Teil
part
des
thegen
Apfels
applegen
‘part of the apple’
b. Teil
part
der
thegen
Äpfel
applesgen
‘some of the apples’
Notice that in German there is yet another interpretation an expression
such as (2.12b) can get. For instance, consider the sentence in (2.13).
It would definitely be true in a situation where there are ten apples
and two of them are completely moldy. Yet, there is also a different
scenario where uttering (2.13) would be truthful, namely if there were
ten apples and each apple had a small patch of mold. At first blush,
this behavior might appear to contradict what I claimed in the previous
paragraph. However, there are good reasons to assume that
this reading is independent from the two discussed above and results
from a silent distributivity operator. Notice that the sentence would
be judged false in a scenario where there were ten apples seven of
which had a small patch of mold, whereas the others were not moldy.
This shows that German explicit set partitives such as (2.12b) have
another interpretation in addition to the subset reading discussed
above. Crucially, this interpretation does not involve quantification
over arbitrary material parts. Rather, the definite seems to have a distributive
interpretation with wide scope over the indefinite. Though
it was important to draw the distinction between the two possible
interpretations, I will ignore this additional distributive reading in
the remaining part of this chapter.8
(2.13) German (Nina Haslinger, p.c.)
Ein
a
kleiner
small
Teil
part
von
of
den
thedat
zehn
ten
Äpfeln
applesdat
ist
is
schimmlig.
moldy
‘A minority of the ten apples are moldy/A small part of each
of the ten apples is moldy.’
8. In fact, this interpretation might turn out to be a German idiosyncrasy since it
seems to be unavailable in other languages, e.g., in Polish. Though it is an interesting
issue on its own, this study is about something different.
24
2. Partitives and part-whole structures
Thus, it appears that the same expression can form either an entity
partitive or a set partitive depending on the number value on its
complement. This phenomenon seems intriguing and might imply a
puzzle with respect to a proper treatment of German explicit partitives.
But is it just a German idiosyncrasy? Or does the data in (2.12) actually
reveal something deep about the way the human mind conceptualizes
the difference between objects and collections of objects? And if yes, to
what extent is such a difference relevant for the semantic distinction
between singularities and pluralities outside the domain of partitivity?
I believe that, if approached in a systematic manner, an attempt to
provide answers to these questions can shed new light on what it
means to be an individuated entity and, as we will see, what it means
to be countable.
Prima facie, it seems that there are two ways how to account for
the phenomenon in question. A radical solution is to conclude that the
data in (2.12) suggest a unified parthood structure for both singular
and plural individuals across the board (Moltmann 1997, 1998). In
other words, it is to posit that singularities and pluralities are in fact
much less distinct than standardly assumed. On such an approach,
the relationship between an object and its parts would be the same
relationship as the one holding between a collection of objects and objects
being part of that collection. Another way to explain the analogy
would be to assume that partitive expressions such as Teil employ a
derived notion of parthood which generalizes over two distinct yet corresponding
part-structures (Barker 1998, Ionin et al. 2006). However,
before I elaborate more on this issue, let us examine to what extent
the analogy between entity and set partitives is valid from a wider
cross-linguistic perspective.
2.2.1 Entity and set partitives across languages
For some reason, the analogy between explicit entity and set partitives
does not hold in English. As observed by Schwarzschild (1996, pp. 165–
166), there is a contrast between (2.14a) and (2.14b), see also (2.15).9
The phrase in (2.15b) is not a well-formed English expression and in
order to refer to a subset of the relevant apples one needs to use a
9. Similar contrasts have been reported by Chierchia (2010).
25
2. Partitives and part-whole structures
partitive construction such as (2.15c) where the existential quantifier
some is employed as the upstairs determiner. Thus, if English were
like German, it would be possible to use the sentence in (2.14b) to
describe a situation where some of the boys were in Texas. But the
sentence is odd. Schwarzschild concludes that since English part of
cannot co-occur with count plurals, it selects for singularity-denoting
complements exclusively.
(2.14) English (Schwarzschild 1996)
a. Part of the car was painted.
b. #Part of the boys were in Texas.
c. Part of the group was in Texas.
(2.15) English (Guy Tabachnick, p.c.)
a. part of the apple
b. #part of the apples
c. some of the apples
Schwarzschild (1996) examines explicit entity and set partitives for
a particular reason. On the basis of the contrast between (2.14b) and
(2.14c) and other facts concerning predicate non-sharing, he argues
against analyses positing that group nouns denote impure atoms.10
However, what is interesting from the perspective of this study are
some possible consequences of the data in (2.14) and (2.15) for the
ontology encoded in that part of natural language semantics that deals
with parthood. Specifically, what is implied by the argument put forward
by Schwarzschild is that singularities and pluralities involve two
distinct part-whole structures or, in other words, there is a fundamental
difference between how we conceptualize the relationship between
parts of an object and that object and members of a set of objects and
that set. In fact, that is the received view in theories of parthood (e.g.,
Link 1983, Simons 1987). On the discussed approach, English part of is
modeled as an existential ‘pieces’ quantifier in a sense reverse of each.
Simplifying a bit, part of only takes arguments denoting singularities
and ensures that the intersection of the sets of parts of a singularity
with the set corresponding to the verbal predicate is non-empty.
10. See also de Vries (2015) for a different approach building on the same conclusion.
26
2. Partitives and part-whole structures
Whatever the reason English does not allow for explicit set partitives,
i.e., partitives headed by part with embedded plural DPs, it
is not an ordinary behavior from a typological point of view. On the
contrary, the analogy introduced in (2.12) holds in multiple languages
displaying the singular/plural distinction including at least representatives
of such diverse language families as Germanic, Slavic, Romance,
Celtic, Ugro-Finnic, Semitic, and Basque, see (2.16)–(2.23). In all the
languages listed below, explicit partitives allow for both singular and
plural DPs. For instance, (2.16a) is an example of the Dutch explicit entity
partitive on a par with the English construction in (2.15a). Similar
to German, see (2.12b), in order to quantify over parts of a plurality,
i.e., particular apples, the same partitive word deel ‘part’ can be used
with the plural DP, as witnessed by the felicity of (2.16b). The same
holds in other Indo-European languages such as Polish, see (2.17),
Russian, see (2.18), Italian, see (2.19), Portuguese, see (2.20), and Irish,
see (2.20). Moreover, the same phenomenon can be observed outside
the Indo-European language family. For instance, in an agglutinative
language such as Hungarian both explicit entity partitives and explicit
set partitives are possible, see (2.22). Similar, the Hebrew explicit entity
partitive construction in (2.23a) refers to a set of parts of the relevant
boy, i.e., his body parts, whereas the phrase with the same xelek ‘part’
expression in (2.23b) would be true of some of the boys. Finally, as
attested in (2.24) the analogy holds in the isolate Basque language.11
(2.16) Dutch (Erlinde Meertens, p.c.)
a. deel
part
van
of
de
the
appel
apple
‘part of the apple’
b. deel
part
van
of
de
the
appels
apples
‘some of the apples’
11. That seems to be the case at least in the Biscayan dialect. The speakers of the
Gipuzkoan dialect I questioned disagree. Many thanks to Aitor Lizardi Ituarte, Gillen
Martinez de la Hidalga, and Itziar Orbegozo Arrizabalaga for their judgments and
to Katie Frasier for putting me in touch with them.
27
2. Partitives and part-whole structures
(2.17) Polish
a. część
part
jabłka
applegen
‘part of the apple’
b. część
part
jabłek
applesgen
‘some of the apples’
(2.18) Russian (Tetiana Kamyshanova, p.c.)
a. čast’
part
jabloka
applegen
‘part of the apple’
b. čast’
part
jablok
applesgen
‘some of the apples’
(2.19) Italian (Schwarzschild 1996, p. 186; adapted)
a. parte
part
del
of-the
muro
wall
‘three parts of the wall’
b. parte
part
dei
of-the
muri
walls
‘some of the walls’
(2.20) Portuguese (Muriel Assmann, p.c.)
a. parte
part
da
the
maçã
apple
‘part of the apple’
b. parte
part
das
the
maçãs
apples
‘some of the apples’
(2.21) Irish (Radvan Markus, p.c.)
a. cuid
part
den
from-the
úll
apple
‘part of the apple’
b. cuid
part
de
from
na
the
húlla
apples
‘some of the apples’
28
2. Partitives and part-whole structures
(2.22) Hungarian (Balázs Surányi, Lilla Pintér, p.c.)
a. az
the
alma
apple
egy
a
része
partposs
‘part of the apple’
b. az
the
almák
apples
egy
a
része
partposs
‘some of the apples’
(2.23) Hebrew (Schwarzschild 1996, p. 184; adapted)
a. xelek
part
me-ha-baxur
from-the-boy
‘part of the boy’
b. xelek
part
me-ha-baxur-im
from-the-boy-s
‘some of the boys’
(2.24) Basque (Aitor Lizardi Ituarte, p.c.)
a. sagarraren
applegen
zati
part
bat
a
‘part of the apple’
b. sagarren
applesgenpart
zati
a
bat
‘some of the apples’
A similar phenomenon can be observed in partitives involving proportional
expressions including half -words and fractions. This time also
English does show the discussed pattern. For instance, half, most, and
two thirds in (2.25a), (2.26a), and (2.27a), respectively, quantify over
parts of a singular entity, i.e., yield a portion of stuff constituting the
whole, whereas in (2.25b), (2.26b), and (2.27b) they quantify over parts
of a plural individual, i.e., indicate a subset of individuals making up
the total plurality.
(2.25) English
a. half of the apple
b. half of the apples
29
2. Partitives and part-whole structures
(2.26) English
a. most of the apple
b. most of the apples
(2.27) English
a. two thirds of the apple
b. two thirds of the apples
Again, this kind of pattern is attested cross-linguistically, as witnessed
in the examples from Polish in (2.28)–(2.29) and German in (2.31)–
(2.33). In all the phrases provided below, the proportional quantifier
can quantify either over material parts of the relevant object, i.e., apple,
or over individual objects, i.e., apples, making up the whole plurality.
(2.28) Polish
a. połowa
half
jabłka
apple.gen
‘half of the apple’
b. połowa
half
jabłek
apples.gen
‘half of the apples’
(2.29) Polish
a. większość
most
jabłka
applegen
‘most of the apple’
b. większość jabłek
most applesgen
‘most of the apples’
(2.30) Polish
a. dwie
two
trzecie
thirds
jabłka
applegen
‘two thirds of the apple’
b. dwie
two
trzecie
thirds
jabłek
applesgen
‘two thirds of the apples’
30
2. Partitives and part-whole structures
(2.31) German (Nina Haslinger, p.c.)
a. Hälfte
half
des
thegen
Apfels
applegen
‘half of the apple’
b. Hälfte
half
der
thegen
Äpfel
applesgen
‘half of the apples’
(2.32) German (Viola Schmitt, p.c.)
a. ein
a
Großteil
most
des
thegen
Apfels
applegen
‘most of the apple’
b. ein
a
Großteil
most
der
thegen
Äpfel
applesgen
‘most of the apples’
(2.33) German (Viola Schmitt, p.c.)
a. zwei
two
Drittel
thirds
des
thegen
Apfels
applegen
‘two thirds of the apple’
b. zwei
two
Drittel
thirds
der
thegen
Äpfel
applesgen
‘two thirds of the apples’
Furthermore, it is well known that in many languages displaying the
singular/plural distinction, there are classes of number-neutral nominals
referring to individuated discrete entities such as object mass
nouns (known also as fake or neat mass nouns; see, e.g., Gillon 1992,
Chierchia 1998a, 2010, Barner and Snedeker 2005, Bale and Barner 2009,
Landman 2011) and pluralia tantum, i.e., plural nouns without singular
counterparts (e.g., Wierzbicka 1988, Corbett 2000, KoptjevskajaTamm
and Wälchli 2001, Wisniewski 2010; see also Wągiel 2015a, 2017).
The examples of the first category in English are jewelry, furniture, and
mail, whereas scissors, eyeglasses, and pants fall into the second class.
Both types of expressions are systematically ambiguous between a
singular and a plural reading, i.e., they can either refer to one individual
or to a sum of individuals. For instance, furniture can denote either
one or many pieces of furniture. Consequently, (2.34a) would be true
if there were, say, one wardrobe in the room as well as if there were
31
2. Partitives and part-whole structures
several chairs or a chair, a table, and a wardrobe in the room. Similar,
since scissors is number-neutral, (2.34b) would be true either if there
were one or many utensils on the table.
(2.34) English
a. There is furniture in the room.
b. There are scissors on the table.
Interestingly, in many languages partitives including number-neutral
nominals such as object mass nouns and pluralia tantum are ambiguous
between a part-of-a-singularity and part-of-a-plurality reading.
For instance, the Czech phrase in (2.35a) would be true of a part of a
singular shoe in question as well as of a subset of the relevant shoes.
Similar, (2.35b) can either refer to a part of one utensil or it can mean a
subset of the total number of the relevant scissors. The same ambiguity
appears in proportional partitives such as (2.36a) and (2.36b).
(2.35) Czech (Hana Strachoňová, p.c.)
a. část
part
obuvi
footweargen
‘part of the footwear/some of the footwear’
(i) part-of-a-singularity reading
(ii) part-of-a-plurality reading
b. část
part
nůžek
scissorsgen
‘part of the scissors/some of the scissors’
(i) part-of-a-singularity reading
(ii) part-of-a-plurality reading
(2.36) Czech (Hana Strachoňová, p.c.)
a. polovina
half
obuvi
footweargen
‘half of the footwear’
(i) part-of-a-singularity reading
(ii) part-of-a-plurality reading
32
2. Partitives and part-whole structures
b. polovina
half
nůžek
scissorsgen
‘half of the scissors’
(i) part-of-a-singularity reading
(ii) part-of-a-plurality reading
A similar effect is reported to appear in some constructions in languages
with general number such as Japanese, as originally observed
for proportional partitives by Sauerland and Yatsushiro (2004). For
instance, (2.37) would be true both in a scenario where John read a
number of books which is greater than a half of the total number of
books and in a scenario where John read more than a half of one book.
(2.37) Japanese (based on Sauerland and Yatsushiro 2004)
John-wa
John-top
hotondo
most
hon-o
book-acc
yomi-oeta
read-finished
‘John finished reading most of the books/most parts of the
book(s).’
a. part-of-a-singularity reading
b. part-of-a-plurality reading
Likewise, Watanabe (2013) discusses pairs of constructions involving
both explicit and proportional partitives such as those provided in
(2.38)–(2.40). Different linearizations in the doublets correlate with
different interpretations. Constructions in which partitive words follow
nominals are ambiguous between a part-of-a-singularity and a
part-of-a-plurality reading, whereas sentences with partitive words
preceding the noun unambiguously refer to parts of a singular object.
In particular, (2.38a) can either mean that a part of a single apple in
question is rotten or that a subset of all the relevant apples is rotten.
Similar, (2.39a) and (2.40a) mean that either more than half of one
apple is rotten or that the number of rotten apples exceeds a half of
the total number. On the other hand, the partitives with the reversed
order in the sentences in (2.38b), (2.39b), and (2.40b) do not give rise to
such an ambiguity and can only make reference to subsets of a larger
set, i.e., some of the apples and most of the apples, respectively.
33
2. Partitives and part-whole structures
(2.38) Japanese (Watanabe 2013; adapted)
a. Ringo-no
apple-gen
ichibu-ga
part-nom
kusatteiru.
is-rotten
‘Part of the apple is rotten/Some of the apples are rotten.’
(i) part-of-a-singularity reading
(ii) part-of-a-plurality reading
b. Ichibu-no
part-lnk
ringo-ga
apple-nom
kusatteiru.
is-rotten
‘Some of the apples are rotten.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
(2.39) Japanese (Watanabe 2013)
a. Ringo-no
apple-gen
hotondo-ga
most-nom
kusatteiru.
is-rotten
‘Most of the apple(s) is/are rotten.’
(i) part-of-a-singularity reading
(ii) part-of-a-plurality reading
b. Hotondo-no
most-lnk
ringo-ga
apple-nom
kusatteiru.
is-rotten
‘Most of the apples are rotten.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
(2.40) Japanese (Watanabe 2013)
a. Ringo-no
apple-gen
dai-bubun-ga
large-part-nom
kusatteiru.
is.rotten
‘Most of the apple(s) is/are rotten.’
(i) part-of-a-singularity reading
(ii) part-of-a-plurality reading
b. dai-bubun-no
large-part-lnk
ringo-ga
apple-nom
kusatteiru.
is.rotten
‘Most of the apples are rotten.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
The data discussed above suggest that in many languages part-words
are able to simultaneously access elements making up pluralities of
individuals as well as subatomic part-whole structures. Of course,
34
2. Partitives and part-whole structures
there is no reason to believe that part-words are simply reducible to the
mereological partood relation, just as there is no reason to believe that
the noun member is the natural language exponent of the set-theoretic
membership relation (see Champollion 2010, 2017). However, it is
tacit to assume that such expressions do employ a notion of parthood
which might be modeled in terms of some primitive formal relation.
For theories that sort domains, and thus distinguish strictly between
individual and material parthood (e.g., Link 1983), the cross-linguistic
behavior of partitive words might be seen as problematic to account
for in a systematic manner. On the other hand, as will be discussed
in Chapter 7 postulating a unified ontology for singular objects and
pluralities thereof may encounter a number of problems to account
for.
I believe that at this point an empirical conclusion can be drawn.
Though the sample of languages and constructions discussed in this
section definitely does not exhaust the cross-linguistic diversity in the
domain of explicit and proportional partitives (nor was it intended
to), I conclude that the pattern involving analogous structures for explicit
and proportional entity and set partitives is robust. In multiple
typologically and genetically different languages the same partitive
word can be used to designate both parts of a single individual and
subsets of objects constituting a larger plurality. And though the question
whether this fact tells us something non-trivial about the way
discrete objects and sums thereof are conceptualized or rather something
interesting about partitive words themselves remains somewhat
open, I believe that systematic patterns should not be neglected but
instead guide development of a proper approach. The next section will
provide a potentially problematic observation concerning different
properties of part-words in explicit entity and set partitives.
2.2.2 Partitivity and countability
At first blush, the cross-linguistically attested distribution of partwords
as well as other proportional expressions in entity and set
partitives discussed in the previous section suggests at least a strong
correspondence between subatomic and superatomic part-whole structures.
The fact that the very same expression can be used to quantify
over parts of a singular individual as well as over parts of a plural-
35
2. Partitives and part-whole structures
ity in multiple typologically different and often unrelated languages
suggests that an ambiguity approach postulating two different lexical
entries for the relevant proportional expressions is quite improbable.
To the contrary, it seems that something more essential is happening
here and a more detailed examination of the phenomenon might shed
new light on the properties of that component of the language faculty
that is dedicated to the interpretation of part-whole relations.
However, Schwarzschild (1996) presents data that might seem to
be a counterargument for an approach postulating a semantics operating
on a unified part-whole structure that would allow for both
quantification over wholes as well as subatomic quantification (cf.
Moltmann 1997, 1998). The crucial observation concerns the fact that
part-words in explicit set partitives have different properties than partwords
occurring in entity partitive constructions. Specifically, such
expressions are countable only if they refer to material parts of a singular
object. For instance, consider the contrast between the phrases
in (2.41). The count explicit entity partitive in (2.41a) comprises the
Italian pluralized part-word parti ‘parts’ modified by the cardinal numeral
tre ‘three’. What the phrase can only denote is a set of pluralities
consisting of three parts of different individual walls, e.g., a set including
a sum of a part of the wall w1, a part of the wall w2, and a
part of the wall w3. Crucially though, (2.41a) cannot refer to parts of a
plurality, i.e., three individual walls or three pluralities of walls. To
illustrate the distinction, let us assume that there are 10 relevant walls:
w1, . . . , w10. In such a scenario, (2.41b) is not felicitous on the reading
where one is counting parts of a plurality, e.g., it would not be true
of the set of three walls {w1, w2, w3} or the set of three subsets of the
walls {{w1, w2}, {w3, w4, w5}, {w6, w7}}.
(2.41) Italian (Schwarzschild 1996, p. 186; adapted)
a. tre
three
parti
parts
del
of-the
muro
wall
‘three parts of the wall’
b. tre
three
parti
parts
dei
of-the
muri
walls
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
36
2. Partitives and part-whole structures
This phenomenon is by no means an Italian idiosyncrasy. To the contrary,
it is attested cross-linguistically, as will be demonstrated in the
the following examples. For instance, the very same pattern can be
found in Portuguese. Similar to the interpretation of (2.41b), the sentence
(2.42b) would be true if, say, a brick of the wall w1 were red,
two bricks of the wall w2 were red, and three bricks of the wall w3
were red. However, given the scenario with the eight relevant walls
w1, . . . , w8 introduced above it would not be considered true if, e.g.,
w1, w2, and w3 were all painted red, nor if three subsets of the walls,
e.g., {w1, w2, w3}, {w4, w5}, and{w6, w7}, were all painted red.
(2.42) Portuguese (Muriel Assmann, p.c.)
a. Três
three
partes
parts
do
the
muro
wall
são
are
vermelhas.
red
‘Three parts of the wall are red.’
b. Três
three
partes
parts
dos
the
muros
walls
são
are
vermelhas.
red
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
Such a pattern is also attested outside Romance. For instance, the
German sentence in (2.43a) is true if there are three green spots on
the relevant apple. However, (2.43b) cannot mean that three subsets
of the apples are green. Rather, the only possible interpretation of the
sentence is that three parts of different apples are green.
(2.43) German (Nina Haslinger, p.c.)
a. Drei
three
Teile
parts
des
thegen
Apfels
applegen
sind
are
grün.
green
‘Three parts of the apple are green.’
b. Drei
three
Teile
parts
der
thegen
Äpfel
applesgen
sind
are
grün
green
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
A similar effect is found in Slavic, e.g., (2.44) illustrates the relevant
contrast in Polish. Again, the count explicit set partitive does not quantify
over sub-pluralities of the apples in question, but rather over parts
37
2. Partitives and part-whole structures
of individual apples. The sentence in (2.44b) does not get a part-of-aplurality
interpretation.12
(2.44) Polish
a. Trzy
three
części
parts
jabłka
applegen
są
are
zielone.
green
‘Three parts of the apple are green.’
b. Trzy
three
części
parts
jabłek
applesgen
są
are
zielone.
green
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
Furthermore, there two distinct constructions in Japanese that display
the same patter, see (2.45). Despite the fact that the noun ringo ‘apple’
is number-neutral and as such it can either refer to a single apple or
to a collection of fruit, the count explicit partitive in (2.45a) where the
part-word bubun is modified by a numeral forces a part-of-a-singularity
reading.13 The sentence simply cannot mean that three different subsets
of whole apples are damaged. Similar, the use of the classifier
construction involving a numeral and the classifier kasyo in oder to
quantify over parts of referents of a number-neutral noun excludes a
part-of-a-plurality reading. As witnessed in (2.45b), the only possible
interpretation involves subatomic quantification, i.e., three parts of a
singularity are damaged.
(2.45) Japanese (Yasutada Sudo, p.c.)
a. Ringo-no
apple-gen
mit-tsu-no
three-cl-gen
bubun-ga
part-nom
itandeiru.
damaged
‘Three parts of the apple are damaged.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
12. Notice that there is yet another reading of modified set partitives with DPs
denoting food terms in Polish. It involves the count-mass shift and indicates a
fraction of volume, e.g., three-fourths of the total volume of the apples. I ignore this
interpretation here, but I will return to it in Section 3.3.
13. Again, for the sake of simplicity I ignore here the attested distributive reading
on which there are multiple apples and three parts of each of them are damaged.
38
2. Partitives and part-whole structures
b. Ringo-no
apple-gen
san-kasyo-ga
three-cl-nom
itandeiru.
damaged
‘Three parts of the apple are damaged.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
The effect is even stronger if we consider partitives with animate DPs.
For instance, notice the contrast between (2.46a) and (2.46b) in Italian.
The first is felicitous if a subset of the relevant boys were in Texas.
However, the second sentence implies an organ trafficking context
since the only possible interpretation one can get is that three body
parts belonging to the boys were in Texas. A similar effect holds in
Polish. While the sentence involving the explicit set partitive in (2.47a)
is perfectly fine and simply means that some of the relevant boys sleep,
its modified counterpart in (2.47b) is nonsensical since it asserts that
three body parts sleep. Unless one considers a macabre scenario in
the spirit of the Addams Family with sentient animate disembodied
hands, body parts cannot be experiencers of a sleeping state, and thus
the awkwardness of the sentence.
(2.46) Italian (Schwarzschild 1996, p. 186; adapted)
a. Parte
part
dei
of-the
ragazzi
boys
erano
were
in
in
Texas.
Texas
‘Some of the boys were in Texas.’
b. #Tre
three
parti
parts
dei
of-the
ragazzi
boys
erano
were
in
in
Texas.
Texas
(2.47) Polish
a. Część
part
chłopców
boysgen
śpi.
sleeps
‘Some of the boys sleep.’
b. #Trzy
three
części
parts
chłopców
boysgen
śpią.
sleep
Moreover, the phenomenon is not restricted to explicit partitives modified
by cardinal numerals but holds also for constructions involving
other types of quantifiers. For instance, let us consider the examples in
(2.48) where the referents of proportional partitives are arguments of
a predicate involving the total adjective czysty ‘clean’ (Yoon 1996, Rot-
39
2. Partitives and part-whole structures
stein and Winter 2004). Similar to adjectives such as full, flat, and closed,
clean is an absolute gradable adjective that indicates the maximal degree
of the property in question (Unger 1975, Kennedy and McNally
2005). In other words, clean is true of an entity that is completely clean
or at least the degree of cleanliness falls near the maximal value on
the scale. Let us now contemplate the contrast between (2.48a) and
(2.48c) on the one hand and (2.48b) and (2.48d) on the other. The first
two sentences involve regular proportional partitives which quantify
over whole individuals. Therefore, if there are six relevant toys t1, ...,
t6 in total, (2.48a) and (2.48c) would be true if three of the six toys, e.g.,
t1, t2, and t3, or at least four of the six toys, e.g., t1, t2, t3, and t4, were
clean, respectively. However, when an explicit set partitive is modified
by proportional quantifiers such as połowa ‘half’ and większość ‘most’
the whole phrase does not quantify over wholes anymore but rather
over parts of singular individuals. Hence, (2.48b) and (2.48d) can be
true even if it is not the case that a half or most of the toys are clean,
respectively. In fact, they might be true if none of the toys is actually
clean. Imagine a scenario where there are six teddy bears and one paw
of each of them is dirty. It follows that none of the teddy bears is clean
though most of their parts are clean.
(2.48) Polish
a. Połowa
half
zabawek
toysgen
jest
is
czysta.
clean
‘A half of the toys are clean.’
b. Połowa
half
części
partsgen
zabawek
toysgen
jest
is
czysta.
clean
‘A half of the parts of the toys are clean.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
c. Większość
most
zabawek
toysgen
jest
is
czysta.
clean
‘Most of the toys are clean.’
d. Większość
most
części
partsgen
zabawek
toysgen
jest
is
czysta.
clean
‘Most of the parts of the toys are clean.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
40
2. Partitives and part-whole structures
Similar, the same distinction can be detected in contexts regarding
universal quantification. Though in such environments it is hard to
distinguish between quantification over parts of singularities and
quantification over parts of pluralities, the difference becomes evident
if an adverbial modifier such as almost is used. Both (2.49a) and (2.49b)
can be true even if none of the toys is completely clean. Since it is
sufficient for an object be considered dirty if only one of its parts
is dirty, one can imagine a scenario where almost every part of the
relevant six teddy bears is clean and yet the teddy bears are dirty. And
again, modified explicit partitives do not quantify over wholes but
rather over parts of wholes despite the fact that the downstairs DP is
plural.
(2.49) Polish
a. Prawie
almost
każda
every
część
part
zabawek
toysgen
jest
is
czysta.
clean
‘Almost every part of the toys is clean.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
b. Prawie
almost
wszystkie
all
części
parts
zabawek
toysgen
są
are
czyste.
clean
‘Almost all parts of the toys are clean.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
A brief investigation into the cross-linguistic behavior of partitive
words suggests that the discussed pattern is robust. Table 2.1 summarizes
the observed semantic behavior. Both unmodified and modified
partitive words c-commanding singular DPs quantify over material
parts of the relevant entity, whereas the domain of quantification for
partitive words c-commanding plural DPs shifts when they combine
with quantifiers. Specifically, although explicit partitives with unmodified
partitive words quantify over parts of a plurality, in the presence
of a quantifier they are no longer able to count whole individuals and
instead operate on the subatomic level. In fact, it appears that there are
no count explicit set partitives since apparently no partitives quantify
over subsets of the set denoted by the downstairs DP in the first place.
Rather, constructions such as (2.41b) in Italian are a special type of
41
2. Partitives and part-whole structures
entity partitive with the only difference that material parts do not
belong to one object as in (2.41a) but are distributed onto a plurality
of individuals. This is puzzling because it suggests that despite the
apparent homophony part-words in explicit entity partitives and count
explicit partitives as well as half -words in corresponding proportional
partitives are in fact different semantic expressions.
singulars plurals
bare count bare count
subatomic quantification *
quantification over wholes * * *
Table 2.1: Properties of partitive words
Based on the discussed contrasts in Italian, Schwarzschild (1996)
concludes that the uncountability of part-words in set partitives reveals
that Italian and English do not disagree with respect to their ontologies.
Since English part of selects for singularity-denoting complements
and cannot co-occur with count plurals, it is modeled as an existential
‘pieces’ quantifier in his system. Presumably Italian parte ‘part’ in
entity partitives and count explicit partitives, see (2.19a) and (2.41a),
respectively, is an expression of the same type, whereas parte in set
partitives, see (2.19b) is a quantifier of a different sort.
As mentioned before, Schwarzschild (1996)’s argument concerns
primarily extensions of group nouns. However, what his novel observation
further corroborated in this section seems to imply is that
in diverse languages singular individuals and pluralities are associated
with distinct mereological structures. This can be seen as a valid
counterargument for an approach attempting to unify part-whole relations
corresponding to extensions of singulars and plurals such as
the theory of Moltmann (1997, 1998) if the main empirical motivation
concerns the morphological parallelisms in quantification in entity,
set, and mass partitives.
However, there are two interrelated issues concerning an approach
that posits multiple lexical entries for part-words to explain their nontrivial
quantificational behavior. One problem has to do with crosslinguistic
correspondences. Since the same part-word is often used
both in entity and set partitives cross-linguistically as we have seen
42
2. Partitives and part-whole structures
in the previous section, a homophony would have to be postulated
in multiple diverse languages. Though of course it is not a logical
impossibility, it seems very implausible that languages in general
do not differentiate morphologically between the two expressions in
question.
Another issue concerns intralinguistic systematicity. As witnessed
by the well-formedness of English proportional and fraction partitives
such as those in (2.25)–(2.27), partitive words such as half and most as
well as fractions such as two thirds can be used both in entity and set
partitives where they get a part-of-a-singularity and part-of-a-plurality
reading, respectively. Again, as we have seen such distribution holds
cross-linguistically. Interestingly, proportional quantifiers pattern with
part-words when modified by numerals. For instance, consider the
Polish sentences in (2.50a) and (2.50b). In the first the count explicit
partitive quantifies over material parts of the relevant wall and the
sentence is truth-conditionally equivalent with a statement that the
whole wall is red. Nonetheless, the count explicit partitive in the
second sentence does not mean that all the walls are red but rather
that among the walls there two such that they are half red-painted.
Despite the downstairs plural DP, the whole phrase is not interpreted
as a set partitive and similar to (2.50a) it quantifies over parts of distinct
wholes.
(2.50) Polish
a. Dwie
two
połowy
halves
muru
wallgen
są
are
czerwone.
red
‘Two halves of the wall are red.’
b. Dwie
two
połowy
halves
murów
wallsgen
są
are
czerwone.
red
‘Two halves of the walls are red.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
Moreover, the data in (2.51) and (2.52) further corroborate the contrast.
Since the quantifier both combines universal quantification with an
inference that the cardinality of counted objects equals two (Barwise
and Cooper 1981, Schwarzschild 1996, pp. 139–145), in most cases
43
2. Partitives and part-whole structures
it can be paraphrased by the definite numeral expression the two.14
Hence, the example in (2.51a) simply states that the two halves of the
wall are red, i.e., that all the halves are red and that there are only two
halves. However, when one tries to process (2.51b), it gets confusing.
Of course, if the modified partitive quantified over whole walls, it
would not but it is not what the phrase means. Instead, obie ‘both’
implies that the total number of halves is two, whereas at the same
time the plural DP infers that there are at least two walls, thus at least
four halves, which results in a contradictory statement.
(2.51) Polish
a. Obie
both
połowy
halves
muru
wallgen
są
are
czerwone.
red
‘Both halves of the wall are red.’
b. #Obie
both
połowy
halves
murów
wallsgen
są
are
czerwone.
red
Similar, since it is only possible for an object to have two halves, the
sentence in (2.52a) is strange because it asserts that the wall has more
than two parts forming 50% of its material constitution each. On the
other hand, there is nothing awkward about (2.52b). The sentence
does not state that there are more walls than there are walls but rather
that among all the walls in question there are three walls half-covered
in red paint.
(2.52) Polish
a. #Trzy
three
połowy
halves
muru
wallgen
są
are
czerwone.
red
b. Trzy
three
połowy
halves
murów
wallsgen
są
are
czerwone.
red
‘Three halves of the walls are red.’
At this point, I conclude that if one pursues an approach explaining
the discussed behavior in terms of ambiguity, one is forced to assume
the cross-linguistic as well as intra-linguistic patterns in question are
just a remarkable coincidence. Nevertheless, my own view is that
14. But see Ladusaw (1982) for discussion of the asymmetry between the two quantifiers
in partitives.
44
2. Partitives and part-whole structures
systematic parallelism is something that should be approached with
serious consideration since it may reveal some deep aspects of the
nature of natural language semantics. For instance, the distinction
between collective, distributive, and cumulative understandings of
sentences involving semantically plural DPs does hold in multiple
environments in multiple languages and the research on its roots has
lead to many prominent theories in the study of pluralities (e.g., Scha
1981, Link 1983, 1984, Hoeksema 1983, Gillon 1987, Landman 1989a,b,
2000, Schwarzschild 1991, 1996, Schein 1993, Lasersohn 1995, Beck and
Sauerland 2000, Winter 2001, Champollion 2010, 2017, Dotlačil 2010,
Schmitt 2013, Zweig 2008). I would like to suggest that the discussed
behavior of part-words modified by cardinal numerals and other quantifiers
can shed new light on the matter of countability. Therefore, I
will not dismiss hastily the morphological evidence concerning the
analogy between entity and set partitives but rather I will argue that
the parallelism is not coincidental.
Crucially, there is an intriguing twist concerning count explicit
partitives suggesting that the phenomenon discussed in this section
results from an independent factor, specifically from the type of entities
plural expressions denote. In particular, the extensions of regular
plurals comprise arbitrary sums of individuals, i.e., scattered entities
which bear no topological commitments with respect to the configuration
of the parts of a plurality and it seems that natural language
does not consider such entities to be units one could quantify over.
But what if there were a language with a plural that similar to, say, the
German plural denotes sums of individuals but in addition asserts
a specific spatial relation holding between individuals making up a
plurality that guarantees that such a sum has object-like properties?
If the claim relating countability with being an integrated entity is
on the right track, one would expect that when part-words heading
partitives involving such untypical plural expressions are modified
by numerals the quantificational behavior of the whole phrase should
differ from what we observed so far. The next section will examine
the interaction between cardinals and partitives involving so-called
irregular plurals in Italian.
45
2. Partitives and part-whole structures
2.3 Italian irregular plurals
There is a relatively small set of irregular nouns in Italian that display
both morpho-syntactic and semantic peculiarities with respect
to the singular/plural distinction. These nouns form an inflectional
class whose defining characteristic is that it exhibits an idiosyncratic
agreement pattern involving a gender shift in the plural. For instance,
let us consider a representative of the class, e.g., the masculine noun
dito ‘finger’. As witnessed by the well-formed phrases in (2.53), the
singular form dito is masculine, whereas the plural dita not only shows
the irregular inflectional marker -a (regular masculine nouns end in -i
in the plural), but also triggers feminine agreement (Acquaviva 2008,
p. 125), as opposed to the regular pattern in (2.54). In particular, while
zii ‘uncles’ takes the regular plural marker -i and combines with the
masculine plural definite article i and masculine plural possessive
pronoun tuoi, dita can only be modified by their feminine equivalents,
i.e., le and tue respectively.
(2.53) Italian (Acquaviva 2008, p. 125)
a. il
thesg.m
tuo
yoursg.m
dito
fingerm
‘your finger’
b. le
thepl.f
tue
yourpl.f
dita
fingersf
‘your fingers’
(2.54) Italian (Acquaviva 2008, p. 125)
a. il
thesg.m
tuo
yoursg.m
zio
unclem
‘your uncle’
b. i
thesg.m
tuoi
yoursg.m
zii
unclesm
‘your uncles’
Forms such as dita are sometimes referred to in the literature as Italian
irregular plurals in -a (see Acquaviva 2008 and references therein) or
collective plurals, as opposed to regular distributive plurals (see Cor-
46
2. Partitives and part-whole structures
bett 2000 and references therein).15 The morpho-syntax and semantics
of such forms have been a subject of intense investigations both within
traditional Italo-Romance linguistics (Hall 1956, Brunet 1978, pp. 30–
90) as well as formally oriented approaches (Ojeda 1995, Acquaviva
2008, Kučerová 2015). Historically, the class arose as a result of the
disappearance of Latin neuters in Italo-Romance (see Löfstedt 1956,
pp. 46–51, Lausberg 1963, Rohlfs 1968), and thus all the nouns that
belong to it are inanimates usually referring to body parts, series of
events, measurement units, and mass concepts.16 In the contemporary
Italian it is restricted to approximately twenty nouns, but the pattern
used to involve many more lexemes in the past (Santangelo 1981). Interestingly,
the marker -a is nowhere else in the language the exponent
of the plural and in at least some cases it seems to be responsible for
an additional semantic effect to be discussed below.
The class of Italian irregular nouns can be divided into two groups.
The first group consists of nouns that take irregular plural forms
exclusively, i.e., do not pluralize in -i altogether. Table 2.2 lists some of
the examples. Although plural forms such as uova ‘eggs’ and centinaia
‘hundreds’ are morphologically irregular, they do not seem to display
any semantic idiosyncrasy. They simply do what regular plurals do,
i.e., refer to pluralities of entities satisfying the property introduced by
the nominal stem. With this respect, there seems to be no difference
between irregular plurals and regular forms such as pomodoro ‘tomato’
∼ pomodori ‘tomatoes’ and miliardo ‘billion’ ∼ miliardi ‘billions’. In both
cases, the pluralized noun merely denotes a collection of eggs or fruits
and a number of hundreds or billions, respectively.
More interestingly, however, there are also a number of nouns with
both regular and irregular plural counterparts such as those listed in
Table 2.3.17
15. Kučerová (2015) uses the terms the relational pattern and atomic pattern.
16. I will not discuss irregular measure nouns and mass terms here. Though I
acknowledge that they might constitute a challenge for a unified semantic interpretation
of the class (cf. Ojeda 1995 and Acquaviva 2008), this fact does not affect the
core observation to be presented in this section.
17. Acquaviva (2008, pp. 124–129) introduces a more subtle classification by distinguishing
also a class which does have the regular -i alternant but it is dialectally
restricted, and thus not available for all speakers of Italian, e.g., lenzuolo ‘sheet’ ∼
%lenzuoli ‘sheets’ ∼ lenzuola ‘sheets (bed linen)’. Moreover, he lists irregular pluralia
tantum, i.e., feminine plural -a forms with no singular nor regular plural counter-
47
2. Partitives and part-whole structures
singular regular plural irregular plural
uovo ‘egg’ *uovi uova ‘eggs’
riso ‘laughter’ *risi risa ‘pearls of laughter’
paio ‘pair’ *paii paia ‘pairs’
centinaio ‘hundred’ *centinai centinaia ‘hundreds’
miglio ‘mile’ *migli miglia ‘miles’
Table 2.2: Italian nouns with irregular plurals exclusively (based on
Acquaviva 2008, pp. 126–127)
singular regular plural irregular plural
muro ‘wall’ murri ‘walls’ murra ‘walls (in a complex)’
osso ‘bone’ ossi ‘bones’ ossa ‘bones (in a skeleton)’
filo ‘thread’ fili ‘threads’ fila ‘threads (in a fabric)’
fondamento ‘basis’ fondamenti ‘bases’ fondamenta ‘foundations’
urlo ‘shout’ urli ‘shouts’ urla ‘shouts (in a series)’
Table 2.3: Italian nouns with both regular and irregular plural coun-
terparts
Though at least to some extent the effect seems to depend on a
particular idiolect and register,18 in the cases of doublets concurrent
irregular feminine -a plurals tend to differ from run-of-the-mill masculine
-i forms in meaning. As witnessed by the translations of irregular
forms given in Table 2.3, there is often a sense of collectivity or cohesion
in addition to the ordinary plural interpretation. For instance,
the irregular form ossa ‘bones’ evokes an interpretation that the bones
belong together as in a skeleton, whereas regular ossi ‘bones’ simply
refers to a plurality of unrelated bones, see (2.55b) (Corbett 2000, p.
153). Similar, the only way to interpret mura ‘walls’ is by picturing a
walled complex, e.g., a perimeter of city walls, see (2.55a) (Acquaviva
2008, p. 150). In both cases, there is a strong intuition that the referents
parts, e.g., vestigia ‘relics’, as well as fossilized irregular plural forms found only in
idiomatic expressions. For sake of clarity of the main argument, I will not discuss
all the nuanced morphological and semantic intricacies here and I refer the reader
to Acquaviva’s study for more details.
18. See Acquaviva 2008 for a detailed discussion.
48
2. Partitives and part-whole structures
of the irregular plural forms are naturally related and due to this kind
of semantic flavor Italian -a plurals have been analyzed as collectivizers
of a particular sort (Ojeda 1995) or expressions inherently encoding
the cohesion of referents (Acquaviva 2008).
Aware of all the subtleties within the class, I will focus my attention
only on expressions following the second pattern in Table 2.4 (Acquaviva
2008). Specifically, plural forms of nouns that do not pluralize
in -i at all get a regular semantic interpretation, i.e., merely refer to
pluralities of individuals. On the other hand, the meaning of -a plural
forms of nouns having doublets differ from the meaning of those in -i.
In such cases, regular plurals are semantically regular, whereas irregular
plurals denote pluralities of objects conceptualized as constituents
of a larger cohesive entity.19
singular plural
regular cohesive
-o -a *
-o -i -a
Table 2.4: Semantic interpretation of Italian plurals
In the first formal semantic treatment of Italian irregular plurals,
Ojeda (1995) proposes an account on which expressions such as braccia
‘arms’ and ginocchia ‘knees’ do not denote mere sets of body parts but
instead make reference to naturally occurring sets of arms or knees, i.e.,
pairs of arms or knees of one body.20 The core idea of the approach is
that nouns denote with a certain indeterminacy of individuation and
that the plural is the unmarked number (Ojeda 1993). Hence, plural
markers are mere identity functions which do not affect the reference
of the stem. On the other hand, the singular restricts the denotation of
the stem to the domain of atomic individuals. Within this approach,
nominal stems in regular/irregular doublets are in fact homophonous
19. Acquaviva (2008, pp. 151–152) argues against separating the two classes due
to the fact that there exist irregular plurals without an -i counterpart that unlike
singulars get a mass interpretation, e.g., midolla ‘marrow’. For the sake of clarity, I
will ignore such intricacies here.
20. In a sense, irregular plurals somewhat resemble duals in that they both tend to
denote natural aggregates (see also Acquaviva 2008, p. 152 for discussion).
49
2. Partitives and part-whole structures
expressions having distinct denotations and the distribution of the
plural markers -i and -a is constrained in that each of them can only
combine with a semantic object of a particular type. In other words, -a
does not have a different semantic value than -i but they differ in their
input requirements. Specifically, the regular marker -i selects for stems
denoting sets of all sets of singular referents including singletons, e.g.,
the set of all sets of individual arms. On the other hand, the irregular
ending -a combines with stems that attribute a collective meaning
associated with the plurality the whole noun refers to. In particular, -a
selects for a homophonous stem denoting the set of natural aggregates
of objects within the universe of discourse, e.g., the set of pairs of
knees of each individual and pluralities formed by such pairs rather
than arbitrary sums of knees.
However, Acquaviva (2008) rejects the view that the collectivizing
effect is the characteristic for the whole class. Instead, he proposes that
the common denominator of irregular plurals is that they denote entities
conceptualized as weakly individual undifferentiated objects and
argues for a lexicalist approach. In fact, Ojeda (1995) himself admits
that the proper analysis of irregular plurals referring to mass concepts
such as cervello ‘brain’ ∼ cervelli ‘brains’ ∼ cervella ‘brains (mass)’ is
challenging for his account. Furthermore, as pointed out by Acquaviva
a straightforward compositional analysis of the relationship between
-a and the stem is problematic. One reason is that irregular plurals that
lack regular -i alternants do not show any semantic peculiarity and a
mechanism explaining a different selectional requirement would have
to be postulated.
In order to avoid controversy I will focus on examples that indisputably
invoke the sense of collectivity, specifically I will test the
behavior of the irregular forms mura ‘walls (in a complex)’ and ossa
‘bones (in a skeleton)’, as demonstrated in the series of alternations in
(2.55).21
21. In (2.55), I use morpho-syntactic glosses. However, in the following part of the
text I will gloss irregular plurals following the pattern as coll.
50
2. Partitives and part-whole structures
(2.55) Italian
a. muro ∼
wallsg.m
muri ∼
wallpl.m
mura
wallpl.f
‘wall ∼ walls ∼ walls (in a complex)’
b. osso ∼
bonesg.m
ossi ∼
bonepl.m
ossa
bonepl.f
‘bone ∼ bones ∼ bones (in a skeleton)’
In this study, I refrain from taking a stand regarding the issue of the internal
syntactic and semantic structure of Italian irregular plurals and
I remain agnostic with respect to the exact compositional contribution
of the irregular marker -a, if any. However, I would like to suggest another
perspective on reflecting the meaning of irregular forms falling
into the pattern when they compete with regular -i plurals. In the next
section, I will discuss novel data suggesting that at least a subset of
irregular plurals is conceptualized as aggregates of entities related to
each other by means of spatial connectedness.
2.3.1 Aggregate meaning
Given the discussed semantic evidence and previous accounts by
Ojeda (1995) and Acquaviva (2008), it is plausible to posit that when an
Italian plural in -a alternates with a regular -i form, its extension does
not simply comprise arbitrary sums of individuals but rather a more
complex type of entities. Specifically, I propose that the topological
notion of connectedness or stable proximity of parts making up a
plurality denoted by the irregular plural is involved in the way how
the referents of such an expression are conceptualized. As already
indicated in the previous section, at least in some cases there is a good
reason to assume that the reported collective flavor is due to the fact
that Italian irregular plurals encode integrated plural individuals, i.e.,
pluralities that unlike referents of regular plurals involve particular
spatial relations holding between individual parts, and thus have the
potential to form cohesive aggregates and even individuated wholes.
For instance, Kučerová (2015) gives an example of the alternation
between dito ‘finger’ ∼ diti ‘fingers’ ∼ dita ‘fingers (of a hand)’.22 If one
22. In some varieties of Italian, the regular plural diti ‘fingers’ is either unavailable
or available only marginally as a stigmatized form (Acquaviva 2008, p. 126).
51
2. Partitives and part-whole structures
refers to a plurality of fingers attached to a hand, then the irregular
form dita would be used. On the other hand, when talking about
detached fingers, e.g., in an anatomy class setting, the regular plural
diti is the appropriate form to use. It seems that in patterns such as dito
∼ diti ∼ dita the irregular form indicates a particular type of relation
holding between individual singular objects referred to by the noun,
namely a topological relation giving rise to a cluster consisting of
fingers.
To illustrate that, let us imagine a military context where a siege
is led by the famous general Garibaldi who attempts to destroy the
defensive fortification of the attacked city. Now, consider the sentences
in (2.56). If (2.56a) is true, then there is no wall any more, just a pile
of stones. Similar, uttering (2.56b) implies that there are no walls any
more, just a number of piles of stones. However, (2.56c) can also mean
something else, namely that the walls are not completely dismantled
and though they probably received some damage during the siege,
they are still recognizable as such, but importantly they are no longer
connected. In other words, there is no complex of walls anymore and
what is left is only a discontiguous collection of dissociated objects,
i.e., independent walls.
(2.56) Italian (Enrico Flor, p.c.)
a. Garibaldi
Garibaldi
ha
has
smantellato
dismantled
il
the
muro.
wall
‘Garibaldi has dismantled the wall.’
b. Garibaldi
Garibaldi
ha
has
smantellato
dismantled
i
the
muri.
walls
‘Garibaldi has dismantled the walls.’
c. Garibaldi
Garibaldi
ha
has
smantellato
dismantled
le
the
mura.
wallscoll
‘Garibaldi has dismantled the walled complex.’
The predicate smantellare ‘dismantle’ is what I call a verb of separation,
i.e., a transitive expression sensitive to the part-whole structure of
its first argument. When applied to a DP, it implies that the spatial
connection between the parts of its referents has been undone or, in
other words, the denoted individual has been disintegrated. Since referents
of count singular nouns are arguably conceptualized as clusters,
52
2. Partitives and part-whole structures
i.e., pluralities of interrelated objects, if the complement of smantellare
in a true sentence is a singular count DP, it follows that the entity
denoted by that noun has ceased to exist as an integrated whole. To
put it another way, the connections holding between its parts have
been unbound and what is left is merely an arbitrary collection of
pieces which do not form an individuated object anymore. On the
other hand, regular plural nouns refer to arbitrary sums of objects, i.e.,
there is no sense in which parts of a plurality are related to each other
other than by being members of a certain set. Denotations of regular
plurals are constituted by scattered collections of singular individuals
and in that sense plural individuals constitute a very different type of
entity than referents of count singulars. Therefore, a verb of separation
such as smantellare cannot disintegrate a plurality simply because
there are no topological relations involved in the first place and instead
it operates on the part-whole structures of particular singular
individuals making up a plurality. As a result, what is obtained is a
number of dismantled singular objects. However, irregular plurals
such as mura denote cohesive aggregates, and thus arguably involve
the notion of spatial connection on top of the plurality meaning. In
other words, entities denoted by such expressions are pluralities but
at the same time they are similar to singular objects in that their parts,
i.e., members of a collection such as individual walls, constitute a
cohesive entity. Consequently, verbs of separation can dissolve the
topological structure connecting parts of a plurality and as a result
another meaning arises, as witnessed by the possible interpretations
of a sentence such as (2.56c).
Contrasts such as those in (2.56) show that the difference between
muri and mura lies in that the first denotes a plurality of unrelated
walls, whereas the latter makes reference to a cohesive aggregate of
walls, i.e., walls making up a particular architectonic unit. In other
words, the irregular plural encodes the cohesion of elements and
often invokes the meaning of an integrated whole with a complex
inner structure of associated parts connected to each other, while the
regular -i form does not, and thus simply refers to an arbitrary sum of
individuals of a particular kind.
This claim seems to be further strengthened by the fact that a significant
number of nouns taking the irregular plural are words referring to
body parts, i.e., entities naturally occurring in cohesive aggregates and
53
2. Partitives and part-whole structures
typically situated in great proximity, involving simultaneous movement,
whose collections are often conceptualized as functional units.
Another conspicuous subset includes nouns denoting objects typically
consisting of multiple elements spatially associated with each other,
e.g., city walls, foundations, and fabrics of threads, as well as a genuine
group noun paio ‘pair’. Yet another subclass involves nominals
referring to series of events such as grida or urla ‘shouts (in a series)’.
Such expressions denote pluralities of events associated with each
other, and thus making up a single complex eventuality. For instance,
urla would be true of a series of cries by a single agent or a plural
individual, e.g., audience, which are perceptually contiguous in time
and/or space (Acquaviva 2008, pp. 149–150). Though the meaning of
this subclass seems to be intuitively closely related to the semantic
effect of connectedness discussed here, I will refrain from speculating
how exactly the proposed interpretation could be extended to
eventualities and simply note the analogy.
Before we move on to the discussion of the interaction between
cardinals and partitives in Italian, let us conclude that a subset of
irregular plurals refers to cohesive aggregates of objects associated
with each other. The evidence concerning verbs of separation suggests
that this association could be understood by means of a particular
spatial configuration.
2.3.2 Interaction with cardinals in partitives
The core observation to be discussed in this section concerns the contrast
between explicit partitives with regular plurals and explicit partitives
with irregular plurals displaying the cohesive effect discussed
above, e.g., muri ∼ mura and ossi ∼ ossa. As already discussed in Section
2.2.1, Italian is similar to many other languages in that it allows
for both singulars and plurals in explicit partitives. Moreover, Italian
displays the same pattern as, e.g., German with respect to the quantificational
properties of partitive words in explicit and proportional
partitives. Specifically, similar to count explicit entity partitives, see
(2.57a), partitives involving plural DPs c-commanded by pluralized
parte modified by a numeral such as in (2.57b) always quantify over material
parts of objects and not over sets of wholes although unmodified
set partitives get a part-of-a-plurality reading. Intriguingly, everything
54
2. Partitives and part-whole structures
changes in count explicit partitives involving -a forms following the
discussed alternating pattern, as in (2.55). When an irregular doublet
is swapped for a regular one, the range of possible interpretations
significantly extends.
(2.57) Italian (Enrico Flor, p.c.)
a. Due
two
parti
parts
del
of-the
muro
wall
sono
are
rosse.
red
‘Two parts of the wall are red.’
b. Due
two
parti
parts
dei
of-the
muri
walls
sono
are
rosse.
red
‘Two parts of the walls are red.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
c. Due
two
parti
parts
delle
of-the
mura
wallscoll
sono
are
rosse.
red
‘Two parts of the walled complex are red.’
(i) part-of-a-singularity reading
(ii) part-of-a-plurality reading
For instance, imagine a scenario of a walled city whose fortifications
consist of six walls w1, . . . , w6 connected to each other in such a way
that they form a hexagon, as depicted in Figure 2.1. Given that scenario,
as one could expect the sentence in (2.57c) would be true if two parts
of two distinct walls making up the complex were red, e.g., a part
of w1 and a part of w3. But surprisingly unlike (2.57b) it would also
be true if two distinct individual walls making up the complex were
red, e.g., the wall w1 and the wall w3, or if two continuous sections of
walls making up the complex were red, e.g., the subsets {w1, w2} and
{w4, w5}. In other words, the meanings that are impossible in explicit
set partitives with regular plurals, are easily available with irregular
plurals. However, as already mentioned there is a crucial constraint
on possible interpretations of (2.57c). Namely, one can count parts of
the plurality consisting of the walls in question as long as they form a
contiguous section of the whole structure, i.e., (2.57c) would not be
true of two arbitrary pluralities constituted by walls that are not in a
55
2. Partitives and part-whole structures
tangential relationship with each other. In particular, it would be false
if, say, the wall w1 as well as the walls {w3, w5} were red.23
w3
w4
w5
w6
w1
w2
Figure 2.1: Sections of city walls
Similar, assume a bit morbid scenario involving a decaying human
bone or skeleton and consider the contrasts between the examples
in (2.58). Given that there is one relevant bone, the entity partitive
in (2.58a) gets a part-of-a-singularity reading as usual, and thus the
sentence would be true if there were three decayed spots on, e.g., a femur.
This is not surprising and similar the interpretation of (2.58b) fits
the discussed semantic shift regarding modified explicit set partitives.
However, the partitive involving the irregular plural ossa in (2.58c) behaves
differently. Seemingly, the most readily available interpretation
of the example concerns three groups of connected bones making up
the skeleton, e.g., it would be true of a decayed femur and knee, a
decayed ulna and radius, and the decayed neck and skull. The sentence
would be felicitous if three separate bones decayed though, e.g.,
a femur, ulna, and the skull. Furthermore, similar to (2.58b) it would
be judged true if three were decayed parts on three distinct bones,
e.g., a part of the femur, a part of the ulna, and a part of the skull. But
again, the reading that (2.58c) cannot get involves three collections of
disconnected bones. For instance, it would be infelicitous in a scenario
where, say, a femur and the skull, a radius and a knee, and a rib and
an ankle bone are decayed. That is because these pairs of bones do
23. I assume here the ‘exact n’ semantics for numerals (e.g., Horn 1992, Geurts 1998,
Breheny 2005; see also Geurts 2006 for an overview). Notice that the judgments do
not change if the numeral is modified by overt exactly.
56
2. Partitives and part-whole structures
not form cohesive wholes, i.e., topologically continuous entities of a
particular sort, and consequently cannot be conceptualized as units
one could quantify over.
(2.58) Italian (Enrico Flor, p.c.)
a. Tre
three
parti
parts
del
of-the
osso
bone
sono
are
cariate.
decayed
‘Three parts of the wall are decayed.’
b. Tre
three
parti
parts
degli
of-the
ossi
bones
sono
are
cariate.
decayed
‘Three parts of the bones are decayed.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
c. Tre
three
parti
parts
delle
of-the
ossa
bonescoll
sono
are
cariate.
decayed
‘Three parts of the skeleton are decayed.’
(i) part-of-a-singularity reading
(ii) part-of-a-plurality reading
Table 2.5 summarizes the patterns we have observed so far. Specifically,
the quantificational behavior of the Italian part-word parte depends
on certain denotational properties of expressions it combines with.
singulars regular pl irregular pl
bare count bare count bare count
subatomic quantification *
quantification over wholes * * *
Table 2.5: Properties of Italian parte ‘part’
The investigation into the interaction between cardinal numerals
and partitives involving Italian irregular plurals leads to a somewhat
surprising conclusion. It turns out that the cross-linguistic observation
concerning different properties of partitive words in entity and set
partitives as discussed in Section 2.2.2 actually does not provide evidence
for distinct part-whole structures for singulars and plurals, as
suggested by Schwarzschild (1996). That is not to say that there is no
difference in how constitution of objects as opposed to sums thereof
57
2. Partitives and part-whole structures
is conceptualized. That is to say that the Italian data suggest that the
difference does not regard the relation between parts and a whole, but
rather that it regards the relation governing how parts are organized
in space with respect to each other. In particular, regular plurals encode
no topological relations between entities making up pluralities
or, in other words, simply denote arbitrary sums of individuals. Consequently,
parts of a plurality do not form an integrated entity that
could be considered a unit one could assign a number which seems
to be a prerequisite for quantification. On the other hand, despite
the plurality inference Italian irregular forms in -a differ significantly
from regular plurals and somewhat resemble singulars in that they
do encode how parts are organized with respect to each other. Specifically,
their extensions consist of cohesive aggregates, i.e., pluralities
arranged in a particular spatial configurations such that individual
objects making up a plurality are either connected or at least stay in a
relatively stable proximity. Given that the domain of quantification
consists of entities conceptualized as such clusters, counting parts
consisting of several singularities is possible as long as they form integrated
fragments or continuous sections of the whole. Therefore, I
posit that the explanation of the patterns summarized in Table 2.1 and
2.5 should be contingent on the manner how we conceptualize topological
relations holding between parts of different types of entities as
well as on the sensitivity of quantificational operations employed in
natural language to such relations.
2.4 Parts vs. subsets
The Italian data provide intriguing evidence but let us hold on for a
minute and reconsider the claim that being countable is a property
of only those entities that are integrated wholes. One could wonder
to what extent the fact that count explicit partitives involving regular
plural DPs cannot quantify over sums and can only refer to parts
of singular objects tells us anything about the nature of countability.
After all, to the extent they are pragmatically admissible sentences in
which the cardinal modifies an expression such as subset instead of
a part-word are definitely compatible with an interpretation where
the domain of quantification consists of collections of individuals, i.e.,
58
2. Partitives and part-whole structures
felicitous when counting subsets of a larger set. For instance, unlike
(2.59b) the sentence in (2.59a) would be true in a situation where there
are, say, ten relevant apples {a1, . . . , a10} and three arbitrary subsets
of the apples got spoiled, e.g., {a1, a2}, {a3, a4, a5} and {a6, a7}.24 In
other words, is it really the case that counting requires units that are
integrated objects? Or maybe there is simply something weird about
partitive words and we should not jump to a conclusion concerning
countability based on the partitive data?
(2.59) Polish
a. Trzy
three
podzbiory
subsets
jabłek
applesgen
zgniły.
rot
‘Three subsets of the apples got spoiled.’
b. Trzy
three
części
parts
jabłek
applesgen
zgniły.
rot
‘Three parts of the apples got spoiled.’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
At first blush, this seems to be a justified objection and I think it should
be taken seriously. However, there are a number of empirical facts
suggesting that what happens in (2.59a) is very different from what
happens in (2.59b). First of all, it is crucial to emphasize that natural
language expressions such as set or subset are not barely linguistic
reflections of set-theoretic concepts involving notions such as {. . . } or
relations ⊆ and ⊂. On the contrary, such expressions have a number
of properties that classify them on a par with group nouns such as
committee, band, or pile.25 On the other hand, similar to bare plurals
set partitives refer to pluralities but part-words are not collectives, as
attested by a number of tests involving predicate non-sharing.
24. This reading is different from a type of reading discussed in Section 3.3 where
reference to three-fourths of the total volume of the apples would be involved. What
is crucial for the discussion of (2.59b) is that the quantified subsets of individuals
cannot be arbitrary.
25. In the following text, I will ignore the evidence for distinguishing between
different classes of group nouns (see Pearson 2011, Henderson 2017, Dočekal and
Wągiel 2018) and assume that for our purposes there are no relevant differences
between nouns such as band and pile.
59
2. Partitives and part-whole structures
It has been proposed in the literature that group nouns and plural
NPs do not co-refer since one can find multiple contrasts concerning
(in)compatibility with certain predicates (Barker 1992, Schwarzschild
1996). For instance, the sentence with a collective DP as a subject is
perfectly fine with the predicate have five members, see (2.60a), but its
altered version involving a plural DP as in (2.60b) is not.
(2.60) English (Lønning 1987, p. 153)
a. The committee has five members.
b. #The men have five members.
Now, let us test the behavior of the Polish part-word część ‘part’ with
respect to plurals, regular collective nouns, and the expression podzbiór
‘subset’.26 The contrasts in (2.61) show that plurals and explicit set
partitives differ from the typical group noun stos ‘pile’ as well as
podzbiór in that they are not compatible with predicates of the type have
ten elements which are assumed to indicate bunches (Schwarzschild
1996).
(2.61) Polish
a. #Jabłka
apples
mają
have
dziesięć
ten
elementów.
elementsgen
b. #Część
part
jabłek
applesgen
ma
has
dziesięć
ten
elementów.
elementsgen
c. Stos
pile
jabłek
applesgen
ma
has
dziesięć
ten
elementów.
elementsgen
‘The pile of apples has ten elements.’
d. Podzbiór jabłek ma dziesięć elementów.
subset applesgen has ten elementsgen
‘The subset of apples has ten elements.’
26. It needs to be admitted that due to the restricted distribution of the noun podzbiór
(often limited to mathematical and computational contexts) the following sentences
involving this expression is not how people usually talk. Rather, these examples
sound nerdy or somewhat like clumsily worded math exercises rather than something
a regular person would say. Nevertheless, despite the slight lexical oddity
the judgments concerning compatibility with different predicates, interactions in
co-variational contexts and with overt distributive operators seem to be very clear.
60
2. Partitives and part-whole structures
The same pattern can be observed when the examined expressions
appear as first arguments of the two-place predicate należy do ‘belongs
to’ which arguably designates assignment of objects to some abstract
entity representing a collection. Though both (2.62c) and (2.62d) are
felicitous, the sentences including plurals and explicit set partitives in
(2.62a) and (2.62b), respectively, cannot saturate the predicate.
(2.62) Polish
a. #Ta
this
papierówka
white-transparent
należy
belongs
do
to
jabłek.
applesgen
b. #Ta
this
papierówka
white-transparent
należy
belongs
do
to
części
partgen
jabłek.
applesgen
c. Ta
this
papierówka
white-transparent
należy
belongs
do
to
stosu
pilegen
jabłek.
applesgen
‘This white transparent apple belongs to the pile of ap-
ples.’
d. Ta
this
papierówka
white-transparent
należy
belongs
do
to
podzbioru
subsetgen
jabłek.
applesgen
‘This white transparent apple belongs to the subset of
apples.’
Another contrast between plurals and collective nouns concerns distribution
with reciprocals, as demonstrated in (2.63).
(2.63) English (Schwarzschild 1996, p. 168)
a. The rocks in that pile are touching each other.
b. #That pile is touching each other.
Polish examples such as those in (2.64) show that there is a contrast
between plurals and explicit set partitives on the one hand and typical
collectives such as stos and podzbiór on the other. The first combine
felicitously with reciprocal expressions, whereas the latter do not. In
particular, (2.61a) and (2.61b) indicate either strong or intermediate
reciprocity (Fiengo and Lasnik 1973, Dalrymple et al. 1998, Beck 2001),
i.e., either each of the apples in question is touching all the other apples
or any two of the apples are connected by a chain of apples that stand
61
2. Partitives and part-whole structures
in the reciprocal relation. Nevertheless, it is impossible to define such
truth conditions for (2.61c) and (2.61d).
(2.64) Polish
a. Jabłka
apples
dotykają
touch
się
refl
nawzajem.
each-other
‘The apples are touching each other.’
b. Część
part
jabłek
applesgen
dotyka
touches
się
refl
nawzajem.
each-other
‘Some of the apples are touching each other.’
c. #Stos
pile
jabłek
applesgen
dotyka
touches
się
refl
nawzajem.
each-other
d. #Podzbiór
subset
jabłek
applesgen
dotyka
touches
się
refl
nawzajem.
each-other
A similar distribution is attested in contexts where the reciprocal is
covert, see (2.65), including co-variational environments such as (2.66).
Specifically, plurals do combine with predicates triggering covariation,
whereas group nouns do not.
(2.65) English (Dougherty 1970)
#The trio collided.
(2.66) English (Schwarzschild 1996, p. 168)
a. The members of group A live in different cities.
b. #Group A lives in different cities.
Let us then consider Polish examples involving plurals, explicit partitives,
group nouns, and phrases c-commanded by podzbiór in subject
position, as provided in (2.67) and (2.68). In (2.67a), the sentenceinternal
different expression w różnych miastach ‘in different cities’ is
bound within a clause to express covariation with a plural argument
due to a built-in distributive operator (Carlson 1987, Beck 2000, Dotlačil
2010, Brasoveanu 2008). As a result, the sentence would be true if
for each of the musicians it was the case that they lived in a city that
is different than the cities other musicians live in. Due to this kind of
meaning, sentence-internal different expressions are good indicators
of distributivity.
62
2. Partitives and part-whole structures
(2.67) Polish
a. Muzycy
musicians
mieszkają
live
w
in
różnych
differentloc
miastach.
citiesloc
‘The musicians live in different cities.’
b. Część
part
muzyków
musiciansgen
mieszka
lives
w
in
różnych
differentloc
miastach.
citiesloc
‘Some of the musicians live in different cities.’
c. #Zespół
band
mieszka
live
w
in
różnych
differentloc
miastach.
citiesloc
d. #Podzbiór
subset
muzyków
musiciansgen
mieszka
lives
w
in
różnych
differentloc
miastach.
citiesloc
Similar, the sentences in (2.68) include the distributive preposition po
which is a marker for distant distributivity in Polish (e.g., Przepiórkowski
2008, 2010, 2014, Przepiórkowski and Patejuk 2013).27 As such, similar
to binominal each (e.g., Safir and Stowell 1988, Zimmermann 2002,
Dotlačil 2012, Champollion 2012a) and distributive numerals (e.g., Gil
2002, Oh 2005, Cable 2014, Hofherr and Etxeberria 2017) po marks the
distributive share and distributes it over the distributive key denoted
by a different phrase, here the subject DP. As a result, the possibility of
a collective reading is ruled out and a sentence in which po occurs is
obligatorily interpreted distributively, e.g., in (2.68a) each of the musicians
played a song. This conflicts with the semantics of group nouns
which in general do not allow for the access to individual members of
a collection, hence (2.68c) is an awkward sentence. As witnessed by
the contrast betwwen (2.68b) and (2.68d), explicit set partitives pattern
with plurals, whereas podzbiór behaves as typical collectives.
(2.68) Polish
a. Muzycy
musicians
zagrali
played
po
distr
piosence.
songloc
‘The musicians played a song each.’
27. For detailed studies on distant distributivity and distributive po in Slavic see
also, e.g., Franks (1994, 1995), Harves (2003), Knežević (2015).
63
2. Partitives and part-whole structures
b. Część
part
muzyków
musiciansgen
zagrała
played
po
distr
piosence.
songloc
‘Some of the musicians played a song each.’
c. #Zespół
band
zagrał
played
po
distr
piosence.
songloc
d. #Podzbiór
subset
muzyków
musiciansgen
zagrał
played
po
distr
piosence.
songloc
Yet another diagnostic involves the VP disjunction test proposed by
(de Vries, 2015, pp. 30–31). When the test is applied, część does not
exhibit the kind of behavior associated with group nouns. To illustrate
this, let us consider (2.69a) where the definite plural DP the children
serves as a subject of the disjunctive sentence. There are two possible
interpretations of (2.69a). On the collective interpretation, see (2.69a-i),
the disjunction singing or dancing holds of the plurality as a whole.
In other words, it is either the case that all the relevant children are
singing or that all the relevant children are dancing. However, there is
yet another reading of (2.69a), namely the distributive interpretation,
see (2.69a-ii), according to which the disjoined VPs hold not of the
plurality of children as a whole but rather of each individual child.
Thus, some of the children might be singing while others might be
dancing. Crucially, when the definite plural is replaced with a group
noun such as team as in (2.69b), the distributive interpretation is no
longer available. The only reading such a sentence can get is that the
whole plurality was either involved in singing or in dancing.
(2.69) English (de Vries 2015, p. 31; adapted)
a. The children are singing or dancing.
(i) collective reading
(ii) distributive reading
b. The team is singing or dancing.
(i) collective reading
(ii) #distributive reading
Now, let us examine the behavior of Polish nouns część and podzbiór
compared to plurals and group nominals in sentences involving VP
disjunction such as those in (2.70). The example (2.70a) can have either
a collective or a distributive reading, i.e., it would be true either when
64
2. Partitives and part-whole structures
all the children are dancing or singing or when some dancing whereas
others are singing. On the other hand, a sentence with a group noun
as a subject like (2.70b) can only be interpreted collectively. That is
also the case in (2.70d), i.e., all the children in the relevant subset need
to be either dancing or singing. However, (2.70c) is felicitous in both
collective and distributive scenarios. Once again, the part word część
patterns with plurals rather than collectives which definitely proves
that it differs from expressions such as subset.
(2.70) Polish
a. Dzieci
children
tańczą
dance
lub
or
śpiewają.
sing
‘The children are dancing or singing.’
(i) collective reading
(ii) distributive reading
b. Grupa
group
dzieci
childrengen
tańczy
dances
lub
or
śpiewa.
sings
‘The group of children is dancing or singing.’
(i) collective reading
(ii) #distributive reading
c. Część
part
dzieci
childrengen
tańczy
dances
lub
or
śpiewa.
sings
‘Some children are dancing or singing.’
(i) collective reading
(ii) distributive reading
d. Pozbiór
subset
dzieci
childrengen
tańczy
dances
lub
or
śpiewa.
sings
‘A subset of children is dancing or dancing.’
(i) collective reading
(ii) #distributive reading
Notice that whatever theory of the relationship between agreement
and collectivity/distributivity one might have (e.g., de Vries 2015,
see also Schwarzschild 1996 and Pearson 2011 for the discussion of
group nouns in British English as well as Bosnić 2016 for experimental
data concerning distributivity and agreement mismatches in BCS), it
cannot explain the semantic difference between part expressions and
group nouns. In Polish, część similar to zespół (and any other group
65
2. Partitives and part-whole structures
noun) triggers singular agreement on the verb. As witnessed by the
glosses in the examples provided above, in sentences with both explicit
partitives and group nouns, see (2.67b)–(2.67c) and (2.70c)–(2.70b), the
verbs agree in number with the subjects. Furthermore, both in (2.68b)
and (2.68c) the past participles agree with część and zespół in gender
by taking the feminine and masculine form, respectively. Therefore, it
is not the agreement pattern but rather lexical properties of część that
are accountable for the semantic difference.
Finally, despite being distinct syntactic categories partitive words
pattern with cardinal numerals in that in general they allow for modification
by class A/B modifiers such as more than and at least (Nouwen
2010, 2016, Brasoveanu 2012), see (2.71).28 This again suggests that
their nature is purely quantificational.
(2.71) Polish
a. Co najmniej
at-least
część
part
murów
wallsgen
jest
is
czerwona.
red
‘At least some of the walls are red.’
b. Co najmniej
at-most
część
part
murów
wallsgen
jest
is
czerwona.
red
‘At most some of the walls are red.’
c. Co najmniej
at-least
połowa
half
murów
wallsgen
jest
is
czerwona.
red
‘At least half of the walls are red.’
d. Co najwyżej
at-most
połowa
half
murów
wallsgen
jest
is
czerwona.
red
‘At most half of the walls are red.’
e. Co najwyżej
at-least
pięć
five
murów
wallsgen
jest
is
czerwonych.
red
‘At least five of the walls are red.’
f. Co najwyżej
at-most
pięć
five
murów
wallsgen
jest
is
czerwonych.
red
‘At most five of the walls are red.’
28. Since there are some restrictions with this respect which I believe result from
the indefinite nature of some partitive words, I present here only parallel examples
with class B modifiers.
66
2. Partitives and part-whole structures
The results of a battery of tests applied to Polish nouns część ‘part’ and
podzbiór ‘subset’ unequivocally indicate that podzbiór is a group noun,29
and thus triggers obligatory collective interpretations of sentences in
which it occurs, whereas część patterns with plurals including being
ambiguous between collective and distributive readings. This is a
significant contrast between the expressions in question suggesting
that their extensions differ in a significant way. In particular, it has
been commonly advocated in the literature that group nouns do not
refer to pluralities but rather to atomic individuals, i.e., somewhat
abstract singularities30 that might be associated with their members
(e.g., Barker 1992, Schwarzschild 1996, Winter 2001, Champollion
2017).31 Whatever extraordinary properties such atoms have, it is
plausible to assume that they are conceptualized as something distinct
from mere sums of objects.
Therefore, I conclude that explicit partitives refer to genuine pluralities
whereas group nouns including expressions such as set and
subset denote abstract entities that are associated with pluralities of
their members but are ontologically distinct from them. Having this
in mind, there is nothing surprising about the fact that count explicit
set partitives do not quantify over sums of individuals and in fact, it
turns out that part-words actually pattern with regular nominals, as
will be shown below.
It has been observed for a long time that the semantics of numeral
phrases in languages such as English poses a compositional puzzle.
On the one hand, cardinal numerals higher than one combine only
with plural nominals, on the other, plural NPs modified by cardinals
are interpreted as singular expressions. Specifically, although plurals
denote sums of entities, the domain of quantification is always a set
of atomic individuals (Kratzer 1989, Chierchia 1998b, Landman 2000;
see also Kobuchi-Philip 2006). Contrary to what one would expect if
29. Schwarzschild (1996, p. 168) actually lists set among bunch nouns.
30. Accidentally, this corresponds to the fact that in set theory sets are considered
as objects in their own right.
31. But see, e.g., Link (1984, 1998), Landman (1989a,b, 2000), de Vries (2015) for an
alternative view on which a group noun denotes a special type of plurality shifted
to an impure atom or group-atom depending on a particular approach. Pearson
(2011), on the other hand, proposes that group nouns are predicates of individual
concepts.
67
2. Partitives and part-whole structures
numerals simply counted elements in a given set (e.g., Barwise and
Cooper 1981) and combined with plural nouns in a straightforward
manner, cardinals ignore sums of objects and assign numbers only
to singular individuals. Actually, this fact has been already realized
in the system of Link (1983) where the extension of a phrase such as
three children is supposed to contain only special types of elements,
namely exactly three atoms. To illustrate why such a restriction is
necessary, let us consider (2.72). Let us assume that there are four
relevant children in the universe of discourse, say, Anne, Betty, Carl,
and Danny, i.e., a, b, c, and d respectively. Then the meaning of the
plural noun children is the set involving both atomic children and all
the sums generated by joining the atoms, see (2.72b). If three simply
counted the elements in the set denoted by the modified noun, among
the possible verifications of the sentence in (2.72a) would be (2.72c)
and (2.72d) since in both cases the set of children that slept consists of
three elements.
(2.72) a. Three children slept.
b. children = {a, b, c, d, ab, ac, ad, bc, bd, cd, abc, abd, acd,
bcd, abcd}
c. three children ∩ slept = {a, ab, abcd}
d. three children ∩ slept = {a, b, ab}
The truth conditions in (2.72c) and (2.72d) predict (2.72a) to be true
both in a scenario where there were four individual children sleeping
and in a scenario where only two individual children slept. However,
this not how the sentence is understood. It seems that what cardinals in
a language such as English do is that they restrict possible verifications
of sentences including numeral phrases in such a way that sums of
individuals are excluded from the domain of quantification.32
With this in mind, let us return to count explicit partitives such
as the Italian examples in (2.41), repeated here as (2.73). Similar to
regular plural DPs unmodified explicit set partitives denote pluralities
or, more precisely, sub-pluralities. However, when combined with a
numeral they do not allow for quantification over sub-pluralities but
only over material parts of individuals making up pluralities. At first
32. Alternatively, one might assume that there is a null classifier responsible for
such a restriction (e.g., Selkirk 1977, Kobuchi-Philip 2006).
68
2. Partitives and part-whole structures
blush, it seems surprising but actually in languages such as English the
same effect can be found in numeral phrases. Specifically, the numeral
combines with the plural noun which denotes a set of pluralities;
however, the domain of quantification does not consist of pluralities
but rather atomic individuals. Hence, as demonstrated in (2.74) the
phrase three walls does not mean three pluralities of walls but rather a
plurality of three walls. Similar, the Italian partitive phrase in (2.73b)
cannot be interpreted as three pluralities of sub-pluralities of walls.
Instead, it would be felicitously paraphrased as a plurality of three
parts of walls.
(2.73) Italian (Schwarzschild 1996, p. 186; adapted)
a. tre
three
parti
parts
del
of-the
muro
wall
‘three parts of the wall’
b. tre
three
parti
parts
dei
of-the
muri
walls
‘three parts of the walls’
(i) part-of-a-singularity reading
(ii) #part-of-a-plurality reading
(2.74) Pluralities of wholes and parts
a. three walls
(i) #three pluralities of walls
(ii) plurality of three walls
b. tre parti dei muri
(i) #three pluralities of parts-of-a-plurality of walls
(ii) plurality of three parts of walls
In this section, I have argued that set partitives differ from collectives
and discussed issues regarding the domain of quantification in count
explicit partitives as well as group nouns and plurals modified by
cardinal numerals. An important question is why natural language
does not allow for counting arbitrary sums of entities. However, before
we discuss more data that might guide us to provide an answer to this
puzzle, let us return to the analogy concerning entity and set partitives
and its potential implications for the part-whole structures singular
and plural expressions employ.
69
2. Partitives and part-whole structures
2.5 Ambiguity or indeterminacy
The data we have begun with in Section 2.2 might be interpreted
as suggesting that both singulars and plurals employ a unified partwhole
structure. Another way of looking at the evidence involves
preserving distinct mereologies and at the same time postulating that
partitive words are either ambiguous between two different meanings
or that they encode a generalized parthood relation which enables
them to operate on both part-whole structures. A question I have not
raised until this moment concerns to what extent there is an empirical
basis broad enough to provide a solid argument in favor of one of
the approaches. The evidence for both cross-linguistically and intralinguistically
systematic behavior of partitive words indicates that
an ambiguity account seems rather implausible. Nevertheless, before
we conclude, let us see whether an independent test can provide an
argument for or against such an approach.
A standard diagnostic to detect ambiguity is the so-called zeugma
test (Lasersohn 1995, see also Zwicky et al. 1975). The test works as
follows. The examined lexical item is put in a syntactic configuration
with a coordination structure. If the expression in question is in fact
ambiguous, then it must take the same interpretation with respect to
both conjuncts. Otherwise the zeugma effect arises, i.e., the sentence
takes on the flavor of a joke. On the other hand, if the tested item is
unspecific or indeterminate, it can take different interpretations with
respect to different conjuncts without any hilarious effect.
To illustrate this, let us consider the two examples in (2.75). The
verb rent must have the same interpretation with respect to both DPs,
a car and a house, i.e., it is either the case that John owned a car and a
house and he rented them out to someone, or it is the case that John
rented a car and a house from someone. But crucially it cannot be the
case that John rented out a car and rented a house as a tenant or vice
versa. Therefore, since rent must have the same interpretation with
respect to the DPs a car and a house, it is ambiguous between the two
described readings. On the other hand, the verb describe in (2.75b) is
indeterminate with respect to whether describing is done in speech
or in writing. Since it is perfectly natural to understand the sentence
in a way implying that John described a car in speech, whereas he
70
2. Partitives and part-whole structures
described a house in writing, describe is not ambiguous but rather
merely unspecific.
(2.75) English (Lasersohn 1995, p. 94; adapted)
a. John rented a car and a house.
b. John described a car and a house.
The zeugma test has been used commonly in the study of natural
language semantics including the research on pluralities (see, e.g.,
Dowty 1987, Lasersohn 1995, Nouwen 2016).33 Let us then examine
whether part-words are ambiguous between two different readings
giving rise to a part-of-a-singularity reading in entity partitives on the
one hand and to a part-of-a-singularity reading in set partitives on the
other or, alternatively, they are merely indeterminate in this respect.
To this end, let us first consider the well-formed and felicitous German
sentences in (2.76). In the explicit entity partitive in (2.76a) the partitive
word Teil ‘part’ takes a structure involving coordination of two singular
DPs as a complement. As witnessed by the translation, the meaning
of the sentence involves material partitivity, i.e., quantification over
parts of the relevant singular objects, namely the apple and the pear.
In other words, (2.76a) would be true if part of the apple and part of
the pear got spoiled. On the other hand, Teil in (2.76b) combines with
a conjoined phrase involving two plural conjuncts. In this case, Teil
quantifies over parts of pluralities, i.e., the whole sentence designates
that a subset of the apples and a subset of the pears got spoiled.
(2.76) German (Viola Schmitt, Martin Prinzhorn, p.c.)
a. Ein
a
Teil
part
des
thegen
Apfels
applegen
und
and
der
thegen
Birne
peargen
sind
are
verfault.
rotten
‘Part of the apple and the pear got spoiled.’
33. However, Lasersohn (1995, p. 95) points out that in fact it does not detect ambiguity,
but rather whether a particular item always makes the same contribution to
the proposition. I will leave this issue open though.
71
2. Partitives and part-whole structures
b. Ein
a
Teil
part
der
thegen
Birnen
pearsgen
und
and
der
thegen
Äpfel
applesgen
sind
are
verfault.
rotten
‘Some of the pears and the apples got spoiled.’
Now let us consider the interaction between Teil and a coordinate
structure conjoining a singular and a plural DP. The assumption behind
(2.77) is that if Teil were merely indeterminate with respect to
a part-of-a-singularity and part-of-a-plurality reading, it should be
able take the first interpretation with respect to the first conjunct and
the latter interpretation with respect to the second conjunct in (2.77a)
and vice versa in (2.77b). On the other hand, if it were ambiguous
between two different meanings such behavior should not be possible.
Interestingly, Teil not only can combine felicitously with a construction
involving coordination of a singular and plural DP, but also irrespective
of the linear order of the conjuncts, see (2.77a) and (2.77b), it can
take simultaneously two different readings.34
(2.77) German (Viola Schmitt, Martin Prinzhorn, p.c.)
a. Ein
a
Teil
part
des
thegen
Apfels
applegen
und
and
der
thegen
Birnen
pearsgen
sind
are
verfault.
rotten
‘Part of the apple and some of the pears got spoiled.’
b. Ein
a
Teil
part
der
thegen
Birnen
pearsgen
und
and
des
thegen
Apfels
applegen
sind
are
verfault.
rotten
‘Some of the pears and part of the apple got spoiled.’
34. Notice, however, that this is not the case in Slavic. For instance, an exact translation
of the sentences in (2.77) in Polish or in Czech would be considered awkward. I
hypothesize that the reason for the infelicity of such examples has to do with the fact
that these are conjoined NPs rather than DPs. Assuming that the silent determiner
sits above the coordinate structure would explain why such partitives are impossible
since DP-internal coordination does not allow for mixing of singulars and plurals
(Heycock and Zamparelli 2003).
72
2. Partitives and part-whole structures
The very same effect arises in (2.78), where one of the conjuncts is
a singulare tantum noun, specifically the object mass noun Geschirr
‘tableware’ can refer either to one piece of tableware or to a collection
thereof. Both (2.78a) and (2.78b) are felicitous and they both can get the
intended meaning, i.e., Teil can simultaneously quantify over material
parts of the kitchen sink and over individual items making up the
tableware. Hence, the mismatch between the values of grammatical
number on conjunct nominals in (2.77) definitely does not play a role
here and the effect seems to be purely semantic.
(2.78) German (Viola Schmitt, Martin Prinzhorn, p.c.)
a. Ein
a
Teil
part
des
thegen
Geschirrs
tablewaregen
und
and
des
thegen
Spülbeckens
sinkgen
sind
are
verrostet.
rusted
‘Some of the tableware and part of the kitchen sink got
rusted.’
b. Ein
a
Teil
part
des
thegen
Spülbeckens
tablewaregen
und
and
des
thegen
Geschirrs
sinkgen
sind
are
verrostet.
rusted
‘Part of the kitchen sink and some of the tableware got
rusted.’
At this point, a conclusion can be drawn. Given the results of the
zeugma test, I conclude that part-words in languages such as German
are not ambiguous between two different interpretations. Rather, they
are semantically unspecific in the sense that their meaning is general
enough to cover configurations involving both material and individual
parthood. As a result a part-of-a-singularity understanding is derived
in entity partitives whereas a part-of-a-plurality understanding arises
in set partitives.
2.6 Summary
The data discussed in this section provide compelling evidence that
cross-linguistically it is common for the same partitive word including
proportional quantifiers such as half -words to appear both in entity
73
2. Partitives and part-whole structures
and set partitives. This fact suggests that such expressions can operate
both at the superatomic and subatomic level depending on what
structure is provided by the embedded DP. However, at the same time
explicit partitives modified by a cardinal numeral can only quantify
over material parts of singular individuals irrespective of whether the
complement DP is singular or plural. At first sight, this fact seems to
be at variance with the claim that there is unified parthood utilized by
partitive words in both entity and set partitives. Nonetheless, an important
empirical finding concerning Italian irregular plurals showed
that if a plural expression encodes a certain topological configuration,
i.e., denotes cohesive pluralities, count explicit partitives can get a
part-of-a-plurality reading in such a case. This fact demonstrates that
the uncountability of explicit partitives with plurals does not indicate
per se different part-whole structures for singularities and pluralities.
Rather, countability results from the interplay between the meaning
of a partitive word and the extension of a singular or plural DP it
combines with and topological constraints play a crucial role in this
interaction.
Furthermore, on the basis of the zeugma test involving German
partitives with conjoined singular and plural DPs I have shown that
it is empirically inadequate to account for the parallelism between
explicit and proportional entity and set partitives in terms of semantic
ambiguity. In addition, given the cross-linguistic and intra-linguistic
evidence the two remaining possibilities are the following. It is either
the case that singulars and plurals employ a unified part-whole structure
or that their mereologies do differ but at the same time partitive
words involve a derived indeterminate notion of parthood which allows
them to quantify over elements of whatever part-whole structure
they are applied to. I argue for the first option. However, by advocating
this claim I do not advance a view that singulars and plurals employ
the same structures. What I claim is that it is not the part-whole relation
that differs but rather that there is another notion involved which
is responsible for how parts are topologically arranged with respect
to each other. Intuitively, this seems correct since we conceptually
distinguish between integrated wholes and arbitrary sums of parts
and quantification in natural language is sensitive to the distinction.
Specifically, only integrated parts of integrated wholes can be assigned
numbers when counting. On the other hand, scattered entities such
74
2. Partitives and part-whole structures
as pluralities are restricted from the domain of quantification cardinal
numerals establish. In addition, Italian irregular plurals provide
evidence that there are natural language expressions involving yet another
type of structure. In particular, such nominals designate entities
that are similar to plurals in that they consist of multiple integrated
objects but at the same time the sum thereof is arranged in a particular
way, i.e., it constitutes a cluster.
So far, I have argued that explicit and proportional partitives provide
evidence that countability is restricted to entities that form an integrated
whole as typically denoted by concrete singular count nouns,
while the arbitrary sums of parts regular plural nouns refer to cannot
be counted. Furthermore, the data concerning subatomic quantification
suggests that this constraint is also applicable at the level of
material parts of individuals. If that is correct, one would expect that
there are natural language expressions dedicated exclusively to quantification
over integrated parts similar to cardinal numerals that count
integrated wholes. In the next section, I will introduce novel data concerning
distinct types of half -words in Polish as well as other partitive
expressions.
75
3 Exploring topological sensitivity
In the previous chapter, I have discussed the data concerning quantification
over parts in explicit entity and set partitives. Specifically, I
have presented evidence suggesting that only integrated parts of integrated
wholes can be subject to counting. As I have already indicated
in the introduction, in English the difference can be captured by the
contrast between bare part and a part. I have argued that expressions
corresponding to the first refer to arbitrary parts of a whole including
discontinuous portions of a substance making up a whole, and thus
are uncountable. On the other hand, expressions corresponding to
the latter refer to continuous integrated parts of a whole and as such
can pluralize and combine with cardinals. In this chapter, I explore
to what extent natural language is sensitive to this distinction. To this
end, I will provide novel evidence from different types of Polish partitive
words that encode the contrast formally. It will turn out that this
phenomenon is not something idiosyncratic but rather cross-linguistic
evidence suggests that it is relatively widespread with different languages
using different means to express it. Moreover, based on the data
concerning whole-adjectives I will demonstrate the linguistic relevance
of two aspects of being whole, namely maximality and integrity.1
3.1 Continuous and discontinuous parts
Given the meaning of the plural in languages such as English, it follows
that expressions such as a part are incompatible with predicates
denoting pluralities. Intuitively, the reason is that the extension of
a phrase headed by a part comprises integrated portions, i.e., parts
that come in one piece, whereas plurals denote arbitrary sums, i.e.,
scattered entities consisting of discontinuous elements. However, if
1. The scope of this chapter would be much narrower if it were not for my informants
with whom I have tested the interpretation of expressions in languages
other than Polish. In particular, I am very grateful to Jonathan Bobaljik, Jeffrey Parrott,
and Guy Tabachnick for their judgments and discussion concerning English,
to Nina Haslinger and Maximilian Prüller for German, Erlinde Meertens and Izabela
Jordanoska for Dutch, Muriel Assmann for Portuguese, and Chang Liu for
Mandarin.
77
3. Exploring topological sensitivity
the plural were augmented with an additional semantic feature guaranteeing
that a plurality of integrated objects is itself an integrated
object, as is arguably the case in at least some Italian irregular plurals,
then an expression corresponding to English a part is able to combine
with such a plural expression. In such a case, it would yield a part of
a plurality whose constituents are in a particular topological relation,
namely they are connected to each other.
But is there any additional reason to assume that natural language
expressions are sensitive to topological relations operating alongside
part-whole structures? And if so, to what extent do the discussed phenomena
really tell us something important about countability? The
claim advanced here is that what counts as one needs to be an integrated
part (proper or improper) of an integrated whole. Though this
might seem intuitively correct, a question arises where the property of
‘coming in one piece’ comes from. After all, countability is the ability
of an expression to appear in morpho-syntactic environments related
to counting, but counting is a semantic operation of assigning numbers
to units of a particular type. Hence, one could wonder whether
there is evidence that there is something in an expression’s meaning
that corresponds to the notion of being an integrated entity, and thus
being countable. One prominent view holds that countability is not a
property of a particular class of nouns but rather of full DPs (e.g., Borer
2005a, see also Allan 1980 and Pelletier and Schubert 1989 and the
following work), and thus arises within a complex nominal syntactic
structure. On the other hand, other approaches posit that it is lexical
meaning that determines countability patterns (e.g., Wierzbicka 1988,
Wisniewski et al. 2003). However, evidence discussed in the literature
is mainly restricted to cases of quantification over wholes and
almost totally ignores constructions in which what is counted are bits
of objects rather than whole individuals.
In this section, I will provide additional evidence for the significance
of the role of spatial integrity, i.e., compactness, of parts in
subatomic quantification. The core data come from the distribution
and semantic properties of distinct classes of Polish half -words.
78
3. Exploring topological sensitivity
3.1.1 Polish half-words
Polish distinguishes lexically between three distinct half -words, as
presented in (3.1). They are morphologically derived from one another
with pół being morphologically the least marked form with no
derivational affixes. As presented in (3.1a), pół consists only of a root
and null inflectional marker. On the other hand, połowa and połówka
are derived forms involving in addition the morpheme -ow-/-ów- as
well as the diminutive suffix -k- in the latter case, see (3.1b)–(3.1c).
(3.1) Polish
a. pół-∅
root-inflectional.marker
‘half1’
b. poł-ow-a
root-derivational.suffix-inflectional.marker
‘half2’
c. poł-ów-k-a
root-deriv.suffix1-deriv.suffix2-infl.marker
‘half3’
In terms of φ-features, pół shows neuter agreement whereas połowa and
połówka are feminine, as demonstrated in (3.2). Furthermore, similar
to vague quantifiers such as dużo ‘many/much’ pół is defective in that
it has only the singular nominative and accusative forms. On the other
hand, połowa and połówka are both well-behaved nouns exhibiting the
full feminine declension including plural forms.
(3.2) Polish
a. To
thisnom.n
czerwone
rednom.sg.n
pół
half1.nom.n
jabłka
applegen.n
zgniło.
rotsg.n
‘This red half of the apple got spoiled.’
b. Ta
thisnom.f
czerwona
rednom.sg.f
połowa
half2.nom.f
jabłka
applegen.n
zgniła.
rotsg.f
‘This red half of the apple got spoiled.’
c. Ta
thisnom.f
czerwona
rednom.sg.f
połówka
half3.nom.f
jabłka
applegen.n
zgniła.
rotsg.f
‘This red half of the apple got spoiled.’
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3. Exploring topological sensitivity
At first blush, the Polish half -words in (3.2) are mere synonyms making
the same contribution to the interpretation of the discussed sentences.
However, closer investigation reveals intriguing differences in their
distribution and meaning. Let us first consider the most frequent and
semantically least marked half expression, i.e., połowa. As witnessed in
(3.3), there are no constraints on its distribution and it can felicitously
combine with all types of entity-denoting predicates including count
singulars, collectives, plurals, and mass terms.2
(3.3) Polish
a. połowa
half2
jabłka
applegen
‘half of the apple’
b. połowa
half2
stosu
pilegen
(jabłek)
(applesgen)
‘half of the pile (of apples)’
c. połowa
half2
jabłek
applesgen
‘half of the apples’
d. połowa
half2
soku
juicegen
‘half of the juice’
On the other hand, pół has a more restricted distribution since it is
incompatible with cumulative predicates such as plurals and mass
nouns, see (3.4). Thus, it can only combine with regular count singulars
and collective nouns.
(3.4) Polish
a. pół
half1
jabłka
applegen
‘half of the apple’
b. pół
half1
stosu
pilegen
(jabłek)
(applesgen)
‘half of the pile (of apples)’
2. I provide the examples with collective nouns only for sake of completeness. In
the following part of the study, I will not investigate the interaction between and
partitives and collectives.
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3. Exploring topological sensitivity
c. #pół
half1
jabłek
applesgen
d. #pół
half1
soku
juicegen
Notice that the constraint on the distribution of pół is not morphosyntactic.
The phrases in (3.5) show that it can felicitously combine
with pluralia tantum. However, in such a case the whole partitive
phrase gets a part-of-a-singularity interpretation. Specifically, (3.5a)
cannot refer to 50% of the relevant utensils but only to a half of one
pair of scissors. Similar, (3.5b) involves the noun bakalie ‘dried fruit’
which is a plurale tantum expression in Polish and as such it can either
denote one piece of dried fruit or a plurality thereof. The half -word pół
can combine with bakalie as long as it quantifies over material parts of
a single foodstuff. For instance, it would be true of a half of a raisin
but not of, say, five out of ten raisins.
(3.5) Polish
a. pół
half1
nożyczek
scissorsgen
‘half of the scissors’
b. pół
half1
bakalii
dried-fruitgen.pl
‘half of the dried fruit’
Furthermore, the claim that pół is incompatible with mass nouns is
somewhat imprecise and requires elaboration. In fact, when it can
combine with mass terms, it enforces a mass-count shift via the Universal
Packager (see, e.g., Bach 1986, Jackendoff 1991, Landman 1991).
In other words, the phrases in (3.6) are only felicitous on the portion
reading, i.e., (3.6a) would only be true of half a glass of juice whereas
(3.6b) refers to half a pint of beer or some other standardized or contextually
salient measure of volume. This shows that pół is sensitive to the
semantics of DPs it c-commands. In particular, cumulative predicates
are disallowed because they denote scattered entities such as arbitrary
sums or amorphous substances.
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3. Exploring topological sensitivity
(3.6) Polish
a. pół
half1
soku
juicegen
‘half of the juice’
(i) #substance reading
(ii) portion reading
b. pół
half1
piwa
beergen
‘half of the beer’
(i) #substance reading
(ii) portion reading
Finally, the distribution of the third Polish half -word, i.e., połówka,
is even further constrained since it is compatible only with regular
concrete singular nouns, as witnessed by the contrasts in (3.7). Prototypically,
it takes nominals denoting solid objects one could easily cut
or divide into separate parts such as food terms or building materials
like bricks. Similar to pół, it is distinctively odd with plurals and mass
terms. Moreover, it does not work with collective nouns.
(3.7) Polish
a. połówka
half3
jabłka
applegen
‘half of the apple’
b. #połówka
half3
stosu
pilegen
(jabłek)
(applesgen)
c. #połówka
half3
jabłek
applesgen
d. #połówka
half3
soku
juicegen
The distributional facts concerning particular Polish half -words are
summarized in Table 3.1. While the distribution of połowa is unconstrained,
i.e., it can appear in all types of proportional partitives including
entity, set, and mass partitives, the distributional potential of
pół and połówka is significantly restricted in that neither of them can
modify cumulative predicates such as plurals and mass terms, and
thus they can only occur in entity partitives. The different distribution
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3. Exploring topological sensitivity
of particular expressions with respect to different types of nominals
suggests distinct semantic properties of those lexical items. Given
that the only type of nominals that is compatible with all the Polish
half -words is singular count nouns, let us discuss in detail intuitions
about the meanings of entity partitive phrases in (3.4a), (3.3a), and
(3.7a).
singulars collectives plurals mass nouns
połowa
pół * *
połówka * * *
Table 3.1: Distribution of Polish half -words
In order to do so, consider two different subdivisions of an entity,
as depicted in Figure 3.1 and 3.2. In both cases, the dashed lines
mark an area designating a portion of the apple that constitutes approximately
50% of the whole. However, there is a crucial difference
between the two depicted situations in terms of topological configuration
of the stuff making up the halves. The share represented in Figure
3.1 constitutes a continuous half of the apple and as such it can be
considered a solid object in its own right within the whole, whereas
Figure 3.2 illustrates just an arbitrary portion that is not contiguous,
i.e., the apple stuff does not form an integrated part of the apple. In
other words, the marked quantity in Figure 3.1 is an integrated part,
whereas the portion in Figure 3.2 is not.
Figure 3.1: Continuous half Figure 3.2: Discontinuous half
Let us now discuss possible extensions of proportional entity partitives
involving pół, połowa, and połówka. The phrases in (3.4a) and
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3. Exploring topological sensitivity
(3.3a) are true of both the object depicted in Figure 3.1 and the entity
in Figure 3.2. Out of the blue, the continuous half reading is definitely
the dominant one, however the discontinuous half interpretation can
become salient in multiple contexts. For instance, it gets extremely
relevant in a cooking scenario where pragmatically halves might be
considered in terms of volume rather than individuated parts. Crucially,
the ambiguity is systematic in entity partitives involving both
połowa and pół and the sentences in (3.8a) and (3.8b) are true both in
a situation where the marked half in Figure 3.1 is red and when the
designated portion in Figure 3.2 is red. Now, it is quite intriguing that
the partitive construction in (3.7a) does not denote entities such as the
one illustrated in Figure 3.2. Therefore, entity partitives with połówka
refer only to continuous integrated halves of an object denoted by the
‘downstairs’ DP. Consequently, the sentence in (3.8c) would be judged
true if the part indicated in Figure 3.1 were red and false with respect
to the portion in Figure 3.2.
(3.8) Polish
a. Pół
half1
jabłka
applegen
jest
is
czerwone.
red
‘Half the apple is red.’
b. Połowa
half2
jabłka
applegen
jest
is
czerwona.
red
‘Half the apple is red.’
c. Połówka
half3
jabłka
applegen
jest
is
czerwona.
red
‘A half of the apple is red.’
A similar phenomenon can be observed in object position. In the
sentences in (3.9), the partitives appear as arguments of a verb of consumption
and as such denote an incremental theme (see, e.g., Dowty
1991, Krifka 1998, Filip 1999, Rothstein 2003). Again, there is a contrast
between (3.9a) and (3.9b) on the one hand and (3.9c) on the other
in terms of truth conditions. The first two sentences denote weaker
propositions since for them to be judged true it would be enough
for Marysia to eat a quantity of the apple in question corresponding
approximately to 50% of the apple’s total volume. However, in many
scenarios in which (3.9a) and (3.9b) would be true the sentence in
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3. Exploring topological sensitivity
(3.9c) would not. This is because the proportional partitive headed
by połówka denotes a contiguous half, i.e., an integrated object within
the whole as opposed to arbitrary portions of the apple mass. This
stronger meaning results in that (3.9c) is true of a scenario such as the
one illustrated in Figure 3.1 and false with respect to Figure 3.2.
(3.9) Polish
a. Marysia
Marysia
zjadła
ate
pół
half1
jabłka.
applegen
‘Marysia ate half the apple.’
b. Marysia
Marysia
zjadła
ate
połowę
half2
jabłka.
applegen
‘Marysia ate half the apple.’
c. Marysia
Marysia
zjadła
ate
połówkę
half3
jabłka.
applegen
‘Marysia ate a half of the apple.’
Furthermore, similar to pół proportional partitives involving połówka
and pluralia tantum get a part-of-a-singularity reading. The phrases
in (3.10a) and (3.10b) refer to halves of one object.3 In addition, similar
to the previously discussed examples a part yielded as a result of
subatomic quantification has to be integrated. Thus, (3.10a) and (3.10b)
would be true of one scissor blade and, say, one continuous half of a
raisin, respectively.
(3.10) Polish
a. %połówka
half3
nożyczek
scissorsgen
‘half of scissors’
b. połówka
half3
bakalii
dried-fruitgen.pl
‘half of dried fruit’
Given the non-trivial truth conditions of the sentence in (3.8c), let
us consider several naturally occurring examples involving entity
3. Some speakers report that (3.10a) is somewhat odd. I suspect that the reason is
that scissors are not a kind of thing one normally divides into separate parts. My
own intuition, however, is that though the phrase is certainly unusual, it is definitely
interpretable.
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3. Exploring topological sensitivity
partitives headed by the half -word połówka from the National Corpus
of Polish (NCP)4 as given in (3.11).
(3.11) Polish (NCP)
a. ...zjadł (...)
he-ate
połówkę
half3.acc
sztokfisza.
dried-codgen
‘...he ate (...) a half of a dried cod.’
b. ...masz
you-have
ochotę
desireacc
na (...)
on
drink
cocktailacc
w
in
połówce
halfloc
kokosa?
coconutgen
‘...would you like (...) a cocktail in a half of a coconut?’
c. ...jesteśmy
we-are
jak
like
te
these
dwie
two
połówki
halves3
jajka.
egggen
‘...we’re like those two halves of an egg.’
d. ...połówki
halves
okna (...)
windowgen
trzasnęły
banged
o
about
ścianę.
wallacc
‘...the halves of a window (...) banged on a wall.’
e. Otworzył
he-opened
portfel
walletacc
i
and
wyjął (...)
he-took-out
połówkę
half3.acc
karty.
cardgen
‘He opened his wallet and took (...) a half of a card out
of it.’
The first three examples in the sample, see (3.11a)–(3.11c), involve reference
to foodstuffs. The sentence in (3.11a) would be true if the man
in question consumed half a dried cod. Importantly, however, it would
not be true in a situation where he ate, say, the tail and took some bites
of the front left side and middle right side of the fish even if the total
amount consumed equaled 50% of the whole issue. The man had to
eat either the left side or the right side of the cod or alternatively the
half starting from the head or the half starting from the tail. Similar,
the question in (3.11b) makes reference to a half of a coconut coming
in one piece, i.e., one would feel deceived if after answering “yes”
they got several pieces of a shell filled with portions of a cocktail they
ordered. On the other hand, the use of połówka in (3.11c) is somewhat
4. The National Corpus of Polish is a representative digital corpus of the contemporary
Polish language consisting of over 1 billion words with a balanced subcorpus
of 300-million tokens.
86
3. Exploring topological sensitivity
metaphorical. The inference here is that the speaker asserts that they
and the addressee are like two individuated halves of an egg that
fit well together. Again, the halves need to be continuous parts one
receives after a clear horizontal or vertical cut. Yet another example is
provided in (3.11d). Here, however, the downstairs DP in the partitive
does not refer to food but to a solid object of another type, i.e., a window
consisting of two casements. The partitive phrase clearly implies
that the halves in question are individuated and easily distinguishable
parts, hence the casement window interpretation. Finally, the sentence
in (3.11e) would be judged true only if the man in question took out a
half of a card in one piece, i.e., it would necessarily be false if he took
out several torn scraps of a card.
The discussion of (3.8c) as well as the genuine examples from the
National Corpus of Polish in (3.11) leaves no doubt that połówka requires
the referents of a partitive it occurs in to constitute an integrated
part. Such a finding is quite remarkable and forces us to abandon the
initial intuition that Polish half -words are semantically indifferent. The
recapitulation of referential properties of explicit partitives involving
the expressions in question is given in Table 3.2. The main contrast
between less marked połowa and pół on the one hand and more marked
połówka on the other lies in that the first two can designate discontinuous
portions of stuff making up a whole, whereas the latter denotes
only such divisions that constitute continuous, i.e., integrated, halves
of an object.
continuous part discontinous part
połowa
pół
połówka *
Table 3.2: Denotations of Polish half -words
The novelty of the data introduced here is twofold. First of all,
Polish half -words provide strong evidence that natural language is
sensitive to the topological arrangement of parts of an entity. Second,
the data imply that individuation is also possible at the subatomic
level, i.e., quantification over parts in natural language reflects the fact
that some parts might be assigned the status of an individual in its own
87
3. Exploring topological sensitivity
right. In general, the distinction between different half -words in Polish
can be described as follows. The least semantically marked expression
połowa is topologically neutral, i.e., the spatial make-up of an entity
denoted by the downstairs DP in a partitive is irrelevant as long as it
constitutes approximately 50% of a whole. That is why połowa does
not discriminate between count singulars, plurals, and mass terms
and can felicitously appear in entity, set, and mass partitives. It simply
separates out a half of whatever entity it is applied to. Furthermore,
in this case subatomic quantification does not encode any constraints
on what kind of entity is yielded. Hence, the result might be either
a scattered entity such as a plurality or a discontinuous portion of
matter. On the other hand, pół and połówka are topologically sensitive.
They both require a DP they c-command to denote an integrated object,
i.e., a cohesive whole. That explains why they cannot combine with
expressions referring to scattered entities such as arbitrary sums of
individuals or portions of a substance as typically denoted by plurals
and mass nouns. The difference between the two is that pół is similar
to połowa in that it does not impose any topological constraints on
the resulting entity, whereas połówka does. In other words, while pół
selects for an integrated object and returns either its continuous or
discontinuous half, połówka yields an integrated half of an integrated
whole.
3.1.2 Inherent vagueness
Similar to other proportional quantifiers expressions such as English
half appear to be inherently vague. Though disregarded by prescriptive
grammars as an oxymoron the concept of a bigger and smaller
half is present in lexicons of many languages, as attested in the English,
German, and Polish example in (3.12)–(3.14). Interestingly, such
phrases cannot be explained as bleached expressions meaning simply
something like part. For instance, for Polish speakers who would use
or at least accept the NP in (3.14a), i.e., speakers who have not internalized
the linguistic prescription in question, it would be definitely
true of a share constituting 55% of a whole. However, it is unlikely
it would be judged true of 75% of a whole and intuitively it seems
definitely false of 90% of a whole.
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3. Exploring topological sensitivity
(3.12) English
a. %bigger half
b. %smaller half
(3.13) German
a. %größere
bigger
Hälfte
half
‘bigger half’
b. %kleinere
smaller
Hälfte
half
‘smaller half’
(3.14) Polish
a. %większa
bigger
połowa
half2
‘bigger half’
b. %mniejsza
smaller
połowa
half2
‘smaller half’
Given the possible interpretations discussed above, it seems plausible
to assume that in natural language proportional quantifiers such as
English half denote a relation of constituting approximately 50% share
of a whole. In other words, their meaning is fuzzy in the sense it allows
for unequal parts as long as the disproportion falls within a particular
range defined contextually.
3.2 More topology-sensitive partitive words
The observations made in Section 3.1 provide novel evidence for certain
semantic properties of natural language expressions that have not
been recognized so far in the formal study of meaning. Nevertheless,
a question might arise whether topological sensitivity is a marginal
issue or whether it constitutes a broader phenomenon associated to
a number of expressions in a language. Interestingly, it appears that
half -words are not the sole category in Polish that displays the distinction
between topological neutrality and topological sensitivity. In
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3. Exploring topological sensitivity
this section, I will examine more data from Polish partitive words
indicating that the pattern is robust.
3.2.1 Quarter-words
Let us first consider the contrasts between two other proportional
partitives involving what I will refer to as quarter-words ćwierć and
ćwiartka ‘quarter’, as provided in (3.15). Similar, to połówka, see (3.1c),
ćwiartka is a morphologically complex expression derived from ćwierć
by the suffix -k-.
(3.15) Polish
a. ćwierć-∅
root-inflectional.marker
‘quarter1’
b. ćwiart-k-a
root-derivational.suffix1-infl.marker
‘quarter2’
In terms of agreement, similar to the distinction between pół and
połówka, see (3.2), ćwierć is neuter, whereas ćwiartka triggers feminine
agreement on adjectives and pronouns, as demonstrated in (3.16).
(3.16) Polish
a. To
thisnom.n
czerwone
rednom.sg.n
ćwierć
quarter1.nom.n
jabłka
applegen.n
zgniło.
rotsg.n
‘This red quarter of the apple got spoiled.’
b. Ta
thisnom.f
czerwona
rednom.sg.f
ćwiartka
quarter2.nom.f
jabłka
applegen.n
zgniła.
rotsg.f
‘This red quarter of the apple got spoiled.’
The distribution of ćwierć and ćwiartka mimics the pattern observed in
the pół and połówka alternation, i.e., both quarter-words appear to be
sensitive to the type of nominal they combine with, see (3.17) and (3.18),
respectively. In particular, as witnessed by the infelicity of the phrases
in (3.17c)–(3.17d) as well as (3.18c)–(3.18d) neither of them occurs
in set and mass partitives and only ćwierć combines with collective
nouns. Similar to połówka, ćwiartka typically combines with food terms
or nominal denoting objects people usually cut into comparable pieces.
90
3. Exploring topological sensitivity
(3.17) Polish
a. ćwierć
quarter1
jabłka
applegen
‘quarter of the apple’
b. ćwierć
quarter1
stosu
pilegen
(jabłek)
(applesgen)
‘quarter of the pile (of apples)’
c. #ćwierć
quarter1
jabłek
applesgen
d. #ćwierć
quarter1
soku
juicegen
(3.18) Polish
a. ćwiartka
quarter2
jabłka
applegen
‘quarter of the apple’
b. #ćwiartka
quarter2
stosu
pilegen
(jabłek)
(applesgen)
c. #ćwiartka
quarter2
jabłek
applesgen
d. #ćwiartka
quarter2
soku
juicegen
This kind of behavior can be contrasted with a regular fraction expression
such as jedna czwarta ‘one-fourth’, see (3.19). Unlike quarter-words
or the half -words pół and połówka, fractions show no distributional
constraints and can freely combine both with quantized predicates
involving count singulars and collectives and cumulative predicates
such as plural nouns and mass terms.
(3.19) Polish
a. jedna czwarta
one-fourth
jabłka
applegen
‘one-fourth of the apple’
b. jedna czwarta
one-fourth
stosu
pilegen
(jabłek)
(applesgen)
‘one-fourth of the pile (of apples)’
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3. Exploring topological sensitivity
c. jedna czwarta
one-fourth
jabłek
applesgen
‘one-fourth of the apples’
d. jedna czwarta
one-fourth
soku
juicegen
‘one-fourth of the juice’
Based on the distributional evidence, I posit that similar to pół and
połówka the quarter-words ćwierć and ćwiartka are topology-sensitive.
Specifically, they quantify only over parts of integrated wholes. To put
it differently, expressions having scattered entities such as arbitrary
portions of matter or sums of individuals are disallowed as their input.
On the other hand, fractions do not show such requirements and can
yield a proportion of an entity of any kind. Table 3.3 summarizes the
observations, namely the possibility of fraction entity, set, and mass
partitives as opposed to the non-existence of proportional set and
mass partitives headed by the quarter-words in Polish.
singulars collectives plurals mass nouns
jedna czwarta
ćwierć * *
ćwiartka * * *
Table 3.3: Distribution of Polish quarter-words
Moreover, the pattern is further corroborated by the differences in
truth conditions of sentences involving fraction partitives and proportional
partitives headed by ćwierć on the one hand and proportional
partitives headed by ćwiartka on the other. Similar, to the distinction
discussed with respect to half -words, see (3.8) and (3.9), there is a
contrast in terms of possible verifications of (3.20a) and (3.20b) and
(3.20c). The first two sentences are true if any proportion of the apple
constituting approximately 25% of the whole apple is red regardless
of whether it forms a discontinuous or continuous part. On the other
hand, (3.20c) would only be true if an integrated quarter of the surface
of the apple were red, i.e., similar to sentences involving połówka the
partitive headed by ćwiartka would not be true of a discontinuous part.
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3. Exploring topological sensitivity
(3.20) Polish
a. Jedna czwarta
one-fourth
jabłka
applegen
jest
is
czerwona.
red
‘One-fourth of the apple is red.’
b. Ćwierć
one-fourth
jabłka
applegen
jest
is
czerwone.
red
‘A quarter of the apple is red.’
c. Ćwiartka
one-fourth
jabłka
applegen
jest
is
czerwona.
red
‘A quarter of the apple is red.’
Again, the same effect appears in object position. The incremental
themes expressed by fraction partitives and proportional partitives
with ćwierć can refer either to continuous or discontinuous entities.
Hence, the sentences (3.21a) and (3.21b) would be true in a scenario
where Marysia ate one piece of the apple as well as in a scenario where
she took several unconnected bites constituting approximately 25% of
the total volume. In contrast, (3.21c) requires the theme of the verb of
consumption to be an integrated object, i.e., it would be judged false
in the second scenario.
(3.21) Polish
a. Marysia
Marysia
zjadła
ate
jedną czwartą
one-fourth
jabłka.
applegen
‘Marysia ate one-fourth of the apple.’
b. Marysia
Marysia
zjadła
ate
ćwierć
quarter1
jabłka.
applegen
‘Marysia ate a quarter of the apple.’
c. Marysia
Marysia
zjadła
ate
ćwiartkę
quarter2
jabłka.
applegen
‘Marysia ate a quarter of the apple.’
Furthermore, examples naturally occurring in the NCP corpus seem
to corroborate the intuitions described above. The sentences in (3.22a)
and (3.22b) involve reference to foodstuffs. The first does not report
that the best meal in the place in question consisted of, say, fried
chicken strips, but rather that the portion of meat was processed in
one piece. Similar, (3.22b) means that there was a continuous part of a
93
3. Exploring topological sensitivity
baguette in a bag and not, e.g., several slices. Finally, (3.22c) would be
judged as false if what slided out were some scraps of a paper sheet.
(3.22) Polish (NCP)
a. Najlepsza
best
tu
here
kolacja:
dinner
ćwiartka
quarter2
kurczaka...
chicken
‘The best dinner here was a quarter of a chicken...’
b. Biorę (...)
I-take
torebkę,
bagacc
w
in
której
whichloc
leży
lies
ćwiartka (...)
quarter2
bagietki...
baguettegen
‘I’m taking a bag in which there is a quarter of a baguette.’
c. Ze
from
środka
interiorgen
wysunęła
slided-out
się
refl
ćwiartka
quarter2
papieru.
papergen
‘A quarter of a paper sheet slided out from inside.’
The comparison of extensional properties of partitives involving fractions,
ćwierć, and ćwiartka with regard to topological relations is given
in Table 3.4. The results show the very same pattern as discussed with
regard to Polish half -words, see 3.2.
continuous part discontinous part
jedna czwarta
ćwierć
ćwiartka *
Table 3.4: Denotations of Polish quarter-words
The comparison of Polish quarter-words with fractions mirrors the
pattern observed for the distinction between topology-neutral and
topology-sensitive half -words. In the next section, I will discuss even
more evidence from partitive words.
3.2.2 Piece-words
Another class of expressions displaying topological sensitivity consists
of piece-words. Polish distinguishes between two such expressions,
namely cząstka and kawałek ‘piece’. The first is derived from the par-
94
3. Exploring topological sensitivity
titive word część ‘part’ by the suffix -k-, compare (3.23a) and (3.23b),
whereas the second appears to be a basic form.5 Similar to English piece,
in many contexts kawałek and cząstka can be used interchangeably with
the part-word część though the first is a more frequent expression than
the latter.6 Nonetheless, similar to other Polish partitive words also
in this case there are some interesting topological properties worth
discussing.
(3.23) Polish
a. część-∅
root-inflectional.marker
‘part’
b. cząst-k-a
root-derivational.suffix-infl.marker
‘piece1’
c. kawałek-∅
root-inflectional.marker
‘piece2’
The distribution of the two expressions in question shows that they
both resist combining with plural DPs, see (3.24b) and (3.25b). Furthermore,
the infelicity of phrases such as (3.25c) and (3.25c) suggests
that both kawałek and cząstka impose the same topological restrictions
on DPs they combine with as the partitive words pół, połówka, ćwierć,
and ćwiartka, i.e., disallow cumulative predicates denoting scattered
entities.
5. In fact, kawałek is also morphologically complex and from a diachronic point of
view it has been formed from kawał ‘large share’. However, from the perspective of
contemporary Polish, the relationship between the two is rather obscure and since
the latter is semantically marked, it appears at it were actually derived by means
of the deletion of -ek- from the first. This seems to be further corroborated by the
fact that it is only possible to derive verbs from kawałek and not from kawał, e.g.,
kawałkować ‘to portion’ ∼ *kawałować.
6. Another meaning of the noun cząstka is ‘physical particle, corpuscule’. This
meaning, however, will not be discussed here.
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3. Exploring topological sensitivity
(3.24) Polish
a. cząstka
piece1
jabłka
applegen
‘piece of an apple’
b. #cząstka
piece1
jabłek
applesgen
c. #cząstka
piece1
soku
juicegen
(3.25) Polish
a. kawałek
piece2
jabłka
applegen
‘piece of an apple’
b. #kawałek
piece2
jabłek
applesgen
c. #kawałek
piece2
soku
juicegen
Nevertheless, under closer inspection it turns out that in fact the pieceword
kawałek is not incompatible with all mass nouns but rather only
with one particular class of such expressions. As witnessed in (3.26),
mass partitives headed by kawałek are possible as long as the downstairs
DP does not involve a liquid term, see (3.26a). In particular,
kawałek can felicitously combine with mass expressions denoting solid
substances as well as granular and artifactual aggregates.7
(3.26) Polish
a. #kawałek
piece1
wody
watergen
b. kawałek
piece1
złota
goldgen
‘piece of gold’
7. Although examples such as (3.26c) and (3.26d) are not frequent and seem quite
unusual, they are definitely interpretable and the intuitions regarding their meaning
are very clear. Notice also that a similar pattern is observed with respect to German
Stück ‘piece’ as well as the diminutive forms such as Stückchen and Stückerl (Nina
Haslinger, p.c.).
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3. Exploring topological sensitivity
c. kawałek
piece1
ryżu
ricegen
‘piece of rice’
d. kawałek
piece1
obuwia
footweargen
‘piece of footwear’
Interestingly, the partitive in (3.26b) does not simply refer to any portion
of gold but rather to to an integrated (though most probably
amorphous) lump, i.e., to a nugget. Similar, the phrases in (3.26c) and
(3.26d) have contiguous entities in their extension. Granular mass
terms denote aggregates of objects such as grains of rice or granules
of gravel that seem too small or not significant enough to be perceived
as individuals in their own right (Grimm 2012b, Sutton and
Filip 2016). Instead, they are conceptualized as clusters of elements.
As it is well-known, those elements cannot be accessed be regular
quantificational expressions such as distributive quantifiers or cardinal
numerals, hence the uncountability of granulars. However, the
piece-word kawałek in (3.26c) can access the part-whole structure of
particular granules making up the extension of a mass noun and trigger
subatomic quantification, i.e., quantification over parts of a single
granule. As a result, the whole partitive can get a part-of-a-singularity
interpretation. In particular, (3.26c) is true of a piece of one grain of
rice. Similar, object mass nouns refer to stable discrete entities such as
pieces of furniture or items of tableware (Gillon 1992, Chierchia 1998a,
2010, Barner and Snedeker 2005, Bale and Barner 2009, Rothstein 2010,
Landman 2011). Despite their denotations, object mass nouns are uncountable,
i.e., do not allow for quantification over individual objects
they refer to. However, kawałek yet again quantifies over parts of such
discrete objects and as a result (3.26d) is interpreted as referring to a
piece of a single shoe.
The findings concerning the distribution of Polish piece-words in
comparison to the part-word część are summarized in Table 3.5. Both
cząstka and kawałek have a selectional restriction prohibiting them from
combining with expressions denoting arbitrary sums of individuals,
i.e., plurals. However, the main difference concerns the distribution
with uncountable nouns. Unlike cząstka which is generally incompatible
with mass terms, kawałek is sensitive to the properties of referents
97
3. Exploring topological sensitivity
of a particular uncountable noun. In particular, it seems to require a
certain stability of form so that parts singled out from a whole can
sustain a fixed constant shape. In other words, kawałek accepts entities
displaying any kind of topological arrangement as long as it
guarantees that the spatial form of pieces is relatively stable. This is
way it rejects liquid terms but can combine with expressions denoting
solid substances, granular aggregates as well as artifacts, i.e., discrete
entities related in terms of similar origin and functionality.
singulars plurals
mass nouns
liquids others
część
cząstka * * *
kawałek * *
Table 3.5: Distribution of Polish piece-words
Another issue concerns the internal structure of objects denoted
by partitives headed by piece-words. As provided in Table 3.6, similar
to the partitive words połówka and ćwiartka, both cząstka and kawałek
yield parts that can be recognized as contiguous integrated entities
within a whole. This contrasts with the standard part-word część which
similar to, e.g., the half -word połowa, can deliver both continuous and
scattered parts. What is especially interesting about kawałek is that
unlike połówka and ćwiartka it does not require the whole to be an
integrated object. As long as it is not an arbitrary sum of individuals
or a constantly deforming substance, kawałek yields a portion of the
whole that is one piece.
continuous part discontinous part
część
cząstka *
kawałek *
Table 3.6: Denotations of Polish piece-words
The properties of the piece-word kawałek seem to relate to the scale
of individuation proposed by Grimm (2012b). However, there seems to
98
3. Exploring topological sensitivity
be an interesting discrepancy. In Grimm’s original proposal, see (3.27),
liquids and solid substances are grouped together with no ordering
relation between them, i.e., water falls into the same category as gold.
Yet the data concerning the distribution of the Polish piece-word kawałek
seem to suggest that an even more fine-grained distinction might be
necessary to account for some grammatical phenomena in natural
language. Since partitive phrases headed by kawałek can only denote
stable objects that come in one piece and are able to sustain their shape,
they cannot combine with liquid terms such as water simply because
fluids continually deform, i.e., lack a given shape. Thus, it is possible
that distinguishing between liquids and solid substances in terms of
ordering on the scale of individuation, as proposed in (3.28), might
appear to be required.8
(3.27) Fragment of the scale of individuation (Grimm 2012b, p. 80;
adapted)
liquid/solid substance < granular aggregate < artifactual
aggregate...
(3.28) Modified fragment of the scale of individuation
liquid < solid substance < granular aggregate < artifactual
aggregate...
Before we move on to discussing how the interplay of quantification
and topology looks like from a cross-linguistic perspective and
whether other languages can shed new light on the issues related to
this phenomenon, let us briefly discuss the somewhat surprising fact
that mass partitives are countable. In the next section, I will consider
the relationship between substances and portions thereof.
3.3 Mass, parts, quantities, and pieces
One of the very few attempts to link the issue of countability with
partitivity and pseudo-partitivity (see, e.g., Selkirk 1977) has been pur-
8. It would be an interesting enterprise to explore whether there are more natural
language expressions sensitive to the distinction between liquid and solid substance
terms and if so whether that fact provides any interesting insights on the semantics
of mass nouns. However, such a research project lies far beyond the scope of this
study.
99
3. Exploring topological sensitivity
sued in Chierchia (2010) who raises an interesting question concerning
how it can be that one cannot count mass, whereas portions of mass
are countable.9 Specifically, how come that expressions such as (3.29a)
and (3.29b) can have a different status with respect to the mass/count
distinction and yet refer to the very same entity.
(3.29) English (Chierchia 2010; adapted)
a. That water is contaminated.
b. Those three quantities of water are contaminated.
Chierchia examines English relational expressions such as quantity,
part, and piece in pseudo-partitives and partitives such as those in
(3.30)–(3.32) and observes some differences in their distribution including
the following. Though quantity is slightly degraded with singular
count nouns, it combines felicitously with bare plurals and mass terms.
On the other hand, as we have already seen in Section 2.2.1 the English
expression a part of is incompatible with bare plurals. Moreover, it
fails to take mass nouns as its complements. Finally, piece seems to
fall somewhere in between the two categories since it can head partitives
involving singular definite DPs and bear mass nouns but cannot
combine with bear plurals.
(3.30) English (Chierchia 2010; adapted)
a. (?) a quantity of that person
b. a quantity of apples
c. two quantities of gold
(3.31) English (Chierchia 2010; adapted)
a. a part of that person
b. #a part of apples
c. #a part of gold
(3.32) English (Chierchia 2010; adapted)
a. a piece of that pizza
b. #a piece of apples
c. a piece of gold
9. See also Landman (2016) for related considerations.
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3. Exploring topological sensitivity
Chierchia attributes the selectional restrictions of quantity, part, and
piece to input requirements of the corresponding functions ranging
over singular entities and/or sets of individuals and analyzes them
in terms of partitions imposing relative atomicity and non-overlap of
members. 10 Although the approach is definitely thought-provoking, it
seems that it misses an important point regarding the contrast between
the expressions in question for at least two reasons. First, it neglects
the role of topological sensitivity as discussed above. But also it seems
to ignore the distinction between counting and measuring (see, e.g.,
Rothstein 2017). To foreshadow, counting is sensitive to topological
characteristics such as being an integrated object, whereas measuring
is not.11 For start, consider the count pseudo-partitive in (3.33a) and
count explicit partitive in (3.33b). It seems that what happens in both
cases is measurement in terms of volume. In (3.33a), the singular
count noun pizza is apparently shifted to the mass interpretation via
the Universal Grinder or a similar operation. Similar, the part-word in
(3.33b) could be paraphrased as portion or proportion and it essentially
designates a particular volume of the rice in question.
(3.33) English (Chierchia 2010; adapted)
a. two quantities of that pizza
b. two parts of that rice
Why the distinction between counting and measuring is of any significance
here? Intuitively, quantities are about volume, pieces involve
some form of individuation, and parts can shift between these two
aspects of quantification. However, to address this question more
properly, let us consider the difference between the Polish part-word
część and piece-word kawałek. Similar to multiple other languages, Polish
does not allow mass terms to combine with cardinal numerals
(modulo coercion), see (3.34). However, as witnessed in (3.35) mass
partitives headed by część are countable. In this case, the part-word
gets a portion interpretation and measures the extension of the mass
noun in terms of volume. For instance, the phrase in (3.35a) designates
two out of some number of more or less equally large portions the
10. I will return to Chierchia’s proposal in Section 7.1.
11. I will discuss the distinction between the two operations in detail in Section
5.2.2.
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3. Exploring topological sensitivity
total quantity of gold was divided into. Crucially, it implies that in
the relevant context there are more than two such portions. Similar,
(3.35b) indicates two comparable portions out of a larger number of
portions of rice.
(3.34) Polish
a. #dwa
two
złota
golds
b. #dwa
two
ryże
rices
(3.35) Polish
a. dwie
two
części
pieces1
złota
goldgen
‘two pieces of gold’
b. dwie
two
części
parts
ryżu
ricegen
‘two parts of rice’
However, the lack of a measuring reading in the minimal pair examples
involving the piece-word kawałek in (3.36) suggests that this expression
resists mere quantification in terms of volume and can only serve as
a basis for counting individuated objects. Neither (3.36a) nor (3.36b)
can get the volume interpretations available for (3.35a) and (3.35b),
respectively. They can only denote pluralities of pieces, i.e., integrated
parts, of gold and rice, respectively.
(3.36) Polish
a. dwa
two
kawałki
pieces1
złota
goldgen
‘two pieces of gold’
(i) quantification in terms of objects
(ii) #quantification in terms of volume
b. dwa
two
kawałki
pieces1
ryżu
ricegen
‘two pieces of rice’
(i) quantification in terms of objects
(ii) #quantification in terms of volume
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3. Exploring topological sensitivity
To better illustrate the contrast, let us consider several natural sentences.
Measuring uses of part-words are especially frequent in recipes
and other cooking-related contexts. For instance, consider the examples
in (3.37). Both in (3.37a) and (3.37b), the count explicit partitives
make reference to a particular volume of water and rice, respectively.
(3.37) Polish
a. Dolej
pourimp
do
to
garnka
potgen
dwie
twoacc
części
partsacc
wody,
watergen
a
and
w
in
trzeciej
thirdloc
namocz
soakimp
fasole.
beansacc
‘Add two thirds of the water to the pot and soak the
beans in the third part.’
b. Wymieszaj
miximp
kaszę
porridgeacc
w
in
proporcji
proportionloc
2:1,
2:1
to
this
znaczy
means
dwie
twoacc
części
partsacc
ryżu
ricegen
na
on
jedną
oneacc
część
partacc
prosa.
milletgen
‘Mix the porridge in the 2:1 ratio, which means two parts
of rice for one part of millet.’
A similar effect can be observed in count partitives with plural nouns.
For instance, the count plural jabłka ‘apples’ in (3.38a) is shifted to the
mass denotation and the part-word część quantifies over portions of
apple mass making up the relevant apples and not over the apples as
separate individuated objects. Whether the ingredients in question are
stored as wholes or are cut into pieces plays no role for the truth conditions
of the sentences. Consequently, if in a response to (3.38a) one
apple was put on the baking tray and a half of an apple was blended
or, alternatively, four apples were put on the baking tray and two
were blended, the instruction would be followed correctly. Crucially,
however, if we swap część in (3.38a) for the piece-word kawałek, we
get a significant contrast. Specifically, the sentence in (3.38b) cannot
be understood as instructing to put four apples on the baking tray
and to blend the remaining two. Instead, it says you should put two
integrated parts of different apples on the baking tray as well as blend
one such part.
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3. Exploring topological sensitivity
(3.38) Polish
a. Dwie
twoacc
części
partsacc
jabłek
applesgen
rozłóż
putimp
równomiernie
evenly
na
on
blasze
trayloc
do
to
pieczenia,
bakinggen
a
and
pozostałą
remainingacc
część
partacc
rozmiksuj.
blendimp
‘Put two thirds of the apples evenly on the baking tray
and blend the remaining part.’
b. Dwa
twoacc
kawałki
piecesacc
jabłek
applesgen
rozłóż
putimp
równomiernie
evenly
na
on
blasze
trayloc
do
to
pieczenia,
bakinggen
a
and
pozostały
remainingacc
kawałek
pieceacc
rozmiksuj.
blendimp
‘Put two apple pieces evenly on the baking tray and blend
the remaining piece.’
(i) quantification in terms of objects
(ii) #quantification in terms of volume
At this point, a preliminary conclusion can be drawn. The structure of
parts can be such that it is merely a scattered entity, an unconnected
configuration of bits related merely by virtue of belonging to the
same whole. However, some parts are arranged in such a manner that
they form a spatially contiguous entity that can be conceptualized
as an individual in its own right. It appears that natural language is
sensitive to this contrast and the Polish data show that this semantic
distinction can be encoded formally in the lexicon. Although not every
partitive word imposes topological restrictions regarding the spatial
form of a portion of an entity the whole partitive construction denotes,
there are some that do. Furthermore, it appears that counting and
measuring differ with respect to topological properties of quantified
entities. While the first operation requires integrated objects, the latter
does not.
The facts discussed above suggest that accounts that neglect the distinction
in question most probably miss something crucial about how
we humans conceptualize part-whole structures and how it relates to
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3. Exploring topological sensitivity
the phenomenon of countability in grammar. In the next section, I will
inspect cross-linguistic evidence that further corroborates this claim.
3.4 Cross-linguistic parallels
Polish gives an important insight into the role of topological notions
in quantification over wholes and parts. However, one could wonder
to what extend the semantic behavior of proportional partitives discussed
above is a Polish idiosyncrasy. In this section, I will discuss
cross-linguistic evidence suggesting that topology-sensitive expressions
are a relatively widespread phenomenon in partitive constructions
though not necessarily expressed by distinct lexical items. For
instance, it is reported that the English sentences in (3.39b) and (3.39a)
have different truth conditions. While the first is simply true in a scenario
where 50% of an apple was eaten by Mary, the meaning of the
latter is stronger since it is true only if Mary ate a continuous half of
an apple such as the one depicted in Figure 3.1.
(3.39) English (Guy Tabachnick, p.c.)
a. Mary ate half an apple.
b. Mary ate a half of an apple.
However, the semantic difference between (3.39a) and (3.39b) is very
slight and in multiple contexts it might be extremely hard to detect.
Therefore, before we discuss novel data from English, German, Dutch,
Portuguese, and Mandarin, let us briefly elaborate on the method
developed to tell an integrated-part and a scattered-part reading apart.
For the sake of clarity, I will limit the discussion of subatomic topological
sensitivity to proportional partitives involving half -words.
Since topological aspects of partitivity can be very subtle and intuitions
with this respect tend to be somewhat obscure, in order to
develop a proper diagnostic to detect topology-sensitive partitive expressions
it seems necessary to come up with plain test objects readily
divisible into comparable continuous and discontinuous parts differentiated
by easily distinguishable properties. For instance, consider
the flag test, as represented in Figure 3.3 and 3.4. The Maltese-like
flag in Figure 3.3 consists of two continuous areas each constituting
50% of the flag’s surface. It is easy to distinguish between the halves
105
3. Exploring topological sensitivity
since the left one is red whereas the right one is white. Therefore, the
spatial arrangement of the parts fit what I will call an AB pattern. On
the other hand, the red area in the Canadian-like flag illustrated in
Figure 3.4 is discontinuous, i.e., it does not constitute an integrated
half. I will call such an arrangement an ABA pattern.
Figure 3.3: Flag AB Figure 3.4: Flag ABA
As we have seen in the previous sections, Polish encodes topological
sensitivity lexically by differentiating between distinct partitive
words. Similar to Polish, Portuguese distinguishes between distinct
half -words, namely metade and meio ‘half’. The first has nominal features
and takes the partitive da-phrase, as demonstrated in (3.40),
whereas the latter is an adjectival expression agreeing with a modified
noun in number and gender. Interestingly, similar to Polish half -words
metade and meio differ with respect to both their distributional and
referential properties.
Let us first consider the difference in the distribution of the two
Portuguese partitive expressions. While metade patterns with połowa
in not being picky about the type of predicate within the da-phrase
it can combine with, see (3.40), meio appears to correspond to pół
and połówka. As demonstrated in (3.41c), it cannot form a partitive
with a mass term. Moreover, when combined with a plural noun it
lacks a half-of-a-plurality reading, i.e., it cannot be interpreted as a
quantifier over individuals making up the denoted plurality. Instead it
can only operate on the subatomic level of particular members of the
plurality denoted by a modified noun, i.e., it gets a half-of-a-singularity
interpretation exclusively. In other words, meio can only occur in an
entity partitive and not in a set partitive. For instance, (3.40b) cannot
refer to a half of the total number of apples. The only interpretation
available regards a plurality of halves of apples.
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3. Exploring topological sensitivity
(3.40) Portuguese (Muriel Assmann, p.c.)
a. metade
half1
da
thesg.f
maça
applef
‘half of the apple’
b. metade
half1
das
thepl.f
maças
applesf
‘half of the apples’
c. metade
half1
do
thesg.m
suco
juicem
‘half of the juice’
(3.41) Portuguese (Muriel Assmann, p.c.)
a. meia
half2.sg.f
maça
applef
‘half of the apple’
b. #meias
half2.pl.f
maças
applesf
c. #meio
half2.sg.m
suco
juicem
It seems plausible to posit that the contrasts between the phrases in
(3.40) and (3.41) follow from topological restrictions imposed by meio
and metade on what they select, similar to the distinction between
Polish pół and połówka on one hand and połowa on the other. In other
words, the nominal half -word metade is topology-neutral, and thus can
combine with DPs that denote integrated objects as well as those that
have scattered entities in their extensions. On the other hand, adjectival
meio appears to be sensitive to the type of topological relations holding
between parts of the referents of NPs it modifies. In particular, it
requires integrated wholes. From the syntactic perspective, meio as
a nominal modifier sits lower in the structure than metade which ccommands
the whole DP, i.e., as an adjective it is merged with the
NP. Assuming that the NumP is higher than the AP (e.g., Ritter 1991,
1992) this might explain scopal properties of meio when it comes to
subatomic quantification with respect to regular plurals, i.e., why
(3.41b) denotes a plurality of halves rather than a half of a plurality.
However, meio can also combine with lexical plurals such as pluralia
tantum, see (3.42), and it has been argued that the plural in lexical
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3. Exploring topological sensitivity
plurals is significantly lower, i.e., inside the nP (see, e.g., Acquaviva
2008, Alexiadou 2011, Smith 2015, 2017). Notice that (3.42) gets a
similar interpretation as the phrase in (3.41b), i.e., denotes a plurality
of halves or rather, due to the number-neutrality of óculos ‘eye-glasses’,
either a plurality of halves or one half of an object. I argue that this fact
alongside the infelicity of phrases where meio modifies mass terms, see
(3.41c), supports the semantic explanation of the observed behavior
in terms of topological sensitivity.
(3.42) Portuguese (Muriel Assmann, p.c.)
meio
half2.m
óculos
eye-glassesm
a. part-of-a-singularity reading
b. #part-of-a-plurality reading
Even more interestingly, the results of the flag test show that meio in
fact patterns with Polish połówka. As reported by my informant, the
sentence with the proportional partitive with metade in (3.43a) is true
both in the flag AB, see Figure 3.3, and in the flag ABA scenario, see
Figure 3.4. However, the sentence involving (3.43b) is judged true only
in the first case and false in the latter. This means that metade does not
impose any topological restrictions on the portion of an entity denoted
by the downstairs DP, wheres meio yields only integrated parts.
(3.43) Portuguese (Muriel Assmann, p.c.)
a. Metade
half1
da
the
bandeira
flag
é
is
vermelha.
red
‘Half the flag is red.’
(i) AB
(ii) ABA
b. Meia
half2
bandeira
flag
é
is
vermelha
red
‘A half of the flag is red.’
(i) AB
(ii) #ABA
A similar pattern is attested in Dutch where the distinction between
the nominal half -word de helft ‘half’ on the one hand, see (3.44a), and
adjectival half ‘half’ on the other, see (3.44b), is used in order to in-
108
3. Exploring topological sensitivity
dicate the discussed contrast. The first expression forms a standard
proportional entity partitive by combining with the prepositional vanphrase
which takes a singular DP as its complement (de Hoop 1997,
2003; see also Cleven 2013, pp. 16–18). The latter, however, resembles
meio in that it shows agreement with the modified NP. The suffix -e
on half shows up when the noun is either definite or common gender,
or both, as in the case of (3.44b). Similar to the Portuguese example,
the flag test shows that de helft licenses both a continuous-half and a
discontinuous-half interpretation, whereas adjectival half disallows
the latter reading. In other words, while the sentence in (3.44a) is reported
to be true of both Figure 3.3 and 3.4, (3.44b) is judged false in
the ABA flag scenario. This fact strongly suggests that Dutch lexicalized
the distinction between topologically neutral and topologically
sensitive partitive words. While de helft is similar to Polish połowa or
Portuguese metade in that it simply returns an entity constituting 50%
of a whole, truth conditions imposed by half are stronger, i.e., the
partitive yields a half that needs to be integrated.
(3.44) Dutch (Erlinde Meertens, Izabela Jordanoska, p.c.)
a. De
the
helft
half1
van
of
de
the
vlag
flag
is
is
rood.
red
‘Half the flag is red.’
(i) AB
(ii) ABA
b. De
the
halve
half2
vlag
flag
is
is
rood.
red
‘A half of the flag is red.’
(i) AB
(ii) #ABA
It turns out that similar to Polish Portuguese and Dutch distinguish
between topology-neutral and topology-sensitive half -words lexically.
However, it is not the only way the distinction can be encoded in natural
language. Thus, let us now consider a few strategies languages use
to differentiate between the two partitive meanings in question I have
identified. Interestingly, different structures give rise to similar semantic
effects concerning topological relations in subatomic quantification.
The sample discussed here is relatively small, but the discussed data
109
3. Exploring topological sensitivity
suggest that the observed semantic phenomenon is systematic and
cross-linguistically valid.
Let us start with the syntactic distinction occurring in English, as
already mentioned in (3.39). From the morphological point of view
there is no contrast between the partitive word half in (3.45a) and
(3.45b). Nevertheless, the syntactic structures in which it appears in
each of the two sentences significantly differ (see Vannestål 2004 for
discussion). In (3.45a), half is a predeterminer since it can occur before
regular determiners such articles and demonstratives (Quirk et al.
1985, pp. 257–258). On the other hand, in (3.45b) half forms a standard
proportional entity partitive by taking the of -phrase with a singular
DP as its complement. In this environment it seems to have nominal
properties since it can be preceded by a determiner and it allows for
numeral and adjectival modification (Huddleston and Pullum 2002,
p. 434). The syntactic distinction translates into a semantic difference.
Specifically, the sentence in (3.45a) is considered appropriate with
respect to both the AB and ABA flag in Figure 3.3 and 3.4, respectively,
whereas (3.45b) would be judged true only in the first scenario. The
contrast shows the relevance of topology in subatomic quantification
even in the absence of distinct lexical items differentiating between
a continuous-part and a discontinuous-part interpretation. In other
words, it appears that natural language can employ purely syntactic
means in order to indicate the notion of topological integrity of parts.
Notice that the half -word in (3.45b) is preceded by by the article. Superficially,
the distinction between half and a half of resembles to some
extent the contrast between the part-expressions part of and a part of.
The appearance of the indefinite article suggests that the latter should
be treated on a par with count nominals and countability seems to
imply spatial integrity.
(3.45) English (Jonathan Bobaljik, Jeffrey Parrott, p.c.)
a. Half the flag is red.
(i) AB
(ii) ABA
b. A half of the flag is red.
(i) AB
(ii) #ABA
110
3. Exploring topological sensitivity
Another strategy is employed by German. At first blush, it appears to
pattern with Portuguese and Dutch since it also distinguishes between
the nominal half -word Hälfte ‘half’ and adjectival halb ‘half’. They show
similar behavior with respect to count singulars, plurals, and mass
terms as the metade/meio and de helft/half alternations. As witnessed
in (3.46a), Hälfte combines felicitously with all the types of expressions
in question, whereas halb is incompatible with mass nouns, see
(3.47c), and obligatorily gets a plurality-of-parts interpretation with
plurals, as in (3.47b). These facts suggest that Hälfte is topology-neutral,
whereas halb is sensitive to the spatial constitution of referents of the
NP it attaches to. Similar to Portuguese meio and Dutch half it rejects
cumulative predicates which denote scattered entities.12
(3.46) German (Nina Haslinger, p.c.)
a. Hälfte
half1
des
thegen
Apfels
applegen
‘half of the apple’
b. Hälfte
half1
der
thegen
Äpfel
applesgen
‘apple-halves’
12. Note that the pattern reported in (3.47) holds only for indefinite DPs. For reasons
unclear to me, when the definite article c-commands NPs modified by halb,
new interpretations emerge in an unexpected way. In particular, (ib) gets a part-ofa-plurality
reading, whereas (ic) is absolutely felicitous. Whatever mechanism is
responsible for the contrast, I assume it is related with some non-trivial properties
of the definite article.
(i) German (Nina Haslinger, p.c.)
a. der
the2.sg.m
halbe
halfsg.m
Apfel
applem
‘the half of the apple’
b. die
the2.pl
halbe
halfpl
Äpfel
apples
‘the half of the apples’
c. der
the2.sg.m
halbe
halfsg.m
Saft
juicem
‘the half of the juice’
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3. Exploring topological sensitivity
c. Hälfte
half1
des
thegen
Saftes
juicegen
‘half of the juice’
(3.47) German (Nina Haslinger, p.c.)
a. ein
am
halber
half2.sg.m
Apfel
applem
‘a half of the apple’
b. #halbe
half2.pl
Äpfel
apples
‘apple-halves’
c. #halber
half2.sg.m
Saft
juicem
However, despite identical selectional restrictions German halb differs
significantly from the corresponding adjectival half -words in Portuguese
and Dutch. As the results of the flag test reveal, unlike in the
case of meio and half there is no contrast between halb and Hälfte with
respect to the the spatial make up of the partial entity they yield. As
witnessed in (3.48a) and (3.48b) sentence with both types of expressions
are judged true in both the AB and the ABA flag scenario. It
turns out that halb patterns with pół rather than with połówka, i.e., it
requires an integrated object as an input and simply returns its half
irrespective whether it is a continuous or discontinuous entity.
(3.48) German (Nina Haslinger, Maximilan Prüller, p.c.)
a. Die
the
Hälfte
half1
von
of
der
thegen
Fahne
flag
ist
is
rot.
red
‘Half the flag is red.’
(i) AB
(ii) ABA
b. Die
the
halbe
half2
Fahne
flag
ist
is
rot.
red
‘Half the flag is red.’
(i) AB
(ii) ABA
However, there is yet another construction in German which displays
the kind of semantic behavior observed in typologically sensitive ex-
112
3. Exploring topological sensitivity
pressions such as Polish połówka, Portuguese meio, and Dutch half. In
particular, consider the subject of the sentence in (3.49a) involving the
half -word Hälfte preceded by both what at first blush appears to be
an indefinite article and the definite article die ‘the’. Notice, however,
that the element eine in this environment actually is not an indefinite
article. The mere fact that it can combine with the definite article die
suggests it is an expression of a different type. Furthermore, unlike
regular indefinite articles, it is always stressed. As shown by the ungrammaticality
of (3.49b), in this construction one cannot put stress
on the half -word.
(3.49) German (Maximilian Prüller, p.c.)
a. Die
the
EINE
a/one
Hälfte
half1
der
thegen
Fahne
flag
ist
is
rot.
red.
‘The half of the flag is red.’
b. *Die
the
eine
a/one
HÄLFTE
half1
der
thegen
Fahne
flag
ist
is
rot.
red.
Interestingly, the truth conditions of sentences with the die eine Hälfte
partitive expression contrast with sentences involving regular proportional
partitives with respect to the flag test. As we have already
discussed, (3.50a) is true of both relevant scenarios, i.e., Figure 3.3 and
3.4. However, (3.50b) is consistently judged false in the flag ABA situation.
This shows that similar to Polish German developed a tripartite
semantic distinction between different proportional partitive expressions,
i.e., topologically neutral partitives headed by die Hälfte on the
one hand and topology sensitive expressions modified by adjectival
halb as well as die eine Hälfte constructions which yield integrated parts
on the other. The expressions are distinguished from each other both
lexically and structurally with the most marked structurally marked
form being at the same time most semantically complex.
(3.50) German (Nina Haslinger, Maximilan Prüller, p.c.)
a. Die
the
Hälfte
half1
von
of
der
thegen
Fahne
flag
ist
is
rot.
red
‘Half the flag is red.’
(i) AB
(ii) ABA
113
3. Exploring topological sensitivity
b. Die
the
eine
a/one
Hälfte
half1
der
thegen
Fahne
flag
ist
is
rot.
red
‘The half of the flag is red.’
(i) AB
(ii) #ABA
Finally, the examined contrast involving yet another two types of
constructions is found in Mandarin. The first kind of structure is
provided in (3.51a). Here, the partitive meaning is conveyed by the
phrase involving the proportional expression yí-bàn ‘half’ related by
the associative marker de to the nominal guó qí ‘national flag’ (see,
e.g., Jing-Schmidt 2005, p. 294, Jin 2018). In this construction, the half word
bàn does not follow a classifier and combines directly with the
preceding cardinal numeral yí ‘one’ forming a constituent. It is possible
that yí in yí-bàn is in fact a grammaticalized expression interpreted as
an indefinite rather than a numeral (cf. Hsieh 2008, p. 127, Jin 2018).13
In any case, this structure is topologically neutral and yields both
integrated and discontinuous halves of entities since sentences such
as (3.51a) are reported to be true of the flag in Figure 3.3 as well as
the one in Figure 3.4. In (3.51b), however, the partitive morpheme
bàn behaves more like a quantifier since it needs to combine with the
classifier miàn dedicated to counting flat and smooth objects such as
mirrors in order to quantify over parts of the national flag denoted by
the nominal. Therefore, it appears that in this case the syntactic status
of the half -word is on a par with cardinals. And again, a different
structure corresponds to a different semantics. Unlike (3.51a), the
sentence in (3.51b) can only mean that an integrated half of the national
flag is red, i.e., it would be judged false in the scenario represented in
Figure 3.2.
(3.51) Mandarin (Chang Liu, p.c.)
a. guó
national
qí
flag
de
lnk
yí-bàn
one-half
shì
cop
hóng
red
de.
lnk
‘Half the national flag is red.’
(i) AB
(ii) ABA
13. See also Hsieh (2008, p. 46) and Zhang (2011) for the discussion of issues regarding
constituency and other types of environments bàn appears in.
114
3. Exploring topological sensitivity
b. bàn-miàn
half-cl
guó
national
qí
flag
shì
cop
hóng
red
de.
lnk
‘A half of the national flag is red.’
(i) AB
(ii) #ABA
The Mandarin example discussed above concludes this brief exploration
of cross-linguistic strategies of encoding sensitivity to topological
notions in partitive expressions. Though the sample of languages
examined here was relatively small and most probably more strategies
await to be discovered within a future thorough cross-linguistic
investigation, the presented evidence proves that natural language
encodes subtle topological distinctions in the domain of partitivity.
The contrasts examined here are often very slight but a careful examination
suggests that the topology-related phenomena in question are
not something idiosyncratic but rather occur systematically across
languages.
In the next section, I will discuss yet another class of expressions
that might shed new light on the issues related to the interplay of
quantification and topology. In particular, I will consider some similarities
and contrasts concerning two whole-adjectives in Polish, namely
cały ‘whole’ and kompletny ‘complete’. To this end, I will explore two
different respects of being entire as well as the interaction between
whole-adjectives and different types of nominals from the perspective
of subatomic quantification and topological sensitivity.
3.5 Whole-adjectives
Due to contrasts parallel to the one illustrated in (3.52), adjectives
such as whole and entire have been analyzed as universal quantifiers
over parts (Moltmann 1997) or maximizing modifiers, i.e., expressions
restricting exception tolerance (Morzycki 2002).14 There is a strong
intuition that compared to (3.52a) the sentence in (3.52b) implies that
no part of the entity in question are exempt from having to satisfy the
main predicate. In particular, (3.52a) would be true if a sufficiently
14. Morzycki (2002) builds on Brisson (1998)’s analysis of all in terms of maximizing
modification.
115
3. Exploring topological sensitivity
significant proportion of the ferret were under the water, i.e., if part
of a paw or a tail were not submerged, under most circumstances the
sentence would not be considered false. However, the proposition
expressed by (3.52b) is stronger. It states that all the parts of the ferret
are submerged.
(3.52) English (Morzycki 2002)
a. The ferret is submerged.
b. The whole/entire ferret is submerged.
In Polish, alongside the standard whole-adjective cały ‘whole’, there
is yet another expression that shares some of its semantic properties
yet differs in an interesting respect, namely the adjective kompletny
‘complete’. Though there is only a partial overlap between the two
expressions in question in terms of their distribution, in principle they
show the same pattern. Let us first consider the distributional potential
of whole-adjective cały. What is interesting about Polish is that it can
be used for universal quantification over parts of stuff denoted both
by singular count nouns and mass terms. Specifically, (3.53a) refers to
all the material parts making up a particular apple, whereas (3.53b)
is true of an entity comprising all the portions of juice in a relevant
context. On the other hand, when modifying a plural expression such
as (3.53c) cały cannot scope over the plural. In other words, (3.53c) is
true of a plurality of whole apples but false of a whole plurality of not
necessarily whole apples.
(3.53) Polish
a. całe
wholesg.n
jabłko
applen
‘whole apple’
b. cały
wholesg.m
sok
juicem
‘all the juice’
c. całe
wholepl
jabłka
apples
‘whole apples’
(i) plural > whole
(ii) #whole > plural
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3. Exploring topological sensitivity
Compared to cały, the distribution of kompletny is significantly restricted.
Intuitively, this is due to the fact that it seems to presuppose
certain complexity in terms of a part-whole structure. In particular, it
can only modify nominals referring to objects consisting of multiple
easily recognizable and distinguishable parts such as utensils and
clothes.15 However, apart from this restriction the whole-adjective
kompletny mirrors the pattern observed for cały. For instance, due to
the fact that computers consist of many detachable elements, (3.54a)
is a meaningful phrase in Polish. Similar, when combined with an
object mass term it implies that all discrete objects comprising the
extension of the noun are in place. For example, (3.54b) would be true
of a collection of equipment where there is no item missing. Moreover,
the whole-adjective kompletny also appears frequently with collective
nouns such as zestaw ‘set’ and kolekcja ‘collection’ which denote groups
of artifacts. On the other hand, when modifying a plural expressions
as in (3.54c) kompletny cannot assert of a plurality as such, but rather
of singular individuals. In other words, (3.54c) means that individual
computers are complete and not the plurality thereof.
(3.54) Polish
a. kompletny
completesg.m
komputer
computerm
‘complete computer’
b. kompletny
completesg.m
sprzęt
equipmentm
‘complete equipment’
c. kompletne
completepl
komputery
computers
‘complete computers’
(i) plural > complete
(ii) #complete > plural
Notice also that both cały and kompletny can combine with pluralia
tantum, see (3.55a)–(3.55b). The plurale tantum nominal nożyczki laparaskopowe
‘laparoscopic graspers’ denotes a specialized object con-
15. There is yet another meaning of kompletny that could be glossed as ‘total’ which
does not impose such a constraint. This use is especially frequent in colloquial Polish;
however, it will not be considered here.
117
3. Exploring topological sensitivity
sisting of multiple detachable and replaceable elements which makes
it a proper expression to be modified by kompletny. In both cases, the
whole phrase is ambiguous between a singular and a plural reading,
i.e., it either refers to one whole or complete object, respectively, or
to a plurality of whole or complete objects. However, the reading on
which (3.55a) or (3.55b) imposes a semantic restriction on a plurality
is not available, i.e., the phrases cannot mean something like a whole
or complete plurality of laparoscopic graspers. This is despite the fact
that the plural in plurale tantum nouns has been argued to be low in
the structure, i.e., inside the nP (see, e.g., Acquaviva 2008, Alexiadou
2011, Smith 2015, 2017). Thus, I assume that the data suggest the reason
for the lack of the interpretation in question is not exclusively due
to the position in structure but also due to some other factor.
(3.55) Polish
a. całe
wholepl
nożyczki
scissors
laparoskopowe
laparoscopicpl
‘whole laparoscopic graspers’
b. kompletne nożyczki laparoskopowe
completepl scissors laparoscopicpl
‘complete laparoscopic graspers’
Table 3.7 gives an overview of the distribution of the Polish wholeadjectives
cały and kompletny as discussed above. While both can cooccur
with singular count nouns and mass terms, the second requires
the modified noun to refer to complex entities consisting of multiple
easily individuated parts. On the other hand, when combined with
regular plurals none of the Polish whole-adjectives can scope over the
plural. In other words, what is targeted is always the structure of
particular singular individuals and not that of a plural entity.
singulars mass nouns plurals
cały #
kompletny #
Table 3.7: Distribution of Polish whole adjectives
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3. Exploring topological sensitivity
The evidence regarding the distribution of the Polish whole-adjectives
in question suggests the distinction between integrated objects as prototypically
denoted by singular count nouns and scattered entities
designated by mass terms on the one hand and arbitrary sums of
individuals referred to plural expressions. To some extent this contrast
resembles what was observed with respect to the distribution of the
piece-word kawałek. Before I will move on to discussing some semantic
differences between cały and kompletny, let us briefly examine a crosslinguistic
context which at first blush might cast some doubt on the
cross-linguistic validity of the pattern observed in Polish. In the next
section, I will consider the data concerning a German whole-adjective
modifying plural nouns.
3.5.1 German ganz and plurals
Similar to Polish, in German the whole-adjective ganz ‘whole’ can be
used as a universal quantifier over parts of matter making up referents
of both singular count nouns as well as mass terms, as demonstrated
in (3.56a)–(3.56b) (see also Moltmann 1997, pp. 123–127).16
(3.56) German (Nina Haslinger, p.c.)
a. der
thesg.m
ganze
wholesg
Apfel
Apfelm
‘the whole apple’
b. der
thesg.m
ganze
wholesg
Saft
juicem
‘all the juice’
However, it has also been reported that at least in some German dialect
the adjective ganz ‘whole’ when modifying plural DPs gives rise to a
universal quantification interpretation (see Moltmann 1997, pp. 123–
127). This is claimed to be possible with plurals referring to sums
constituting natural wholes, i.e., sums whose members are considered
as having less individuality, and thus do not appear to be prominent
with respect to the whole (Moltmann 1997, p. 125). For instance, in
the sentences (3.57a) and (3.57b) children are part of a more or less
anonymous group, whereas bees are perceived as constituents of a
16. Similar to what was discussed in footnote
119
3. Exploring topological sensitivity
swarm. In both cases, the modifier ganz seemingly quantifies over
individuals making up the pluralities and the whole DP yields the
total number of children and bees involved in coming and stinging,
respectively.
(3.57) German (Moltmann 1997, p. 125; adapted)
a. Die
thepl
ganzen
wholepl
Kinder
children
sind
are
gekommen.
come
‘All the children have come.’
b. Die
thepl
ganzen
wholepl
Bienen
bees
haben
have
Maria
Maria
gestochen.
stung
‘All the bees have stung Maria.’
However, under more rigorous investigation it appears that at least
in one Austrian dialect showing the discussed behavior there is an
important distinct between ganz and the universal quantifier alle ‘all’.
Whatever the exact semantic contribution of ganz, it seems that since
unlike alle it allows for exceptions. For instance, consider the contrast
given in (3.58). While the sentence in (3.58a) can be supplemented
with another clause stating that not all of the children have come,
(3.58b) obviously excludes such a continuation.
(3.58) German (Maximilian Prüller, p.c.)
Context: My neighbors have 10 children and I don’t like them
but have to invite them.
a. Dann
then
sind
are
die
the
Nachbarn
neighbors
mit
with
ihren
theirdat
ganzen
wholedat.pl
Kindern
childrendat
gekommen.
come
Zwei
two
waren
were
krank
ill
zuhause,
at-home
und
and
es
it
waren
were
immer noch
still
acht!
eight
‘Then the neighbors have come with their children. Two
of them were ill and stayed home, but there were still
eight!’
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3. Exploring topological sensitivity
b. #Dann
then
sind
are
die
the
Nachbarn
neighbors
mit
with
allen
alldat
ihren
theirdat
Kindern
childrendat
gekommen.
come
Zwei
two
waren
were
krank
ill
zuhause,
at-home
und
and
es
it
waren
were
immer noch
still
acht!
eight
Although definitely more systematic research is required to establish
what precisely is going on, I take the contrast in (3.58) as evidence suggesting
that German does not pose a lethal threat for the generalization
regarding the topological sensitivity of whole-adjectives based on the
Polish data. In the next section, I will return to the alternation between
cały and kompletny and examine two different aspects of wholeness.
3.5.2 Integrity vs. maximality
There is yet another difference between the Polish whole-adjectives
cały and kompletny which also intuitively appears to be the most significant
one. The difference relates to an ambiguity between what I
will call an integrity reading and a maximality reading. The first is
topological in nature and indicates that the parts of a whole are connected
in such a way that they form an intact integrated object.17 The
latter, on the other hand, simply employs universal quantification over
parts or mereological exhaustivity, and thus indicates that no part of
a whole is missing.18 Under ordinary circumstances these two aspects
of wholeness are interrelated. Hence, the contrast between the two
interpretations might seem somewhat subtle and in many contexts it
17. See also Moltmann (1997, p. 127) for a remark on the integrity reading of German
ganz in predicate position as, e.g., in (i).
(i) German (Moltmann 1997, p. 127; adapted)
Das
the
Glas
glass
ist
is
noch
still
ganz.
whole
‘The glass is still intact.’
18. However, see Morzycki (2002) for arguments against treating whole-adjectives as
universal quantifiers over parts based on the anaphoric and scope properties they
give rise to. Instead, he proposes to treat them as maximizing modifiers. Since the
main point here concerns the distinction between quantification over wholes and
subatomic quantification, I only signal the problem here and leave it unaddressed.
121
3. Exploring topological sensitivity
is probably difficult to detect. However, it is relatively easy to come
up with examples where it becomes significant. For instance, consider
the scenario described in (3.59a). Notice that there are two kinds of
model aircraft Jan might have bought for Marysia, namely either a
pre-built model that has already been put together or a construction
kit with a collection of separate parts that require gluing. Then, depending
on whether Marysia got the first or the latter her brother
Piotruś’s question can be understood along the lines of topological
integrity or mereological exhaustivity. In particular, (3.59b) provides
an answer to a question implying a maximality reading, i.e., whether
no part is missing, whereas (3.59c) is a reply in terms of an integrity
interpretation, i.e., whether the parts are connected. This shows that
though the two ways in which being a whole can be understood are
frequently related, they are in fact separate semantic phenomena and
in some contexts only one of them can become prominent if not the
only relevant aspect.
(3.59) Polish
Context: Jan bought his daughter Marysia a model aircraft.
After Marysia unwrapped the gift and opened the box, her
brother Piotruś asks:
a. Czy
int
ten
this
samolot
model-aircraft
jest
is
cały?
whole
‘Is this model aircraft whole?’
b. Nie,
no,
brakuje
it-lacks
śmigła.
propellergen
‘No, it lacks the propeller.’
c. Nie,
no,
trzeba
it-needs
go
itacc
skleić.
stick-together
‘No, you need to stick it together.’
Now, let us compare what has been observed in (3.59a) with what
happens in the same context when the question in (3.60a) is slightly
modified. Specifically, instead of cały what appears in predicate position
is the whole-adjective kompletny. Here, the only meaningful answer
is the one in (3.60b). The reply in (3.60c) is just remarkably odd. That is
because unlike cały kompletny is not ambiguous and implies exclusively
the maximality aspect of wholeness. Therefore, the question in (3.60a)
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3. Exploring topological sensitivity
can only be interpreted as whether the box includes all the relevant
parts of the model aircraft.
(3.60) Polish
Context: Jan bought his daughter Marysia a model aircraft.
After Marysia unwrapped the gift and opened the box, her
brother Piotruś asks:
a. Czy
int
ten
this
samolot
model-aircraft
jest
is
kompletny?
complete
‘Is this model aircraft complete?’
b. Nie,
no,
brakuje
it-lacks
śmigła.
propellergen
‘No, it lacks the propeller.’
c. #Nie,
no,
trzeba
it-needs
go
itacc
skleić.
stick-together
‘No, you need to stick it together.’
The contrast between the number of possible answers in (3.59a) and
(3.60) shows that cały is ambiguous between an integrity and maximality
reading, whereas kompletny can be interpreted exclusively in terms
of maximality. Though the distinction is delicate, it is systematic and
can be detected in multiple environments. Even out of the blue the
intuitions regarding the contrast between the sentence in (3.61a) and
(3.61b) are very clear. The first states that all the relevant parts of the
toy are in place or that the parts are connected, or both, whereas the
latter implies only that no relevant part of the toy is missing.
(3.61) Polish
a. Ta
this
zabawka
toy
jest
is
cała.
whole
‘This toy is whole.’
(i) maximality reading
(ii) integrity reading
b. Ta
this
zabawka
toy
jest
is
kompletna.
complete
‘This toy is complete.’
(i) maximality reading
(ii) #integrity reading
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3. Exploring topological sensitivity
Table 3.8 recapitulates the findings. Distinguishing between a maximality
and integrity interpretation allows us to conclude that while
the Polish whole-adjective kompletny is topology-neutral, cały is not.
maximality integrity
cały
kompletny *
Table 3.8: Interpretations of Polish whole-adjectives
The contrast concerning possible interpretations of the Polish wholeadjectives
discussed in this section provides yet another piece of evidence
that some natural language expressions are sensitive to topological
relations holding between particular entities. Specifically, the
whole-adjective cały can be used to imply that all the parts of an object
remain in such a spatial configuration that the object in question constitutes
an integrated whole. In the next section, I will focus on the
maximality aspect of wholeness. In particular, assuming that the semantics
of cały involves universal quantification over entities of some
sort I will examine what exactly can constitute the domain of such
quantification.
3.5.3 Universal quantification over parts and wholes
Though the Polish whole-adjective cały can combine with a wide range
of expressions including singular count nouns, prototypical mass
terms, granulars, object mass nouns, and plurals, in each case the
interpretation of the whole NP depends on the type of referents denoted
by the modified noun. The difference becomes apparent when
we distinguish between subatomic quantification, i.e., quantification
over material parts making up elements of the denotation of a certain
noun, and quantification over wholes, i.e., quantification over
entire elements. To illustrate the distinction, let us first consider the
interpretative contrast between singular count nouns and liquid terms.
For instance, the phrase in (3.62a) involves subatomic quantification,
whereas (3.62b) does not. The whole-adjective cały in (3.62a) quantifies
universally over material parts of an apple rather than over an apples
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3. Exploring topological sensitivity
as such. On the other hand, in (3.62b) cały triggers universal quantification
over entities in the extension of the modified mass noun. The
translations of the sentences in (3.63) illustrate the difference.
(3.62) Polish
a. całe
wholesg.n
jabłko
applen
‘whole apple’
(i) subatomic quantification
(ii) #quantification over wholes
b. cały
wholesg.m
sok
juicem
‘all the juice’
(i) #subatomic quantification
(ii) quantification over wholes
(3.63) Polish
a. Marysia
Marysia
zjadła
ate
całe
wholeacc
jabłko.
appleacc
‘Marysia ate the whole apple.’
b. Marysia
Marysia
wypiła
drank
cały
wholeacc
sok.
juiceacc
‘Marysia drank all the juice.’
The distinction between subatomic quantification and quantification
over wholes becomes more evident in the case of granular and object
mass nouns since these expressions allow for both, see (3.64). In particular,
the NP in (3.64a) can either refer to the total amount of rice or
to rice involving whole grains. Similar, (3.64b) is either true of whole
shoes or of the total number of shoes in a given context.
(3.64) Polish
a. cały
wholesg.m
ryż
ricem
‘all the rice’
(i) subatomic quantification
(ii) quantification over wholes
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3. Exploring topological sensitivity
b. całe
wholesg.n
obuwie
footwearn
‘all the footwear’
(i) subatomic quantification
(ii) quantification over wholes
The contrast can be clearly demonstrated by considering non-trivially
different truth conditions between (3.65a) and (3.66a) on the one hand
and (3.65b) and (3.66b) on the other. The most prominent interpretation
of (3.65a) is that there is no rice left. On the other hand, the
sentence in (3.65b) does not infer that Marysia bought all the rice, but
rather that she bought whole grain rice, i.e., rice that has not been
milled. Similar, (3.66a) is true only if there was no shoe left in the
shop last year, whereas the context rules out this interpretation in
(3.66b). Instead, the available reading states, that the factory produces
complete shoes.
(3.65) Polish
a. Marysia
Marysia
zjadła
ate
cały
wholeacc
ryż.
riceacc
‘Marysia ate all the rice.’
b. Marysia
Marysia
kupiła
bought
cały
wholeacc
ryż,
riceacc
natomist
whereas
Jan
Jan
mączkę
flouracc
ryżową.
riceadj.acc
‘Marysia bought whole rice, whereas Jan bought rice
flour.’
(3.66) Polish
a. W
in
tym
thisloc
roku
yearloc
nasz
our
sklep
shop
wyprzedał
sold-out
całe
wholeacc
obuwie.
footwearacc
‘This year, our shop sold out all the footwear.’
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3. Exploring topological sensitivity
b. Nasza
our
fabryka
factory
produkuje
produces
całe
wholeacc
obuwie,
footwearacc
natomiast
whereas
konkurencja
competitors
jedynie
only
podeszwy.
solesacc
‘Our factory produces whole footwear, whereas our competitors
produce only soles.’
Finally, as it has already been mentioned cały cannot scope over the
plural. Consequently, a phrase such as (3.67a) denotes a plurality of
whole objects and not a whole plurality of objects. This behavior is
irrespective to whether the modified nominal is a regular plural or a
plurale tantum noun, see (3.67b).
(3.67) Polish
a. całe
wholepl
jabłka
apples
‘whole apples’
(i) subatomic quantification
(ii) #quantification over wholes
b. całe
wholepl
nożyczki
scissors
‘whole scissors’
(i) subatomic quantification
(ii) #quantification over wholes
The quantificational behavior of cały modifying plurals is well illustrated
in the sentence in (3.68). The following clause make it clear that
in this case the whole-adjective indicates that no part of each relevant
apple is exempt from being consumed.
(3.68) Polish
Marysia
Marysia
zjadła
ate
całe
wholeacc
jabłka,
applesacc
natomiast
whereas
Jan
Jan
zostawił
left
ogryzki.
apple-coresacc
‘Marysia ate the whole apples, whereas Jan left the apple
cores.’
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3. Exploring topological sensitivity
The quantificational properties of the Polish whole-adjective cały ‘whole’
are summarized in Table 3.9. The crucial observation is that the domain
of quantification depends in a non-trivial manner on the extension of
the modified nominal. Specifically, cały quantifies over material parts
of an entity denoted
singulars plurals object mass mess mass
subatomic quantification *
quantification over wholes * *
Table 3.9: Quantificational properties of Polish cały ‘whole’
This concludes the investigations into the relevance of topological
sensitivity with respect to part-whole structures in natural language.
3.6 Summary
In this section, I have provided additional evidence for the relevance
of the notion of spatial integrity for subatomic quantification. In particular,
novel evidence shows that the distinction between topologyneutral
and topology-sensitive partitive words is lexicalized in Polish.
Specifically, Polish has three morphologically distinct half -words, i.e.,
połowa, pół, and połówka. What they share is that when applied to an
entity they return part constituting approximately 50% of a whole.
They differ, however, regarding the type of entity they select for as
well as the type of part they yield. Given different distributional and
referential properties the data could be explained as follows. The half
word połowa is topology-neutral, i.e., it measures halves of any type
of entity, be it a solid individual, mass substance, aggregate, or an
arbitrary plurality of individuals. The outcome of quantification is
a portion of a whole which is topologically indeterminate. In other
words, połowa can either designate an integrated entity within the partwhole
structure of an individual or merely an arbitrary discontinuous
sum of pieces making up the whole. Unlike połowa, pół is sensitive to
the type of entity denoted by its complement. Specifically, it selects
only for integrated individuals, i.e., individuated objects that come
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3. Exploring topological sensitivity
in one piece.19 Nonetheless, similar to połowa the resulting partitive
phrase can refer either to a continuous or discontinuous half but in this
case it has to be part of an integrated whole. Finally, the meaning of
połówka is the most constrained. It shares selectional restrictions with
pół, i.e., it excludes scattered entities such as substances and granular
aggregates as denoted typically by mass terms as well as pluralities of
individuals, but in addition it imposes topological constraints on the
extension of the resulting partitive construction. To put it in a different
way, połówka cuts out an integrated part constituting approximately
50% of the volume of a continuous individuated object.
Similar, the alternation between fractions and the quarter-words
ćwierć and ćwiartka ‘quarter’ corresponds to the pattern observed
in half -words. In particular, fractions are topology-neutral, whereas
ćwierć patterns with pół and ćwiartka mirrors the behavior of połówka
in that it requires a portion of an object to form a contiguous entity.
Polish further distinguishes between distinct piece-words one of
which, i.e., kawałek, yields an integrated object within a part-whole
structure as long as the whole does not denote an arbitrary sum of
individuals. Furthermore, the contrast between two types of Polish
whole-adjectives, i.e., cały and kompletny, makes it easier to determine
two distinct aspects of wholeness, namely maximality and integrity.
The cross-linguistic investigation revealed that the phenomenon
observed in Polish partitives is not a peculiarity of one language but
rather appears in at least several other languages. The distinction is not
always expressed lexically and sometimes it manifests itself in different
interpretations assigned to different syntactic structures. It turns out
that Portuguese and Dutch distinguish between the topology-neutral
half -words metade and helft on the one hand and the topology-sensitive
adjectives meio and half semantically resembling Polish połówka on the
other. In contrast, English does not differentiate between distinct lexical
items but rather distinguishes two constructions, i.e., the topologyneutral
half DP phrase as opposed to the a half of DP structure which
again shows the semantic behavior similar to połówka. The same contrast
is reported to hold between the yí-bàn and bàn-CL constructions
19. This is a slight simplification since pół can also combine with collective nouns.
However, as declared above I will ignore this issue here.
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3. Exploring topological sensitivity
in Mandarin. Finally, the German data suggest a tripartite distinction
similar to Polish involving the die Hälfte, halb, and die eine Hälfte.
The significance of the data discussed here lays primarily in revealing
the relevance of topological relations holding between parts
of individuals forcing us to recognize that natural language semantics
is sensitive to whether parts come in one piece or constitute discontinuous
entities. In the following section, I will introduce yet another set
of novel evidence concerning subatomic quantification. In particular,
I will show that similar to cardinal numerals which are dedicated to
counting wholes there are numerical expressions in natural language
dedicated to counting parts.
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4 Multipliers
In the previous chapters, I have provided robust linguistic evidence for
the relevance of subatomic quantification in natural language as well
as for the significant role the topological notion of integrity plays with
respect to that phenomenon. I have examined multiple types of partitive
constructions including count explicit partitives and topologysensitive
proportional partitives. What these two types of expressions
have in common is that they impose a certain constraint on how the
denoted parts of objects are conceptualized. Specifically, they require
the parts to constitute integrated continuous elements. On the other
hand, we have seen that subatomic quantification is not restricted to
partitive constructions and is also attested in the adjectival domain,
e.g., German and Romance adjectival half -words as well as different
types of whole-adjectives. At this point, one could expect that given
the amount of evidence for the relevance of part-whole structures
in different types of constructions there should be quantificational
expressions in natural language that are specialized for counting parts
of entities. It turns out that in fact in many languages there are such
expressions.
In this chapter, I will present novel data concerning subatomic
quantification in the adjectival domain. In particular, I will examine semantic
properties of numerical expressions such as English double and
triple which I will refer to as multipliers (following Quirk et al. 1985,
Huddleston and Pullum 2002). In particular, I will focus mostly on
Slavic multipliers exemplified by Polish podwójny, Czech dvojitý, Russian
dvojnoj, and Bosnian-Croatian-Serbian (BCS) dvostruki all ‘double’
which all seem to have identical properties. Unlike cardinal numerals,
multipliers do not count entities, but rather their particular parts. I
have chosen Slavic multipliers due to their morphological complexity
which suggests a rich semantic structure that is not expressed formally
in a language such as English. To foreshadow, I will argue that since
multipliers in Slavic are derivationally complex, they are in fact compositional.
Though the distribution of multipliers is relatively broad, I
will mainly concentrate on a subset of environments in which they can
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4. Multipliers
occur that allows us for novel insights with respect to the phenomena
discussed in this study.1
4.1 Significance of multipliers
Since the early days of formal semantics (starting with Montague
1973) the meaning of quantificational expressions such as numerals
has been constantly receiving a lot of attention. Over the last several
decades, extensive and extremely important work has been done
in this area leading to a number of influential theories of cardinals
(e.g., Barwise and Cooper 1981, Scha 1981, Landman 2004, Ionin and
Matushansky 2006). Nevertheless, it seems that there is still a vast
territory left unchartered since natural languages exhibit multiple
classes of numerical expressions that did not receive nearly as much
recognition as cardinal numerals. In this study, I will contribute to the
study of such somewhat neglected classes of quantifiers by providing
novel evidence considering multipliers. Though such expressions are
cross-linguistically common, see (4.1), as for now their semantic properties
are surprisingly understudied and I am not aware of any formal
attempt to account for their behavior and meaning.2
(4.1) a. English
double
b. German
doppelt
c. Italian
doppio
d. Lithuanian
dvigubas
e. Finnish
kaksinkertainen
1. Most of the data discussed in this chapter come from linguistic corpora. However,
I would like to sincerely thank Mojmír Dočekal, Pavel Caha, and Markéta Ziková,
Tetiana Kamyshanova, Boban Arsenijević, Guy Tabachnick, and Viola Schmitt for
confirming my intuitions concerning some of the Czech, Russian, BCS, English, and
German phrases, respectively.
2. An exception is Wągiel to appear a on which the whole chapter is loosely based.
132
4. Multipliers
f. Hungarian
dupla
g. Mandarin
shu¯ang
The fact that multipliers have been somewhat overlooked is even more
striking given that they exhibit non-trivial quantificational behavior
that differs significantly from the one observed in cardinal numerals.
For instance, consider examples such as those in (4.2). The noun
phrase in (4.2a) simply denotes a set of singular entities having the
property of being a crown. Interestingly, although the phrase in (4.2b)
is true of a plurality of two entities, (4.2c) can only be interpreted as
referring to a single individual similar to (4.2a). There is, however, a
crucial difference between the two expressions since (4.2c) seems to
involve subatomic quantification. In particular, the multiplier appears
to quantify over elements within the inner structure of a denoted entity.
In other words, double in (4.2c) restricts the denotation of the noun
to only those crowns that have a particular complex form.
(4.2) English
a. crown
b. two crowns
c. double crown
In the following parts of this chapter, I will discuss distribution and
semantic properties of Slavic multipliers with a special focus on Polish.
I will argue that such expressions are compositional and involve
quantification over objects that are conceptualized as integrated parts
of integrated wholes. Furthermore, I will suggest a generalization
concerning the meaning of a representative subset of data, i.e., multiplier
phrases involving concrete singular nouns, and point out some
non-trivial consequences as well as discuss other types of nominals
multipliers combine with and how it relates to subatomic quantifica-
tion.
4.1.1 Morphological complexity of Slavic multipliers
Since it is widely recognized that the derivational morphology in
Slavic languages is particularly rich, in recent years increasing at-
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4. Multipliers
tention has been drawn to the semantics of Slavic morphologically
complex numerical expressions. The research has lead to a number
of insightful investigations concerning such constructions in Czech
(Dočekal 2012, 2013, Dočekal and Wągiel to appear), Polish (Wągiel
2014, 2015b, to appear a, to appear b), and Russian (Khrizman 2015).
However, while previous research focused mainly on the impact of
numeral morphology on the collective/distributive alternation with
respect to different types of numerals, in this study I will examine the
adjectival domain and present evidence of the significance of multipliers
for subatomic quantification, i.e., quantification over cognitively
salient parts of entities. In Section 3.1 and 3.2, we have seen how formal
complexity correlates with topology-sensitive semantics in partitive
words in Polish. In this section, I will focus on the morphology of
Polish, Czech, Russian, and BCS multipliers which unlike their English
counterparts exhibit a complex derivational structure. I believe
this choice of data will be instructive since it will allow us to confront
multipliers with cardinals in a straightforward manner that will reveal
intriguing similarities as well as differences between the two classes.
The examined lexical items in question will be exemplified in
the following sections by expressions derived from the numeral root√
dw/
√
dv corresponding to the number 2. Let us assume the morphological
make-up of the basic cardinals and matching multipliers
as in (4.3)–(4.6).3 As can be witnessed in the morpheme orderings in
(4.3b)–(4.6b), all the multipliers in question consist of numeral roots
shared with corresponding cardinals and some additional morphology
including the affixes po ... n-/-it-/-n-/-struk- and the morphemes
-ój-/-oj-/-o- which mark a non-cardinal stem as well as inflectional
markers. Notice that the morphemes -ój-/-oj-/-o- appear in multipliers
derived from numeral roots referring to numbers 2–3. In multipliers
derived from roots referring to the number 4, the suffix -ór-/-er-/-oro-
3. For convenience, I provide only unmarked forms for cardinals in (4.3a)–(4.6a).
Similar, in (4.3b)–(4.6b) masculine forms represent whole declensional paradigms.
For BCS in (4.6b), the derivationally complex adjective dvostruki ‘double’ has been
chosen for sake of exposition despite the fact that there is also another BCS multiplier,
namely dupli ‘double’. Although it seems that there is a difference in the distribution
of the two forms, I assume that there is no relevant semantic distinction between
them with respect to the phenomena discussed here.
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4. Multipliers
occurs instead, e.g., Polish po czw-ór n-y, Czech čtv-er-n-ý, Russian
četv-er-n-oj, and BCS četv-oro-stuk-i all ‘quadruple’.
(4.3) Polish
a. dw-a
numeral.root-inflectional.marker
b. po dw-ój n-y
circumfix numeral.root-non.card.stem circumfix-infl.marker
(4.4) Czech
a. dv-a
numeral.root-inflectional.marker
b. dv-oj-it-ý
numeral.root-non.cardinal.stem-deriv.suffix-infl.marker
(4.5) Russian
a. dv-a
numeral.root-inflectional.marker
b. dv-oj-n-oj
numeral.root-non.cardinal.stem-deriv.suffix-infl.marker
(4.6) BCS
a. dv-a
numeral.root-inflectional.marker
b. dv-o-struk-i
numeral.root-non.cardinal.stem-deriv.suffix-infl.marker
In general, Slavic multipliers exhibit adjectival morphology and agree
with modified nouns in case, number, and gender, as in (4.7). I will refer
to the derivational circumfix po ... n- in Polish and suffixes -it-, -n-,
and -struk- in Czech, Russian, and BCS, respectively, as multiplicative
morphemes. Moreover, I will assume here that inflectional markers
and morphemes constituting non-cardinal stems have no contribution
to the semantics of multipliers. Therefore, in the analysis to come I will
focus only on the interaction between numeral roots and multiplicative
morphemes.
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4. Multipliers
(4.7) Polish
Ta
thisnom.sg.f
podwójna
doublenom.sg.f
szyba
glassnom.sg.f
się
refl
stłukła.
brokesg.f
‘This double glazing broke.’
An approach neglecting the semantic role of some of the examined
morphemes in composition might seem in fact a considerable simplification
since affixes marking non-cardinal stems are attested not only
in multipliers, but also in a variety of other numerical expressions.
For instance, consider Polish denumeral group nouns such as dwójka
‘(group of) two’, derived numerals presupposing that a plurality is
heterogeneous with respect to the natural gender of referents such
as dwoje ‘two (one male and one female)’, taxonomic numerals like
dwojaki or dwoisty both ‘twofold’ as well as denumeral verbs such as
podwoić ‘to double’.4 At first blush, it seems appealing to assume that
the appearance of non-cardinal stem markers in all the above cases
is non-coincidental from the semantic point of view. Therefore, an
alternative approach could assign, e.g., Polish -ój-/-oj-/-oi- a meaning
general enough to cover the whole variety of considered expressions,
i.e., an underspecified operation on numbers that is different from the
one employed in cardinals. Nevertheless, in this study I will remain
agnostic with respect to that possibility. Instead, I will assume that
the discussed markers are semantically vacuous and that their sole
function is purely structural, specifically they simply form stems. Similar,
for the sake of simplicity I will ignore all the intricacies related to
the semantics of gender as well as its relationship with quantification
(see, e.g., Arsenijević 2016 and Fassi Fehri 2016, 2017) and the role of
inflectional markers in coding such an interaction.
4.1.2 Subatomic quantifiers
In the previous section, we have seen that in Slavic the morphological
make-up of multipliers resembles to that of cardinals with the crucial
difference that the first consistently involve more structure. Let us
now consider the meaning of phrases in which multipliers modify
a noun. I will start with a set of examples in (4.8)–(4.11) which are
4. See Wągiel (2014, 2015b) for a discussion and possible analysis of expressions
such as dwójka and dwoje. Notice also that -ój-, -oj-, and -oi- are allomorphs.
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4. Multipliers
semantically analogous and equivalent to the English phrase in (4.2c).
These examples will be considered as illustrative since they are all
attested in representative text corpora such as the National Corpus
of Polish (NCP), Czech National Corpus (CNC), Russian National
Corpus (RNC), and Croatian National Corpus (CrNC). Furthermore,
the Polish noun korona ‘crown’ ranks at the 13th place on the list of
collocation candidates within two-words collocations for the form
podwójna ‘double’ in the balanced sample of the National Corpus
of Polish whereas the Czech noun koruna ‘crown’ ranks at the 7th
place for the form dvojitá ‘double’ in a corresponding Czech sample
in the Czech National Corpus. Moreover, the example is of historical
relevance since it resembles the famous case of the complex papal
crown discussed in the philosophical literature (Wiggins 1980) which
I will also briefly address.
(4.8) Polish (NCP)
podwójna
doublesg.f
korona
crownf
‘double crown’
(4.9) Czech (CNC)
dvojitá
doublesg.f
koruna
crownf
‘double crown’
(4.10) Russian (RNC)
dvojnaja
doublesg.f
korona
crownf
‘double crown’
(4.11) BCS (CrNC)
dvostruka
doublesg.f
kruna
crownf
‘double crown’
As already mentioned, noun phrases such as those in (4.8)–(4.11) denote
sets of singular (and not plural) entities despite the fact that in
the morphological make-up of multipliers modifying a noun there
are numeral roots which involve reference to the number 2. Given the
classical perspective on the meaning of numerals, at first sight this fact
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4. Multipliers
might appear somewhat puzzling. Nevertheless, it seems clear that in
all the multiplier examples discussed in this chapter quantification is
involved albeit it is not quantification over wholes. To illustrate this
claim, let us consider the historical example in (4.12).5 The sentence in
(4.12a) indicates a complex inner structure of the Pschent, and thus
entails the sentence in (4.12b). If it is true of an object that it is a double
crown, it has to be a crown and it has to consist of two elements.
(4.12) Polish
a. Pszent
Pschent
to
this
podwójna
double
korona.
crown
‘The Pschent is a double crown.’
b. |= Pszent
Pschent
składa
consists
się
refl
z
from
dwóch
twogen
części.
partsgen
‘The Pschent consists of two parts.’
However, it is not sufficient that the object referred to by the noun
Pszent consists of any two elements since it is not the case that every
crown consisting of two parts is a double crown. Rather, the two relevant
elements need to have a particular property, i.e., a property comparable
to the property of the whole. In other words, a double crown
consists of two parts which can be considered crowns themselves. For
instance, the reader familiar with ancient history will recognize the
Pschent as a crown worn by rulers of Ancient Egypt. Historically, as
demonstrated in Figure 4.1 the Pschent combined two parts, i.e., the
Red Deshret Crown of Lower Egypt, see Figure 4.2, and the White
Hedjet Crown of Upper Egypt, see Figure 4.3. Its design represented
the pharaoh’s power over all of the unified kingdom. Due to the discussed
features of the relevant parts of the Pschent, one could say it is
a two-in-one crown.
5. In the following text, the phenomena will be discussed mainly on the basis
of Polish examples. However, the generalizations are intended to hold for every
language in question.
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4. Multipliers
Figure 4.1: Pschent Figure 4.2: Deshret Figure 4.3: Hedjet
Note that there is only one cut that divides the Pschent into the
Deshret and Hedjet and this characteristic can be generalized to any
hypothetical double crown. In other words, not every part of a double
crown is a crown itself. On the contrary, in this and similar cases within
the set of all the parts of a whole there are exactly two parts of which a
property comparable to the property of the whole holds. Therefore, the
semantics of Slavic predicates such as those in (4.8)–(4.11) and others
corresponding to the English phrase double crown does not involve the
property of divisive reference which is sometimes assumed to hold
for mass nouns (e.g., Cheng 1973).6 Notice also that the referents of
multiplier phrases differ from things designated by nouns like twig,
rock, and fence. Though such entities also involve parts that have the
property of a whole, e.g., parts of a twig are also twigs and parts of
a rock are also rocks (e.g., Zucchi and White 2001, Rothstein 2010),
the crucial distinction lies in the arbitrariness of subdivisions. While
objects in the extensions of twig-like expressions can be partitioned in
an arbitrary manner and the resulting parts will still have the property
of a whole (assuming some level of granularity), complex individuals
denoted by multiplier phrases typically require a specific subdivision
in order to identify the salient parts. This property makes them on a
par with the already mentioned example of the papal tiara which is a
complex object constituting a triple crown. What I believe to be the
signature property of such individuals is that they are simultaneously
6. I do not share this view. For an overview of the problems related to treating
divisive reference as the hallmark of mass terms see, e.g., Grimm (2012b, pp. 111–
119).
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4. Multipliers
perceived as one integrated entity and as a configuration of parts that
could be considered independent objects in their own rights.
At this point, one might object that the double crown example is
somewhat fancy and does not really reveal the meaning of multipliers.
However, the characteristic described above is also valid for many
other examples including nouns ranked relatively high as collocation
candidates in the NCP. For instance, consider (4.13a). The noun drzwi
‘door’ ranks at the 23rd place on the NCP collocation list for the nonvirile/neuter
form podwójne.7 An object referred to by the phrase is a
door made of two independently moving leafs, i.e., entities that could
be perceived as self-sufficient doors. Similar, a double garage denoted
by the expression in (4.13b) is a building with two entrances leading
to separate parking spaces. Again, it appears that the structure of the
entity in question involves parts that can be considered as having a
property similar to the property of the whole.
(4.13) Polish (NCP)
a. podwójne
doublepl
drzwi
doors
‘double door’
b. podwójny
doublesg.m
garaż
garagem
‘double garage’
The same way of thinking about multiplier phrases also applies to
examples that at first blush appear to be less obvious than those discussed
above. For instance, Polish multipliers frequently occur in
constructions such as those in (4.14). This is corroborated by the fact
that, e.g., the noun warstwa ‘layer’ ranks at the 7th place on the NCP
collocation list for the feminine form podwójna. At first sight, it seems
that expressions such as (4.14a) and (4.14b) refer to homogeneous
entities and as such do not designate objects involving a complex
structure. However, under closer inspection it turns out that even in
these cases division into parts is not arbitrary. Specifically, a double
layer is normally taken to be a layer consisting of two layers merged
together. Since layers are flat entities spread over some surface, only
a cut alongside a certain plane can divide the two parts constituting
7. Notice that in Polish the noun drzwi ‘door’ is a plurale tantum.
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an entity denoted by the phrase in (4.14a). Similar, the expression in
(4.14b) designates an entity whose structure can be partitioned only in
a particular manner in order to separate the salient parts. Intuitively,
a double coating consists of two parts that themselves are coverings
applied to the surface of some object.
(4.14) Polish (NCP)
a. podwójna
doublesg.f
warstwa
layerf
‘double layer’
b. podwójna
doublesg.f
powłoka
coatingf
‘double coating’
Certainly, examples such as those in (4.8)–(4.11) and (4.13)–(4.14) do
not exhaust the combinatorial potential of Slavic multipliers and some
more problematic cases will be discussed subsequently in Section 4.2.
Nevertheless, the data discussed above indicate that at least a subset of
constructions involving multipliers indicates clear cases of subatomic
quantification. Noun phrases such as those in (4.8)–(4.11) and (4.13)–
(4.14) denote sets of things conceptualized as singular objects with
a complex inner structure. The role of multipliers in the discussed
constructions is to quantify over particular parts of such structures, i.e.,
parts that have a property comparable to the property of the whole.
I will refer to such elements as self-sufficient parts. However, before
we turn our attention to some less obvious examples and see how far
the generalization proposed above can get us, let us dedicate some
space to examine to what extent the described semantic properties are
representative for multiplier phrases in general.
4.1.3 Cross-linguistic similarities
In this section, I will discuss corpus-based evidence suggesting that
the semantic behavior of multipliers discussed in the previous section
is not a Slavic idiosyncrasy but it is actually widespread in languages
such as English and German. In particular, I will inspect a number
of phrases mostly involving some of the most frequent collocates of
double as provided by the Corpus of Contemporary American English
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4. Multipliers
(COCA) and supplement them with German equivalents found in the
German Reference Corpus (GRC). This brief investigation will both
provide a slightly broader cross-linguistic perspective and prepare
the ground for a discussion of more problematic cases in Section 4.2.
Let us start with a somewhat metalinguistic example. Specifically,
the symbol . . . used to indicate the semantic evaluation function
every formal semanticist knows so well, is referred to by the pluralized
phrase in (4.15a), i.e., double brackets. The structure of the entity is
such that each of the two parts of the, say, opening bracket can be
considered a bracket itself. However, since they are spatially arranged
in such a way that they form a perceptibly one integrated object, under
ordinary circumstances it would be rather awkward to refer to a double
bracket as two brackets. An analogous effect can be observed with
respect to the meaning of the phrase in (4.15b) which refers to a basin
designed in such a way that it includes two parts each of which is a
sink itself. Similar, (4.15c) is true of a cooking device that consists of
two merged ovens, i.e., two parts that have a property comparable
with the property of the whole. The expression (4.15d) refers to a knot
wound twice which typically results in an object consisting of two
recognizable parts that are knots themselves.
(4.15) English (COCA)
a. double bracket
b. double sink
c. double oven
d. double knot
Other examples of the discussed sort involve multiplier phrases such
as those in (4.16). Each of these expressions refers to an entity perceived
as a complex object involving two salient parts with a property
comparable to the one of the whole. Specifically, (4.16a) and (4.16b)
denote a staircase consisting of two connected staircases and a tomb
built in such a way that it comprises two tombs in one structure, respectively.
Similar, the phrases in (4.16c)–(4.16e) designate entities
conceptualized as one object consisting of two merged individuals in
their own right. Finally, the frequent expression in (4.16f) refers to a
body part that due to the excessive amount of fat or skin looks as if it
were a two-layered chin.
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4. Multipliers
(4.16) English (COCA)
a. double staircase
b. double tomb
c. double canoe
d. double flute
e. double needle
f. double chin
Similar to the phrases such as (4.14), the COCA provides examples of
homogeneous entities involving self-sufficient parts, see (4.17). The
same observations I have discussed with respect to the Polish data
also apply to the English expressions.
(4.17) English (COCA)
a. double layer
b. double glazing
c. double stitching
Sometimes, parts of an individual denoted by a multiplier phrase are
not actually connected but remain in such proximity that they are
perceived as forming one complex entity. For instance, the phrases
in (4.18a) and (4.18b) refer to close-set pairs of meteorological and
astronomical objects, respectively. In such cases, the perceived distance
between individual rainbows and stars is considered so small that a
pair is conceptualized as a complex entity consisting of two parts each
of which has a property comparable to the one of the whole. Similar
examples include, e.g., double nebula and double cluster which also refer
to astronomical objects perceived as complex dual entities.
(4.18) English (COCA)
a. double rainbow
b. double star
The results of several quick searches in the German Reference Corpus
(GRC) consulted with native speakers demonstrate that the reported
effects are also attested in German. The phrases in (4.19) exemplify different
types of referents of multiplier phrases including homogeneous
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entities, see (4.19b), and phenomena perceived as wholes consisting
of similar objects remaining in relative proximity, see (4.19c).8
(4.19) German (GRC)
a. Doppelkrone
double-crown
‘double crown’
b. Doppelverglasung
double-glazing
‘double glazing’
c. Doppelregenbogen
double-rainbow
‘double rainbow’
The quick overview of some of the frequent contexts multipliers appear
in shows that the double crown example discussed in Section 4.1.2
is not some extravagant case. Rather, it represents common semantic
behavior of multipliers. I conclude that the English and German evidence
based on the collocation list from COCA and and searches in
GRC suggests that the pattern is robust.
4.1.4 Multipliers as degree modifiers
Briefly, I will return to the data both from Slavic and English to examine
somewhat more problematic examples. Nonetheless, before
I move on to the discussion of cases where at least at the first sight
intuitions concerning the subatomic structure of entities denoted by
multiplier phrases are not that clear, let us discuss yet another use
of the expressions in question. In English, multipliers can be used
as predeterminer modifiers, i.e., external modifiers of the whole DP
occurring before the central determiner such as the definite article
(see, e.g., Quirk et al. 1985, pp. 257–258, Huddleston and Pullum 2002,
pp. 433–436).9 Examples of such uses are given in (4.20). The nominals
8. In fact, German distinguishes between two distinct though morphologically
related multiplier expressions, e.g., adjectival doppelt and suffixal Doppel-. Though it
seems that their distribution differs, for the sake of brevity I will not examine the
alternation here and leave such an investigation for future research.
9. Actually, the term predeterminer modifier is used only by Huddleston and
Pullum (2002). Quirk et al. (1985) use the term predeterminer instead.
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provided in (4.20) appear to be degree-related (cf. Morzycki 2009). For
instance, the meaning of the noun size in (4.20b) could be thought of
as corresponding to some degree of volume. Similar, the noun fun is
clearly gradable since it allows for degree modification as witnessed
by the well-formedness of the comparative phrase more fun. Consequently,
I conclude that multipliers used as predeterminer modifiers
involve degree modification and as such are expressions of a different
type.
(4.20) English (Quirk et al. 1985, p. 257, Huddleston and Pullum
2002, p. 434; adapted)
a. double the amount
b. triple the size
c. quadruple the fun
At some abstract level it might seem appealing to try to associate
the use of multipliers as degree modifiers with the use related to
subatomic quantification we are interested here. However, I will not
pursue such an attempt here for two reasons. First, the phrases in (4.20)
have a different structure than, e.g., those in (4.15). In the first, multipliers
combine with NPs whereas in the latter they modify whole DPs.
I argue that this fact indicates that there are good reasons to believe
that in each case they introduce a semantic operation of a different
type. The second reason has to do with cross-linguistic distribution
of multipliers. For instance, Czech distinguishes lexically between
subatomic multipliers such as dvojitý ‘double, consisting of two parts’
and dvojnásobný ‘double, twice as large’ (see Dočekal and Wągiel to
appear for a related discussion). This fact further suggests that the
semantics of English multipliers used as degree modifiers differs from
what is the main interest of this study.
4.2 Less obvious cases
In the previous section, I have discussed model examples of multiplier
phrases denoting complex entities with self-sufficient parts. I have
signaled that the distribution of multipliers in the examined Slavic
languages as well as in English and German is richer. In this section, I
will discuss other types of nominals that can be modified by numerical
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4. Multipliers
expressions such as double and show how the perspective posited
above allows us to capture the meaning of less obvious cases.
4.2.1 Essential parts
So far, I have discussed examples of multiplier phrases denoting objects
involving self-sufficient parts, i.e., elements that have a property
comparable to the property of the whole. However, one does not need
to search for a long time to come across phrases in which the multiplier
does quantify over parts of an individual, but crucially those parts
are not self-sufficient. Examples of such expressions are frequent in
the fast food culture and involve, e.g., Slavic phrases such as those in
(4.21)–(4.24). In each of these examples, it is not the case that its referent
consists of two parts that are hamburgers themselves, but rather
that it consists of two patties in a bun. This seems to differ significantly
from the cases discussed so far. Notice, however, the fact that parts
multipliers quantify over are not self-sufficient does not mean that
they are arbitrary. In fact, they appear to be the most salient parts of
the whole thing since in the fast food context it is commonly assumed
that the essential element of a hamburger is a patty whereas other
parts of a sandwich are considered to be merely a garnish.
(4.21) Polish
podwójny
doublesg.m
hamburger
hamburgerm
‘double hamburger’
(4.22) Czech
dvojitý
doublesg.m
hamburger
hamburgerm
‘double hamburger’
(4.23) Russian
dvojnoj
doublesg.m
gamburger
hamburgerm
‘double hamburger’
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4. Multipliers
(4.24) BCS
dupli
doublesg.m
hamburger
hamburgersg.m
‘double hamburger’
The fact that a certain part of an entity is perceived as more salient than
others corresponds to what is sometimes called structured parthood
(Champollion and Krifka 2016). As mentioned in Chapter 1, the term
has been postulated to account for the fact that some parts appear
to have a more significant status than others or, in other words, are
perceptibly distinguished within a whole.10 The difference between
structured and unstructured elements is that the former are cognitively
salient parts of a whole whereas the latter are not. Table 4.1 gives some
examples of structured part relations.
whole part
a (certain) man his head
a (certain) tree its trunk
a house its roof
a mountain its summit
a battle its opening shot
an insect’s life its larval stage
a novel its first chapter
Table 4.1: Examples of structured parthood (Simons 1987, p. 10)
For some time, it has been recognized that structured parthood
plays an important role in organizing lexicons of natural languages,
specifically via semantic relations such as meronymy and hyponymy
(see, e.g., Cruse 1986). However, in contemporary research on natural
language, the impact of structured part relations on compositional
semantics seems to be somewhat neglected (e.g., Champollion 2012b
and Champollion and Krifka 2016). However, it appears that multipliers
provide evidence that cognitively salient parts are relevant from
the quantificational perspective. In particular, I would like to suggest
that counting self-sufficient parts is a particular instance of a more
10. For an overview and discussion of related philosophical issues see, e.g., Simons
(1987) and Varzi (2016).
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4. Multipliers
general subatomic quantification over particular elements within a
make-up of an entity. Consequently, I posit that the inner structure of
objects such as hamburgers is in fact linguistically encoded in some
way and that multipliers are sensitive to its most cognitively salient
parts. In particular, they select for such essential elements within the
part-whole structure of a certain whole and quantify over them. Within
such a view, self-sufficient parts constitute a particular case within a
class of structured parts, specifically a subclass of cognitively most
salient elements, i.e., essential parts.
However, what counts as an essential part is somewhat vague
and can be subject to different conceptualizations under different
circumstances. To illustrate this claim, let us consider another sample
from the COCA collocate list for the English multiplier double. For
instance, consider the term in (4.25a) one can stumble across in the
context of typography. Though in the Unicode standard the expression
refers to the symbol ⇒, it is commonly used to designate also the
symbols ↔ and . In the Unicode terminology, what appears to be
considered the most salient part, and thus multiplied is the horizontal
line. However, in the more common and lax use of the expression, it
seems that it is the arrowhead that is taken to be the essential part.
Similar, in typography the phrase in (4.25b) refers to the symbol ‡.
Its design indicates that the parts worth quantification over are the
handles. However, in a military, martial arts, or gaming context the
very same expression or a similar phrase such as (4.25c) is much more
likely to denote thrusting weapons with two blades. On the other
hand, there are some examples that display a stable meaning. For
instance, the phrase in (4.25d) refers to a gun with two joint barrels.
From a ballistic point of view, the barrel ensures functionality of a
gun, whereas in terms of size it also constitutes its most significant
portion. Consequently, it is perceived as the essential part of a weapon
and when the noun shotgun combines with a multiplier, the resulting
phrase designates objects consisting of two merged barrels.
(4.25) English (COCA)
a. double arrow
b. double dagger
c. double sword
d. double shotgun
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At this point, it is clear that the generalization concerning multipliers
as quantifiers over self-sufficient parts requires an adjustment. In fact,
multipliers count essential, i.e., cognitively most salient, elements
within the subatomic part-whole structure encoded by the modified
noun. Self-sufficient parts constitute only a subset of such entities. In
the next section, I will discuss the case of mass nouns where at first
blush talking about essential parts does not make any sense.
4.2.2 Portions
In the previous sections, I have discussed examples involving count
nouns. Nonetheless, similar to equivalent expressions in other languages
Slavic multipliers co-occur also with mass terms. At first sight,
it might appear to be problematic for the proposed perspective since it
is somewhat odd to think about referents of prototypical mass nouns in
terms of consisting of essential parts. For instance, consider the phrases
in (4.26)–(4.29). Prima facie, it may seem that there is no straightforward
correlation between the semantic behavior of multipliers we
have observed so far and the examples introduced above since none of
the expressions refers to some cognitively salient parts of coffee, e.g.,
coffee grounds. However, a more careful examination reveals some
non-trivial facts about such examples. More specifically, the contrast
appears to fade when we realize that multiplier phrases involving
mass terms in fact are not mass. Notice that the expressions in (4.26)–
(4.29) neither denote a scattered entity, i.e., a substance in general, nor
do they refer to some arbitrary portion thereof. Instead, they designate
a standard amount, i.e., a big cup of coffee, which consists of two
standard amounts of coffee or, in other words, it is equivalent to two
regular cups of coffee.
(4.26) Polish
podwójna
doublesg.f
kawa
coffeef
‘double coffee’
(4.27) Czech
dvojitá
doublesg.f
káva
coffeef
‘double coffee’
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4. Multipliers
(4.28) Russian
dvojnoj
doublesg.m
kofe
coffeem
‘double coffee’
(4.29) BCS
dupla
doublesg.f
kahva/kava/kafa
coffeef
‘double coffee’
Let us consider the relationship between bear mass nouns and mass
nouns modified by multipliers more closely on the basis of Polish
data. As already mentioned in Section 3.3, Polish mass terms are generally
incompatible with cardinal numerals unless they are coerced
to a portion reading.11 Consequently, a phrase such as (4.30a) lacks a
substance interpretation and can only refer to a standard or contextually
determined portion of coffee. As I have already stated, the same
is true of a multiplier phrase such as that in (4.26). With this respect
multipliers pattern with cardinals in that they coerce the meaning of
the modified mass noun. Furthermore, as can be witnessed by the felicity
of (4.30b) multiplier phrases involving mass terms are countable.
When combined with a cardinal, they denote a plurality of portions
that are some number larger than standard.
(4.30) Polish
a. trzy
three
kawy
coffees
‘three coffees’
(i) portion interpretation
(ii) #substance interpretation
b. trzy
three
podwójne
doublepl
kawy
coffees
‘three double coffees’
Given the portion interpretation of examples such as (4.26)–(4.29),
describing their meaning in terms of an interaction between quantifi-
11. Another possible coercion involves a taxonomic interpretation where the cardinal
quantifies over subkinds of a particular kind. For the sake of brevity, I ignore
this phenomenon here.
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4. Multipliers
cation over self-sufficient parts and some sort of a mass-count shift
appears as a plausible possibility. For this purpose, one can adopt the
standard Universal Packager operation (see, e.g., Bach 1986, Jackendoff
1991, and Landman 1991) which yields quantized portions or packages
of substance for a given mass entity.12 Assuming that the mass noun
kawa ‘coffee’ is shifted by the Universal Packager operation before it
is modified by the multiplier,13 the resulting count noun refers to a
standard or contextually determined portion of coffee, specifically
a cup of coffee. The multiplier then asserts that within the package
there are two standard or contextually determined portions, i.e., two
cups of coffee. As a result, we obtain a double coffee, i.e., a standard
portion consisting of two smaller parts that have a property of being a
standard portion of coffee as well. This is actually analogous to the
cases provided in (4.8)–(4.11). Furthermore, note that in the discussed
coffee scenario the portion of beverage is defined pragmatically and at
least to some extend fixed. In other words, there are exactly two parts
of the amount of coffee constituting double coffee that are standard
portions. Smaller or bigger amounts are simply not standard with
respect to coffee. Yet of course the standard can differ with respect to
other beverages or food-stuffs. For instance, multiplier phrases such
as those in (4.31) refer to different amounts of substance.
(4.31) English (COCA)
a. double vodka
b. double martini
c. double ice cream
Interestingly, it seems that there is a deep relationship between multipliers
and packaging. One could easily imagine that multipliers would
also interact with another mass-count shift, specifically with Universal
Sorter (see, e.g., Bunt 1985) which converses the mass noun into a
count denotation referring to quantized subkinds. However, it seems
that at least in Slavic there is no such interaction since phrases such
12. An alternative approach to be taken is to enrich the structure with a classifier.
As suggested by Selkirk (1977) and subsequent work, it is plausible to assume that
possibly every natural language uses classifiers which are often null.
13. There are good reasons to presume that it is in fact the case since only after the
shift mass nouns can combine with number words.
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4. Multipliers
as those in (4.26)–(4.29) lack the relevant interpretation. In particular,
(4.26) is not true of a substance consisting of two kinds of coffee, say
a drink brewed from a mixture of Coffea arabica and Coffea canephora
beans. This fact seems to suggest that multipliers operate only on
mereologies involving object-level entities, i.e., tokens, and cannot
count parts of kind-level individuals, i.e., types.14
Up to this point, I have focused on multiplier phrases involving
concrete nouns since they are the subject of this study. However, as can
be witnessed by the collocation lists in the examined corpora many of
such constructions comprise also nominals of different sorts. For the
sake of completeness, in the next two sections I will briefly discuss
how the observations collected so far can inform us about the meaning
of two other types of multipliers phrases.
4.2.3 Events
I believe that the discussion of the novel data discussed so far gives
some intriguing insights into issues concerning subatomic quantification
in the adjectival domain. The proposed generalization seems
to cover a significant subset of constructions in which multipliers appear
including examples such as those in (4.8)–(4.11) and many others
provided in this chapter. Nonetheless, as attested in the examined
corpora multipliers can co-occur also with nouns that at first sight
might appear to be problematic since they do not refer to concrete
entities such as crowns, hamburgers, or portions of coffee. For instance,
as indicated by examples such as those in (4.32) Polish multipliers
can also combine with deverbal nouns which do not involve reference
to entities, but rather to some abstract objects. Neither murder nor
victory are spatial objects as opposed to all previously discussed cases.
(4.32) Polish (NCP)
a. podwójne
doublesg.n
morderstwo
murdern
‘double murder’
b. podwójne
doublesg.n
zwycięstwo
victoryn
‘double victory’
14. I leave aside the question what a part of a kind could be.
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4. Multipliers
It is usually assumed that deverbal nominalizations with the exception
of participant and result nominals denote properties of events (e.g.,
Grimshaw 2011), i.e., spatiotemporal particulars with a location and
time (Davidson 1967). A standard test reveals that a noun such as
Polish morderstwo ‘murder’ refers to an event since it can be felicitously
modified by event-modifying predicates such as mieć miejsce ‘take
place’ and być świadkiem ‘witness’ (e.g., Grimshaw 1990), as indicated
in (4.33a) and (4.33b), respectively.
(4.33) Polish
a. Morderstwo
murder
miało
had
miejsce
placeacc
w
in
piątek.
Fridayacc
‘The murder took place on Friday.’
b. Jan
Jan
był
was
świadkiem
witnessins
morderstwa.
murdergen
‘Jan witnessed the murder.’
If multipliers can count essential parts of not only entities but also
events, the proposed view naturally extends to examples such as
those in (4.32). An intuitive way to understand subatomic quantification
within the structure of an event involves temporal characteristics.
Within this view, a self-sufficient part of an event is a subevent of which
the same property as the one of the whole event holds. For instance,
in a murder scenario in which a maniac stabbed to death two victims
with a knife one after another there would be two relevant parts of
the whole murdering event, i.e., stabbing victim 1 and stabbing victim
2, and each of those subevents can be considered a murder as well,
see Figure 4.4. In such a case the multiplier in, e.g., (4.32a) quantifies
over self-sufficient parts of the murdering event, i.e., subevents that
are murders themselves. Since there is no murder without a victim,
the phrase infers that there were two individuals murdered during
the complex murdering event which seems to be the expected result.
Nonetheless, one could object that such an approach cannot account
for situations such as a bomb scenario in which an intended
explosion affected two victims leading to a double homicide. In such
a case, quantification over temporally subsequent subevents is not
possible since the explosion had an immediate effect and there was
virtually no running time of the event. However, I believe there is no
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4. Multipliers
Figure 4.4: Parts of a double murder event in a knife-stabbing scenario
reason to assume that temporal chunks are the only possible parts that
can constitute an event. Since events are complex peculiarities involving
not only times, but also locations, in a bomb scenario described
above it is still possible to differentiate between parts of an event in
spatial terms. Consider the situation depicted in Figure 4.5. Even with
no temporally defined subevents it is quite straightforward that one
could specify two areas within the space in which the explosion took
place, as indicated by the dashed line. As a result, we obtain two parts
of the murdering event each of which turns out to have the property
of the whole and since individuals are in general assumed to occupy
space, it should be possible to draw divisions like that with respect to
any event.
Figure 4.5: Parts of a double murder event in a bomb scenario
All things considered, it seems that the way of thinking about the
meaning multipliers I argue for can be easily extended to phrases involving
reference to abstract peculiarities such as eventualities. Adopting
a typal distinction between entities and events as typically assumed
in standard Neo-Davidsonian frameworks (e.g., Carlson 1984, Dowty
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1989, Parsons 1990) along with the proposal that pluralities of events
are obtained similar to pluralities of entities (Bach 1986) would simply
require to ensure that the semantics of multipliers is general enough
to cover both types of objects. In the next section, I will inspect yet
another type of expression that is often modified by multipliers and
does not necessarily denote concrete things.
4.2.4 Social roles
The final piece of data that appears to be problematic for the claim that
multipliers are in fact subatomic quantifiers is constituted by phrases
with nouns referring to social roles in examples such as those in (4.34).
Under ordinary circumstances, the phrase in (4.34a) does not refer to
an individual who consists of two parts, e.g., Siamese twins working
as spies, but rather they seem to involve a relationship between an
individual and two other entities. To be an agent means to be an agent
for some intelligence agency, i.e., one cannot be an agent without an
employer for whom they spy. Similar, (4.34b) refers to one person who
is a champion in two disciplines. In both cases, the multiplier seems
to quantify over entities denoted by the argument of the noun which
at first blush is not expected given the previously discussed semantic
behavior.
(4.34) Polish (NCP)
a. podwójny
doublesg.m
agent
agentm
‘double agent’
b. podwójny
doublesg.m
mistrz
masterm
‘double champion’
At this stage, it is not entirely clear how to integrate the ideas developed
in this paper with the treatment of examples such as those
in (4.34). However, a possible extension of the proposed view is to
assume that nouns such as agent involve some kind of an abstract association
to entities related to denoted individuals, e.g., an association
between agents and intelligence agencies, and that such an association
is subject to part-whole relations. A promising perspective to pursue
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involves adopting the notion of roles into semantic theory (see Sowa
1984, Steimann 2000, Zobel 2017). Intuitively, roles are functions or
capacities of individuals, i.e., social constructs independent of the individuals
that bear them. Crucially for our purposes, for an individual
to bear a particular role, it must stand in a certain relationship to other
individuals. Following Zobel (2017), one could distinguish between
class nouns such as man, i.e., properties of individuals of type e, t ,
and role nouns such as agent by adding a new type r referring to roles
and a shifting mechanism relating roles and individuals. Role nouns
would then essentially denote properties of roles at type r, t , but they
could be shifted to refer to properties of entities as well. Within this
view, multipliers would quantify over essential parts of roles rather
than individuals and the whole phrase could then be shifted to refer
to entities if required. This would account for the fact that expressions
such as those in (4.34) denote individuals performing a complex
role, i.e., individuals involved in activities and bearing responsibilities
related to two distinct intelligence agencies. However, exploring
the interaction between subatomic quantification and roles is rather
challenging and goes beyond the scope of this study. Instead, in the
next section I will return to the relationship between multipliers and
cardinals.
4.3 Multipliers vs. cardinals
As we have seen in Section 4.1.1, Slavic multipliers display morphological
complexity which suggests semantic compositionality. For instance,
the Polish multiplier podwójny ‘double’ consists of the numeral root√
dw corresponding to the number 2 present also in other types of numerical
expressions, e.g., the cardinal dwa ‘two’, as well as additional
morphemes including a special multiplicative circumfix po . . . n-.
This fact indicates that what multipliers and cardinals have in common
is some sort of reference to integers. Though they both indicate
that the number of quantified things equals 2, they differ in that they
are devised to count entities of a distinct type. In particular, cardinals
are semantically foreordained to count wholes. It is true that in
Chapter 2 and 3 we have seen that they can be used in count explicit
partitives in order to quantify over parts of objects, but notice that it
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is only possible when they modify a partitive word. In such a case,
entities denoted by the whole partitive are treated as objects in their
own right. Hence, the source of subatomic quantification is in fact
the partitive and the cardinal simply counts provided entities in their
relative entirety. Bare cardinals can never access elements within a
part-whole structure of a given entity. Rather, they always count entire
objects in the denotation of a modified noun. On the other hand,
multipliers are essentially subatomic quantifiers. They always count
parts of objects referred to by a modified nominal.
To better illustrate the claim introduced above, let us closely consider
the contrasts between truth conditions of the sentences involving
numerical expressions in (4.35). Under normal circumstances, there
is no way the statement in (4.35a) could be judged true if there were
only two parts of a crown or crowns, e.g., an orb and a jewel, in the
vault. The numeral phrase simply cannot be understood as referring to
two parts instead of two entire objects.15 In other words, cardinals are
semantically devised to counting wholes. This is not in contradiction
with the fact that in a sentence including a count explicit partitive
construction as in (4.35b) the cardinal does count parts of a whole
object. Consequently, (4.35b) would be true if there were only, say, an
orb and a cross in the vault. Notice, however, that it is the partitive
word that employs subatomic quantification. The cardinal here simply
counts provided parts as if they were considered objects in their own
right. In other words, it does not operate on the part-whole structure
of a given part. This contrasts with the meaning of (4.35c) which
would be true in a scenario where there were just one object in the
vault. Crucially, however, that object would have to have two essential
parts or, to put it differently, two parts having a property of being sort
of a crown. Finally, the felicity of the sentence in (4.35d) shows that
both types of quantification can co-occur within one phrase. Here, the
multiplier triggers subatomic quantification constrained as discussed
above, whereas the cardinal quantifies over wholes having the inner
structure imposed by the multiplier. Consequently, the sentence would
be true in a situation where in the vault there were three crowns such
15. Of course, what portion of an entity counts as an entire object is rather vague
and might differ with respect to a particular context. However, this is not at issue
here.
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4. Multipliers
that each of them consisted of two self-sufficient parts. Notice that
the relative order of the cardinal and multiplier as well as adjectival
properties of the latter further suggest that subatomic quantification
takes place first and only after the resulting wholes are counted.
(4.35) English (Guy Tabachnick, p.c.)
a. There are two crowns in the vault.
b. There are two parts of the crown in the vault.
c. There is a double crown in the vault.
d. There are three double crowns in the vault.
The discussed contrast between cardinals and multipliers becomes
even more salient if we consider what happens when the latter modify
partitive words. I have already dedicated a lot of attention to count
explicit partitives such as (4.36a), but let us now contemplate the
meaning of an equivalent multiplier phrase such as (4.36b). Crucially,
the multiplier quantifies over parts of a part of the adapter. The whole
expression would be true of an entity that is an adapter part and
consists of two comparable parts. Yet again, the multiplier employs
quantification within the part-whole structure of a thing denoted by
the nominal no matter what that thing is. Though such a use is for sure
very rare, the phrase is definitely interpretable.16 A natural example
of a sentence with such a construction from the NCP is provided in
(4.37).
(4.36) Polish
a. dwie
two
części
parts
adaptera
adaptergen
‘two parts of the adapter’
b. podwójna
double
część
part
adaptera
adaptergen
‘double part of the adapter’
16. The example was actually found on a website of a producer of adapters with a
picture of a described part.
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4. Multipliers
(4.37) Polish (NCP)
Skrzynkę
boxacc
horyzontalną
horizontalacc
otwiera
opens
się
refl
podobnie
similar
jak
how
podwójną
doubleacc
część
partacc
przedniej
frontgen
ścianki...
wallgen
‘The horizontal box can be opened similar to the double part
of the front wall...’
I conclude that the purpose of cardinals is essentially to count whole
entities in the extension of a modified nominal. If they modify a regular
noun, they simply quantify over objects in its denotation. Similar,
if they appear in a count explicit partitive, they count provided parts
in their entirety as if they were wholes. At the same time, natural language
developed a special type of expression dedicated to numerical
subatomic quantification, namely multipliers. Multipliers differ from
cardinals in that they always count essential parts of objects denoted
by a modified nominal be it a whole or a part. In other words, they are
equipped to count parts within a whole. The quantificational properties
of the two numerical expressions in question are summarized in
Table 4.2.
cardinals multipliers
subatomic quantification *
quantification over wholes *
Table 4.2: Properties of cardinals and multipliers
This concludes the examination of some non-trivial properties of
the neglected class of multipliers with respect to the issues concerning
subatomic quantification.
4.4 Summary
In this chapter, I have focused on a key topic concerning subatomic
quantification. Specifically, I have presented novel evidence demonstrating
that in natural language there are numerical expressions
devised for the purpose of quantification over parts, namely multipliers
such as English double. The fact that such lexical items are
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4. Multipliers
cross-linguistically widespread indicates the relevance of part-whole
structures of singular objects for the interaction between parthood
and quantification in natural language semantics. The morphological
evidence from Slavic demonstrates that Polish, Czech, Russian,
and Bosnian-Croatian-Serbian multipliers are formally complex expressions
derived from numeral roots of corresponding cardinals. I
have argued that the morphological complexity of Slavic multipliers
suggests semantic compositionality.
Multipliers are sensitive to the internal part-whole structure of
objects denoted by nouns they modify and involve quantification
over cognitively most salient elements of those objects. Often, such
elements are self-sufficient parts, i.e., entities that have a property
comparable with the property of the whole. Sometimes, they are not
self-sufficient but rather they are considered the most significant parts
within the inner structure of an object. I have generalized that multipliers
count essential parts, i.e., parts perceived as crucial for an entity
to be considered as having a particular property, with self-sufficient
parts constituting a subset thereof. Moreover, I have examined multiplier
phrases involving mass terms. Interestingly, such expressions
receive a portion interpretation and as such are countable. This fact
indicates that multipliers either perform or require a mass-count shift
in terms of packaging. Furthermore, I have proposed how the view
argued for here could be extended to cover cases of multiplier phrases
involving other types of NPs, namely event nominals and role nouns.
Incorporating abstract objects such as eventualities and roles into
the ontology would allow us to capture the meaning of virtually all
multiplier phrases in terms of one unified mechanism. Finally, I have
discussed how multipliers differ from cardinals. The crucial distinction
between the two boils down to what type of entity they count in
terms of part-whole structure. In particular, while cardinals always
quantify over wholes, multipliers always count parts of a singular
object which, similar to complex entities or events they constitute,
have themselves a property considered essential for an entity to fall in
the denotation of a modified noun.
This chapter concludes the empirical part of this thesis. The most
important insights could be summarized as follows. First, singulars
and plurals share a unified part-whole structure. Second, natural language
is sensitive to topological relations holding between parts of
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4. Multipliers
entities in the denotations of nominal expressions and allows for quantification
at the subatomic level. Third, only entities that constitute
integrated wholes are countable. Finally, though different types of numerical
expressions specialize in quantification either over wholes or
over parts, the underlying mechanism is the same. In the next chapter,
I will introduce a conceptual background that will serve as an informal
notional basis for a formal account for subatomic quantification.
161
5 Conceptual background
In the previous three chapters, I have discussed novel data concerning
the interaction between partitivity and countability. Several novel
observations have been made. In particular, I have presented linguistic
evidence suggesting that singulars and plurals do not differ with
respect to the part-whole relation they employ but rather that parts of
singularities remain in a particular topological configuration ensuring
that the whole constitutes an integrated object, whereas parts of pluralities
do not, and thus form a scattered entity. This claim is motivated by
two linguistic facts. First, cross-linguistically the same partitive words
can appear both in entity and set partitives. Second, the contrast between
regular plurals and Italian irregular plurals reveals the crucial
topological distinction between the two. Furthermore, I have shown
that natural language does not allow for counting arbitrary sums of
topologically unrelated parts. Only entities that are conceptualized as
constituting continuous objects can be subject to counting. This restriction
appears to hold both on the level of wholes and on the subatomic
level, as the evidence concerning topology-sensitive partitive words as
well as the semantic behavior of multipliers indicate. In other words,
quantification in natural language is sensitive to topological relations
between parts of entities it operates on.
In this chapter, I will discuss the conceptual background concerning
countability and subatomic quantification I assume here. In particular
I will present three claims regarding the relevance of topological
notions in nominal semantics, general counting principles, and their
significance for subatomic quantification. This informal conceptual
framework will motivate the formal account for the core phenomena
explored in this study that I will develop in Chapter 6 and 7. However,
before I present in detail the notional core of this study, let us consider
a more general cognitive context. Though the linguistic evidence
presented so far is compelling and entirely motivates the claims, I believe
it is useful to confront it with what we know from psychological
research in order to provide supplementary support.
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5. Conceptual background
5.1 Psychological evidence
In this section, I will review psychological evidence that demonstrates
several factors concerning human cognition that correlate with the
linguistic evidence introduced in Chapter 2, 3, and 4. First, I will
discuss the relevance of how spatial categories such as solidity, shape,
and most importantly integrity of entities are conceptualized with
respect to the mass/count distinction in grammar. Furthermore, I will
consider the significance of the part/whole distinction for language
acquisition and review the evidence that humans have simultaneous
perception of an object as a whole and as a collection of parts. Finally, I
will discuss what cognitive studies tell us on number sense in humans,
especially on the relation between integrity and counting in young
children. The literature on each topic is abundant. For the sake of
brevity,in each case I will only refer to a few representative studies.
5.1.1 Object/substance distinction
I will start with a brief overview of the research in cognitive psychology
relating individuation with the difference in perception of solid
objects and amorphous substances that suggests that this distinction is
relevant to the phenomenon of countability in natural language. In the
past 30 years, convincing evidence has been presented that indicates
that the distinction between objects and substances is not merely an
alternation based on grammar. Contrary to Quine (1960)’s influential
claim that it is language what provides the means for individuation
of objects (see also Pelletier and Schubert 1989), it appears that count
and mass syntax reflect on how we see entities in the world rather
than the other way round. In other words, the research to be discussed
here shows that the distinction between objects and substances is not
something conventional, formal, or arbitrary imposed by the way a
particular language is devised, i.e., a way of speaking so to say. To the
contrary, it appears to be a deeply embedded component of human
cognition manifesting itself in non-verbal infants at a few months of
age and shared with non-human species.
Multiple experiments demonstrate that children associate count
nouns with solid discrete objects at an early age (see, e.g., Landau
et al. 1988, Soja et al. 1991, Imai and Gentner 1997). The evidence come
164
5. Conceptual background
from the results of the so-called word extension task which concerns
learning new words by subjects. The procedure is as follows. First,
a child is presented with a novel entity and instructed on what it is
called. That entity can be either a solid object or a shapeless portion
of a substance. Next, two other entities resembling the initial one with
respect to different perceptual features are introduced. One would
match the already introduced entity in shape, whereas the other would
be of the same material. Interestingly, the extensive research shows
that children extend the name of a novel item to items of the same
shape only if the initial entity was a solid discrete object. On the other
hand, if they were first presented an amorphous portion of a novel
substance, they do not extend its name to items of a similar shape, but
rather to objects made out of the same material.
Based on such evidence, Soja et al. (1991) argue that certain ontological
commitments are prior to the linguistic mass/count distinction,
contrary to Quine’s claim. Additional support for such a conclusion
comes from the fact that the results were replicated in populations
using languages with syntax significantly different from English. For
instance, Imai and Gentner (1997) demonstrate parallel evidence based
on the same experimental paradigm using the word extension task in
the Japanese speaking environment. Though Japanese lacks a straightforward
morpho-syntactic equivalent of the distinction between mass
and count nouns present in languages such as English, Japanese speaking
children distinguish between objects and substances just as well
as their English speaking peers.
A number of studies provide varied and carefully controlled evidence
that preverbal children are endowed with certain assumptions
concerning the nature of objects (e.g., Carey 1985, Spelke 1990, Soja
et al. 1991, Carey and Spelke 1996). For instance, they expect objects
to be bounded and cohesive. The first assumption reflects on an entity
having natural boundaries, whereas the latter translates into an
anticipation that objects have parts that stick together. Furthermore,
objects are expected to move across space as wholes along continuous
paths and to retain identity upon collisions with other entities. On the
other hand, substances lack any of those properties. An exemplary
experimental paradigm is as follows. An infant is introduced either to
a solid object, e.g., a teddy bear, or a portion of substance, e.g., clay,
displayed in front of them. Next, the entity is covered by a screen and
165
5. Conceptual background
the child sees that a second item is placed behind the screen. Finally,
depending on an experimental condition after the screen was removed
either there were two items revealed or due to the manipulation by an
experimenter only one item could be seen. Interestingly, if the initially
introduced entity was a discrete solid object, children reacted with
great surprise if after removing the screen they could find only one
item. On the other hand, no such reaction was recorded if they were
introduced to a portion of a substance at the beginning. The conclusion
is that preverbal children appear to know the different between
objects, pluralities thereof, and substances. They expect that if one
adds a teddy bear to a teddy bear, the result should be two teddy bears,
whereas if one combines clay with clay, one does not get a plurality of
clay. Since similar results were also found in non-human primates (see,
e.g., Hauser and Carey 2003, Hauser and Spaulding 2006 on rhesus
monkeys), it seems plausible to assume that this property of human
cognition has some evolutionary history. I will come back to the issue
of number in Section 5.1.4.
However, though the difference between discrete solid objects
and amorphous substances has a significant status in cognitive and
linguistic development of children, it turns out that the correlation
between this distinction and mass/count syntax is imperfect. A corpus
study on toddler vocabulary pursued by Samuelson and Smith (1999)
shows a significant asymmetry between particular syntactic categories.
Specifically, while the large majority of nouns that English speaking
children learn early are count nouns (74% of the vocabulary), only a
small portion involves mass nouns (10% of the vocabulary). There is
also a third class of expressions that considered ambiguous between
count and mass (16% of the vocabulary). The results of the examination
in order to find out whether the object/substance distinction correlates
with the mass/count distinction are somewhat surprising. In general,
a correspondence between solid objects and count nouns on the one
hand and non-solid entities on the other was in fact observed. However,
it was imperfect and and interesting mismatches were recognized.
Surprisingly, solidity and shape turn out to be good predictors of
166
5. Conceptual background
count nouns, whereas mass nouns appear to be less correlated with
non-solidity and shapelessness.1
Yet another strand of research provides support for the findings
discussed above. In particular, Prasada et al. (2002) demonstrate that
categorization of a noun as countable or uncountable does not reduce
to establishing an immutable ontological distinction between an object
and a substance. Rather, it seems that speakers of languages such as
English categorize nouns in terms of countability based on whether
they construe entities they refer to as objects or as substances. A series
of experiments utilizing novel vocabulary tasks show that a novel
item that has an irregular shape is less probable to be referred to
by a novel count noun when contrasted with entities of a regular
shape. However, the same item is significantly more probable to be
considered countable when displayed among other similar entities
having such an irregular shape. This means that it is not the case that
some objective properties of things influence the way grammar is
directly. Instead, the mass/count distinction appears to reflect how
humans conceptualize entities in the world.
To conclude, the psychological research reviewed in this section
indicates that contrary to Quine’s claim, human beings apparently do
have certain ontological commitments prior to acquiring syntax of a
particular language. Specifically, the way how children are biased towards
perceiving objects as opposed to substances correlates with the
mass/count distinction in grammar. However, this correspondence is
not of a one-to-one type and there are well-known cases of mismatches.
Yet, it is crucial that the distinction in grammar is not due to different
ontological categories in some objective sense, but rather it reflects
on how entities in the world are conceptualized. In the next section, I
will discuss the relevance of part-whole structures in the process of
language acquisition. In particular, I will briefly review how a certain
bias concerning what it means to be a whole guides children in how
they learn lexical meanings of nouns.
1. Notice that probably the main reason is the existence of the category of object
mass nouns such as furniture.
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5. Conceptual background
5.1.2 Whole object assumption
Since at least early 1990s, language acquisition studies have shown that
vocabulary development is constrained by certain mechanisms that
govern it in order to ensure its effectiveness. In particular, children are
guided by a number of word learning biases i.e., endowed assumptions
that allow us to eliminate unlikely alternatives in order to efficiently
process and learn new lexical meanings (see, e.g., Markman 1990,
Hansen and Markman 2009, Hollich et al. 2007). Such assumptions
begin to manifest around the age of 18 months when rapid expansion
of a child’s vocabulary starts. They are crucial in deciding what the
reference of a new noun is, e.g., what aspect of an object it designates,
as well as solving the problem of indeterminacy. One of the word
learning biases that will interest us here is the so-called whole object
assumption (Markman 1990).
The whole object assumption allows children to constraint the
meaning of novel words by guaranteeing that a child relates a new
noun with an object in its entirety rather than with some arbitrary part
or property of that object (Markman 1990). For instance, if an adult
points to an item and labels it, e.g., a doll, a child assumes that the
noun doll is meant to refer to the whole object and not to its part or
characteristic. Though in principle the adult’s verbal behavior might
have been interpreted as referring to the doll’s head, leg, dress, color,
or size, children intuitively rule out such possibilities and associate the
label with the whole item.2 This bias holds even in situations where
object’s color or a certain dynamic activity are made salient to a child
(Hansen and Markman 2009).
Though the original research regarded 18-month-year old children
and older, later studies have demonstrated that in fact infants can
associate nouns with whole objects already at the age of 12 months
(Hollich et al. 2007). Crucially, the effect was attested despite the fact
that test items could be viewed as two separate objects and even when
a certain part of an item was made salient. Interestingly, similar results
were obtained in experiments where participants were adult. Hence,
the mechanism appears to be deeply embedded. Its relevance relies
in that it enables children to determine which of numerous logically
possible meanings a word could have is actually the correct one.
2. See Quine (1960) for a related problem of the inscrutability of reference.
168
5. Conceptual background
In this section, we have seen that children distinguish intuitively
between parts and wholes. In language acquisition, when exposed to
a novel noun, they do not attribute its meaning to a part of an object
it refers to but rather immediately assume it is true of a whole thing.
The whole object assumption has often been related to the findings
concerning infants’ perception of objects discussed in the previous
section. It has been hypothesized that the constraint in question reflects
the non-linguistic status of objects (Hollich et al. 2007). This might
suggest that from the perspective of a child the fact that wholes involve
parts is somewhat discriminated. However, the evidence is much more
complex. In the next section, I will provide a brief overview of the
study of part-whole perception in children.
5.1.3 Part-whole perception
It is a cognitive fact that we often conceive entities as being made
up of smaller entities related to each other in a particular manner.
Experimental evidence shows that humans have two simultaneous
perceptions concerning part-whole structures of entities (e.g., Witkin
1950, Meili-Dworetzki 1956). On the one hand, we possess the ability
to discriminate parts from a whole, i.e., decompose an entire object
into distinct elements making it up. On the other hand, we are able
to integrate the total sum of the parts into a complete whole. This
phenomenon is usually referred to as part-whole perception and it is
standardly attributed to an ability of an individual to decenter, i.e., to
shift attention from parts to the whole or vice versa (see Piaget and
Morf 1958).
Though adults in general show part-whole perception, it has been
subject to controversy in the psychological literature at what age it
emerges. In the early Piagetian studies, the experimental results suggested
that this ability is absent in young children (Elkind et al. 1964).
In a typical experiment, a child was presented a drawing of a whole
consisting of parts that were independent objects in their own right,
e.g., a person made out of fruit as in Figure 5.1, and was instructed
verbally to report what they see. According to Elkind et al., children
report that they perceive simultaneously a whole and parts attributed
to the same form, e.g. both a man and fruit, only at the age of 8. Before
that age, subjects failed in complete decentration with 5-to-6-year-
169
5. Conceptual background
olds showing complete centration, i.e., they reported that they saw
only parts, e.g., fruit but not a person, and older children gradually
improved.
Figure 5.1: Part-whole perception (Elkind et al. 1964)
However, later studies suggest that part-whole perception develops
much earlier. For instance, Kimchi (1993) provides evidence that 5years-old
children show sensitivity to parts and to part-whole relations
and that this sensitivity improves with age. Even more interestingly,
Boisvert et al. (1999) show that 3-year-olds demonstrate good performance
in part-whole perception tasks using multiple-choice tests
instead of Piagetian verbal tasks used in earlier studies. Finally, Quinn
et al. (1993)’s results indicate that 3-month-old infants can group elements
of visual pattern information into larger perceptual units based
on lightness similarity in a way that suggests at least some component
of part-whole perception.
I interpret the experimental results presented above as suggesting
that human cognition is devised to be sensitive to part-whole structures
since even very young individuals can decompose a whole into
pieces by discriminating its parts as well as represent individual elements
as making up an integrated collection that is a whole. Since
these two perceptions are simultaneous, one could expect that we intuitively
categorize entities as certain configurations of smaller things
that despite their often complex inner structure are singular objects. At
the same time, parts of such objects remain cognitively salient and if
required, can be easily accessed. If the experiments reported above are
on the right track, this property of human mind appears to be either
innate or at least emerge at an early stage of cognitive development.
170
5. Conceptual background
5.1.4 Number sense
The final piece of cognitive evidence to be discussed here concerns human
number sense, i.e., an intuitive understanding of what numbers
mean and how they can be affected by various operations (see, e.g.,
Dehaene 1997 for an overview). This mental mechanism allows for a
special type of sensory perception through which the cardinality of a
given set of entities can be perceived with similar ease as their size,
color, shape, or position and as such provides the basis for various
forms of calculation. In humans, number sense is based on two different
cognitive systems, specifically the approximate number system
and the object tracking system (see, e.g., Hyde 2011). The first is a
cognitive system supporting the estimation of the magnitude of a
collection of objects without relying on symbolic representations (see,
e.g., Feigenson et al. 2004, Nieder and Dehaene 2009 for an overview).
It manifests already in infants and gets more developed with age
(Cantlon et al. 2006). On the other, the parallel object tracking system
is the mental ability to immediately enumerate small sets without
counting by means of individuation (see, e.g., Carey 1998, 2009 for an
overview). It allows us to rapidly recognize values from 0 to 4, whereas
the approximate number system is credited with the non-symbolic
representation of all greater numbers (Piazza 2010).
It appears that there is evidence indicating that humans are at
least to some extent predisposed to develop certain numerical abilities
such as the concept of exact number and simple arithmetic based on
their number sense. For instance, Gelman and Gallistel (1978) argue
that children are endowed with innate principles of counting and
have intuitive understanding of the cardinality of a set of entities and
its conservation under changes that do not affect quantity. It seems
that a lot of knowledge concerning how to put objects in one-to-one
correspondence with numbers is intuitively understood by children
though it is never taught or formulated in an explicit way. In particular,
when learning to count children are not taught that each entity must be
counted once and only once or that one number cannot be associated
with more than one entity, but this principle is taken for granted.
Similar, children know intuitively that number words are supposed
to be recited in fixed order and that the last numeral represents the
cardinality of the whole enumerated set. According to this view, innate
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5. Conceptual background
knowledge regarding counting principles precedes and facilitates the
acquisition of that part of lexicon that includes number words and
guides its application in a particular situation of counting.3
The above introduced hypothesis seems to be corroborated by
further experimental evidence that suggests that at the age of 2,5 years
children understand that counting is an abstract procedure that can
be applied to different kinds of objects including concrete entities
as well as events (Wynn 1990). 3,5-year-olds know that the order in
which they recite numerals is crucial, whereas the order of pointing
at counted items is irrelevant as long as each item is counted and
none item is counted twice. Furthermore, children are able to indicate
and correct subtle errors that result from violations of basic counting
principles such as reciting numerals out of order, counting the same
object twice, or omitting an item while counting (Gelman and Meck
1983, Gelman et al. 1986). By the age of 4 years, children show that they
have already mastered the basics of counting and they can generalize
the procedure to novel situations. Older children often spontaneously
reinvent arithmetic developing new strategies for calculation best fit
for a particular problem (Dehaene 1997, p. 119).
However, probably the most intriguing finding from the perspective
of the interest of this study is a tight link between spatial integrity
and numerical information observed in young children. Specifically,
Shipley and Shepperson (1990) provide a particularly striking demonstration
of the relevance of objects understood as integrated discrete
physical entities for cognition of 3-to-4-year-olds. In the experiment,
children were presented sets of items and instructed specifically what
they were supposed to count. For instance, an array of forks with one
item broken in two pieces as illustrated in Figure 5.2 was displayed in
front of a child who was then asked to count the forks. Unlike adults
and older children, children between 3 and 4 years of age did not
answer that there are five forks, but rather that there six of them, i.e.,
they counted each detached part of a fork as a separate entity. Similar,
when instructed to count kinds, e.g., different kinds of animals, or
3. However, see Fuson (1988) for an opposing view claiming that children learn
counting by imitation. For instance, Fuson argues that children recite strings such
as onetwothreefourfive...as uninterrupted sequences and only later on they segment
them in order to delimitate numerals, learn to extend them to larger values and
apply to particular situations.
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5. Conceptual background
properties, e.g. different colors, in a setup where each kind or property
was exemplified by a number of distinct discrete items, children
included each separate object in their count.
Figure 5.2: Relevance of integrity in counting (Dehaene 1997, p. 60;
adapted from Shipley and Shepperson 1990)
The results demonstrate that the canonical countable entity for a
young child is a discrete physical object that comes in a single piece.
The role of spatial integrity is crucial in determining what counts as
one and this bias precedes the process of learning how to count. In
other words, at an early stage of development of arithmetic abilities
young children simply cannot avoid counting each discrete integral
object as one unit. It is very likely that the assumption that number is
a property of sets consisting of discrete spatially contiguous objects is
a deeply embedded principle that facilitates mastery of counting.
The findings of Shipley and Shepperson were further confirmed
for other forms of linguistic quantification in experiments based on
quantity judgment tasks involving comparative constructions and pluralization
(Melgoza et al. 2008). In particular, when presented an object
divided into three parts, e.g., a broken fork, on the one hand and two
whole objects, e.g., two integral forks, on the other, 4-year-old children
judged the first to be more objects than the latter. Furthermore, in
an elicitation task children often labeled broken objects using plural
morphology. For instance, a phrase some forks was used to refer to three
separated pieces of a broken fork. The results presented by Melgoza
et al. not only confirm the findings of Shipley and Shepperson but also
indicate the existence of a spatial bias for many forms of quantification
in natural language. In all tested cases, the spatial criterion of integrity
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was preferred over other factors in order to establish what counts as
one. In other words, when children quantify sets, they seem to define
individual set members in terms of spatial contiguity in the first place.
I take these results to be evidence suggesting that countability as a
property of sets of discrete separate entities is a deeply embedded
principle. Though as we know from every-day life experience adults
can develop a mechanism to retrieve the original part-whole structure
of counted entities, e.g., reconstruct a whole fork by combining
separate pieces in thought, I hypothesize that the underlying quantificational
procedure is based on a principle relating numbers with
things that come in one piece. However, before I conclude, I will briefly
discuss how human quantification differs from number sense in other
animals.
Massive evidence indicates that number sense is not exclusive to
human mind. Well-established research in etiology shows that apprehension,
comparison, and even approximate addition of quantities is
also present in animals (see, e.g., Davis and Pérusse 1988, Gallistel 1989,
and Dehaene 1997, pp. 13–40 for overview). The most well-known
evidence concerns the ability to represent and discriminate quantities
of relative sizes in primates (e.g., Woodruff and Premack 1981,
Matsuzawa 1985, Rumbaugh et al. 1987, Washburn and Rumbaugh
1991, Boysen et al. 1996), but it is also found in other mammals such
as dolphins (Mitchell et al. 1985), cats (Thompson et al. 1970), and
rats (Capaldi and Miller 1988) as well as birds (Pepperberg 1987) and
even fish (Agrillo et al. 2012). Furthermore, recent studies in botanics
suggest that there are reasons to also assume plant arithmetic (Böhm
et al. 2016). The evidence indicates either that number sense is an
evolutionary ancient part of cognition or that there were multiple convergent
evolution events. In any case, it is a widespread phenomenon
shared by a wide range of species.
However, despite the large body of evidence indicating that many
non-human animals have the approximate number system, their number
sense seems to differ significantly from that of humans. Though
there are well-documented cases of approximate quantification in animals,
no case of symbolic addition is known in any species other than
the chimpanzee and only after long training dedicated to learning a
small set of digits and with frequent errors in computation (Dehaene
1997, p. 39). On the other hand, young children spontaneously count
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5. Conceptual background
on their fingers, often up to 10 before the age of three and acquire
syntax of complex numerals with ease. Thus, the more lax concept
of numerosity is often used with respect to animal quantification in
order to distinguish it from the symbolic and verbal representation
of number in humans. To conclude, though some forms of quantification
appear to be frequent in the animal kingdom, they seem to
differ significantly from what humans do when they count. While
non-human animals estimate quantities on a regular basis, it seems
that the ability to establish one-to-one correspondence between an
object and a discrete symbol which allows for computation of the exact
number of entities in a given set is an extremely rare property among
species.
I conclude that counting as performed by humans is a quite unique
ability which associates discrete entities with abstract representations
of number. I hypothesize that the core mechanism underlying all
counting manifests in early childhood and concerns establishing oneto-one
correspondence between integral physical objects that come in
one piece and integers. In other words, I assume that the topological
notion of spatial integrity is the basis for what counts as one and only
later humans develop ways of abstracting away from this principle,
e.g., by extending it to quantification over abstract entities such as
kinds and properties or being able to retrieve the original connection
between detached parts. In the following sections, I will attempt to
relate the linguistic evidence presented in Chapter 2, 3, and 4 with the
conclusions of cognitive psychology in order to develop the three main
claims of this study that will serve as a conceptual background for the
formal account for subatomic quantification in natural language.
5.2 The three claims
I have already presented a broad range of data demonstrating both
the relevance of subatomic structures in natural language and the
important role the way entities including parts of a whole are spatially
related with each other plays with respect to quantification. Furthermore,
in the previous section I have reviewed additional psychological
evidence suggesting that these issues are deeply embedded in human
cognition. Now is the time to discuss in detail what I believe to be the
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5. Conceptual background
conceptual core of this study, namely the three claims I make with
respect to the interaction between parthood, topology, and quantification
in natural language. I will start with postulating that natural
language semantics is sensitive to topological relations holding between
elements within part-whole structures designated by nominal
expressions. Next, I will posit a set of general counting principles including
constraints concerning non-overlap, maximality, and integrity
of counted things. Finally, I will postulate that the very same universal
mechanism applies irrespective whether one quantifies over entire
entities or over parts of a whole.
5.2.1 Relevance of topological notions with respect to part-whole
structures
It is well-known that topological notions play an important role in natural
language. The existence of locative expressions involving prepositions
such as inside, near, under, between, and far shows that relations
concerning how we conceptualize position and space are deeply
rooted in grammar. Arguably, they constitute universal components
of human cognition and natural language semantics. Though the
research on locative PPs is well-established and contributed to our
understanding what means language uses to encode information regarding
location of objects with respect to each other (e.g., Clark 1973,
Herskovits 1985, Zwarts and Winter 1997, and Kracht 2002), the question
concerning spatial constitution of entities remained somewhat
elusive in the study of meaning. One line of argumentation justifying
why those kind of issues should remain unaddressed is that they simply
stem from every-day world knowledge, and thus as extra-linguistic
factors are not supposed to be incorporated into the semantic theory.
For instance, Schwarzschild (1996) famously argues that the fact that
(5.1a) entails (5.1b) has nothing to do with referential properties of
the subject DPs in both sentences. Rather, the inference simply results
from facts concerning what we know about where brains are located.
This might seem plausible for examples such as (5.1). However, in
previous chapters of this study we have seen that there are a number
of natural language expressions that are sensitive to topological
properties of part-whole structures corresponding to their referents.
My view is that if we want to account for the meaning of nouns in
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5. Conceptual background
its entire complexity, topology-related phenomena deserve serious
consideration.
(5.1) English (Schwarzschild 1996, p. 187)
a. Bill is in Texas.
b. Bill’s brain is in Texas.
It has been acknowledged for a long time that nominal semantics differentiates
between count singulars referring to integrated objects on
the one hand and mass terms and plurals denoting scattered entities
and arbitrary sums of individuals, respectively, on the other. As we
have seen in Section 5.1.1, there are good reasons to assume that this
contrast correlates with some fundamental properties of human cognition.
Soon, we will see that the distinction can be captured in terms
of how parts constituting a whole are spatially arranged. However,
before I even start considering how to develop a proper account, it
is crucial to realize that the extent to which topology plays a role in
part-whole structures associated with referents of nominals is highly
underestimated, if not neglected, in the contemporary mainstream research
on pluralities and countability. One of the important empirical
contributions of this study is about compensating this deficit.
The data presented in Chapter 2 and 3 show that certain classes of
nouns encode certain types of spatial configurations of entities making
up a whole. In particular, at least some Italian irregular plurals
discussed in Section 2.3 denote integrated pluralities, i.e., sums of
individuals that form cohesive wholes. On the other hand, different
types of topology-sensitive partitive words explored in Section 3.1
and 3.2 involve reference to continuous parts whereas extensions of
their topology-neutral counterparts comprise also from discontinuous
portions of a whole. Though the evidence explored in this study emphasizes
specifically the role of integrity in subatomic quantification,
recent research on different types of collective nouns and mass terms
points to a similar conclusion regarding the relevance of topology in
nominal semantics on independent grounds. In particular, Grimm
(2012b) proposes that a subset of mass nouns that denote aggregates
of granular objects, hence granular mass terms like rice and gravel,
involve reference to clustered individuals, i.e., bundled entities transitively
connected to each other. Similar, Henderson (2017) argues that
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5. Conceptual background
swarm nouns such as grove, horde, and swarm denote large pluralities
whose constituents remain in proximity within a certain spatial
configuration. Moreover, Grimm and Dočekal (to appear) discuss a
particular class of Slavic derived mass nouns such as Czech list ‘leaf’
∼ listí ‘foliage’ that refer to collections of singular entities that are
easily distinguishable yet conceptualized as aggregates.4 Finally, as
proposed by Grimm (2012b) referents of concrete count nouns can be
viewed as entities whose part-whole structure is constrained in such
a way that it forms an integrated object in its own right.
The psychological evidence discussed in Section 5.1 emphasizes
the role of spatial integrity of objects on the one hand and our ability to
simultaneously perceive individual parts of such cohesive wholes as
entities in their own right on the other. Given the amount of linguistic
evidence supplemented with findings concerning human cognition, I
argue that semantic theory needs to accommodate to those facts. In
other words, my first claim is as follows. There is more to the meaning
of common nouns than usually assumed and without a proper
approach incorporating the insights concerning the role of topological
notions in part-whole structures many phenomena will be left
unexplained or even unnoticed. I posit that a good starting point is to
try to capture the difference between count singulars, regular plurals,
and mass nouns in terms of different topological relations encoded
within corresponding part-whole structures. Specifically, count singulars
refer to entities that constitute integrated wholes whose parts stick
together rather than mere collections of spatially unrelated portions
of matter. On the other hand, plural nouns denote arbitrary sums of
such individuals, i.e., they require their parts to be constrained by
spatial integrity but impose no topological restrictions on configurations
in which they appear. Finally, a proper treatment of mass terms
should account for the fact that their prototypical referents are scattered
unbounded entities. An additional advantage of such a novel
perspective on is that it suggests that different part-whole structures
do not emerge due to distinct parthood relations. Instead, it seems
that different part-whole structures arise as a result if the interaction
between one unified parthood relation with distinct topological no-
4. For experimental investigation into the meaning of such expressions in Czech
and Polish see Dočekal and Wągiel (2018).
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5. Conceptual background
tions. In the next section, I will posit that the postulated view on what
it means to be a whole can shed new light on countability.
5.2.2 General counting principles
In general, I follow a well-established linguistic tradition assuming
that the mass/count distinction corresponds to the contrast concerning
how different types of entities are conceptualized (e.g., Wierzbicka
1988, Chierchia 1998a,b, 2010, 2015, Grimm 2012b). As we have seen
in Section 5.1, a substantive body of research in cognitive psychology
provides evidence that human beings categorize stable bounded
things that preserve shape differently than amorphous substances.
The first are considered delimited objects whose spatial identity can
be tracked easily throughout time, whereas the latter are merely unindividuated
portions of matter. On the other hand, there appears to be
rich intuitive knowledge concerning what counting is that seems to be
part of human cognitive endowment. Though the exact nature of how
cognitive facts relate to natural language semantics and grammar, i.e.,
the structure and meaning of certain syntactic constructions, might
not be straightforward, I propose a most probably simplified view that
will nevertheless reveal what I argue is an advantageous perspective
on what it means to be countable.
Usually in the study of countability, at some point when it comes
to defining what counting is something like ‘counting is quantification
over what counts as one’ is stated. Often, ‘what counts as one’ is
understood in terms of atomicity. In particular, an atom in a technical
sense is an object that has no proper parts. Though at first blush this
notion seems rather counterintuitive (see Champollion 2010, 2017 for
discussion),5 it has been very influential in the research on countability.
To the extent that counting is often implicitly assumed to be simply
quantification over atoms. In this section, I will attempt to elaborate
a bit more on what counting is and how it differs from other forms
of quantification. For this purpose, I will consider three conditions
entities need to satisfy in order to be able to be put in one-to-one correspondence
with numbers based on what we know what humans do
when they count. In particular, I will propose three general counting
5. I will come back to this issue in Section 7.1.
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5. Conceptual background
principles that will allow us to better understand countability in a
more general manner. As we will soon see, this novel perspective will
prove of great significance with respect to subatomic quantification.
Counting is usually understood as establishing one-to-one correspondence
between what is being counted and numbers. This kind of
association is necessary to determine the cardinality of a particular set
of entities. Though there are a number of aspects of counting as well
as many morphologically distinct numerical expressions dedicated
to particular quantificational purposes, including cardinal numerals,
ordinals, fractions, percentage expressions, nominals such as dozen,
approximators like hundreds, and numerical adverbials, e.g., twice, it is
usually assumed that at the end their interpretation is based on number
(see Rothstein 2017).6 In particular, different types of numerical
expressions appear to represent enumerations of different types of
sets, e.g., sets of entities as opposed to sets of events.
However, though the definition introduced above involves a significant
component of counting, it is incomplete, and thus fails to capture
what counting really is and how it differs from other quantificational
operations such as measuring. As we will soon see, there are a number
of situations where despite successfully establishing one-to-one
correspondence between entities and numbers we would intuitively
reject a particular enumeration as counting. Therefore, a more specific
characterization is required. In particular, I take counting to be
a quantificational operation that is governed by three general principles
restricting what kind of object can be assigned a number. I will
refer to those constraints as the principle of non-overlap, maximality,
and integrity. It is not unlikely that their knowledge is inherent, and
thus when talking about counting, their existence is taken for granted.
That is precisely why I believe it might be useful to formulate them
explicitly in order to get better understanding of the phenomenon of
countability.
The principle of non-overlap ensures that things that count as one
need not overlap, i.e., do not share a part (cf. Landman 2011, 2016).
Guaranteeing disjointness of units of counting is necessary to avoid
6. Notice also that both complex numerical expressions are necessary if a language
is to be able to enumerate the infinite series of numbers since in order to do so it
requires a recursive system (Rothstein 2017, pp. 12–13).
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5. Conceptual background
a possibility of an entity being counted twice. For instance, assume
portions of matter a, b, c, and d arranged in such a way that c overlaps
with both a and b, specifically c = a b. Now, one could imagine an
operation that would assign numbers to all a, b, c, and d. Summing
them up would yield number 4 but this result is incorrect if one wanted
to count how many portions of matter there are. The reason is that c
is not disjoint from a and b, and thus should not be associated with a
number. To put it differently, when counting it is disallowed to count
a thing two times.
The second principle concerns maximality understood in terms of
mereological exhaustivity. It states that counting requires that what
is associated with a number needs to be a maximal entity of which a
certain property holds. In other words, objects need to be counted in
their entirety, i.e., all parts of a thing need to be put in correspondence
with a particular number and it is disallowed to leave some of them
out. To illustrate this, let us consider a situation where there are three
distinct entities a, b, and c such that c consists of two parts d and
e. Now, assume a quantificational operation that satisfies the nonoverlap
constraint but is not restricted by the principle of maximality.
When applied to the set of entities in question, it might very well
yield 4 since a, b, d and e are disjoint, whereas c shares a part with
both d and e. However, such an operation would not be of great help
if one wanted to know how many entities are there since it fails to
differentiate between wholes and their parts. Counting, on the other
hand, is about recognizing that though what counts as one might be
constituted of smaller elements, for the purpose of quantification the
maximal sum of those elements is considered a whole unit. Again,
counting ensures that an entity is not assigned two numbers. Notice
also that this constraint accounts for how we count homogeneous
entities such as twigs and rocks. Given a particular counting situation,
what counts as one is always the maximal entity irrespective how its
part-whole structure is construed in that situation.
Finally, the principle of integrity requires what counts as one to
be an integrated whole. It is crucial that for an individual to be assigned
a number it is not enough to be disjoint from other entities
and mereologically maximal. In addition, its parts need to be in a
particular spatial configuration, i.e., they need to be connected. This
means that scattered entities such as substances and arbitrary sums
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5. Conceptual background
of individuals normally are not assumed to count as one. In the next
chapter, I will introduce a theory of wholes spelling out those concepts
in a formal way. Notice, however, that the use of the words ‘integrated’
and ‘connected’ is somewhat liberal here and should be rather understood
as ‘conceptualized as integrated/connected’. If the connection
between parts is easily retrievable, e.g., parts are physically dissected
but remain in proximity to the original place of attachment, then with
great probability the principle of integrity would be satisfied. Thus,
individuation in terms of integrity is about how human beings categorize
objects in the world rather than about objective properties of
mind-external entities. As we will see in the next section, this property
is crucial for distinguishing counting from another quantificational
operation, namely measuring. However, before I discuss the difference
between the two in detail, let us consider examples of proper counting
as opposed to what I call illegal counting.7
Given the counting principles of non-overlap, maximality, and integrity
discussed above, we expect that counting works as illustrated
schematically in Figure 5.3. Each of the apples can be associated with
integers since they all satisfy the requirements in question. First, the
apples are disjoint from each other, i.e., they do not share parts. Second,
in each case what is assigned a number is a whole apple, i.e., a
maximal sum of parts making up an object. This means that counting
is complete. In other words, after the procedure is finished, there
are no entities left that have not been associated with an integer. Finally,
individual apples constitute objects that are conceptualized as
integrated wholes, i.e., configurations of parts that connected to each
other. This fact guarantees that each apple can be tracked in space and
time, and thus is easily recognizable as an object distinct from other
entities. The interplay of the three factors in question results in that
the procedure illustrated in Figure 5.3 satisfies conditions on counting.
Given the depicted set of apples, after performing it we would get an
integer corresponding to the total number of apples.
7. Notice that in accordance with the focus of this study the proposed principles
are intended to account for quantification over concrete physical objects. In fact,
they might be more abstract in order to be extendable to quantification over abstract
entities such two ideas and three proposals. However, I will not pursue this possibility
here.
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5. Conceptual background
Figure 5.3: Counting
According to the general counting principles, counting is devised
to be what we see in Figure 5.3. However, one could easily imagine an
operation where, e.g., assigning a number to less than a whole entity
or summing up complementary parts of corresponding entities to
make up what counts as one is not prohibited. For instance, consider a
quantificational operation that satisfies the general counting principle
of non-overlap, but violates the principle of integrity. To illustrate this,
let us assume illegal counting as depicted in Figure 5.4. In contrast to
Figure 5.3, in this case numbers are associated with arbitrary sums of
parts of distinct entities. Notice that those sums can be guaranteed to
cover the total volume of all the relevant apples. If this were somehow
ensured, in the discussed scenario we could obtain the same number
as in Figure 5.3. However, intuitively this operation is very different
from counting. It is simply not what we do when we count.
Figure 5.4: Illegal counting
Though the procedure illustrated in Figure 5.4 is not counting,
it does not mean that it does not represent another quantificational
operation human beings utilize for different purposes. In the next
paragraphs, I will discuss what I believe to be the crucial difference
between counting and measuring.
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5. Conceptual background
Intuitively, the difference between counting and measuring relies
essentially on that the first is about specifying how many objects of
a certain kind are there in a particular context, whereas the latter
concerns determining the quantity of a particular substance in relevant
measure units. In principle, there are three possible views on
the relationship between the two operations in question. The first one
reduces measuring to a particular type of counting (e.g., Gil 2013). The
intuition behind such an approach is that measure words individuate
in terms of quantity (Lyons 1977). Consequently, instead of quantification
over objects measuring is about counting units determined be
measure words. An opposing approach is to view counting as a form
of measuring. In particular, counting can be understood as measuring
a quantity of an object in terms of natural units (Krifka 1989, 1995).
Finally, the third option is to postulate that counting and measuring
are two independent operations (Rothstein 2017). In this study, I adopt
the third perspective and present novel evidence in its favor. In particular,
I propose that the distinction between counting and measuring
can be reformulated in terms of quantificational principles introduced
in the previous section. Though both counting and measuring obey
the principle of non-overlap and maximality, only the first satisfies the
principle of integrity. Measured quantities of a substance cannot overlap
in order to ensure that things are not measured twice. Furthermore,
a unit of measurement needs to correspond to the maximal quantity
of matter to guarantee that measuring is exhaustive. However, only
counting is sensitive to the topological make-up of entities it applies
to.
To better understand the essence of the distinction, let us consider
the following two scenarios. In the first scenario, illustrated in Figure
5.5, someone has spilled some liquid on the table in such a way that
there are two separate blobs a and b whose volume is one and a half
milliliters each. In the second scenario, see Figure 5.6, someone has
simply put two cubes c and d on the table. Let us also assume that
apart from the liquid and the cubes there is nothing else on the table
in each of the cases, respectively.
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5. Conceptual background
a b
Figure 5.5: Measuring and integrity
c d
Figure 5.6: Counting and integrity
Now, let us examine the sentences in (5.2a) and (5.2b) describing
the situations depicted in Figure 5.5 and 5.6, respectively. The statement
in (5.2a) is true despite the fact that one of the three milliliters
must be split between a portion of a and a portion of b since each of the
blobs consists of one and a half milliliters of liquid. On the other, hand
(5.2b) is simply false. This contrast shows that units of measurement
such as milliliters are not objects. In other words, unlike counting
measuring does not care about individuation in terms of integrity and
it indeed appears to be a distinct operation.
(5.2) English (Guy Tabachnick, p.c.)
a. There are three milliliters of liquid on the table.
b. #There are three objects on the table.
I argue that this is because counting is topology-sensitive whereas
measuring is not. In other words, in measuring units of measurement
are not required to be assigned only to contiguous entities. There are
multiple ways how to assign particular milliliters u1, u2, and u3 to
particular portions of a and b and Figure 5.7 illustrates one possible
distribution with u2 corresponding to the volume of liquid of a sum
of one-third of a and one-third of b. As long as the total volume of a
and b equals three milliliters (5.2a) is true with respect to Figure 5.5.
185
5. Conceptual background
u1 u2 u3
Figure 5.7: Measuring in units
Unlike measuring, counting is sensitive to what kind of topological
structure entities in its domain have. In parallel to 5.7, one could
imagine that two-thirds of c correspond to an object o1, one-third of
c plus one-third of d make up o2, and the remaining two-thirds of d
correspond to o3, as depicted in Figure 5.8. However, this is not how
counting works. The statement in (5.2b) is false with respect to Figure
5.6 because counting assigns numbers to integrated individuated
entities rather than arbitrary portions thereof.
o1 o2 o3
Figure 5.8: Counting objects
I argue that what the contrast between truth-conditions of (5.2a)
and (5.2b) discussed above shows is that counting and measuring are
in fact two distinct semantic operations, as proposed by Rothstein
(2017). The core difference between the two boils down to the fact that
counting is topology-sensitive, whereas measuring is not. In other
words, it is misleading to think of measuring in terms of counting
measure units (pace Gil 2013). If it were the case, similar to 5.6 we
would expect (5.2a) to be false with respect to Figure 5.5, contrary to
fact.
The distinction gets even more salient if we realize that many numeral
phrases can get a measure interpretation.8 For instance consider
the contrast between the scenarios described in (5.3) and (5.4). In
8. See Rothstein (2017, p. 3) for an extensive discussion of ambiguities another sort,
namely the alternation between individuating and content readings of container
words such as glass of water.
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5. Conceptual background
(5.3a), the numeral phrase three apples denotes a plurality of integrated
wholes, i.e., distinct individuated objects, and thus it can be felicitously
continued by (5.3b). In other words, since the referents of the
phrase satisfy conditions specified by the general counting principles,
they can be put in one-to-one correspondence with numbers. On the
other hand, the same phrase in the context of (5.4) gets a measure
reading. Given that the slices in the bowl are placed in such a way
that it is impossible to retrieve the original part-whole structures of
individual apples, three apples does not refer to distinct objects but
rather to the volume corresponding to three apples. Consequently, it
is distinctively odd to continue the sentence in (5.4a) with (5.4b). All
things considered, (5.3a) presumes an operation depicted in Figure
5.3, whereas (5.4a) does not.
(5.3) English (Guy Tabachnick, p.c.)
Context: John is cooking with his child. They put three whole
apples on a table. John says:
a. There are three apples on the table...
b. Let’s count them together: one, two, three.
(5.4) English (Guy Tabachnick, p.c.)
Context: John is cooking with his child. They sliced three
apples and put the slices into a bowl. John says:
a. There are three apples in the bowl...
b. #Let’s count them together: one, two, three.
The contrasts discussed above prove that counting indicates integrity
or at least easily retrievable traces of integrity of entities it applies to
whereas measuring ignores this factor. Though monotonic systems of
measurement track part-whole relations (Schwarzschild 2002), they
do not seem to be sensitive to topological notions. On the other hand,
counting does care about the spatial arrangement of parts making up
wholes. This strongly suggests that the two operations in question
are distinct from each other. Perhaps, one could argue that despite
syntactic as well as semantic differences between numeral phrases
and measure phrases (see Rothstein 2017) counting is a very special
case of measuring (see Krifka 1989, 1995). In particular, measuring
could be thought of as a more general quantificational procedure. In
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5. Conceptual background
principle, I remain agnostic on this issue and am open to an argument.
However, for the sake of what this study is about I will assume that
measuring and counting are two distinct semantic operations. In the
next section, I will propose how general counting principles extend to
subatomic quantification.
5.2.3 Subatomic quantification
In Chapter 2, 3, and 4, we have seen that there is substantial evidence
showing that natural language semantics is sensitive to the fact that
entities we refer to by means of nominal constructions consist of parts
and that there are linguistic means to quantify over such parts. On the
other hand, in the previous section I have postulated general counting
principles of non-overlap, maximality, and integrity. So far, we have
seen that there are good reasons to believe that these quantificational
constraints capture what humans do when they count objects. In this
section, I argue that the proposed set of rules constitutes a universal
mechanism that can be applied not only to whole individuals but also
when counting parts of objects. In other words, I posit that subatomic
quantification is subject to the very same constraints as quantification
over wholes.
Similar to what we have seen in Figure 5.3, counting at the subatomic
level presumes mereological maximality and topological integrity
of entities subject to quantification. Typically, counting is sensitive
to the fact that some parts are cognitively more salient within the
part-whole structure of an object than others. This seems to correspond
to what Champollion and Krifka (2016) call structured parthood as
well as to the distinction between specific and arbitrary subdivisions
of a whole into parts introduced in philosophical considerations, as
already mentioned in Chapter 1 (e.g., Krecz 1986, Markosian 1998, Jennings
2010). Notice that what counts as a cognitively salient part can
vary with respect to the context. For instance, assume John’s daughter
has a teddy bear called Fuzzy Wuzzy. In a situation where she has
covered Fuzzy Wuzzy’s entire left half with red paint and its entire
right half with black paint, John might want to say (5.5a). In such
a case, the total volume of matter making up the teddy bear is partitioned
in such a way that the cognitively salient parts are its left
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5. Conceptual background
half and its right half.9 On the other hand, in a scenario where John’s
daughter painted Fuzzy Wuzzy’s left paw red and right paw black,
the count explicit partitive in (5.5b) refers to the teddy bear’s body
parts. Nevertheless, what cognitively salient parts have in common is
that given a particular context they are disjoint. This brings us to the
relevance of the principle of non-overlap in subatomic quantification.
(5.5) English (Guy Tabachnick, p.c.)
a. Both parts of the teddy bear are painted.
b. Two parts of the teddy bear are painted.
Since there are numerous ways how to divide an object into parts,
there are multiple portions of matter of which the property of being
part of that object holds. For instance, Figure 5.9 illustrates a number
of entities the phrase part of the apple could refer to. Notice that none
of them is disjoint from the others, i.e., all of them share a part with at
least one other part. However, the principle of non-overlap states that
such entities cannot be counted. Similar to arbitrary portions of mass,
e.g. juice, arbitrary parts of the apple abide individuation since they
are not well-defined bounded integrated objects and it is virtually
impossible to distinguish them from other parts. This means that
not all parts are equal with respect to countability. Some of them are
countable, whereas others are not. Crucially only those divisions of a
whole into parts can be enumerated that involve only non-overlapping
parts.
Figure 5.9: Overlapping parts
9. The sentence in (5.5a) appears to be slightly degraded due to the fact that instead
of part a more accurate partitive word, i.e., half, could have been used. Many thanks
to Guy Tabachnick for discussing the English examples with me.
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5. Conceptual background
Another issue concerns the principle of maximality. What counts
as part of an object can also consist of smaller parts. Notice that such
parts of parts also satisfy the property of being part of that object.
However, the general counting principles require that only entities in
their mereological entirety can can be associated with a number. This
means that once a particular division of an individual into parts has
been executed in a given counting context, those parts are immutable
and treated as objects in their own right. Consequently, the principle of
maximality applies as it would in a situation when one counts whole
individuals. In other words, given a partition non-overlapping parts
are assumed to be maximal with respect to how the whole has been
divided.
Finally, as we have seen in the previous section countability is also
governed by the principle of integrity. However, as we have seen in
Chapter 3 extensions of expressions referring to parts of objects do
not necessarily involve topological commitments, i.e., parts need not
be continuous. For instance, let us consider explicit entity partitives.
There is definitely a sense in which two or more separated portions of
matter within an object are part of that object, and thus a topologyneutral
explicit entity partitive construction would be true of such a
configuration. Nevertheless, similar to any arbitrary sum such entity is
not countable since associating it with a number would clearly violate
the principle of integrity. Therefore, only sets including parts that are
mereologically maximal integrated entities that do not overlap can be
enumerated. In other words, only a subset of possible divisions of an
object is fit for counting.
Having this in mind, let us see how the general counting principles
apply in a context of subatomic quantification. For this purpose, let us
assume John’s daughter wants to count parts of her teddy bear Fuzzy
Wuzzy. Figure 5.10 represents an exemplary situation that intuitively
fits what we expect from counting. Specifically, the illustrated operation
assigns numbers to cognitively salient non-overlapping parts
of Fuzzy Wuzzy, namely its ear, leg, and paw. Those parts constitute
mereologically maximal and integrated entities, i.e., though they are
elements of a larger whole, when counted they are treated as objects
in their own rights. This means that, e.g., number 1 is associated with
Fuzzy Wuzzy’s whole ear and there is no part of that ear that is left
out. Also, an ear is a continuous part that comes in one piece. To sum
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5. Conceptual background
up, since the depicted division of the teddy bear consists of elements
that are disjoint, maximal, and integrated, all the counting principles
are satisfied and the set of parts can be enumerated.
Figure 5.10: Counting of parts
On the other hand, one could imagine another operation, e.g., one
such as that illustrated in Figure 5.11. Though it is definitely logically
possible, it seems weird and would intuitively be rejected as counting.
There are two reasons why it is so. The first reason is that the indicated
ear and leg do not constitute a continuous integrated part that would
be cognitively salient. Therefore, assigning them number 1 violets the
principle of integrity. In other words, there is a sense in which an ear
and a leg are part of the teddy bear, but there are not a part of it and
only entities that can be considered as such can satisfy the countability
condition. The second reason is that the marked paw is not disjoint
with the right half of the teddy-bear which of course violates the
principle of non-overlap. Given the division in Figure 5.11, the paw
in question would be counted twice and since counting requires that
an entity can be associated with a number once and once only, the
depicted operation fails to satisfy the general counting principles and
represents what I call illegal counting at the subatomic level.
All things considered, counting at the subatomic level presumes
assigning numbers to salient parts of a whole constituting disjoint
integrated entities that are maximal with respect to a given partition.
In other words, once it is decided what counts as a part, it is treated as
an individuated object in its own right and all discussed constraints
concerning counting apply. The key conclusion is that there is one
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5. Conceptual background
Figure 5.11: Illegal counting of parts
universal mechanism governing quantification over wholes as well
as subatomic quantification and exploring the latter can reveal some
fundamental properties of countability in general that otherwise might
be difficult to recognize.
5.3 Summary
In this chapter, I have provided a general conceptual framework that
will serve as a basis for developing a formal account for subatomic
quantification in natural language. I have suggested how the linguistic
evidence presented in Chapter 2, 3, and 4 correlates with findings in
cognitive psychology. In particular, in the first part of the chapter I
have reviewed several representative studies in the research indicating
the following. First, there is compelling evidence that humans
possess an innate ability to perceptually distinguish between objects,
i.e., bounded integrated entities, and substances, i.e., shapeless scattered
portions of matter, and that this contrast correlates with the
mass/count distinction in grammar though the correspondence is
imperfect. Specifically, integrated solid things are predictably referred
to by count nouns, but there is a class of mass terms that also refer
to objects. Importantly, the object/mass distinction is not based on
ontological properties of entities in some objective sense but rather it is
construed by human cognition. Second, the ability to simultaneously
perceive parts as elements of a whole and a whole as a collection of
parts manifests itself in early childhood. The capacity to intuitively dis-
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5. Conceptual background
tinguish between parts and wholes guides vocabulary acquisition. Finally,
human number sense appears to be sensitive to whether counted
items are conceptualized as integrated objects. Experimental evidence
shows that young children always count each separate physical entity
as one. Even if there are instructed to do otherwise or given clues that
two elements might be considered one broken thing, they ignore them
and simply cannot avoid counting contiguous entities. This suggests
that further development of quantification in older humans rests on a
mechanism individuating in terms of spatial integrity.
In the second part of this chapter, I have presented the three claims
that constitute the conceptual core of this study. I have postulated that
natural language is sensitive to how the spatial relationship holding
between parts of a particular entity is conceptualized. The relevance
of the topological notion of integrity manifests itself primarily in how
nominal lexicon is classified into different grammatical categories. In
particular, count singular nouns prototypically designate integrated
wholes, i.e., encode information concerning a particular spatial configuration
of parts their referents consist on, whereas plurals denote
arbitrary sums thereof, i.e., presuppose integrated wholes as parts but
impose no topological constraints on them.
The second claim regards what I call the general counting principles.
I posit that counting is a special kind of quantificational operation
that presumes certain properties of entities that are put in
one-to-one correspondence with numbers. Specifically, the principle
of non-overlap guarantees that enumerated entities are disjoint, and
thus no thing is counted twice. On the other hand, the principle of
maximality requires that numbers are associated with entities in their
mereological entirety, i.e., no part is left out. Finally, the principle of integrity
ensures that what can be counted needs to be conceptualized as
something that comes in one piece. This constraint excludes arbitrary
sums of individuals as well as scattered entities such as substances.
Altogether, the general counting principles guarantee that sets that
can be enumerated consist only of elements that are discrete object.
The final claim extends the general counting principles to subatomic
quantification. In other words, I postulate the quantificational
mechanism described above is a universal mechanism that can apply
both at the level of wholes and at the level of parts. In the next chapter,
I will introduce a theory of wholes called mereotopology in which
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5. Conceptual background
the formal account for subatomic quantification will be grounded.
In particular, it will enable us to capture various subtle topological
distinctions including the intuitive notion of an integrated whole as
opposed to an arbitrary sum of parts. This will provide means to
model not only the fact that some entity is part of something else
but also how individual parts are spatially arranged within a whole
configuration.
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6 Theory of parts and wholes
Given the conceptual background described in the previous chapter,
an account for subatomic quantification is supposed to capture the
intuitive notions of parthood and integrity. In this chapter, I will introduce
a theory of parts and wholes that provides a formal apparatus
devised to model both concepts. This theory is usually referred to as
mereotopology and as suggested by the name it involves two interrelated
components. In the following sections, I will first introduce
standard mereology, its axioms, and advantages. Next, I will turn to
its limitations and discuss how it can be extended with topological notions
such as connectedness. As a result, the sophisticated mereotopological
framework to be presented will allow us to model integrated
wholes as opposed to other types of entities. The mereotopological
distinctions to be developed will play a crucial role in the formal account
for subatomic quantification in natural language I will propose
in the next chapter.
6.1 Mereology
Mereology (from the Greek µ ρoζ ‘part’) is the study of parthood, i.e.,
relations between part and whole as well as between parts within a
whole. It has been proposed by Leśniewski (1916) and further developed
by Leonard and Goodman (1940) and Goodman (1951) as an
alternative for set theory. In particular, since set theory is founded on
two distinct part-whole notions of set membership ∈, i.e., the relation
between an element and a set, and subset relation ⊆, i.e., the relation
between sets, it distinguishes between a singleton set and its member,
i.e. {a} = a, and postulates the empty set ∅. In order to avoid such
assumptions mereology does not postulate abstract entities such as
sets, and thus does not distinguish between ∈ and ⊆. Instead, only
one primitive relation of parthood is postulated. Therefore, there is no
need to draw the distinction between a singleton set and its member
nor to assume the notion of the empty set.
The part-whole relation can hold between portions of masses, parts
of individuals, members of groups, locations, events, times, and even
abstract entities (Simons 1987, Winston et al. 1987). It is a prominent no-
195
6. Theory of parts and wholes
tion in how the world appears to be structured, how human beings perceive
things and talk about things. Therefore, mereology plays an important
role in ontology, cognitive psychology, and natural-language
semantics. Since Quine (1960)’s discussion of mass nouns as scattered
objects (a clearly mereological concept), considering countability in
mereological terms became standard in philosophy (see especially
Sharvy 1980 for a treatment of mass definite descriptions in this spirit).
However, it was Link (1983) who introduced formal mereology to
linguistics in his lattice-theoretic approach to pluralities.
There are different ways how to spell out mereological intuitions
and axiomatize the parthood relation (see, e.g., Simons 1987, Casati
and Varzi 1999, and Varzi 2016 for thorough surveys of the field).
However, in this chapter I will discuss only a system that gives rise to
algebraic structures, i.e., sets with binary operations defined on them,
that is often referred to as classical extensional mereology. For brevity,
however, I will simply call it standard mereology since it is considered
to be a standard in mereology (though see, e.g., Rescher 1955 for an
alternative proposal) and is most commonly used in formal semantics
for natural language (e.g., Link 1983, Krifka 1989, Landman 2000, and
Champollion 2017, to name just a few prominent approaches). The axiomatization
of standard mereology presented here largely follows the
extensive discussions provided by Simons (1987) and Casati and Varzi
(1999) as well as the encyclopedic entry on linguistic applications of
mereology by Champollion and Krifka (2016). To avoid terminological
confusion I will follow Grimm (2012b) in distinguishing between individuals,
i.e., a pre-theoretic notion regarding well-defined physical
entities, and m-individuals, i.e., objects in the mereological sense.
The most received view on the axiomatization of mereology takes
the concept of part to be the basic notion. The part-of relationship 1 is
the primitive relation which is reflexive, transitive, and antisymmetric.
The axioms given in (6.1)–(6.3) constrain to be a partial order.
(6.1) Reflexivity
∀x[x x]
(Every thing is part of itself.)
1. I use the symbol following Landman (1989a,b, 2000). Other authors use ≤
instead. The main motivation behind this notational decision is to avoid potential
confusion with the ‘less than or equal to’ relation between numbers.
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6. Theory of parts and wholes
(6.2) Transitivity
∀x∀y∀z[(x y ∧ y z) → x z]
(Any part of any part of a thing is itself part of that thing.)
(6.3) Antisymmetry
∀x∀y[(x y ∧ y x) → x = y]
(Two distinct things cannot both be part of each other.)
Given that the primitive relation states that an m-individual is part
of itself, it might be useful to introduce an auxiliary notion of parthood
which is not reflexive. The formula in (6.4) provides the definition of
proper part.
(6.4) Proper part
x y
def
= x y ∧ ¬(y x)
(A proper part of a thing is a part of a thing that is distinct
from it.)
Furthermore, one might also want to account for the fact that entities
can share parts. For this reason, an ancillary relation overlap can be
defined, as in (6.5).2 From the mereological point of view, identity can
be understood as a special case of overlap.
(6.5) Overlap
x ◦ y
def
= ∃z[z x ∧ z y]
(Two things overlap if and only if they share a part.)
The parthood relation together with the derived notions of proper
part and overlap constitute the core of standard mereology. However,
without any further constraints the system developed so far gives rise
to structures which have been traditionally dismissed as objects that a
theory of part-whole relations should represent. For instance, Figure
6.1 illustrates a mereological relation between the m-individuals a and
b such that b is a proper part of a and there is no other proper part of a.3
Similar, Figure 6.2 depicts a model in which two distinct m-individuals
2. I use the x ◦ y notation following Simons (1987), Link (1983), and Champollion
and Krifka (2016). Other authors use o(x, y) instead.
3. The line represents with the m-individual corresponding to the lower node
being part of the m-individual corresponding to the higher node.
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6. Theory of parts and wholes
a and c both share all parts. Though such models are allowed within
the system, intuitively they are not what a theory of parts and wholes
should account for. To remedy this, further constraints have to be
introduced.
a
b
Figure 6.1: Object with a solitary proper part
a
b
c
d
Figure 6.2: Objects sharing all parts
Standard mereology can be devised in order to restrict how an mindividual
can be decomposed into parts. In particular, an additional
axiom can be added that constrains mereological objects in such a way
that every proper part must be supplemented by another disjoint part.
In other words, there is always a mereological difference between a
whole and its proper parts. An extension known as remainder principle
or supplementation, see (6.6), guarantees that m-individuals
cannot consist of a single proper part, and thus models such as that in
Figure 6.1 are ruled out.
(6.6) Supplementation
x y → ∃z[z y ∧ ¬(z ◦ x)]
(Whenever a thing has a proper part, it has more than one.)
However, supplementation is insufficient to exclude structures like the
one in Figure 6.2, and thus another extension needs to be implemented
in order to ban them. Given , the notion of sum can be defined. Sums
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6. Theory of parts and wholes
are devised to capture a pretheoretical concept of collections, i.e.,
the result of grouping several entities together. In natural language,
conjoined terms and definite descriptions like the water have been
analyzed in terms of sums. In particular, in his influential paper Link
(1983) treats expressions such as John and Mary as referring to the
sum of two individuals, namely John and Mary. Similar, Sharvy (1980)
proposes that a definite expression such as the water denotes the sum of
all water. I will follow here the classical definition of sum due to Tarski
(1929) as presented in (6.7) (for alternative definitions see Simons 1987
and Casati and Varzi 1999).
(6.7) Sum
sum(x, P)
def
= ∀y[P(y) → y x] ∧
∀z[z x → ∃z [P(z ) ∧ z ◦ z ]]
In prose, a sum of (the things in) a set P is a thing that consists of
everything in P and whose parts each overlap with something in P.
Since the part structures in standard mereology are closed under sum
formation, for any two individuals there is also a sum of those two
individuals. This is ensured by uniqueness of sums which requires two
things composed of the same parts to be identical, see (6.8). This principle
excludes structures such as that in Figure 6.2. Since b and d form
two different sums, namely a and c, uniqueness of sums is violated
and the model is ruled out, as desired. Notice also that introducing
uniqueness of sums makes the axioms of reflexivity and antisymmetry
redundant since any transitive relation that satisfies uniqueness of
sums is provably reflexive and antisymmetric (see Hovda 2009 and
Champollion 2017). The axioms have been discussed for completeness
though.
(6.8) Uniqueness of sums
∀P[P = ∅ → ∃!z sum(z, P)]
(Every non-empty set has a unique sum.)
Finally, the additional notions binary sum and generalized sum can be
derived, as given in (6.9) and (6.10), respectively.4 The first operation
4. Again, I follow Landman (1989a,b, 2000) in the use of the symbols and ,
respectively, which correspond to the symbols ⊕ and one can often find in the
literature.
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6. Theory of parts and wholes
allows us to refer explicitly to the sum of two m-individuals, whereas
the latter to the sum of an arbitrary set.
(6.9) Binary sum
x y
def
= ιz sum(z, {x, y})
(6.10) Generalized sum
X
def
= ιz sum(z, X), where X is any non-empty set
Consequently, the meaning of the coordinate phrase John and Mary
can be represented as j m, whereas the semantics of the definite
description the water as water.
Alongside philosophical arguments for standard mereology, a
significant advantage of this framework is that it involves a wellunderstood
algebraic structure. In particular, as demonstrated by
Tarski (1935) models delivered by mereology are essentially isomorphic
to Boolean algebras with their bottom element removed, or equivalently
complete semi-lattices without the null element (see also Pontow
and Schubert 2006).5 Figure 6.3 gives an exemplary mereological
structure licensed by standard mereology. Note that it is isomorphic
to the powerset of the set {a, b, c} with the bottom element removed. I
will follow the well-established tradition in the semantic literature and
refer to such models as complete semi-lattices, however, as pointed out
by Champollion (2017) due to the lack of the null element the use of
the term complete is rather inadequate since it deviates from standard
mathematical practice (see also Landman 1989a). The definition of
such a structure is given in (6.11).
(6.11) Let S be a set and be a relation from S to S. A pair S,
is called a complete semi-lattice iff satisfies the axioms of
transitivity and uniqueness of sums.
Before we move on to the discussion of the limits of standard
mereology that seem to have posed some serious problems in the study
of natural language semantics adopting lattice-theoretic approaches,
5. The reason is that standard mereology does not allow for an object that is part
of everything and since the empty set is a subset of every other set, it must be
eliminated.
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6. Theory of parts and wholes
a b c
a ca b b c
a b c
Figure 6.3: Semi-lattice
let us briefly contemplate some correspondences between mereology
and set theory.
6.1.1 Mereology vs. set theory
It can be easily noted that the notion of parthood in standard mereology
has basically the same properties as the subset relation in standard
set theory. Therefore, since and ⊆ are very much alike, there a number
of deep correspondences between the two axiom systems, some
of which are illustrated in Table 6.1. Consequently, in many contexts
it might be convenient to regard sums as sets and a question arises on
why mereology is preferable over set theory in modeling pluralities
in natural language semantics.
Though early approaches to the semantics of plural expressions
were grounded in set theory (Bennet 1974, Hausser 1974), a substantive
body of contemporary research embraces systems based on the
notion of parthood. The original motivation behind preferring mereology
over set theory in the logic of plurality developed by Link (1983,
1998) was formulated on philosophical grounds. In particular, Link
argues that the traditional set-up of semantics taking the domain of
individuals to be simply a non-empty set fails to represent various
relations between objects. In other words, it is devised to capture only
flat domains and cannot account for structured domains, as in modeling
natural language. In terms of ontology, Link describes his position
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6. Theory of parts and wholes
property mereology set theory
reflexivity x x x ⊆ x
transitivity x y ∧ y z → x z x ⊆ y ∧ y ⊆ z → x ⊆ z
antisymmetry x y ∧ y x → x = y x ⊆ y ∧ y ⊆ x → x = y
interdefinability x y ⇔ x y = y x ⊆ y ⇔ x ∪ y = y
unique sum/union P = ∅ → ∃!z[sum(z, P)] ∃!z[z = ∪P]
associativity x (y z) = (x y) z x ∪ (y ∪ z) = (x ∪ y) ∪ z
commutativity x y = y x x ∪ y = y ∪ x
idempotence x x = x x ∪ x = x
unique separation x y → ∃!z[x z = y ∧ ¬(x ◦ z)] x ⊂ y → ∃!z[z = y − x]
Table 6.1: Correspondences between mereology and set theory (Champollion
and Krifka 2016; adapted)
as “relative nominalism” and argues that a set-theoretic approach is
not desirable because by treating pluralities as sets, it commits us to
assuming additional abstract objects. On the other hand, since in a
mereological framework pluralities are treated on par with singularities
as individuals, such an approach bears no additional ontological
commitments, and thus is to be preferred.6
Another arguable advantage of using mereology rather than set
theory in plural semantics is that it conveniently allows us to distinguish
type-theoretically between the denotations of common nouns,
i.e., set-denoting expressions of type e, t , and plural individuals,
i.e., individual-denoting expressions of type e (see also Vaillette 2001,
Champollion 2017). Due to the lack of sums set-theoretic approaches
to pluralities conflate the meanings of these two types of expressions
which can be considered an undesired result.
A prominent linguistic application of standard mereology is the
theory developed by Champollion (2017) who adopts the order-theoretic
perspective and treats as primitive. On the other hand, Krifka (1998)
derives from . Other influential theories that were intended to
characterize standard mereology were developed by Link (1983, 1998)
and Landman (1989a, 1991, 2000).7
6. But see Landman (1989a) for discussion challenging such a view.
7. But see a scrupulous review by Hovda (2009) who demonstrates in detail that
these systems contain certain flaws which result in that they fail to describe standard
mereology, as intended.
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6. Theory of parts and wholes
6.1.2 Limits of mereology
As it has been pointed out by many authors, there is independent
motivation for extending standard mereology with auxiliary notions.
One factor concerns the criticism mereology faced with respect to
its inadequacy in modeling objects in the real world. In particular,
mereology is committed to unrestricted sum formation and as such it
is insufficient to capture what it means to be a whole. This results in
diametrical discrepancies between intuitions regarding the nature of
entities in the world and objects mereology actually delivers. To use
Casati and Varzi (1999)’s example, imagine a cup and broken glass.
Intuitively, the first constitutes an object, i.e., an individuated whole,
something that counts as one, whereas the other is just a collection of
shards. However, this distinction cannot be captured by describing
entities purely in terms of parthood. This is because in mereology for
every whole there is a set of parts and to every arbitrary collection of
parts there is their sum, i.e., a complete whole. As a result, a cup and
broken glass have the very same mereological status. In other words,
the allowance of scattered entities makes it impossible to differentiate
between them and individuals constituting integrated wholes. Consequently,
mereology has faced criticism that in principle it fails as a
theory of individuals.
As suggested by Grimm (2012b), it is very likely that at least some
of the shortcomings of mereological approaches to pluralities and
countability in natural language may be due to the general flaws
of standard mereology in attempting to capture what it means to
be a whole. Arguably, extending standard mereological frameworks
with additional notions in order to develop a better theory of objects,
as proposed, e.g., by Casati and Varzi (1999), may provide superior
tools for semantic treatments of quantification in natural language. As
already signaled, a challenging problem concerns the requirement of
unrestricted sum formation which states that for every two elements
in the domain, there is a plural individual corresponding to the sum
of those two elements. Specifically, the ontological status of such a
plural entity has often been disputed. As wittily put by Landman
(1989a)8, if three children messed up the living room, the correct
answer to the question “How many individuals were involved in
8. See also Cresswell (1985).
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6. Theory of parts and wholes
messing up the living room?” is “three”, whereas “seven” clearly
seems incorrect. However, if pluralities were conceptualized as having
the same ontological status as singular individuals, we would expect
this answer to be adequate since that number includes all distinct sums
of the children in question. In other words, if the children were Anne,
Betty, and Carl, i.e., a, b, and c respectively, then 7 is the number of the
elements of the set consisting of all entities that are part of the total sum
of the children, i.e., {a, b, c, a b, a c, b c, a b c}. The fact that
natural language does not allow for quantification over arbitrary sums
suggests that the way individual objects are conceptualized differs
from how we see pluralities of objects. However, standard mereology
lacks notions fine-grained enough to distinguish between objects that
come in one piece and entities that do not.
The next section will introduce an extension of standard mereology
within which more fine-grained notions can be defined. These notions
will enhance mereology in that in addition to parthood they will allow
us to capture also the topological configuration of entities making up
a particular sum, i.e., their spatial arrangement. In other words, the
topological extension will make it possible to discriminate between
scattered entities and wholes that come in one piece.
6.2 Mereotopology
Topology (from the Greek τoπoζ ‘place’) is the study of those properties
of space that are unaffected by continuous deformations of shape
or size of figures such as stretching or bending as opposed to tearing or
gluing. Important contributions to the early development of the field
were made by Fréchet (1906), Hausdorff (1914), and Kuratowski (1922)
and two of the crucial notions studied in topology are connectedness
and compactness.
Theories which extend standard mereology with topological relations
are known as mereotopology. Though early attempts to formalize
such a system trace back to Whitehead (1920, 1929), for many decades
there was no continuation in the systematic study of mereotopological
issues until relatively recent work in artificial intelligence began
to investigate the interaction between mereological and topological
notions in order to develop formal representations of spatial relations
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such as near or inside. Consequently, new developments in philosophy
and ontological modeling (e.g., Clarke 1981, Smith 1996, Roeper
1997) urged to extend standard mereology with topological relations
in order to advance an improved theory of objects.
Though topological notions have been widely used within the
study of locatives (e.g., Clark 1973, Herskovits 1985, Zwarts and Winter
1997, and Kracht 2002), it was not until Grimm (2012b,a) that
mereotopology was introduced to natural language semantics.9 In this
chapter, I will define basic topological concepts and review how they
interact with mereological notions as discuss by Casati and Varzi (1999)
and adopted for the semantic treatment of countability by Grimm
(2012b,a).
There are two main approaches with respect to relating mereology
and topology: i) choosing mereology as a basic level and augmenting
it with topological notions or ii) choosing topology as a basic level
and adding mereological relations. I will follow Grimm (2012b) who
employs a technique simply extending mereology with topological
notions. In particular, he adopts the approach developed in Casati and
Varzi (1999) which I will describe in the following paragraphs.
The crucial topological notion for the purpose of this study is
connectedness (c). This relation is introduced in such a way that it
interacts with other definitions and axioms of standard mereology.
Connectedness is reflexive and symmetric, as defined in the axioms
(6.12) and (6.13).
(6.12) Reflexivity
∀x[c(x, x)]
(Every thing is connected to itself.)
(6.13) Symmetry
∀x∀y[c(x, y) ↔ c(y, x)]
(If x is connected to y, then y is also connected to x.)
9. Mereotopology has been further applied to particular topics concerning countability
in Lima (2014) and Grimm and Dočekal (to appear). The theory has also
inspired approaches that do not develop full-fledged mereotopological accounts
but employ some spatial relations in order to capture certain issues relating to individuation
such as Scontras (2014, 2017), Sutton and Filip (2017a,b), and Henderson
(2017)
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6. Theory of parts and wholes
Notice, however, that unlike the parthood relation , as defined in
standard mereology, the connectedness relation c does not need to be
transitive. For instance, consider the configuration depicted in Figure
6.4. Although a and b are connected to each other and b and c are
connected to each other, it is not the case that a and c are connected to
each other.
a b c
Figure 6.4: Connectedness and transitivity
As discussed by Casati and Varzi (1999), there are a number of intuitive
interactions between the topological notion c and mereological
relations and ◦. These intuitive interactions can be formalized as
the so-called bridging principles which interrelate the mereological
and the topological component of the theory (Varzi 2007). The main
aim of the bridging principles is to secure that irrespective of their full
characterization and c are related in such a manner that a whole and
its parts are firmly connected. In particular, the principle of integrity
ensures that connectedness is implied by parthood, see (6.14), and
consequently mereological overlap is in fact a form of connection as
guaranteed by the principle of unity, see (6.15). Finally, the principle
defined in (6.16) secures monotonicity.10
(6.14) Integrity
∀x∀y[x y → c(x, y)]
(If an m-individual is part of another m-individual, then they
are connected.)
(6.15) Unity
∀x∀y[x ◦ y → c(x, y)]
(If two m-individuals overlap, then they are connected.)
10. In fact, (6.16) implies both (6.14) and (6.15) (Varzi 2007). However, I provide all
the bridging principles for the sake of completeness.
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6. Theory of parts and wholes
(6.16) Monotonicity
∀x∀y x y → ∀z[c(z, x) → c(z, y)]
(If an m-individual is part of another m-individual, then
whatever is connected to the first, is also connected to the
latter.)
Given these extensions, a number of mereotopological properties like
internal part (ip), internal overlap (io), and tangential overlap (to)
can be defined. Such notions enable us to draw subtle distinctions
to account for different spatial configurations that entities may be in.
For instance, the individual b in Figure 6.5 is an internal part of the
individual a since every entity that is connected to b overlaps with a
or, in other words, a includes b. On the other hand, though part of b
is an internal part of a in Figure 6.6, there is also part of b which is
not included in a. Finally, Figure 6.7 illustrates a configuration where
there is overlap between a and b but there is no internal overlap, i.e.,
the individuals only share their edges. The relevant definitions are
given in (6.17)–(6.19).
a b
Figure 6.5: Internal part
a b
Figure 6.6: Internal overlap
a b
Figure 6.7: Tangential overlap
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6. Theory of parts and wholes
(6.17) Internal part
ip(x, y)
def
= x y ∧ ∀z[c(z, x) → o(z, y)]
(6.18) Internal overlap
io(x, y)
def
= ∃z[ip(z, x) ∧ ip(z, y)]
(6.19) Tangential overlap
to(x, y)
def
= o(x, y) ∧ ¬io(x, y)
In addition, within the mereotopological framework standard topological
notions can be defined. (6.20)–(6.23) give definitions for interior,
exterior, closure, and boundary, respectively, whereas Figures 6.8–
6.10 illustrate the intended configurations. In particular, the shaded
area in the Figure 6.8 represents the interior of an object which is taken
to be the sum of m-individual’s internal parts. The dashed line is the
boundary of the object which is not comprised in its interior. On the
other hand, Figure 6.9 represents the exterior of an object which again
does not include the boundary, whereas the solid line in Figure 6.10
indicates that both the interior and boundary make up the closure of
an entity.
(6.20) Interior
ix
def
= X where X = {y : ip(y, x) = True}
(6.21) Exterior
ex
def
= i(−x)
(6.22) Closure
cx
def
= −(ex)
(6.23) Boundary
bx
def
= −(ix ex)
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6. Theory of parts and wholes
a
Figure 6.8: Interior
a
Figure 6.9: Exterior
a
Figure 6.10: Closure
The framework described so far extends standard mereology with
the primitive topological relation of connectedness and several derived
notions which allow us to talk about different configurations
that entities can be in. Such a system provides means to define individuals
as integrated wholes, i.e., objects characterized in terms of
different degrees of connectedness. The following section will discuss
a mereotopological approach to such individuals.
6.2.1 Integrated wholes
A great advantage of the mereotopological approach to natural language
semantics is that by means of the topological relations defined
so far it is possible to capture what it means to be an individual understood
as an integrated whole. In particular, a distinction can be drawn
between entities which come in one piece as opposed to pluralities, i.e.,
scattered entities which bear no topological commitments. To distinguish
the first from the latter, it is essential to introduce the property
of self-connected (sc). Unlike arbitrary sums, sc entities cannot be
divided into separated parts. The definition of sc is provided in (6.24).
(6.24) Self-connected
sc(x)
def
= ∀y∀z[∀w(o(w, x) ↔ (o(w, y) ∨o(w, z))) → c(y, z)]
(An entity is self-connected if and only if any two parts that
form the whole of that entity are connected to each other.)
Given the definition in (6.24), it is possible to differentiate inseparable
individuals from arbitrary sums of entities, i.e., disconnected configurations
of objects. To illustrate this let us assume a model containing
four entities, specifically suppose a and b are two halves of a cube,
whereas c and d are a pyramid and sphere, respectively, see Figure
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6. Theory of parts and wholes
6.11. Intuitively, there is a difference between the sum s1 = a b and
the sum s2 = c d. While s1 forms an integrated whole, the plurality
s2 is just an arbitrary collection of disconnected objects, and thus it
does not make up a solid whole. In other words, sc(s1) is true, whereas
sc(s2) is false.
a b c d
Figure 6.11: Wholes vs. sums
Self-connectedness is a great improvement in the theory of wholeness.
However, this notion is still insufficient to capture what it means
to be a one-piece object since it allows for configurations involving
only an external connection holding between individuals. In other
words, cases when the closure of one object overlaps the other or vice
versa are not ruled out. For instance, Figure 6.12 depicts two spheres
a and b which only touch each other, i.e., their boundaries are connected
at a single point. Given the definition of sc in (6.24), a and b
are self-connected but intuitively such a spatial configuration is not
sufficient to count as an integrated whole.
a b
Figure 6.12: External connection
In order to rule out cases such as the one illustrated in Figure 6.12,
a stronger relation is required. Ostensibly, what is necessary is that not
only boundaries of parts of a whole are connected to each other, but
also that their internal parts are shared. The basic io relation as defined
in (6.18) allows for modeling strong connection by means of internal
overlap but it is too weak to capture the nature of what it means to
be an integrated whole since it does not exclude certain undesirable
configurations. For instance, consider two partially-merged spheres a
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and b in Figure 6.13. Though their interiors are connected, there are
parts of a that are not shared with b and vice versa. Consequently, in
spite of the fact that the sum s = a b satisfies io, it is not what would
intuitively count as a one-piece individual.
a b
Figure 6.13: Strong connection
To solve this problem one needs to define an additional restriction
on the concept of interior given in (6.20). Accordingly, Casati and Varzi
(1999) postulate the property of being strongly self-connected (ssc),
as defined in (6.25). This refinement will get us much closer to the
type of entities we would like to single out, i.e., integrated wholes.
(6.25) Strongly self-connected
ssc(x)
def
= sc(x) ∧ sc(ix)
(An m-individual is strongly self-connected if it is self-connected
and its interior is self-connected.)
The restriction introduced in (6.25) guarantees that not only boundaries
of parts are shared but also that the interior of an entity is selfconnected.
Therefore, the property of ssc rules out objects that merely
touch each other, e.g., the spheres in Figure 6.12, as well as objects that
are partially merged, e.g., Figure 6.13. However, the fact that an entity
is strongly self-connected still does not ensure that such an entity is
an integrated whole in the sense a theory of wholeness is supposed
to capture. For instance, consider the entity a, i.e., the left half of the
cuboid represented in Figure 6.14. Although a qualifies as a strongly
self-connected object since any two parts that form its interior are
connected, intuitively it cannot be characterized as a whole. Obviously,
the reason is that there is yet another half of the cuboid and only
the two together make up the entire individual. Consequently, ssc is
insufficient for our purposes and an even stronger notion is required
to capture what is perceived as an integral whole.
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6. Theory of parts and wholes
a
Figure 6.14: Strongly self-connected part
Describing wholes is about capturing an intuitive notion of unity.
This means that a proper definition should accommodate both topological
integrity and mereological exhaustivity. For this purpose, Casati
and Varzi (1999) introduce the property of being maximally strongly
self-connected (mssc), as defined in (6.26).
(6.26) Maximally-strongly-self-connected
mssc(x)
def
= ssc(x) ∧ ∀y[ssc(y) ∧ ◦(y, x) ↔ y x]
(An m-individual is maximally strongly self-connected if (i)
every (interior) part of the individual is connected to (overlaps)
the whole (strongly self-connected) and (ii) anything
else which overlaps it and is strongly self-connected is once
again part of it (maximality).)
More generally one could single out entities that are maximally strongly
self-connected relative to a particular property. The relativized mssc
property can be characterized as in (6.27) which provides the final
mereotopological definition of what a whole is.
(6.27) Maximally strongly self-connected relative to a property
mssc(P)(x)
def
= P(x) ∧ ssc(x) ∧
∀y[P(y) ∧ ssc(y) ∧ ◦(y, x) ↔ y x]
(An m-individual is maximally strongly self-connected relative
to a property if (i) every (interior) part of the individual
is connected to (overlaps) the whole (strongly self-connected)
and (ii) anything else which has the same property, is strongly
self-connected, and overlaps it is once again part of it (maxi-
mality).)
If an entity satisfies mssc, then it is the largest entity satisfying that
property which is self-connected. For instance, if P is the property of
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6. Theory of parts and wholes
being a cuboid, then the definition in (6.27) will select the largest such
entities among those that come in one piece. In particular, it will single
out the whole cuboid in Figure 6.14 as opposed to, e.g., its left half a
though a itself also satisfies the property of being a cuboid. That is
because a is part of the cuboid whereas the whole cuboid is not part
of any other strongly self-connected object.
Though the structure of the mereotopological framework of Casati
and Varzi (1999) discussed above is relatively simple, the distinctions
developed within it are fine-grained enough for the representation of
spatial objects. Similar to standard mereology, mereotopology gives
rise to algebraic structures isomorphic to Boolean algebras with their
bottom element removed, however such lattice representations are further
endowed with regions indicating which elements are associated
with the connectedness relation c. For instance, consider Figure 6.15.
Given a model with 3 entities, namely two halves of a cube a and b and
a pyramid c, the semi-lattice represents all the possible sums generated
by the set of corresponding m-individuals. The vertical axis illustrates
the parthood relation with individual lines between particular nodes
symbolizing , whereas the horizontal axis represents connectedness.
The shaded area then covers those entities in the structure which are
connected. In other words, the mereological component of the theory
defines part-whole relations between a, b, and c and all the possible
sums thereof, whereas the topological component singles out those
parts that form self-connected wholes. In particular, while a in the
model is part of a b (as well as of itself due to the axiom of reflexivity),
it is not part of b. However, it is connected to b (as well as to a b
due the axiom of integrity and to itself – again reflexivity).
Moreover, as discussed by Grimm (2012b) the subtle niceties between
different types of connectedness, provide means to capture differences
between distinct types of nominals such as object, substance,
and aggregate nouns as well as eliminate old problems concerning
cumulative singular count nouns such as fence or twig (see, e.g., Zucchi
and White 2001, Rothstein 2010). Specifically, among multiple parts
that have a property of being, say, a rock only the maximal (improper)
part counts as a whole object. Hence, I conclude that compared to
standard mereology mereotopology proves to be a superior theory of
parts and wholes and its application in natural language semantics
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6. Theory of parts and wholes
a b c
a ca b b c
a b c
connectedness
parthood
Figure 6.15: Parthood and connectedness (based on Grimm 2012b, p.
136)
should be considered advantageous. In the next section I will discuss
several other kinds of connection.
6.2.2 Other types of connection
In the previous section, we have seen that given the primitive relation
c it is possible to derive a notion that can capture what it means to be
an integrated whole as opposed to an arbitrary sum, namely the mssc
property. This, however, does not deplete the potential of mereotopology
in modeling different types of objects and spatial configurations.
Based on c, many other types of topological relations representing
distinct varieties of connectedness may be defined. In this section, I
will discuss some of them.
The first auxiliary mode of connection to be discussed is the property
of firmly connected (fc), as defined in (6.28).11 This property
holds between two entities when they overlap in a substantive sense.
In other words, two m-individuals are firmly connected if their sum is
11. I use the term following Varzi (2007), whereas Casati and Varzi (1999) and
Grimm (2012a,b) talk about strong connection. My motivation is mainly to avoid
potential confusion with the ssc property which plays a crucial role in defining fc
but is definitely distinct from it.
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6. Theory of parts and wholes
strongly self-connected. Such a definition excludes cases of tangential
overlap, i.e., configurations where entities only touch each other, since
the ssc property implies that not only the boundaries but also the interiors
of m-individuals are connected. With an additional restriction
regarding the local scope of connectedness, the notion of fc seems to
be suitable to capture what it means to be a substance as opposed to
an integrated individual that we count as one (Grimm 2012a,b).
(6.28) Firmly connected
fc(x, y)
def
= ∃w∃z[w x ∧ z y ∧ ssc(w z)]
(Two m-individuals are firmly connected if a sum of their
parts is strongly self-connected.)
In contrast to fc, the property externally connected (ec), as defined in
(6.29), concerns entities that are merely tangentially connected, i.e., it
is only their boundaries that are connected, whereas their interiors do
not overlap. This notion enables us to model configurations of entities
that are not merged but simply touch each other.
(6.29) Externally connected
ec(x, y)
def
= c(x, y) ∧ ¬c(ix, iy)
(Two m-individuals are externally connected if they are connected
but it is not the case that their interiors are connected.)
Another topological notion it might be useful to derive is a one describing
a configuration of objects touching each other in such a way
that they form, say, a row. This variety of connectedness can be captured
by the property by-connected (bc), as provided by Varzi (2007).
It holds between entities that are not connected to themselves but are
associated by virtue of being connected to another mediating entity.
The definition in (6.30) describes bc as a three-place relation establishing
indirect connectedness between the outermost entities within a
configuration.
(6.30) By-connected
bc(x, y, z)
def
= c(x, z) ∧ c(z, y)
(Three m-individuals x, y, and z are by-connected if x is
connected to z and z is connected to y.)
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6. Theory of parts and wholes
A related notion is called mediately connected (mc). This is a binary
relation that holds between entities that are not necessarily connected
to each other, but are part of a by-connected configuration, e.g, a row
of individuals. In other words, there is an object that mediates the
connection between two such entities. The formal definition of bc is
given in (6.31).
(6.31) Mediately connected
mc(x, y)
def
= ∃z[bc(x, y, z)]
(Two m-individuals are mediately connected if they are byconnected
through a third m-individual.)
To illustrate the bc and mc relations, consider the configuration of two
spheres a and c and a cube b, as depicted in Figure 6.16. The three
entities are by-connected since a is externally connected to b and b is
externally connected to c. At the same time, the spheres are mediately
connected since both a and c are connected to the cube b, i.e., b in a
way mediates the connection between a and c.
a b c
Figure 6.16: By-connected and mediately connected
As pointed out by Grimm (2012b), the notion of mc may prove
useful in modeling natural language expressions denoting entities
such as eyes or fingers. Such objects seem to be conceptualized in a
particular way which is further reflected in grammar. In particular, in
some languages they belong to a category of inherently plural or dual
expressions.
Furthermore, the bc relation can be generalized to hold for any
number of entities. This gives rise to the property transitively connected
(tc). As defined in (6.32) (cf. Grimm 2012a,b), it determines
whether two individuals are connected through a series of mediating
entities. For instance, it would hold of two opposite entities in a scenario
similar to the one illustrated in Figure 6.16 with the exception
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6. Theory of parts and wholes
that between the spheres a and c there is not only the cube b, but also
a pyramid d such that it touches both b an c.
(6.32) Transitively connected
tc(x, y, P, C, Z)
def
= ∀z ∈ Z[P(z) ∧ (x = z1 ∧ y = zn) ∧
Cz1z2 ∧ Cz2z3 · · · ∧ Czn−1zn]
where Z = {z1, z2, . . . zn}
(Entities x and y are transitively connected relative to a property
P, a connection relation C, and a set of entities Z, when
all members of Z satisfy P and x and y are connected through
the sequence of zis in Z.)
Finally, the weakest variety of connectedness can be captured by the
notion of proximately connected (pc).12 This relation concerns entities
that are neither contiguous, nor do they touch each other, but rather
they are ‘very close’. The formula in (6.33) defines pc in terms of the
distance function d which yields the distance between two entities it
is applied to. For instance, if a sphere a is 1 meter away from a cube b,
the distance function d would yield 1 meter. The value n is determined
with respect to the predicate in question so that entities that satisfy it
are conceptualized as being sufficiently near each other relative to the
relevant property.
(6.33) Proximately connected (Grimm 2012b, p. 135; adapted)
pc(x, y, P)
def
= d(x, y) ≤ n(P)
(Two m-individuals are proximately connected if the distance
between them is lesser than or equal to the value determined
for a given predicate.)
The overview provided here does not by any means exhaust possible
configurations of objects and different types of connectedness.
Nonetheless, for our purposes the notions introduced in this section
will be absolutely sufficient. The introduced distinctions allow us to
distinguish between several types of entities. First, due to the mssc
relation it is possible to differentiate between arbitrary sums and inte-
12. In order to refer to a similar concept, Casati and Varzi (1999) use the term quasiconnected.
However, I will follow Grimm (2012a,b)’s terminology and formalism
here.
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6. Theory of parts and wholes
grated wholes. On the other hand, the fc property might be useful in
capturing the nature of substances, i.e., entities that come in multiple
instances which overlap with each other. Furthermore, the tc relation
enables us to single out entities consisting of connected objects arranged
in configurations such as rows and piles, whereas pc provides
means to describe entities involving multiple parts remaining in a
proximate distance from each other such as clusters and swarms (cf.
Henderson 2017).
6.3 Summary
In this chapter, I have presented a theory of parts and wholes called
mereotopology. The described system involves a mereological core
devised to capture the intuitive notion of parthood. This allows us to
relate elements that are part of certain larger entities with those entities.
However, what standard mereology cannot account for are certain
relations between parts that result in different spatial configurations.
This is where the topological component introduces the notion of
connectedness. Based on this primitive concept more sophisticated
properties can be derived. From the perspective of this study, the
most prominent is the property of being a maximally strongly selfconnected
individual. This notion enables us to capture integrated
wholes, i.e., objects that come in one piece. In the next chapter, I will
demonstrate how the mereotopological framework introduced here
can be used to model individuals in term of integrated wholes as well
as continuous fragments of such wholes as integrated parts. Such
treatment will allow us for a better understanding of countability and
developing a novel account for subatomic quantification.
218
7 Mereotopological account for subatomic quan-
tification
Given the mereotopological framework introduced in the previous
chapter, let us now attempt to account for some of the relevant expressions
employing quantification over parts such as partitives and
multiplier phrases as discussed in Chapter 2, 3, and 4. The conceptual
basis of the general mechanism of subatomic quantification outlined
in Chapter 5 can now be spelled out formally with the notions of
parthood and connectedness playing the central role in modeling the
meaning of expressions that provide objects to be counted as well as
of those that do the counting.
In this chapter, I will propose an analysis of different types of partitives
and multiplier phrases. I will start with the general discussion of
the implausibility of atomicity based approaches to subatomic quantification.
Next, I will turn to my own proposal spelled out in in terms of
the mereotopology supplemented with additional components such as
measure functions, partitions, and the individuation operation. First,
I will address the mass/count distinction by distinguishing between
different types of nominals as well as give the meaning of the plural
in the spirit of Grimm (2012b). Subsequently, I will turn to defining
other semantic objects I assume in the structure of different types of
partitives and multiplier phrases including the meanings of different
types of partitive words as well as numerical expressions which I treat
as complex constituents comprising a classifier component. Finally, I
will propose a compositional analysis of the expressions in question
by showing how the pieces fit together.
7.1 Doing without atoms
In standard theories of pluralities and countability the mass/count
distinction is often formulated in terms of atomicity (e.g., Link 1983,
Landman 1991, 2000, Chierchia 1998a, 2010). Though particular theories
differ significantly with respect to the exact character of the
alternation, the contrast between count and mass nouns usually boils
down to the existence or lack of minimal building blocks in their de-
219
7. Mereotopological account for subatomic quantification
notation or, alternatively, to a distinct nature of those building blocks.
The approach developed here rejects the view that what counts as
one is best represented as an atomic entity. Instead, I build on the
mereotopological notion of mssc introduced in Section 6.2.1 to capture
what can be an object of counting. There are three main motivations
behind this decision. The first one is the linguistic relevance of topological
relations holding between parts of a whole object as empirically
attested in multiple constructions in various languages discussed in
Chapter 2 and 3. The second stems from the general counting principles
described in Section 5.2.2 which I believe constitute a deeply
rooted part of human cognition and as such interact with that part of
the language faculty that generates grammar of countability. Finally,
the last reason for abandoning atomicity is simply that it does not seem
helpful with respect to subatomic quantification. Before we move on
to the proper proposal, let us briefly consider why that is.
Things that count as one are typically defined as atomic individuals,
i.e., entities that have no proper parts, see (7.1) for the definition of
atom and (7.2) for an optional mereological axiom of atomicity that
delivers atomic domains, i.e., domains consisting of atoms.
(7.1) Atom
atom(x) ↔ ¬∃y[y x]
(An atom is an entity which has no proper parts.)
(7.2) Atomicity
∀x∃y[y x ∧ ¬∃z[z y]]
(For any element, there is a part for which there does not exist
a proper part.)
At first blush, such an approach to what it means to count as one seems
counterintuitive since we know from our everyday life experience that
things we count as one very often do have multiple parts. However,
one should not confuse intuitive concepts with formal notions devised
to represent the semantics of natural language expressions, e.g., the
meaning of the English partitive word part with . Thus, my view is
that there is nothing wrong in dissociating intuitive entities from their
intuitive parts in terms of mereological modeling as long as it serves a
certain purpose. For instance, Champollion (2010, 2017) assumes that
the relation between, say, a teddy bear named Fuzzy Wuzzy and its
220
7. Mereotopological account for subatomic quantification
paw is not mereological parthood. In other words, despite the fact that
Fuzzy Wuzzy consists of a number of material parts it is considered a
mereological atom, i.e., an inseparable unit in the domain of entities.
This might seem plausible since it corresponds to yet another intuition,
namely that though we know that things have parts, we seem to ignore
this fact when we count those things. However, a significant problem
for such an account arises when we decide to quantify not over whole
things but over parts of those things. Thus, I will argue for two claims.
First, having a notion of atomicity is not enough for a full analysis
of quantification of parts. Second, atomicity is actually not needed to
analyze quantification over parts since it can be reduced to topological
notions which are required independently.
To illustrate this, let us consider the following case. Assume we
wanted to count parts of Fuzzy Wuzzy, e.g., its paws. Of course, natural
language allows for that since we can say (7.3a) and it would be true
if, say, Fuzzy Wuzzy’s paws were brown whereas all of its other parts
were of a different color. However, if referents of proper names are
mereological atoms, there are no parts to be counted since the entity
designated by the expression Fuzzy Wuzzy cannot be split. Without
saying anything else, this is untenable if we want to account for (7.3a).
Consequently, one could follow Link (1983) and propose to distinguish
between a domain of entities and a domain of portions of matter over
which a different parthood relation would be defined, i.e., the material
parthood relation m as opposed to the individual parthood relation
i. Assuming the domains are related by a function from entities to
portions of matter, one could argue that subatomic quantification is
about triggering this mapping and operating on portions of matter
corresponding to the mereological atom in the domain of entities.
However, such a proposal raises questions. If the domain of portions
of matter is atomless, which at first blush might seem intuitive, then
how it is possible to count entities of such a domain? Since atomicity
is a necessary condition for countability,1 we need to ensure that at
least some portions of matter are atomic. However, this is problematic.
1. Notice that it is not a sufficient condition due to the existence of object mass
nouns which are also assumed to have atomic reference yet are uncountable (see,
e.g., Chierchia 1998a, Barner and Snedeker 2005).
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7. Mereotopological account for subatomic quantification
(7.3) English
a. Exactly two parts of Fuzzy Wuzzy are brown.
b. Exactly two parts of one part of Fuzzy Wuzzy are brown.
Let us imagine that for some reason we wanted to count parts of a
part of an individual. Again, natural language does allow for that.
The sentence in (7.3b) is not a common thing to say but it is definitely
interpretable and it would be true if, say, exactly two digits of Fuzzy
Wuzzy’s left paw were brown and the remaining part of the paw were
not. Notice that truth-conditionally (7.3b) is not equivalent to (7.3a).
Consider, for instance, a scenario where the brown parts of Fuzzy
Wuzzy included only its head, right leg, and two digits of its left paw.
In such a scenario, (7.3b) would be true but (7.3a) would not. Now,
if countable portions of matter are atomic, then the same problem
as discussed with respect to atoms in the domain of entities arises.
Specifically, if a portion of matter corresponding to Fuzzy Wuzzy’s
left paw is modeled as having no proper parts, then how can we
quantify over its parts? To account for that we would be forced to
postulate another domain and relate it with the domain in which
the material part corresponding to Fuzzy Wuzzy’s paw is an atom.
However, this seems weird and even if we did it, the same problem
would arise again if we wanted to count parts of a digit of Fuzzy
Wuzzy’s paw. Consequently, distinguishing between just two distinct
parthood relations is not enough. In fact, since there are multiple
possible divisions of matter, we might end up establishing numerous
or even potentially infinitely many domains in order to be able to
define parts as atomic objects in a particular domain. This of course
is far from desirable and it seems to me that the discussed example
strongly suggests that sorting domains does not offer a tenable solution
to the issues concerning subatomic quantification and atomicity.
Since sorted domains proved unsatisfactory for our purpose, one
might think of another idea. As discussed in Section 3.3, one of very
few attempts to propose a solution to the problem of countability of
portions of matter (as opposed to uncountability of matter) preserving
atomicity was developed in Chierchia (2010).2 In this system, an
2. Another account worth mentioning was proposed by Landman (2016) but this
theory also abandons atomicity.
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expression such as English part is modeled as a variable over partitions
of an entity, i.e., an expression of type e, e, t that selects an
entity and returns a set of relative atoms that are ‘spatio-temporally
included’ in that entity. Since in principle such partitioning can be
applied recursively, we can obtain parts of parts of entities etc. This
seems to be a significant improvement compared to an attempt to
account for subatomic quantification in terms of sorted domains and
distinct parthood relations. However, there are still several issues with
this approach. First, as already discussed an analysis along the lines
of Chierchia’s proposal fails to account for the fact that parts can be
either continuous or discontinuous and, crucially, natural language
turns out to be sensitive to this distinction. Second, the notion ‘spatiotemporally
included’ introduced above is very loose. Compared to the
fine-grained concepts discussed in the previous chapter, it appears to
be very unsatisfactory. In general, I agree with Chierchia’s ‘mereotopological’
intuition, however its formulation leaves a lot to be desired.
For instance, imagine that Fuzzy Wuzzy was left in a car. Does it mean
that for linguistic purposes it is now part of that car? Or if an elephant
accidentally swallowed Fuzzy Wuzzy in a zoo only to throw it up
after some time, would the poor teddy bear be considered part of that
elephant in the period between swallowing and throwing up? Intuitively,
it does not seem so. Finally, from my point of view postulating
partitions of entities is an attempt to compensate deficits of atomicity
rather than a genuine development. On the one hand, it seems like
circumventing the ban on having proper parts by building blocks of
denotations of count nouns. On the other, though it seems to point in
a desirable direction, it somewhat stops halfway. As we will see, given
the potential of a mereotopological account the vague concept of a
partition of entities does not make a tenable alternative. In particular,
if maintaining one unified parthood relation proved successful, then
partitions of entities could be reformulated in terms of such unified ,
as we will see in Section 7.5.5. Though I will return to partitioning as
a useful tool in modeling partitive words denoting continuous parts, I
do not find Chierchia’s proposal an overall satisfactory analysis.
I believe that at this point it is plausible to conclude that in order to
account for subatomic quantification it is desirable to find a substitute
for atomicity. One might think that a plausible alternative would be to
adopt frameworks that model model building blocks of denotations
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in terms of natural or object units as proposed by Krifka (1989, 1995).
At first blush, this might seem attractive because what counts as one,
i.e., an entity to which we can assign the number 1, is not defined in
terms of not having proper parts. Thus, in principle accounting for
quantification over parts of such an object would not have to face the
issues described above. Nonetheless, such an approach would run
into a nagging conceptual problem. Intuitively, it is quite straightforward
what a natural unit of, say, entities that have the property of
being an apple is or, alternatively, what kind of concept they realize.
But what is a natural unit of a part? Or what kind of concept a part
instantiates? Given the systems mentioned above, it those notions
would have to refer in some way to the denotations of partitive words.
But it seems to me that this is a very strange way of thinking about
parts. However, even if one accepts the existence of natural units or
concepts of this sort, other questions arise. Since there are numerous
ways how to divide portions of matter making up an entity, does it
mean there are numerous natural units or concepts corresponding to
such portions, e.g., a natural unit/concept of a tiny part, half, great
part, and not-that-small-but-not-that-big part? Can both continuous
and discontinuous parts be considered as natural units? Are they realizations
of the same concept or two different concepts? Is there a
difference between natural units or concepts corresponding to parts
of a singularity and those corresponding to parts of a plurality? If
yes, then how do quantificational operations know that one is to be
selected over the other? If no, then how to distinguish between the semantic
subtleties the meaning of partitive words are associated with?
It feels like all of those are quite odd questions but they would need
to be addressed seriously in order to account for the data presented
in Chapter 3. Therefore, it appears to me that trading atomicity for
natural or object units does not make a satisfactory alternative with
respect to subatomic quantification.
Given the problems considered above, I conclude that an approach
based on mereotopology proves more advantageous for accounting for
subatomic quantification. Not only it provides means to better capture
what it means to be a whole but also as we will see it turns out to be
extremely useful in modeling those entities that can be counted. In the
following sections, I will introduce a minimal set of tools necessary
to model two different types of constructions concerning quantifi-
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cation over parts, namely multiplier phrases and partitives. I treat
this repertoire of means alongside the standard principle of function
application as the first attempt to develop a compositional analysis
of the syntactic structures in question that could shed new light on
the somewhat neglected phenomenon of subatomic quantification as
well as on countability in general. At the same time I believe the real
journey begins afterwards.
7.2 Common nouns
I will start my proposal with a brief discussion of the semantics for
different types of concrete nouns, i.e., expressions denoting sets of
individuals such as apples and teddy bears as well as scattered entities
such as juice and rice. Though everything I have to say about the
semantics of nouns extends naturally to NPs, for the sake of brevity I
will focus only on nouns. In general, I assume that all common nouns
of the sort discussed in this study as well as nominals derived from
them, e.g., by adjectival modification, are standard e, t predicates, i.e.,
functions from entities to truth values. Among such expressions there
some that are countable and some that are not. Though countability
is, of course, a grammatical category, I assume that the mass/count
distinction correlates with cognitive factors determining how human
beings conceptualize things in the world, as discussed in Chapter 5.
In particular, I take the counting principles of non-overlap, integrity,
and maximality as the benchmark for deciding what is countable and
what is not. In other words, only nouns whose referents satisfy all
three criteria are countable. In the next section, I will propose how the
meaning of such predicates can be captured.
7.2.1 Count nouns
Building on Grimm (2012b), I model the distinction between count
and mass nouns in terms of mereotopological distinctions developed
in the previous chapter. In particular, I propose that defining countable
predicates is about ensuring that their extensions include only
entities conceptualized as objects, i.e., discrete integrated wholes that
are disjoint from each other. Intuitively, the core distinction between
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count nouns on the one hand and mass nouns including substance
terms, granulars, and object mass nouns on the other can be reduced
to the fact that the first denote only individuated objects, whereas the
latter either do not have integrated wholes in their extension at all,
or they do but in addition they also refer to other types of entities.
For instance, the noun apple simply denotes a set of individuals, i.e.,
distinct apples. On the other hand, the referents of juice do not have
properties integrated wholes have, i.e., they are not constrained as
bounded non-overlapping discrete objects. The case of granular terms
such as rice is somewhat more complicated. Arguable, those predicates
do denote integrated wholes since one can point at a single grain of
rice and truthfully call it rice. Intuitively, grains of rice have similar
properties as apples, i.e., they are disjoint, integrated, and mereologically
maximal, however rice does not refer exclusively to individual
grains of rice. In addition, it can also denote arbitrary sums as well as
clusters of rice which lack those properties. Similar, object mass nouns
such as footwear have both integrated wholes, i.e., individual shoes, as
well as groups of such individuals, i.e., pluralities of shoes, in their
extensions. Thus, though there are important differences between particular
types of mass nouns, they all contrast with count nouns with
respect to what kinds of entities fall into a denoted set.
Since mereotopology provides powerful means to distinguish formally
between objects conceptualized as integrated wholes and other
types of entities, I will use the notion of mssc defined in (6.26) in order
to capture the main intuition behind the contrast discussed above.
Specifically, I will employ the variant of mssc relativized to a property
as provided in (6.27) and repeated here as (7.4). As discussed in
Section 6.2.1, this allows us to distinguish between integrated objects
such as individual apples and scattered entities like juice on the one
hand and arbitrary sums such as pluralities of apples on the other.
(7.4) Maximally strongly self-connected relative to a property
mssc(P)(x)
def
= P(x) ∧ ssc(x) ∧ ∀y[P(y) ∧ ssc(y) ∧ ◦(y, x) ↔
y x]
(An m-individual is maximally strongly self-connected relative
to a property if (i) every (interior) part of the individual
is connected to (overlaps) the whole (strongly self-connected)
and (ii) anything else which has the same property, is strongly
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self-connected, and overlaps it is once again part of it (maxi-
mality).)
Given the mereotopological notion of mssc, we can now characterize
the subset of predicates that refer exclusively to integrated wholes. In
the following text, I use the symbol Pmssc to refer to such predicates.
As stated in (7.5), there is nothing in the denotation of Pmssc that is not
an mssc entity.3 This means that, e.g., apple, is Pmssc, whereas juice, rice,
and furniture are not since either they do not refer to mssc individuals
at all or they also denote entities that are not mssc.
(7.5) Predicate of mssc individuals
Pmssc ↔ ∀x[P(x) → mssc(P)(x)]
(Any m-individual of which a Pmssc predicate is true is an mssc
m-individual relative to the relevant property.)
Let us now consider the semantics of the English count noun apple
provided in (7.6). What it says is that apple is an expression of type e, t
which yields the truth value True for entities that are mssc with respect
to the property apple. Given the definition in (7.4), mssc(apple)(x)
entails apple(x) which means that the discussed predicate denotes a
set of apples that are integrated wholes. Therefore, the proposed semantics
captures the intuition that referents of count nouns constitute
discrete objects without any reference to the notion of atomicity.
(7.6) Count noun
apple = λx[mssc(apple)(x)]
Since the main topic of this study concerns subatomic quantification,
I will refrain from discussing the exact semantics for different types
of mass nouns. Nonetheless, the meaning of all of the expressions
mentioned above can be captured in terms of mereotopological distinctions.
In particular, substance terms such as juice can be modeled
in terms of m-individuals having a certain property that are firmly
connected to distinct m-individuals with the same property, formally
involving the fc relation introduced in (6.28) (see Grimm 2012b, pp.
140–142). On the other hand, granular mass nouns such as rice can be
3. For convenience, I use the term ‘mssc entity/individual’ to refer to objects that
are maximally strongly-self connected relative to a relevant property.
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accounted for by treating them as aggregates, i.e., expressions that
can denote mssc individuals, their sums as well as clusters thereof,
i.e., configurations of mssc entities transitively connected via the tc
relation defined in (6.32) (see Grimm 2012b, pp. 142–148).4 Finally,
object mass nouns such as footwear can be thought of as denoting both
mssc entities and sums thereof though definitely more work needs to
be done with respect to this category.
The mereological approach to nominal semantics allows us to
distinguish countable nouns without postulating atomicity. As we will
see, this will prove crucial in accounting for subatomic quantification.
However, before I move to the discussion of the machinery behind
measuring and counting, the issue of number morphology on nouns
needs to be addressed. In the next section, I will take up the meaning
of the plural.
7.2.2 Pluralization
Following Grimm (2012b), I assume that number morphology is sensitive
to the mereotopological structure of referents of the basic, i.e.,
unmarked, form of a noun.5 Specifically, as we have seen in the previous
section in a language such as English only a subset of nouns
can be characterized as denoting single objects conceptualized as mssc
individuals. For the purpose of this study, I assume that in general
this property corresponds to countability and thus the compatibility
of such nouns with the plural morpheme.6 In particular, the plural
marker attaches only to predicates that have exclusively mssc entities
4. Clusters can be further relativized to different types of connection such as
external connection, see et in (6.29), and proximate connection, see pc in (6.33)
(Grimm 2012b).
5. Alternatively, one could argue that this kind of information is already encoded
in the semantics of the stem (pace, e.g., Pelletier and Schubert 1989, Borer 2005a).
However, for the sake of brevity I will not explore this hypothesis here.
6. This is, of course, a simplification. As mentioned before, for the sake of simplicity
I ignore here collective nouns as well as count abstract nouns. However, I assume that
in principle it is possible to extend the core mereotopological concepts to develop a
framework based on notions derived from or inspired by mssc that would account
for phrases such as two ideas and three committees (see also Grimm 2014).
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in their extensions.7 Having this in mind, the plural can be analyzed
as an operation which selects a set of mssc individuals and applies the
strict pluralization operation +, see the semantics in (7.8). The definition
of + provided in (7.9) employs the classical * operator introduced
by Link (1983) which is defined as a closure under sum formation, see
(7.7). Specifically, what + does is that it selects a predicate, applies *
to it and then removes all the mssc individuals from the pluralized
set. The selectional requirement of the plural marker is introduced
by what I refer to as the individuation presupposition represented
by Pmssc introduced after a dot following immediately the relevant
λ.8 The individuation presupposition requires an argument of the λ
operator to be a predicate denoting only mssc individuals as defined
in (7.5).
(7.7) Algebraic closure (cf. Link 1983)
*P
def
= {x|∃P ⊆ P[x = P ]}
(7.8) Plural
PL = λP . Pmssc[+P]
(7.9) Strict pluralization
+P
def
= *P − P
As indicated by (7.8), I adopt here the so-called exclusive analysis of
the plural. Within this view, plurals are assumed to refer only within
the domain of sums, i.e., singularities are excluded from the denotation,
and thus the plural is treated as designating more than one
(cf. Hoeksema 1983, Chierchia 1998a,b, Grimm 2013; see also Wągiel
2017).9 Hence, what the plural does is that it selects a set including
7. Without any additional assumptions, this approach fails to account for inherently
plural mass nouns such as leftovers (Acquaviva 2008) as well as pluralized mass
nouns as observed, e.g., in Greek (Tsoulas 2009). However, since the general topic of
the meaning of the plural lies beyond the scope of this study, I leave it for future
research.
8. For notational conventions used in this study see Section 1.3.2.
9. This decision is mainly motivated by independent reasons and as far as I can see
for the most part nothing hinges on this from the perspective of the phenomena I
attempt to account for here. However, the exclusive analysis will prove advantageous
in the context of set partitives. Notice also that many arguments have been proposed
in favor of the inclusive view on which singularities are included in the meaning of
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only mssc individuals and yields a new set consisting of all the sums
formed by joining the members of the input set. For instance, let us
consider the semantics of the bare plural apples, see (7.10). In this case,
the plural morpheme -s combines successfully with the noun since the
individuation presupposition is satisfied, i.e., apple is a predicate of
mssc entities. As a result, the set denoted by apple is first closed under
sum and then all the mssc individuals are removed. Hence, we obtain
a new predicate which is true only of entities that are pluralities of
apples.
(7.10) Plural NP
apples = PL ( apple ) = λx + λy[mssc(apple)(y) (x)
To see how this works, consider the following example. Let us assume
a model with three individual apples a, b, and c. Given such a model,
the common noun apple denotes the set of singular apples as in (7.11a).
On the other hand, its plural counterpart apples in (7.11b) denotes the
set comprising all the sums obtained from the singular individuals, i.e.,
pluralities of apples. In both cases, the semantic type of the expression
in question is e, t .
(7.11) Plural NP
a. apple = {a, b, c}
b. apples = {a b, a c, b c, a b c}
An important advantage of such treatment of the plural is that it explains
why the plural marker does not occur on mass nouns without
triggering any additional semantic effect such as a portion interpretation.
Or, alternatively, it is not the plural marking that triggers that
effect but rather the application of the Universal Packager is a prerequisite
for the plural morpheme to be able to attach to a mass noun.10
the plural mainly based on the data concerning downward-entailing environments
(cf. Link 1983, Schwarzschild 1991, Landman 2000, Sauerland et al. 2005, Spector
2007, Zweig 2009). However, see Grimm (2013) for an analysis of such contexts
maintaining the exclusive view.
10. I restrain here, from the discussion of the role of the Universal Sorter since it is
unclear whether kind-level entities can be considered as subject to topological relations.
Potentially, subkind readings constitute a problem for the described approach.
For sure, more research is required on this topic.
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7. Mereotopological account for subatomic quantification
Since packaging restricts the extension of a mass term in such a way
that the resulting portion has properties similar to integrated objects,
it is plausible to postulate that it does so by applying an mssc constraint
or at least some related restriction. In any case, non-coerced
mass nouns fail to satisfy the individuation presupposition simply
because either they do not refer to mssc individuals at all as in the
case of substance terms, or though their denotations include mssc
individuals, they also include other entities such as sums (object mass
nouns) and clusters (granulars).
Apart from the individuation presupposition, the approach to
the meaning of the plural provided here is essentially one of the two
standard analyses. However, there is an important comment to be
made concerning the morpho-syntax/semantics interface. Namely,
I assume that not all morphological plurals get the interpretation
in (7.8). Leaving aside pluralia tantum which would require much
unrelated consideration, I posit that only bare plurals are semantic
plurals, i.e., expressions on which the plural marker is interpreted. To
foreshadow, I will claim that plural morphology in quantificational
constructions such as numeral phrases in languages such as English
makes by contrast no semantic contribution and is merely triggered
syntactically by agreement (cf., e.g., Krifka 1989, 2007).
7.2.3 Consequences
Adopting the mereological account for nominal semantics has some
interesting consequences. First, as we have seen in Section 7.2.1 the
notion of mssc allows us to model individuals denoted by singular
count nouns without reference to atomicity. Therefore, there is no
need to postulate a special category of entities that have no proper
parts. To the contrary, unlike atoms mssc objects are viewed as certain
configurations of parts, specifically configurations forming integrated
wholes. This is a radically different approach from purely mereological
theories. In my opinion, its great advantage is that it succeeds in
capturing two ontological intuitions concerning individuals at the
same time. As we have seen in Chapter 5, there are good reasons to
believe that cognitive structures underlying conceptualization of solid
objects differ from those corresponding to scattered entities. On the
other hand, human beings are able to perceive wholes simultaneously
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as discrete units and as configurations of parts. Unlike atomicity based
theories, a mereotopological account enables us to model natural
language expressions in such a way that the first cognitive aspect can
be captured without sacrificing the latter. From my point of view, it is
a great achievement.
The second favorable consequence of mereotopology is that it provides
a very simple and intuitive way how to distinguish between
integrated sums of parts, i.e., singular individuals, and arbitrary sums,
i.e., plural entities. The distinction simply boils down to whether topological
notions are involved in the part-whole structure of an entity or
not. As mssc individuals, referents of singular count nouns consist of
connected elements, whereas pluralities encode no topological relations
holding between their parts. Intuitively, the sum of two apples a
and b remains the same entity a b irrespective of whether a and b
touch each other, stay in proximity, or are situated in distant locations.
On the other hand, a would cease to exist if its halves were separated
from each other by some contextually significant distance.11 In other
words, while plural entities can be modeled in purely mereological
terms, singular individuals require an additional topological layer. It
seems that such a distinction accounts for an intuition that contrary
to what standard mereology posits singularities and pluralities are
very different types of creatures. However, this approach does not
exclude yet another ontological possibility, namely the existence of
sums of solid entities that are topologically arranged in a particular
way. This fact is of considerable significance since there is linguistic
evidence indicating that introducing such type of entities might be
necessary for the meaning of certain expressions in natural language.
As we have seen in Section 2.3, referents of Italian irregular plurals
make a good candidate for such a class. 12
Another welcome result is that the fact that the theory adopted
here allows to distinguish between different types of entities in terms
11. Often, when parts remain in proximity it is still possible to retrieve the original
part-whole structure from the context. Though the constraints on such a ‘reconstruction’
operation are potentially an interesting topic in cognitive science, I refrain here
from any speculations on this issue and focus only on clear cases.
12. Another kind of expressions that are arguably of this sort are certain types of
Slavic derived collectives (see Dočekal and Wągiel 2018, Grimm and Dočekal to
appear).
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of the distinction between mereotopological and purely mereological
configurations allows us to reduce the number of domains. In particular,
there is no need to distinguish between the domain of entities
and the domain of portions of matter as postulated by Link (1983).
Since referents of count nouns are not considered mereological atoms,
i.e., entities without proper parts, but rather mereotopologically constrained
sums of parts they have additional properties compared to
arbitrary collections of portions of matter. Those properties are formulated
in terms of connection, specifically mssc. Therefore, it is sufficient
to assume only one domain of concrete things, namely the domain
of entities and different types of objects populating this domain can
be distinguished in terms of different types of either mereological or
mereotopological part-whole structures.
I believe that at this point it is useful to readdress the classical
paradox in (7.12) which led Link to distinguish between the domain
of entities and the domain of portions of matter. Assume that (7.12a)
points at a ring which was recently forged from some old Egyptian
gold. If the ring is nothing more than a sum of portions of gold, it
is surprising that (7.12b) is not contradictory. To the contrary, in the
described scenario it is true. In other words, the problem concerns the
fact that an object appears to have different properties than the sum
of its parts, but in mereological terms the two are identical (see also
Rothstein 2010, 2017 for discussion).
(7.12) English (Link 1983; adapted)
a. This is a gold ring.
b. The ring is new but the gold is old.
Mereotopology offers a new explanation for the paradox. What is
crucial here is that the ring is not just the sum of portions of gold
it is made of. Rather, to be perceived as a ring an entity needs to
remain in a particular spatial configuration, i.e., its parts need to be
related by a topological relation. Therefore, the reason why (7.12b)
is not contradictory is that it says that while the parts are old, the
configuration is new. For instance, this intuition could be captured by
attributing different topological properties to the definite descriptions
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the ring and the gold even at a point in time at which of (all of) the
gold.13
To sum up, the view developed here offers a radically different
perspective on nominal semantics compared to standard approaches
which utilize the notion of atomicity. Given the above discussion, I
conclude that it turns out to be advantageous both from the conceptual
and practical point of view because it better captures intuitions
concerning the nature of individuals as well as allows for simpler models
with less domains. Soon we will see that since it is free from the
deficits of atomicity based approaches discussed in Section 7.1, it also
proves significantly favorable in modeling subatomic quantification.
In the following section, I will make a first step towards developing an
analysis along those lines. In particular, I will introduce the theoretical
background for a unified analysis of numerical expressions.
7.3 Measure functions
Following Krifka (1989, 1995), I model quantification over both parts
and wholes in numeral and measure constructions in terms of extensive
measure functions, i.e., operations that directly relate entities to
numbers. The core mechanics behind a measure function is that when
applied to a plurality of individuals or quantity of substance it maps
it onto a real number corresponding to the number of individuals or
units making up the plurality or quantity in question. Such operations
need to satisfy what we expect from counting, i.e., they need to be
additive. The definition in (7.13) ensures that no entity will be counted
twice.
(7.13) Additive measure function
µ is an additive measure function iff it satisfies the following
requirement
∀x∀y ¬x ◦ y → [µ(x y) = µ(x) + µ(y)]
(The measure of a sum consisting of non-overlapping parts
equals the arithmetic sum of the measures of parts.)
13. Of course, it would be a topic for another study to work out what these properties
are.
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7. Mereotopological account for subatomic quantification
Assuming supplementation for the join operation as defined in (6.6)
in Section 6.1 and repeated here in (7.14) alongside the Archimedean
property of measure functions, see (7.15), it can be demonstrated that
an additive Archimedean measure function is monotonic, see (7.16)
(cf. Schwarzschild 2002).
(7.14) Supplementation
∀x∀y[x y → ∃z[z y ∧ ¬(z ◦ x)]]
(Whenever a thing has a proper part, it has more than one.)
(7.15) Archimedean property
∀x∀y[µ(x) > 0 ∧ y x → µ(y) > 0]
(If a measure of a thing is greater than zero, then a measure
of its part is also greater than zero.)
(7.16) Monotonicity
∀x∀y[x y → µ(x) < µ(y)]
(A measure of a whole is greater than a measure of its proper
part.)
Examples of extensive additive measure functions include liter, meter,
and calorie, whereas measure functions such as degree-celsius and
carat are non-additive.14 For instance, for an entity x the measure
function liter yields a number of liters corresponding to the volume of
that entity. In other words, it measures the quantity of space occupied
by its argument in units of liters. Similar, meter and calorie return
the number of meters and calories of a particular entity, respectively.
7.3.1 Counting operation
With this machinery in place, we can now start developing an account
for counting. Let us begin with an operation to which I will refer
as µ#. Its formal definition is provided in (7.17).15 In Section 5.2.2, I
have demonstrated that counting and measuring differ in that the first
is topology-sensitive, whereas the latter is not. Despite this fact, for
14. This distinction corresponds to the monotonicity/non-monotonicity distinction
in Schwarzschild (2002). In the following part of the chapter, I will focus only on
additive/monotonic measure functions.
15. The symbol # in the subscript indicates quantification in terms of number of
elements.
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convenience I will talk about measure functions when referring to both
operations yielding a cardinality of objects as well as operations giving
a measure in terms of, e.g., volume. Given the definitions introduced
above, the counting operation µ# defined in (7.17) is a kind of measure
function. Nevertheless, it is crucial to emphasize that it is a special
kind of measure function. Similar to other additive measure functions,
it cares about non-overlap, but in addition it is also sensitive to the
mereotopological structure of entities it applies to. In particular, as
specified in (7.17) it counts exclusively mssc individuals.16 This fact
makes it distinctively different from measure functions such as liter
which ignore whether their arguments refer integrated entities that
come in one piece.
(7.17) Measure function µ#
µ# is an additive measure function standardized by the following
requirement
∀x[µ#(x) = 1 iff mssc(x)]
To see the measure function µ# at work, let us consider the example
provided in (7.18). Assume that there are two distinct mssc individuals
a and b, i.e., a = b. In such a case, after we apply µ# to the sum of a
and b, the returned value will be 2 which of course is a correct result.
(7.18) Counting via µ#
mssc(a) ∧ mssc(b) ∧ ¬(a ◦ b) → µ#(a b) = 2
(If both a and b are mssc and they do not overlap, i.e., they
are not identical, then the measure function µ# applied to the
sum of a and b yields the number 2.)
The measure function µ# seems to do what we would expect an operation
intended to capture the core intuitions concerning counting to
do. However, there is yet another improvement to be implemented.
As we have discussed in Section 6.2.1, it is desirable to relativize mssc
to a particular property in order to avoid ending up with the single
16. In the original theory of Krifka (1989) the equivalent of µ# measures in terms of
natural units, hence the nu operation. On the other hand, Krifka (1995) introduces
the ou and ku operations (for ‘object unit’ and ‘kind unit’, respectively), whereas in
newer versions of the theory (e.g., Krifka 2007) the ac measure function (for ‘atomic
count’) is defined to count atoms.
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7. Mereotopological account for subatomic quantification
integrated whole corresponding to the entire material universe. Therefore,
a proper device intended to account for counting should also
be relativized to a property so that it does not count absolutely mssc
entities but rather objects that are mssc with respect to a particular
characteristic, e.g., apples. This is can be ensured by introducing a
new operation to which I will refer as #. As can be seen in the definition
in (7.19), it is similar to µ# in that the notion of mssc plays here a
crucial role. However, # differs in that it takes a property P and yields
a measure function that returns a number of mssc individuals relative
to P. For instance, assume there are two distinct apples a and b. Then,
# can yield a measure function that will count entities that are mssc
with respect to the property of being an apple. As a result, we obtain
the number 2.
(7.19) Measure function #(P)
#(P) is an additive measure function standardized by the
following requirement
∀P∀x[#(P)(x) = 1 iff mssc(P)(x)]
Before we move on to introducing another measure function that
will prove useful for modeling multiplier phrases, let us discuss the
mechanism of contextual conditioning which will allow us to account
for proportional partitive words.
7.3.2 Contextual conditioning
The evidence presented in Chapter 2 shows that cross-linguistically
partitive words can appear both in entity and mass partitives on the
one hand and set partitives on the other. In general, in the first two
cases the partitive word quantifies over portions of matter constituting
either a particular individual or a quantity of substance, whereas in
the third situation it quantifies over integrated wholes making up
a plurality. To account for this double compatibility a procedure is
required which will determine what is to be measured or counted
when. For this purpose, I build on Bale and Barner (2009)’s proposal
who assume a generalized context-dependent measure function µ.
Such an approach posits a mechanism of contextual conditioning
along the lines defined in (7.20). The core of the idea is that particular
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measure functions are ordered and depending on a context those that
are ranked higher in the series are favored over those that are ranked
lower.
(7.20) Contextual conditioning
µ is interpreted as one of the measure functions mz in the
series m1, m2, m3 . . . mn such that the argument for µ is in
the range of mz; furthermore, contextually mz is preferred to
my if z < y
Bale and Barner use the mechanism in (7.20) to account for various
meanings of more observed in comparative constructions such as X
has more NP than Y. Specifically, depending on the NP comparison
may be specified in terms of cardinality, volume, intensity etc. For the
purpose of this study, I will make use of the very same mechanism
for a procedure determining that partitive words can quantify over
portions of matter in entity and mass partitives and over individuals
in set partitives. In particular, I propose a partial ordering of measure
functions as provided in (7.21). The first measure function in the series,
i.e., the one ranked highest, is the measure function #(P) which is
devised to count integrated wholes. I remain agnostic with respect to
the exact position in the ranking other measure functions occupy, but
for our purposes establishing the ordering between #(P) and measure
functions measuring in terms of volume is certainly sufficient.
(7.21) Partial ordering of measure functions
m1 = #(P) < mn ∈ volume
where volume is a set of measure functions measuring entities
in terms of volume including m3, liter, pint etc.
To foreshadow, the contextually conditioned generalized measure
function µ can cover the meanings of partitive words in entity and mass
partitives on the one hand and set partitives on the other. In particular,
µ is interpreted as a measure function quantifying over units of volume
when the partitive word combines with a DP involving a singular count
noun which refers to an individual as well as with a DP with a mass
term referring to a quantity of substance. On the other hand, when the
partitive word c-commands a DP comprising a regular plural noun, µ
is interpreted an operation counting individuated integrated wholes.
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7. Mereotopological account for subatomic quantification
This is because whenever the context allows for it, #(P) is preferred
over the functions from the set volume, e.g., cm3. An additional positive
consequence of the adopted mechanism of contextual conditioning
is that it allows for quantification over individuals in partitives with
object mass nouns in the embedded DP (cf. Barner and Snedeker 2005,
Bale and Barner 2009).
7.3.3 Counting essential parts
The final component of the analysis related to measure functions
has to do with modeling multipliers. As we have seen in Chapter
4, multipliers quantify over cognitively salient parts of individuals.
In order to capture this intuition, I propose an additional operation
.17 This operation takes a property and yields an additive measure
function (P) which counts parts of a whole. The formal definition of
(P) is provided in (7.22). Notice that (P) quantifies only over parts
of mssc individuals as ensured by the first conjunct of the requirement.
Furthermore, the second conjunct imposes that a part (P) maps
onto a number needs to be essential for the whole. The parthood
relation captures the intuition that only entities within the part-whole
structure of an object are subject to quantification and the the use of
improper parthood, i.e., , rather than is to account for cases of
homogeneous entities without any distinguishable salient parts such
as standardized portions of coffee or martini. In particular, one might
want to talk about a single portion as opposed to a double portion
rather than to unrelated two portions since there is a clear semantic
contrast between single coffee ∼ double coffee ∼ two coffees.
(7.22) Measure function (P)
(P) is an additive measure function standardized by the
following requirement
∀P∀x[ (P)(x) = 1 iff mssc(P)(x)
∧ ∃y[y x ∧ essential(P)(y) ∧ #(y) = 1]]
17. Here is a mnemonic. The symbol is a square constituted by four smaller
squares each of which is a part of the whole with comparable properties. Hence,
represents a type of object multiplier phrases usually refer to.
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7. Mereotopological account for subatomic quantification
As witnessed by comparing (7.19) and (7.22), the main difference
between #(P) and (P) lies in what type of objects they map onto
numbers. The first simply associates any mssc object with a number,
whereas the latter yields 1 only for what I refer to as an essential part
of an entity, i.e., a cognitively salient element. The notion essential in
(7.22) is relativized to a property and can be defined as in (7.23). I keep
the definition somewhat vague on purpose since what is considered
essential for an entity to be perceived as having a certain property
is often not entirely clear and may differ with respect to a particular
context. For instance, as we have discussed in Chapter 4 multiplier
phrases such as double garage can denote objects of a different internal
structure as long as they can hold two vehicles. Nevertheless, one
important feature of an essential part is that it is an integrated and
recognizable entity within a whole that can be easily individuated
against other parts.
(7.23) Essential parts
essential(P) is true of an mssc part of an entity in the extension
of P that is essential for that entity to be considered as
having a property P
An example of an essential part would be a patty in a hamburger. Intuitively,
it makes the most important element of the whole sandwich.
Similar, out of the blue it seems that the crucial part of a bed is an
area covered with a mattress on which a person can sleep. Finally, in
a natural context the most salient piece of, say, a shotgun is its barrel
since it guarantees functionality as well as constitutes a significant
portion of the weapon. However, notice that the broad definition in
(7.23) also covers self-sufficient parts, i.e., parts that have a property
comparable, i.e., very similar or identical, to the property of a whole. I
argue that a self-sufficient part can be considered an extreme case of an
essential part. If for some reason it is considered that what is essential
cannot be reduced to one particular element, then essential(P) yields
an mssc part that resembles the object it is a part of. For instance, let
us once again consider the case of double crown discussed in Chapter
4. Although there can be multiple cognitively salient parts within
the make-up of a crown, e.g., jewels, orbs, crosses, or fleurs-de-lis, intuitively
no such element is essential for a thing to be considered a
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crown. On the other hand, an object such as the Pschent or papal tiara
counts as a double or triple crown, respectively, because it comprises
parts with comparable properties as the whole. Similar cases arguably
involve objects such as doors and layers, thus a double door involves
two connected door leaves and a double layer is an entity consisting of
two merged layers. For the reasons discussed above, I leave the issue
concerning when an essential part is a self-sufficient part vague and
dependent on how a particular artifact is conceptualized.
All things considered, given (7.22) and (7.23) the (P) measure
function returns a number of essential parts in each object being a
member of a given set. The properties of such a device might help us
to account for the interpretation of multiplier phrases. With the set-up
and all the relevant ingredients necessary to account for the numerical
expressions I am interested here in place, let us now propose the
semantics for cardinals and multipliers that will allow us to account for
the observed phenomena in count explicit and proportional partitives
as well as in multiplier phrases.
7.4 Numerical expressions
In the history of formal semantics spanning over almost half a century
there were not that many expressions that received as much attention
as numerals. Though a lot of revealing research has been done in this
field (see, e.g., Horn 1972, Barwise and Cooper 1981, Scha 1981, Krifka
1999, Landman 2003, 2004, Hofweber 2005, Ionin and Matushansky
2006, Geurts 2006, Nouwen 2010, Kennedy 2013, Rothstein 2013, 2017
to name just a few influential studies), in my opinion there is a shared
misconception concerning numerals since they are commonly treated
as simplex, i.e., non-decomposable, expressions.18 On the other hand,
it has been observed that in a language such as English cardinals
have multiple uses and therefore it is misleading to search for “the”
meaning of numerals but rather it is more appropriate to talk about
multiple meanings associated with such expressions (e.g., Bultinck
2005, Geurts 2006). The interplay of these two aspects results in treating
18. A notable exception is the theory of Kennedy (2013) who proposes that numerals
are generalized quantifiers over degrees and discusses their possible semantic
decomposition.
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7. Mereotopological account for subatomic quantification
cardinal numerals either as ambiguous or postulating some shifting
mechanism to account for at least some of the uses. However, I believe
there are good reasons to claim that the assumption that numerals are
simplex is incorrect.
In my opinion, the prevailing view most probably stems from the
limited scope of the mainstream research on numerals. In particular,
most work on numerical quantification has been done on the basis
of English data and since English lacks a rich morphology, certain
semantic distinctions are not marked by means of different formal
exponents. However, even in English there are complex numericals
such as twice, twofold, twosome, and adjectival two-time but this set
of expressions for the most part has been surprisingly neglected in
the study of numerical quantification. On the other hand, evidence
from languages that have a broader repertoire of derivational means
suggests that numerals are in fact compositional. For instance, recent
research on cardinals as well as derivationally more complex numerical
expressions in Slavic and Semitic indicates that different numerical
affixes correspond to distinct semantic operations responsible for deriving
various meanings (see, e.g., Dočekal 2012, 2013, Dočekal and
Wągiel to appear for Czech, Wągiel 2014 for Polish, Khrizman 2015
for Russian and Fassi Fehri 2016, 2017 for Arabic).
In the following sections, I posit a compositional account for numerical
expressions in natural language. I assume that a prerequisite
for such an approach is that it accounts for the three counting principles
discussed in Section 5.2.2. In particular, it needs to be guaranteed
that the numerical expression ignores overlapping, non-integrated,
and non-maximal entities. In other words, it needs to be devised in
such a way that it counts right.
7.4.1 Numeral roots
Many numerical expressions in numerous languages are morphologically
complex and can be sequenced into separate morphemes
including numeral roots and various additional affixes. In this study,
I follow my previous work in arguing that it is plausible to treat numeral
roots as names of natural numbers, i.e., expressions referring
to abstract objects of type n, see (7.24a) (Wągiel 2015b, to appear a,
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7. Mereotopological account for subatomic quantification
to appear b). For instance, the English root
√
tw as in, e.g., two and
twenty, simply names the integer 2, as specified in (7.24b).
(7.24) Numeral root
a.
√
Numeral = n
b.
√
tw = 2
Furthermore, I assume that virtually all meanings numerical expressions
can have can be derived from this basic semantics by application
of additional operations encoded by different types of classifiers. Those
classifiers can be either introduced overtly by a particular morpheme
or silent. For the sake of coherence and in order to stay as close to
the main argument as possible, in the following sections I will focus
on derivations of only two kinds of numerical expressions, namely
cardinals and multipliers.19
7.4.2 Cardinals
I propose that cardinals are born as singular terms, specifically as
names of numbers at type n. This meaning is preserved in contexts
clearly calling for numerical arguments such as mathematical statements
in (7.25) (see Rothstein 2017 for an extensive discussion of such
environments). However, the singular term semantics can also serve
as a basis to derive expressions that are more abstract than number-
denoting.
(7.25) English (Rothstein 2013)
a. Two plus two is four.
b. Two is the only even prime number.
Since this study concerns subatomic quantification, in the following
part of this section I will restrict my proposal to the use of numerals as
nominal modifiers. In particular, I posit that in attributive position the
meaning associated with the numeral root is shifted to an expression
of type e, t , e, t , i.e., the type of a predicate modifier (cf. Ionin
and Matushansky 2006). I postulate that this shift is performed by
19. For some preliminary proposals of how to treat compositionally other complex
numerical expressions in Slavic see Wągiel (2015b, to appear a), to appear b.
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a classifier element which I will refer to as CL# (see also Sudo 2016
for a similar proposal based on Japanese data). As suggested by the
abbreviation and provided in (7.26), CL# is an expression of type
n, e, t , e, t , i.e., a function from numbers to predicate modifiers,
that introduces the # operation. However, it is not the only thing it
does. In order to ensure that counting goes right, the classifier imposes
a particular constraint on predicates it can select, specifically via the
individuation presupposition it selects only predicates that denote
exclusively mssc individuals. Consequently, the resulting cardinal will
not be compatible with mass nouns as well as pluralia tantum since
they do not fulfill this requirement. Notice also that a predicate P
in the first conjunct is pluralized by the classical * operator. Such a
semantics guarantees that an integer n will be associated with the
number of mssc individuals making up a plurality in the extension
of P, see (7.17)–(7.19). The use of * rather than the strict pluralization
operator + postulated for semantic plurals, see (7.9), is motivated by
the existence of the numeral one which requires singularities.
(7.26) Classifier#
CL# = λnλP. Pmssc λx[*P(x) ∧ #(P)(x) = n]
Given the semantics of CL# proposed above, it is straightforward how
particular cardinals are derived. When the classifier combines with
a numeral root, e.g.,
√
tw, the number variable in (7.26) is saturated
by a particular integer and this gives rise to a full-fledged attributive
numeral as exemplified in (7.27). Therefore, two in a phrase such as
two apples takes a property of being an apple and yields a set of pairs
of objects that are mssc with respect to that property, i.e., a set of pairs
of apples. This is a welcome result, however an important comment is
required. For this semantics to work, it needs to be stipulated that the
morphological plural on the noun the cardinal modifies is not interpreted
semantically since otherwise the pluralized property would
fail to satisfy the individuation presupposition. In other words, the
source of plurality is the * operator introduced by CL#, whereas the
number marker on the noun is a mere agreement plural triggered
syntactically with no semantic contribution (cf. Krifka 1989, Ionin and
Matushansky 2006). At first sight this might seem implausible, but
there are well-documented cases of mismatches between the plural
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7. Mereotopological account for subatomic quantification
as a morpho-syntactic category and the semantic notion of plurality
(see Nouwen 2016 for an overview) as well as examples of languages
such as Finnish, Hungarian, and Turkish that despite having plural
morphology do not employ it in numeral phrases.20
(7.27) Cardinal numeral
two = CL# (
√
tw ) = λP. Pmssc λx[*P(x) ∧ #(P)(x) = 2]
The treatment postulated above builds on a well-established way of
thinking about cardinals since the idea that numerals are in fact names
of numbers dates back to Frege (1884). An early formal semantic account
for cardinals proposed by Scha (1981) assumes a complex syntactic
structure for numerical determiners decomposing them into
the bottom-most Number projection which is interpreted in terms of
reference to an integer and the Numeral and Det projections above
which transform a singular term into a predicate based upon that
integer and generalized quantifier, respectively. Finally, in a recent
theory of counting and measuring by Rothstein (2012, 2013, 2017) it
has been acknowledged that when used as singular terms in examples
such as (7.25) cardinals seem to refer to abstract objects rather than
anything else and this meaning should be accounted for a special
shifting operation.
Apart from an explanation for why cardinals are incompatible
with mass terms, an important advantage of the semantics developed
here is it also allows for a unified analysis of numerical expressions
in classifier and non-classifier languages.21 In the next section, I will
extend the proposed classifier semantics for numerals to multipliers
7.4.3 Multipliers
As we have seen in Chapter 4, multipliers are similar to cardinals
in that they both do what a counting expression does, i.e. establish
20. I simplify here for the sake of brevity. For a general discussion on different
strategies languages use to combine cardinals with nouns and a proposal of an
alternative approach see Bale et al. (2011). One thing that should be mentioned as
a potential problem for the proposed approach concerns collective modification
below the numeral, e.g., two parallel streets and two similar objects. However, since
this study does not focus on such expressions, I leave this issue for future research.
21. See, e.g., Kobuchi-Philip (2006) for some arguments why this is desirable.
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7. Mereotopological account for subatomic quantification
one-to-one correspondence between entities and natural numbers. On
the other hand, the two expressions in question differ significantly in
that the first do not count whole individuals but rather quantify over
certain parts thereof. Furthermore, they appear to be merged with
the modified noun before other quantifiers, and thus their scope is
significantly limited since they allow for modification by numerals
and by the universal quantifier. In other words, they do not infer
a plurality of objects, instead they imply that an object involves a
plurality of parts having particular features. These facts suggest that
the analysis of multipliers should resemble that of cardinals and at
the same time differ in such a way that would capture the relevant
distinctions.
The morphological evidence from Slavic presented in Chapter
4 strongly suggests that similar to cardinals multipliers should be
treated as complex compositional expressions. Just like in the previous
section, I propose that multipliers are decomposable into two
elements. In a language such as Polish, the first is the name of a number
corresponding to a numeral root present in the morphological
make-up of the multiplier, whereas the second is a special classifier
introduced by the multiplicative affix. I will refer to that classifier
as CL . As indicated by the superscript, this element introduces the
operation defined in (7.22) Section 7.3.3 above, see (7.28). Similar
to CL#, CL is a function from numbers to predicate modifiers, i.e.,
an expression of type n, e, t , e, t . It also involves the individuation
presupposition which accounts for the fact that multipliers
can only modify expressions denoting mssc individuals such as count
nouns and mass terms coerced by the Universal Packager. However,
unlike CL# the element CL does not pluralize predicates it applies
to. This is an important difference between the two classifier since it
explains why multiplier phrases do not denote pluralities and allow
for numeral modification.
(7.28) Classifier
CL = λnλP. Pmssc λx[P(x) ∧ (P)(x) = n]
Again, it is straightforward how to get a multiplier by combining
the meaning of a numeral root, see (7.24), with the semantics of CL
in (7.28) above. For instance, consider the denotation of the Polish
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multiplier podwójny ‘double’ in (7.29). The number referred to by the
numeral root
√
dw simply saturates the first argument slot in (7.28)
and as a result an expression of type e, t , e, t is obtained, see
(7.29). The derived predicate modifier selects for a set that has only
mssc individuals in their extensions and returns a subset of such mssc
individuals that have two essential parts. Such a predicate can further
serve as an argument for another quantificational expression, e.g., a
cardinal numeral.
(7.29) Polish multiplier
podwójny = CL (
√
dw ) =
= λP. Pmssc λx[P(x) ∧ (P)(x) = 2]
The proposed semantics appears to successfully account for the subset
of data I focus on here, namely multiplier phrases involving concrete
nouns. Furthermore, as indicated in Section 4.2 it also offers a way
of thinking about examples I do not deal here with. In particular, after
extending the ontology with primitive semantic types for events
and roles a very similar mechanism can be proposed to account for
the meaning of NPs like double murder and double president. Such considerations,
however, reach far beyond the main topic of this study
and I will refrain from discussing an exact implementation of the
idea. Instead, in the following section I will present a brief overview
on how the proposed semantics fits into a bigger typological picture
and how cross-linguistic data supports the treatment of cardinals and
multipliers as complex expressions.
7.4.4 Cross-linguistic support
The structure of cardinal numerals proposed above is further supported
by several empirical findings made in linguistic typology. First,
it has been observed that cross-linguistically numerals and classifiers
are always adjacent (Greenberg 1972). Table 7.1 provides the four attested
orderings of the numeral, classifier, and noun out of six possible
patterns. Notice that in the missing two the noun would separate the
numeral and classifier.
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7. Mereotopological account for subatomic quantification
language ordering
Vietnamese [num-cl]-n
Thai n-[num-cl]
Ibidio [cl-num]-n
Bodo n-[cl-num]
Table 7.1: Attested relative orderings of numerals and classifiers
(Greenberg 1972)
Second, in classifier languages classifiers are often suffixes on numerals
as witnessed, e.g., in (7.30) and (7.31) (see Aikhenvald 2000 for
more data).
(7.30) Japanese (Sudo 2016)
ichi-rin-no
one-cl-gen
hana
flower
‘one flower’
(7.31) Yucuna (Aikhenvald 2000, p. 106)
pajluhua-na
one-cl
yahui
dog
‘one dog’
Furthermore, intriguing data from partly classifier languages such as
Mi’gmaq and Chol show that it can be the case that in a single language
some cardinals require classifiers in order to combine with nouns
whereas others do not, see (7.32) (Bale and Coon 2014). This further
suggests that the classifier makes a constituent with the numeral and
compensates its semantic deficits.
(7.32) Mi’gmaq (Bale and Coon 2014)
a. na’n
five
(*te’s)-ijig
cl-agr
ji’nm-ug
man-pl
‘five men’
b. asugom
six
*(te’s)-ijig
cl-agr
ji’nm-ug
man-pl
(Mi’gmaq)
‘six men’
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7. Mereotopological account for subatomic quantification
Finally, it has been observed that in genetically and typologically
diverse languages there exists an alternation between attributive and
specialized so-called counting cardinals, i.e., referential expressions
that cannot be used as modifiers, see 7.2 (Hurford 1998, 2001). This
fact demonstrates that in some cases an exponent of the name of a
number can be formally distinct from a related attributive expression.
All things considered, I take the data discussed here to indicate that
the analysis postulated for cardinals is on the right track.
language number attributive counting
German 2 zwei zwo
Maltese 2 żewg tnejn
Chinese 2 liˇang èr
Hungarian 2 két kettö
Basque 2 bi biga
Table 7.2: Attributive and counting numerals across languages (Hurford
2001)
Moreover, the semantics postulated in (7.29) can be further supported
by the fact that in many languages the numeral root can be
easily distinguished in the morphological make-up of multipliers. As
we have seen in Chapter 4, Slavic multipliers are morphologically complex
expressions derived from numeral roots by means of affixation
which I argue encodes the classifier element CL . Though many European
languages borrowed their multipliers from Latin, the existence
of multiplicative affixes is by no means a Slavic idiosyncrasy since
similar patterns can be observed in Baltic and Finnic, as witnessed by
the correspondences given in Table 7.3.
language number cardinal multiplier
Russian 2 dva dvojnoj
Lithuanian 2 du dvigubas
Finnish 2 kaksi kaksinkertainen
Table 7.3: Multipliers across languages
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7. Mereotopological account for subatomic quantification
I conclude that the proposed account is plausible both in terms of
what multipliers mean, i.e., what kind of semantic effects they give rise
to, and how they often look like, i.e., their morphological complexity
in some languages suggests their semantic decomposability. The system
developed here is advantageous in that the same compositional
mechanism allows for accounting for different numerical expressions.
Though in this study I focus only on cardinals and multipliers, after
extending it with additional classifiers a number of other numerals
can be derived by its means (see Dočekal 2012, 2013, Wągiel 2014,
2015b, to appear b, Dočekal and Wągiel to appear for discussion of
other types of complex numerals in Slavic). This unified treatment
also explains why only one ordering of cardinals and multipliers is
possible. However, before we discuss how a multiplier phrase modified
by a cardinal is composed step by step from the pieces defined
here, let us introduce additional tools to account for explicit and set
partitives.
7.5 Partitives
The main data set presented in Chapter 2 concerned different types
of partitives in various languages. In this study, I do not attempt to
provide a detailed analysis of the syntax-semantics interface regarding
partitive constructions. Rather, based on the cross-linguistic evidence I
focus on general issues concerning the interaction between partitivity,
topological sensitivity and countability. In particular, I will postulate
a minimal set of ingredients necessary for an attempt to explain what
happens in count explicit partitives and different types of proportional
partitives involving topology-neutral as well as topology-sensitive partitive
words. However, before I propose particular denotations of selected
expressions of that type, there are several semantic components
to comment on.
7.5.1 Partitivity
An important component of partitive constructions considers what
kind of DP can be c-commanded be the partitive word. As already
mentioned in Section 2.1, partitives are subject to the so-called Partitive
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Constraint which disallows certain expressions from the embedded
DP (e.g., Jackendoff 1977, Selkirk 1977, Barwise and Cooper 1981,
Ladusaw 1982). I will follow here the semantic reanalysis of the Partitive
Constraint, as proposed by de Hoop (1997), see (7.33), which
states that the whole downstairs DP is an expression of type e.
(7.33) Partitive Constraint (de Hoop 1997; adapted)
In a partitive, the embedded DP must be entity-denoting, i.e.,
definite or specific.
The formulation in (7.33) means that the semantics of the node ccommanding
the downstairs DP needs to be devised in such a way
that it does not take sets as its input but rather individual things.
Consequently, since Barker (1998) it is commonly assumed that the
preposition of in English partitives is a function from entities to sets of
entities, i.e., type e, e, t , expressing partialness by means of proper
parthood.22 Specifically, the semantics of partitive of establishes the
relation between two entities stating that one is a proper part of
another.
However, as we have seen in Chapter 2 and 3 there are languages
that do not employ prepositions in partitive constructions. For instance,
Russian, Hungarian, and Basque make use of case marking
instead, whereas Japanese and Chinese can wield a number of strategies
to express partitivity including classifier structures. Moreover, in
German and Portuguese NPs can be modified by adjectival half -words
in order to yield proportional meaning. These facts seem to suggest
that from the cross-linguistic perspective the mapping between semantics
and morphology is not that obvious. After all, given that
is introduced by some other element, it is not clear what the exact
semantic contribution of part-words is since they seem to do precisely
the same thing. Therefore, for convenience I incorporate the general
partitive semantics into the meaning of partitive words themselves,
see (7.34). Since I will focus here on the composition of German and
Polish partitives, the reason behind this move is mainly to avoid an
unnecessary detour into the semantics of case.23 However, if required
22. But see Ionin et al. (2006) for a treatment based on improper parthood.
23. See Kagan (2013) for a detailed discussion of some aspects of the meaning of
genitive in Russian.
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the distribution of particular components to be discussed below could
be rearranged so that, say, the genitive case in a language such as
German and Polish is associated with the semantics in (7.34), whereas
particular partitive words contribute additional meaning or are treated
as void.
(7.34) Partitivity
PART = λyλx[x y]
Notice that since I assume only one domain, i.e., the domain of entities,
with one parthood relation defined over it, the partitivity
semantics in (7.34) simply employs the basic mereological notion of
proper parthood which unlike is not reflexive. As a result, there
is no need to distinguish between entity and set partitives in terms
of different parthood relations. Consequently, it is straightforward to
expect that in general partitive words are able to combine both with
expressions referring to singularities and those denoting pluralities.
As witnessed in Section 2.2, this kind of behavior is exactly what is
observed cross-linguistically.
As we have seen, the semantic re-formulation of the Partitive Constraint
in (7.33) urges us to ensure that the downstairs DP position
in partitives is occupied by an expression of type e. Putting aside
the straightforward case of proper names, in the next sections I will
discuss two mechanisms how to guarantee adequate extensions of
definites and specific indefinites, namely maximization and choice
functions, respectively.
7.5.2 Maximization
In order to account for definite DPs, I assume the standard maximization
operation max, see (7.35) (cf. Sharvy 1980, Link 1983), which yields
the unique maximal entity in the denotation of a predicate.24 In other
words, max maps any set which may include singularities, pluralities,
or both onto the element that all other elements in that set are part of.
In cases it selects a pluralized predicate, it returns the supremum, i.e.,
the largest plurality in the set. On the other hand, if max is applied
to a singular predicate, it returns a singularity, provided that that
24. In the literature, the symbol σ is often used instead of max.
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7. Mereotopological account for subatomic quantification
predicate denotes a singleton, i.e., there is only one relevant entity in
a given context. Maximality thus allows us for a unified account for
both singular and plural definite DPs.
(7.35) Maximization operator
max(P) = ∀y[P(y) → y x]
defined if there is unique x in P, otherwise undefined
(If defined, the element in P which all other things in P are
part of.)
In article languages such as English and German, the max operator is
introduced by the definite article, e.g., the and der/die/das/die, respectively.
On the other hand, for article-less languages such as Polish I
assume a DP projection over NP headed by a silent determiner that
can be interpreted as a definite article (e.g., Veselovská 1995, Progovac
1998, Rutkowski 2002). The semantics for such an overt or covert
definite DP head is given in (7.36).
(7.36) Definite article
DEF = λP[max(P)]
Let us now contemplate how the proposed semantics delivers the
meaning of definite singulars and plurals. Assuming a model for
(7.37) with a unique apple a, the noun apple denotes a singleton set
involving a, see (7.37a). After the NP is merged with the determiner
the resulting DP the apple in (7.37b) simply refers to a. On the other
hand, in a model with three apples a, b, and c the pluralized noun
apples denotes the set of a, b, c, and all the derived pluralities, see
(7.38a), whereas the definite the apples has only the maximal sum in its
extension, see (7.38b). This shows that the mechanics of max captures
the intuition that singular definites such as the apple are true of a
unique apple, whereas a definite plural DP like the apples corresponds
to a plurality consisting of all the apples.
(7.37) Definite singular DP
a. apple = {a}
b. DEF apple = a
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(7.38) Definite plural DP
a. PL apple = {a b, a c, b c, a b c}
b. DEF [PL apple] = a b c
The max operation allows us to model the meaning of both singular
and plural definites in partitives in accordance with the Partitive Constraint.
In order to account for specific indefinites, I assume choice
functions which will be briefly described in the next section.
7.5.3 Choice functions
A choice function is an operator selecting a member from a set (Reinhart
1997, Kratzer 1998). On the adopted view, the choice function
variable remains free at LF and its value is provided by the context, i.e.,
the choice of a particular member of a given set may vary depending
on extra-linguistic circumstances. In general, choice functions can be
applied to any kind of set. As a result, they have been widely used in
the cross-linguistic research on different types of specific indefinites
(see, e.g., Matthewson 1998, Kratzer and Shimoyama 2002, AlonsoOvalle
and Menéndez-Benito 2003, Yanovich 2005, Wągiel to appear
b). In this study, however, I will limit my focus to the domain of entities.
In particular, I embrace an approach that a choice function ch
over entities is an expression of type e, t , e which if applied to a
non-empty set of entities P, it yields a specific entity from P relative
to a particular context, see (7.39).
(7.39) Choice function
ch e,t ,e is a choice function if P e,t (ch(P e,t )) = 1
I assume that choice functions are introduced by a specificity element
which can either be expressed formally or silent. For instance, in Russian
it is encoded by a special suffix on indefinite pronouns (see, e.g.,
Yanovich 2005) but lack any exponent in the case when an indefinite
DP including a common noun gets a specific interpretation. In either
way, the specificity element takes a predicate as its input and applies
ch on that predicate as specified in (7.40).
(7.40) Specificity element
SPEC = λP[ch(P)]
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Provided a model with three apples a, b, and c, the denotations of
singulars and plurals are exactly as such cases discussed above, see
(7.41a) and (7.42a), whereas the extensions of specific indefinites can
be as follows. For instance, in a given context apple can be interpreted
as referring to a certain apple, say, b in (7.41b). Similar, the plural apples
can be understood to designate a specific though indefinite plurality
of apples, e.g., a b in (7.41b).
(7.41) Specific indefinite singular DP
a. apple = {a, b, c}
b. SPEC apple = b (in a certain context)
(7.42) Specific indefinite plural DP
a. PL apple = {a b, a c, b c, a b c}
b. SPEC [PL apple] = a b (in a certain context)
The discussed mechanism will allow us to account for specific DPs in
partitives. However, for the sake of brevity in the following sections I
will focus only on constructions involving definite DPs downstairs. For
the treatment of specific indefinites in the embedded DP, it is sufficient
to substitute the choice function for the maximization operator.
7.5.4 Set partitive constraint
Having provided means to ensure that the complement of the partitive
component is of type e, let us return to the meaning of partitive
words in different constructions. One of the observations discussed in
Chapter 2 concerned the cross-linguistic analogy between entity and
set partitives. In particular, in languages such as German and Polish
the same partitive word can combine with both singular and plural
DPs. In the first case, it gives rise to a part-of-a-singularity reading,
whereas in the latter configuration a part-of-a-plurality interpretation
is inferred. This suggest that either the meanings of partitive words
are general enough to cover both readings or such expressions are ambiguous
cross-linguistically. However, the latter option seems unlikely.
Furthermore, in Section 2.5 I have shown that at least in German it is
implausible to postulate ambiguity since partitive words can give rise
to both discussed readings simultaneously in partitives with coordinated
structures involving singular and plural DPs. Therefore, the data
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suggest that the proper treatment should be formulated in terms of
one general meaning. Given that no sorted domains are postulated and
singular individuals are distinguished from plural entities in terms
of the distinction between mereotopological and purely mereological
configurations, the partitivity semantics introduced in (7.34) is exactly
what we need. However, utilizing one unified parthood relation for
singularities and pluralities in both entity and set partitives turns out
to be problematic in one respect.
The issue is the following. If there is only one domain with one
parthood relation, there is nothing that constrains set partitives to
refer to an arbitrary material part of any singular entity making up
a plurality. For instance, given the plurality of apples a b c d a
part-word taking that plurality as an argument could return some
part of, say, b. However, this is not how set partitives are interpreted.
To the contrary, under ordinary circumstances they denote a set of
individual parts of a plurality, e.g., in the above example the extension
could include a b, b c, a b c etc. In order to exclude material
part interpretations, I postulate a special restriction on the meaning
of set partitive constructions, namely the set partitive constraint as
formulated in (7.43). The strict pluralization operation + in the formula
defined in (7.9) is responsible for the exclusive interpretation
of the plural. In particular, +Pmssc represents a pluralized predicate
with all the mssc individuals removed from its extension, e.g., for the
set {a, b, c} the + operation would yield {a b, a c, b c, a b c}.
Hence, the set partitive constraint formulated in (7.43) ensures that an
entity that is part of a plurality is either an mssc individual or a sum
of such individuals. In other words, it excludes all material parts from
the denotations of set partitives.
(7.43) Set partitive constraint
∀Pmssc∀y∀x[(+Pmssc(y) ∧ x y) → *Pmssc(x)]
The set partitive constraint guarantees the proper interpretation of partitives
involving plural DPs. It seems that postulating this restriction
is a necessary move for an analysis of partitive constructions building
on the unified parthood relation. However, I believe that it is not a big
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price to pay compared to the gains such an approach can offer.25 In
the next section, I will readdress the issue of partitioning discussed
briefly in Section 7.1 in order to provide a background for analysis of
particular types of partitive words. The notion of partition will turn
out useful in modeling continuous parts.
7.5.5 Partitions
As we the evidence discussed in this study indicates, quantification
over both wholes and parts can only operate on sets consisting of
disjoint members, i.e., entities that do not share a part. However, since
there are numerous ways how to divide a thing into parts, many
of such divisions would yield multiple entities that do overlap. But
such divisions are undesirable since only those sets can serve as the
domain of quantification that comprise separate, i.e., non-overlapping,
continuous parts. For instance, if one counted one half of a teddy bear
as one, another half as two, and its head and right paw as three and
four, one would count some things multiple times. By doing so they
would violate one of the crucial counting principles and, as a result, fail
to establish a one-to-one correspondence between discrete entities and
numbers. In order to avoid that, a mechanism ruling out undesirable
divisions is required. Such a mechanism is supposed to guarantee
that the domain of quantification does not include overlapping parts.
In other words, what we want is an operation that will be able to
carve up a whole into distinct separate entities. For this purpose, I will
make use of a device well-known in the study of pluralities, namely
partitions (see, e.g, Schwarzschild 1996, Chierchia 2010).
The partitioning operation π is a function of type e, t , e, t
which selects a set of entities, i.e., a predicate P, and yields its subset
π(P), i.e., a set of those elements in P that do not overlap. Given the
standard mereological definition of overlap, see (6.5), partitioning
imposes a condition that no two members of a partition π(P) share a
25. One consequence worth noting is that adopting (7.43) we might have to abandon
the unconditioned transitivity of parthood. Though it is definitely not a mainstream
assumption, it might turn out empirically correct as far as natural language is
concerned (see also Moltmann 1997 for related arguments).
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part. The formal definition of the partitioning operation π is provided
in (7.44).26
(7.44) Partitioning function π
π is a function such that for any property P and any x and y
in π(P) the following requirement is satisfied
¬∃z[z x ∧ z y]
(No two members of a partition overlap.)
To see how partitioning works let us consider what happens when
π is applied to the set A in (7.45a). As a result, we get (7.45b) which
includes only those members of A that do not share a part. On the
other hand, the set B in (7.45c) does not satisfy the condition defined in
(7.44) since a b and a c share a part, namely a. Therefore, although
B is a subset of A, it is not its partition.
(7.45) Partitioning
a. A = {a b, a c, a d, b c, b d, c d, a b c,
a b d, a c d, b c d, a b c d}
b. π(A) = {a b, c d}
c. B = {a b, a c}
It is important to emphasize that (7.45b) represents only one of numerous
possible partitions. For many sets there are multiple subsets that
would satisfy the non-overlap requirement after applying π. Depending
on a context some of those possible partitions may be considered
more natural than others since a particular arrangement of parts making
up an entity may suggest a particular division. For instance, a
teddy bear consists of cognitively salient elements recognizable as
legs, paws, a head, and a body. The fact that we perceive the whole in
such a structured way may result in partitioning the set of material
parts of the toy in question accordingly. However, different partitions
are not excluded, e.g., into its left and right half or its brown and white
part. Crucially, once a particular partition is established in a given
context it is fixed. This seems to be a welcome result since it satisfies
26. Notice that this definition diverges from what is usually expected from partitions,
i.e., not only non-overlap but also relative atomicity. In that sense, π does not
deliver proper partitions. However, I will continue to use the term for the sake of
convenience.
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the intuition that despite multiple possible divisions of a whole into
parts once a choice concerning a particular division has been made, it
cannot be altered. Figuratively speaking, one cannot change what’s
being counted while already counting.
7.5.6 Individuation
The partitioning function π guarantees that a reference set is free of
entities sharing a part. This is an important improvement since it accounts
for the ban of overlap in the domain of quantification. However,
as discussed in Chapter 5 there are two more principles that need to
be accommodated within a theory of countability, namely the conditions
of integrity and maximality. Since in this study I am primarily
concerned with subatomic quantification, I will focus on what makes
part a part, i.e., on how to model the contrast between an arbitrary
share and a stable integrated and recognizable entity within a whole.
In order to do so, we need a more fine-grained device than mere partitioning.
For this purpose, I assume an additional semantic element
which I will refer to as the individuating element (IND). This operation
is a function of type e, t , e, t which selects a set of entities and
returns a subset of the input consisting of non-overlapping integrated
objects. This can be achieved by incorporating into the semantics of
IND two components, specifically the partitioning function π and
the mereotopological mssc relation. To be precise, I propose that IND
introduces the mssc restriction relative to a partitioned property, as
defined in (7.46). Consequently, IND turns a set consisting of different
types of entities into a set including only mssc individuals.
(7.46) Individuating element
IND = λPλx[mssc π(P) (x)]
In general, an idea of an individuating operation is by no means new.
In fact, theories postulating similar operators in order to account for
the mass/count distinction date back at least to Sharvy (1979). For
instance, one of the most prominent and innovative systems was developed
by Borer (2005a) who proposes that count syntax involves a
classifier element which is absent in mass expressions. It is introduced
by a null head merged in the region between the NP and DP and
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expresses division by partitioning the reference domain of encoded
by the nominal stem. Similar, Bale and Barner (2009) posit a count
noun functional head interpreted as a function from semi-lattices to
individuated semi-lattices. Admittedly, a closely related idea to IND
was proposed by Scontras (2014, p. 97) for atomizers such as grain in,
e.g., grain of rice. On his view, partitioning is defined in such a way that
it applies to a kind and the result is a set of mssc instantiations of that
kind. However, what is new about the notion I propose is that as we
will see it allows for the individuation of material parts of individuals,
something that Scontras’s system fails to account for unless postulating
kinds corresponding to denotations of particular partitive words
which seems both conceptually undesirable and empirically inappropriate.27
Given the linguistic relevance of subatomic quantification
discussed in Chapter 2, 3, and 4 I argue that the semantics proposed
in (7.46) proves to be more advantageous.
Despite the fact that in many cases the individuating element is
silent, postulating it is well motivated by the restricted meaning of
partitive words in count explicit and proportional partitives as we
have seen from the contrasts provided in Chapter 2. Though topologyneutral
expressions of that kind can denote both contiguous and discontiguous
entities such as scattered portions of matter and arbitrary
sums of individuals, the moment they combine with the cardinal numeral
they can refer exclusively to integrated objects. Furthermore,
as discussed in Chapter 3 Polish and German provide evidence that
there are expressions that encode the individuating element formally.
In particular, as witnessed by the morphological complexity of Polish
topology-sensitive partitive words such as połówka and ćwiartka as
opposed to topology-neutral połowa and ćwierć there are good reasons
to postulate that the suffix -k- is the exponent of IND. Similar, German
stacked eine in proportional partitives of the the type die eine Hälfte DP
also appears to express the semantics in (7.46).
The next section will provide the semantics of some of the various
types of partitive words discussed in Chapter 2 and 3.
27. An independent problem for approaches relating countability to kinds is that
at least in some languages predicates that according to standard tests proposed
by Carlson (1977, 1980) do not correspond to kinds, e.g., predicates with certain
indexicals, do not seem to behave differently when it comes to counting.
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7.5.7 Partitive words
Having defined the generalized contextually conditioned measure
functions µ in (7.20) as well as the individuating element IND in (7.46),
we are ready to propose the semantics of both topology-neutral and
topology-sensitive partitive words. Let us start with German Teil. As
discussed in Section 7.5.1, for convenience I assume that parthood
or, more precisely, proper parthood is encoded in the semantics of
partitive words. Thus, the meaning of Teil is given in (7.47). The purely
mereological nature of makes it a topology-neutral expression which
can denote a set of scattered entities, arbitrary sums, or integrated
individuals. This characteristic corresponds to the compatibility of
Teil with explicit, set, and mass partitives reported in Section 2.2. On
the other hand, the constraint on set partitives postulated in (7.43)
guarantees that when it selects a plural genitive DP, it does not yield
an arbitrary material part or parts of an mssc individual making up
a plurality. However, when Teil combines with a count singular DP,
the resulting set can consist of various parts of the relevant object
since the proposed semantics bears no topological commitments with
respect to the to spatial arrangement of portions of matter the whole
phrase denotes. In other words, explicit entity partitives can refer to
continuous as well as discontinuous parts.
(7.47) German topology-neutral part-word Teil
Teil = λyλx[x y]
However, as we have seen in Section 2.2.2, in count explicit partitives
the numeral in accordance with the counting principle of integrity
can quantify only over contiguous portions of matter. This means
that in such a syntactic configuration the possible denotation of Teil is
significantly restricted. In other words, the part-word that is normally
topology-neutral is shifted into a topologically sensitive expression
that yields a set of those parts of the relevant whole that are spatially
integrated. I propose that this shift is performed by a silent IND element.
As specified in (7.46), IND applies mssc to a partition of the set
denoted by the input predicate. As a result, all of those parts denoted
by the phrase c-commanded by Teil that are overlapping as well as discontinuous
are excluded from the extension of the shifted expression.
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All the details concerning composition will be discussed in Section
7.6.
Let us now turn to different types of Polish half -words. In general,
the semantics of topology-neutral połowa given in (7.48) is quite similar
to the one of Teil. Both are expressions of type e, e, t , i.e., functions
that take an entity and yield a set of its parts. Furthermore, similar to
Teil nothing restricts połowa from selecting any particular type of thing,
i.e., it is perfectly compatible with individuals, scattered entities, and
pluralities of objects. Finally, the resulting set involves both continuous
and discontinuous halves. The main difference between the two is that
połowa is a proportional expression which denotes only those parts
that constitute a certain portion of a whole. This is captured by the
generalized measure function µ. Given the mechanism of contextual
conditioning defined in (7.20), µ returns an integer corresponding to
a number of individuals if its argument is a plurality, whereas when
applied to a singular individual or scattered entity, it yields a value
corresponding to the measure of its volume in some contextually
salient units, e.g., cm3. Since as discussed in Section 3.1.2 half -words
are inherently vague and can indicate parts that are either smaller
or greater than an exact half, the approximately equal relation ≈ is
preferred over = in the formula in (7.48). Overall, what połowa does is
that it selects an entity of any sort and returns a set of its continuous
and discontinuous parts such that they constitute approximately 50%
of the total cardinality or volume of that entity.
(7.48) Polish topology-neutral half -word połowa
połowa = λyλx[x y ∧ µ(x) ≈ µ(y) × 0.5]
Compared to połowa, its topology-sensitive counterpart pół fails to
combine with with predicates denoting scattered entities and arbitrary
sums of individuals. Given the framework developed here, it
is possible to account for its distributional restrictions discussed in
Section 3.1.1 in terms of mereotopological distinctions. In particular, I
propose that the semantics of pół, see (7.49), is almost identical to (7.48)
with the exception that it involves a special selectional requirement
which I refer to as the integrated individual presupposition, see (7.50).
Specifically, the symbol ymssc in (7.49) stands for an individual that is
mssc relative to some property. This explains why pół does not com-
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bine with plural and mass DPs since these expressions do not refer
to mssc individuals in their extensions. Therefore, pół can only head
proportional entity partitives and after it takes an integrated object, it
yields a set consists of various continuous as well as discontinuous
halves of that object.
(7.49) Polish topology-sensitive half -word pół
pół = λy . ymssc λx[x y ∧ µ(x) ≈ µ(y) × 0.5]
(7.50) Integrated individual presupposition
xmssc ↔ ∀y[y = x → ∃P[mssc(P)(y)]]
Finally, it is time to account for the most formally as well as semantically
complex of the Polish half -words, namely połówka. As we have
seen in Section 3.1.1, it is a morphologically complex expression involving
a special suffix -k- shared with a number of derived partitive
words in Polish such as ćwiartka ‘quarter’ and cząstka ‘part’. All of those
expressions are similar to pół in that they require mssc individuals
as their inputs. However, they differ in that their denotation consist
only of continuous parts. Therefore, I propose that the suffix -k- in fact
introduces the semantics of the IND element specified in (7.46), see
(7.51).28 Thus, the half -word połówka is a complex expression derived
from a simpler topology-sensitive partitive word.
(7.51) Polish individuating suffix -k-k-
= IND = λPλx[mssc π(P) (x)]
Given both the morphological complexity as well as semantic behavior
of połówka discussed in detail in Section 3.1, I postulate that it is derived
from pół via the derivational suffix -k-.29 The resulting expression is a
topology-sensitive half -word, see (7.52), that not only selects entities
28. An additional argument suggesting that such an analysis is on the right track is
the fact that the suffix -k- also appears in the make-up of Polish denumeral group
nouns such as dwójka ‘group of two’ (see Wągiel 2015b). However, I leave the exact
nature of the relationship between -k- in połówka and -k- in dwójka for future
investigations.
29. More precisely, I assume that połówka is derived from pół rather than from połowa
and that the element -ów- in połówka is semantically vacuous. Such an analysis seems
to be corroborated by the fact that the morpheme -ów- appears sometimes as a
linking element in Polish deadjectival nominalizations such as złoty ‘golden; złoty
(Polish currency)’ sim złotówka ‘1-złoty coin’. Notice the ungrammaticality of *złotowy.
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that are mssc individuals but also denotes only parts that are spatially
continuous. The first aspect is ensured by the integrated individual
presupposition encoded in pół, whereas the semantics of -k- guarantees
the latter. In particular, the partitioning function π provides a
contextually salient partition of the set delivered by pół. Thus, the
mssc operation is relativized to a property from whose denotation all
overlapping parts have been eliminated as the result of partitioning.
Ultimately, the resulting set involves only disjoint integrated parts
exactly as desired.
(7.52) Polish topology-sensitive half -word połówka
połówka = -k- ( pół )
Having developed all the machinery necessary to account for subatomic
quantification in natural language, let us see how the pieces fit
together. In the next section, I will walk through the derivation of different
types of partitives including German explicit and set partitives
and Polish proportional partitives as well as multiplier phrases. The
section will be concluded by a proposal how the system developed
here could be extended to Italian count explicit partitives involving
irregular plurals.
7.6 Composition
In the previous sections, I have provided all the ingredients necessary
for an analysis of natural language expressions utilizing subatomic
quantification. Now, let us see how those elements interact. I will start
with the simplest cases of explicit entity and set partitives. Next, I
will walk through the most complex case of count explicit partitives
which involves the interplay of almost all postulated components.
Subsequently, I will discuss the difference between topology-neutral
and topology-sensitive proportional partitives. Finally, I will present
the derivation of multiplier phrases modified by numerals. Similar to
the previous sections, I will illustrate how the proposed system works
on the German and Polish data. For all partitives, I provide examples
involving embedded definite DPs. Though for the sake of brevity I do
not provide explicit derivations, I assume that for specific DPs nothing
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changes except from the use of a choice function, see (7.39), instead of
the maximization operator.
7.6.1 Explicit entity partitives
I begin with a relatively simple example of a German explicit entity
partitive as the one provided in (7.53). The analysis of this case will
allow us to see how the essential components introduced in previous
sections interact with each other and as such it will serve us as a
reference point for other more complex cases. All things considered,
I assume that what gets interpreted at LF is the structure in (7.54).30
The whole phrase is an expression of type e, t which gets a partof-a-singularity
reading. The description of the whole derivation is
provided in (7.55).
(7.53) Explicit entity partitive
Teil
part
des
thegen
Apfels
applegen
‘part of the apple’
(7.54) Structure
e, t
e, e, t
Teil
e
e, t , e
DEF
e, t
Apfel
(7.55) Interpretation
a. Apfel = λx[mssc(apple)(x)]
b. DEF = λP[max(P)]
c. Function Application
DEF Apfel = max( Apfel ) =
= max λx[mssc(apple)(x)]
d. Teil = λyλx[x y]
e. Function Application
Teil [DEF Apfel] = λx[x DEF Apfel ] =
= λx x max λy[mssc(apple)(y)]
30. For the sake of simplicity, I ignore the genitive case marking at LF.
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7. Mereotopological account for subatomic quantification
The predicate Apfel in (7.55a) denotes a set of mssc individuals that
have a property of being an apple. If in a given context this set is a
singleton, the maximization operation max introduced by the definite
article in (7.55b) selects the unique maximal element from that set,
e.g., it turns {a} into a, see (7.55c). Subsequently, the part-word Teil
with the semantics specified in (7.55d) picks that entity and returns a
set of its material parts. Since there are no additional restrictions, the
resulting phrase represented in (7.55e) denotes all possible divisions
of the apple in question including various overlapping continuous as
well as discontinuous parts. This is exactly what we expect from the
extension of a topology-neutral explicit entity partitive.
7.6.2 Explicit set partitives
The next example to be discussed is the German explicit set partitive
(7.56). An analysis of such a construction should account for the fact
that it can get only a part-of-a-plurality interpretation. I assume the
semantic structure in (7.57) for the phrase in question. The only difference
between (7.57) and (7.54) from the previous section is that the
set partitive tree involves the PL node which introduces the pluralization
of the predicate whereas the entity partitive tree does not. How
the structure corresponding to the phrase in (7.56) is interpreted is
demonstrated in (7.58).
(7.56) Explicit set partitive
Teil
part
der
thegen
Äpfel
applesgen
‘some of the apples’
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7. Mereotopological account for subatomic quantification
(7.57) Structure
e, t
e, e, t
Teil
e
e, t , e
DEF
e, t
e, t , e, t
PL
e, t
Apfel
(7.58) Interpretation
a. Apfel = λx[mssc(apple)(x)]
b. PL = λP . Pmssc λx[+P(x)]
c. Function Application: presupposition satisfied
PL Apfel = λx[+ Apfel (x)] =
= λx + λy[mssc(apple)(y)] (x)
d. DEF = λP[max(P)]
e. Function Application
DEF [PL Apfel] = max( PL Apfel ) =
= max λx + λy[mssc(apple)(y)] (x)
f. Teil = λyλx[x y]
g. Function Application
Teil [DEF [PL Apfel]] = λx x DEF [PL Apfel] =
= λx x max λy + λz[mssc(apple)(z)] (y)
Similar to (7.55), the noun in (7.58a) denotes a set of individuals
that are mssc relative to the property apple. Since the plural NP in (7.56)
is semantically interpreted, the plural marker contributes the meaning
in (7.58b). Specifically, it takes the predicate Apfel and applies the
strict pluralization operation +. As a result, the extension of (7.58c)
consists of sums of apples. Next, the definite article yields the maximal
sum plurality from the set which will serve as the argument of Teil,
see (7.58d)–(7.58f). Finally, after the entity variable is saturated as in
(7.58g), the explicit set partitive denotes a set of overlapping parts
of the largest plurality of apples in a given context. The set partitive
constraint defined in (7.43) ensures that this set consists only of parts
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7. Mereotopological account for subatomic quantification
that are either mssc entities or sums thereof, i.e., material proper parts
of particular individuals are not included. This is a welcome result.
7.6.3 Count explicit partitives
The last type of the explicit partitive constructions to be considered
here concerns the significantly more complex example in (7.59). In
general, I assume the semantic structure along the lines in (7.60) for
count explicit partitives. Notice that in accordance with the semantics
of cardinals proposed in Section 7.4.2, the numeral root and CL# form
a constituent. Furthermore, unlike in the explicit partitive examples
in (7.54) and (7.57) in order to ensure counting is possible, I postulate
that there is a special IND node present in the tree at LF, see Section
7.5.6. In the discussed German example, IND has no formal exponent.
However, I argue that the semantic evidence implies that though null
it is interpreted. In particular, count explicit partitives refer only to
pluralities of integrated parts. This of course is in accordance with
the counting principles discussed in Chapter 5 and requires a formal
account. The exact step-by-step derivation is given in (7.61).
(7.59) Count explicit partitive
zwei
two
Teile
parts
des
thegen
Apfels
applegen
‘two parts of the apple’
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7. Mereotopological account for subatomic quantification
(7.60) Structure
e, t
e, t , e, t
n√
zw
n, e, t , e, t
CL#
e, t
e, t , e, t
IND
e, t
e, e, t
Teil
e
e, t , e
DEF
e, t
Apfel
(7.61) Interpretation
a. Apfel = λx[mssc(apple)(x)]
b. DEF = λP[max(P)]
c. Function Application
DEF Apfel = max( Apfel ) = max λx[mssc(apple)(x)]
d. Teil = λyλx[x y]
e. Function Application
Teil [DEF Apfel] = λx[x DEF Apfel ] =
= λx x max λy[mssc(apple)(y)]
f. IND = λPλx[mssc π(P) (x)]
g. Function Application
IND [Teil [DEF Apfel]] =
= λx mssc π( Teil [DEF Apfel] ) (x) =
= λx mssc π λz z max λy[mssc(apple)(y)] (z) (x)
h.
√
zw = 2
i. CL# = λnλP. Pmssc λx[*P(x) ∧ #(P)(x) = n]
j. Function Application√
zw CL# = λP. Pmssc λx[*P(x) ∧ #(P)(x) =
√
zw ] =
= λP. Pmssc λx[*P(x) ∧ #(P)(x) = 2]
k. Function Application: presupposition satisfied
[
√
zw CL#] [IND [Teil [DEF Apfel]]] =
= λx * IND [Teil [DEF Apfel]] (x)
∧ #( IND [Teil [DEF Apfel]] )(x) = 2 =
= λx * λw mssc π λz z max λy[mssc(apple)(y)] (z) (w) (x)
∧# λw mssc π λz[z max λy[mssc(apple)(y)] (z)] (w) (x) = 2
269
7. Mereotopological account for subatomic quantification
The steps (7.61a)–(7.61e) are exactly the same as in the case of the
explicit entity partitive derived in (7.55), i.e., starting from the predicate
of mssc individuals we end up with the node denoting a set of
overlapping continuous and discontinuous parts of a contextually
unique apple. Given the counting principles of non-overlap, integrity,
and maximality, that expression cannot serve as the domain of quantification
since its extension resembles those of mass terms. In particular,
it includes both scattered entities and entities that are not disjoint. In
Chapter 5, I argued that the deeply rooted mechanism of counting is
based upon an algorithm that establishes one-to-one correspondence
with numbers only for discrete objects, i.e., non-overlapping integrated
entities in their mereological maximality, and that the same applies to
subatomic quantification. Therefore, if parts are to be counted, it needs
to be ensured that they constitute entities of that sort. This is what the
individuating element IND introduced in (7.61f) is for. IND combines
with the set of overlapping parts and partitions it in such a way that
the new set consists only of disjoint parts. However, non-overlap itself
does not guarantee that parts are integrated. This is why IND restricts
the denotation of (7.61e) even further by applying the mssc constraint.
As a result, the expression in (7.61g) denotes a set of objects that are
mssc with respect to the property of being a disjoint proper part of
a contextually unique apple. In other words, (7.61g) refers to a set of
integrated parts that can be put in one-to-one-correspondence with
numbers.
As proposed in Section 7.4.2, the cardinal numeral zwei is a complex
expression consisting of a numeral root simply naming integer 2,
see (7.61h), which serves as an argument for the classifier element CL#,
see (7.61i). As indicated in (7.61j), the whole cardinal is a predicate
modifier that for a predicate of mssc entities returns a set of pairs of
such individuals. This is ensured by the following elements. The individuation
presupposition restricts the predicate arguments to those
that denote only mssc entities. On the other hand, the first conjunct of
the assertion of CL# pluralizes the predicate by means of the classical *
operator, and thus provides sums for the #(P) measure function which
yields the number of individuals that are mssc relative to the property
the whole cardinal modifies. After the predicate variable gets saturated
by the entity partitive individuated by IND, the resulting count
explicit partitive in (7.61k) returns a set of pairs of non-overlapping
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7. Mereotopological account for subatomic quantification
integrated parts of a contextually unique apple. And that is exactly
what we want.
Though the computation in (7.61) gets quite complex, I believe
that the underlying mechanism deriving count explicit partitives is
quite simple. Its great advantage is that it allows us to account for
the semantic subtleties concerning topological issues highlighted by
the novel data I presented in Chapter 2 and 3 via the interplay of
the semantics of partitives, the individuating element, and cardinals.
In the following sections, I will go in detail through the proposed
account for the contrast between Polish topology-neutral and topologysensitive
half -words discussed in Section 7.5.7.
7.6.4 Topology-neutral proportional partitives
Let us now turn to Polish proportional partitives. First, I will discuss
the step-by-step derivation of the least complex case, i.e., entity partitives
involving the topology-neutral half -word połowa exemplified by
the phrase in (7.62). In general, such expressions are very similar to
German explicit entity partitives such as those discussed in Section
7.6.1. The semantic tree corresponding to (7.62) is provided in (7.63)
and (7.64) demonstrates how the structure is interpreted.31
(7.62) Topology-neutral proportional entity partitive
połowa
half2
jabłka
applegen
‘half of the apple’
(7.63) Structure
e, t
e, e, t
połowa
e
e, t , e
DEF
e, t
jabłko
31. Similar to German explicit partitives, I ignore case marking at LF.
271
7. Mereotopological account for subatomic quantification
(7.64) Interpretation
a. jabłko = λx[mssc(apple)(x)]
b. DEF = λP[max(P)]
c. Function Application
DEF jabłko = max( jabłko ) = max λx[mssc(apple)(x)]
d. połowa = λyλx[x y ∧ µ(x) ≈ µ(y) × 0.5]
e. Function Application
połowa [DEF jabłko] = λx[x DEF jabłko
∧ µ(x) ≈ µ( DEF jabłko ) × 0.5] =
= λx x max λy[mssc(apple)(y)]
∧ µ(x) ≈ µ max λy[mssc(apple)(y)] × 0.5
Similar to German Apfel, the predicate jabłko in (7.64a) is true of
entities that are mssc with respect to the property apple. Though Polish
lacks articles, as indicated in Section 7.5.2 I assume the silent definite
element DEF specified in (7.64b) which applies the maximization operator
max to the set denoted by jabłko. If it is a singleton, then the
definite DP refers to the unique individual from that set, see (7.64c).
The denotation of the half -word połowa is very similar to that of a
part-word with the exception that it in indicates how big the parts it
denotes are, see (7.64d). It takes the mssc entity denoted by the definite
DP as it first argument and the resulting proportional partitive denotes
a set of parts of a contextually unique apple. The mechanism of
contextual conditioning accounts for the fact that the generalized measure
function µ measures its argument in terms of volume. Thus, each
part in the denotation of the whole proportional partitive constitutes
approximately 50% of the volume of that apple. Notice that among
the members of the set there are continuous as well as discontinuous
halves many of which overlap. Such a denotation is incompatible with
cardinal numerals. In order to deliver countable entities, the individuating
element IND needs to be applied similar as in count explicit
partitives discussed in Section 7.6.3.
Since the half -word połowa is topologically neutral, there are no
restrictions concerning its distribution. Hence, it can also combine
with plurals as in (7.65). I assume the semantic structure of the phrase
in (7.66) and interpretation in (7.67).
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7. Mereotopological account for subatomic quantification
(7.65) Topology-neutral proportional set partitive
połowa
half2
jabłek
applesgen
‘half of the apples’
(7.66) Structure
e, t
e, e, t
połowa
e
e, t , e
DEF
e, t
e, t , e, t
PL
e, t
jabłko
(7.67) Interpretation
a. jabłko = λx[mssc(apple)(x)]
b. PL = λP . Pmssc λx[+P(x)]
c. Function Application: presupposition satisfied
PL jabłko = λx[+ jabłko (x)] =
= λx + λy[mssc(apple)(y)] (x)
d. DEF = λP[max(P)]
e. Function Application
DEF [PL jabłko] = max( PL jabłko ) =
= max λx + λy[mssc(apple)(y)] (x)
f. połowa = λyλx[x y ∧ µ(x) ≈ µ(y) × 0.5]
g. Function Application
połowa [DEF [PL jabłko]] = λx[x DEF [PL jabłko]
∧ µ(x) ≈ µ( DEF [PL jabłko] ) × 0.5] =
= λx x max λy + λz[mssc(apple)(z)] (y)
∧ µ(x) ≈ µ max λx + λy[mssc(apple)(y)] (x) × 0.5
The sole difference between topology-neutral proportional entity
and set partitives is that in the case of the latter the noun is pluralized.
The plural marker in (7.67b) applies to the predicate of mssc
individuals in (7.67a) and yields a set consisting of sums formed from
those singularities. Subsequently, given the set partitive constrained
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and the mechanism of contextual conditioning the measure function
µ in (7.67g) is interpreted as #, i.e., counts the number of mssc parts
of its arguments. As a result, the whole construction denotes plural
entities whose cardinality corresponds to approximately 50% of the
total number of individuals of the maximal sum apples.
Despite the straightforward differences resulting from distinct semantics
of part- and half -words, the denotations of topology-sensitive
explicit and proportional partitives are very much alike. If not modified
by a cardinal, both types of expressions can refer to overlapping
continuous and discontinuous parts of a singularity as well as to overlapping
parts of a plurality. On the other hand, topology-sensitive
partitives are significantly more restricted. In the following section, I
focus on demonstrating how exactly.
7.6.5 Topology-sensitive proportional partitives
An important insight concerning the Polish topology-sensitive half word
pół presented in Section 3.1.1 is that it does not appear in set
and mass partitives. It is not a weird property of one partitive word
since the same behavior is shared, e.g., with the quarter-word ćwierć.
In any case, I argue that this distributional constraint results from
a particular topological restriction imposed on the arguments of pół.
An exemplary topology-sensitive proportional partitive is provided
in (7.68), whereas the structure I assume at LF is given in (7.69). The
complete derivation is presented in (7.70).
(7.68) Topology-sensitive proportional entity partitive
pół
half1
jabłka
applegen
‘half of the apple’
(7.69) Structure
e, t
e, e, t
pół
e
e, t , e
DEF
e, t
jabłko
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7. Mereotopological account for subatomic quantification
(7.70) Interpretation
a. jabłko = λx[mssc(apple)(x)]
b. DEF = λP[max(P)]
c. Function Application
DEF jabłko = max( jabłko ) =
= max λx[mssc(apple)(x)]
d. pół = λy . ymssc λx[x y ∧ µ(x) ≈ µ(y) × 0.5]
e. Function Application: presupposition satisfied
pół [DEF jabłko] = λx[x DEF jabłko
∧ µ(x) ≈ µ( DEF jabłko ) × 0.5] =
= λx x max λy[mssc(apple)(y)]
∧ µ(x) ≈ µ max λy[mssc(apple)(y)] × 0.5
The derivation in (7.70) is almost identical to its twin counterpart
in (7.64). However, notice that in (7.70d) there is a tiny but crucial
difference in the semantics of pół as proposed in (7.49). Specifically,
pół incorporates the integrated individual presupposition defined in
(7.50) which requires its first argument to be an integrated entity. This
constraints the distribution of pół to count singulars since plurals and
mass terms denote arbitrary sums and scattered entities, respectively,
i.e., things distinct from mssc objects. However, pół does not impose any
topological restriction on parts of the relevant individual it yields. The
semantics of the whole partitive in (7.70e) thus states that the denoted
set includes both continuous and discontinuous halves. However, as
we have seen in Chapter 3 there are also partitive words that are
restrictive with respect to the topological properties of entities they
deliver.
This brings us finally to the most intriguing case of Polish proportional
partitives headed by the topology-sensitive half -word połówka
such as the one in (7.71). Though I restrict my focus here to the semantic
structure and interpretation of this example, see (7.72) and (7.73),
respectively, as we have seen in Section 3.2 there are also other expressions
that behave similarly, e.g., the part-word cząstka and quarter-word
ćwiartka.
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7. Mereotopological account for subatomic quantification
(7.71) Topology-sensitive individuating proportional partitive
połówka
half3
jabłka
applegen
‘half of the apple’
(7.72) Structure
e, t
e, t , e, t
-ke,
t
e, e, t
pół
e
e, t , e
DEF
e, t
jabłko
(7.73) Interpretation
a. jabłko = λx[mssc(apple)(x)]
b. DEF = λP[max(P)]
c. Function Application
DEF jabłko = max( jabłko ) = max(λx[mssc(apple)(x)])
d. pół = λy . ymssc λx[x y ∧ µ(x) ≈ µ(y) × 0.5]
e. Function Application: presupposition satisfied
pół [DEF jabłko] =
= λx[x DEF jabłko ∧ µ(x) ≈ µ( DEF jabłko ) × 0.5] =
λx x max λy[mssc(apple)(y)]
∧ µ(x) ≈ µ max λy[mssc(apple)(y)] × 0.5
f. -k- = IND = λPλx[mssc π(P) (x)]
g. Function Application
IND [ pół [DEF jabłko]] =
= λx[mssc π( pół [DEF jabłko] ) (x)] =
= λx mssc π λz z max λy[mssc(apple)(y)]
∧ µ(x) ≈ µ max λy[mssc(apple)(y)] × 0.5 (x)
The only difference between (7.70) and (7.73) lies in that in the first
the half -word pół is used whereas the latter involves derivationally
complex połówka. Thus, the composition in (7.73a)–(7.73c) remains
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7. Mereotopological account for subatomic quantification
unchanged and I will start the discussion from the step (7.73d). As
proposed in (7.52), the morphological complexity of połówka translates
into its semantic complexity. In particular, I assume that the root pł is
a topologically sensitive expression introducing the integrated individual
presupposition, see (7.73d), which ensures that that the half -word
combines only with mssc individuals. On the other hand, the suffix
-k- is interpreted as the individuating element IND, see (7.73e), i.e., it
restricts the denotation of the partitive to non-overlapping continuous
parts. As a result, the whole phrase refers to integrated halves of a
contextually unique apple that can be subject to counting.
This concludes the analysis of distinct types of partitive constructions.
The relevance of Polish topology-sensitive individuating proportional
partitives is that they show that IND can be formally expressed.
As demonstrated in Section 3.4, there are more constructions in different
natural languages that can be considered as involving formal
exponents of IND. Though I will refrain here from further investigation,
if the analysis proposed here is on the right track, I expect to find
even more such expressions cross-linguistically. In the next section,
I will move on to another type of construction involving subatomic
quantification, namely multiplier phrases.
7.6.6 Multiplier phrases
In the previous sections, I have discussed the composition of different
kinds of partitives. The only construction involving numerical
expressions discussed so far was the German count explicit partitive
in (7.59) where the cardinal modified an entity partitive headed by
the part-word Teil. In this section, I will discuss in detail the semantics
of constructions involving derived numerical expressions specialized
precisely for subatomic quantification, namely multipliers. In order to
present the overall picture, I will consider multiplier phrases modified
by cardinals exemplified by (7.74). I posit that the semantic structure
underlying such expressions is essentially as in (7.75), whereas in (7.76)
I provide an exact step-by-step derivation of the main example.
(7.74) Multiplier phrase modified by a cardinal
trzy
three
podwójne
doublepl
hamburgery
hamburgers
277
7. Mereotopological account for subatomic quantification
‘three double hamburgers’
(7.75) Structure
e, t
e, t , e, t
n√
trz
n, e, t , e, t
CL#
e, t , e, t
n√
dw
n, e, t , e, t
CL
e, t
hamburger
(7.76) Interpretation
a. hamburger = λx[mssc(hamburger)(x)]
b.
√
dw = 2
c. CL = λnλP. Pmssc λx[P(x) ∧ (P)(x) = n]
d. Function Application√
dw CL = λP. Pmssc λx[P(x) ∧ (P)(x) =
√
dw ] =
= λP. Pmssc λx[P(x) ∧ (P)(x) = 2]
e. Function Application: presupposition satisfied
[
√
dw CL ] hamburger =
= λx[ hamburger (x) ∧ ( hamburger )(x) = 2] =
= λx[mssc(hamburger)(x) ∧ mssc(hamburger) (x) = 2]
f.
√
trz = 3
g. CL# = λnλP. Pmssc λx[*P(x) ∧ #(P)(x) = n]
h. Function Application√
trz CL# = λP. Pmssc λx[*P(x) ∧ #(P)(x) =
√
trz ] =
= λP. Pmssc λx[*P(x) ∧ #(P)(x) = 3]
i. Function Application: presupposition satisfied
[
√
trz CL#] [[
√
dw CL ] hamburger] =
= λx * [
√
dw CL ] hamburger (x)
∧ #( [
√
dw CL ] hamburger )(x) = 3 =
λx * λy[mssc(hamburger)(y) ∧ mssc(hamburger) (y) = 2] (x)
∧# λy[mssc(hamburger)(y) ∧ mssc(hamburger) (y) = 2] (x) = 3
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In analogy to the previous cases, the noun hamburger is a predicate
of type e, t denoting a set of entities that are mssc with respect
to the property hamburger, i.e., singular hamburgers. On the other
hand, as suggested by morphology the multiplier in (7.76d) consists
of the classifier element CL of type n, e, t , e, t whose number
argument, see (7.76c), is saturated by the number 2 referred to by the
numeral root in (7.76b). Due to the individuation presupposition in
the semantics of CL the resulting numerical expression is a predicate
modifier selecting only predicates of mssc individuals. Since the noun
hamburger satisfies this requirement, it can serve as the argument for
the multiplier. After the two combine in (7.76e), the resulting multiplier
phrase, i.e., a function from entities to truth values, denotes a
set of singular hamburgers such that each includes two patties. This
meaning stems from the fact that the (P) measure function defined
in (7.22) returns 2 as the number of parts that are essential for the
property hamburger and given the relevant extra-linguistic factors
such a part happens to be a patty. Notice also that the fact that the first
conjunct in (7.76e) is not pluralized, unlike in cardinals, results in that
the multiplier phrase is a predicate of mssc entities, i.e., it denotes singular
double hamburgers. As such it can be modified by the cardinal
numeral.
Given the compatible expression, the cardinal in (7.76h) is ready
for the saturation of the predicate variable. It itself is a complex expression
of type e, t , e, t consisting of the numeral root in (7.76f)
whose denotation, i.e., the number 3, serves as the argument for the
classifier element CL#. After the cardinal composes with the multiplier
phrase in (7.76i), what we obtain is a set of triples of hamburgers with
two essential parts each. Counting is done by the measure function
#(P) which puts mssc individuals in one-to-one correspondence with
numbers and yields the integer corresponding to the total number of
objects in a plurality delivered by the * operator introduced in the first
conjunct. The result is exactly what we wanted to derive, i.e., a set of
pluralities consisting of three double hamburgers.
This concludes the analysis of some of the expressions involving
subatomic quantification. In this study, I have focused on several different
types of partitives as well as multiplier phrases. I have proposed
that though both types of constructions quantify over parts of entities,
they employ different means to achieve the goal. While partitives em-
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ploy the proper parthood relation and in more complicated cases
also the individuating element IND and cardinals, multipliers utilize
a special measure function (P) dedicated to counting essential parts
of integrated wholes. In the next section, I will briefly suggest how
the analysis I proposed could be extended to cover more peculiar
examples such as Italian partitives involving irregular plurals.
7.7 Possible extensions
The last part of this chapter concerns a brief informal discussion of
possible extensions of the system developed here to other types of
constructions in different languages. In principle, a number of distinct
kinds of proportional partitives discussed in Section 3.4 could be
accounted for by incorporating the meaning of the individuating
element IND into the semantics of some morpheme. For instance, if
the approach proposed here is on the right track and there are no
additional facts I have not reported here concerning phrases such
as (7.77), a straightforward application of the analysis appears to be
plausible. It seems that the morpheme eine is what encodes IND which
accounts for the reported results of the flag test.
(7.77) German topology-sensitive proportional partitive
die
the
eine
a
Hälfte
half
der
thegen
Fahne
flag
‘the half of the flag’
Similar, in principle the account developed here could be extended
to English constructions such as (7.78). However, most probably this
case would be more challenging since it might be necessary to capture
interactions between the partitive word and determiner on the one
hand as well as with the preposition of on the other.
(7.78) English topology-sensitive proportional partitive
a half of the flag
Furthermore, it might turn out that a similar analysis is appropriate
for the Mandarin classifier construction in (7.79). Here, the classifier
miàn is arguably responsible for introducing IND.
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7. Mereotopological account for subatomic quantification
(7.79) Mandarin topology-sensitive proportional partitive
bàn-miàn
half-cl
guó
national
qí
flag
‘a half of the national flag’
Of course, much more work needs to be done to verify whether the
proposed in fact accounts for the data from German, Mandarin, and
other languages displaying a similar pattern. However, I believe that
at this stage of the research it is a plausible hypothesis to postulate
that what makes topology-sensitive partitives different from their
topology-neutral counterparts is the fact that the first involve a formal
exponent of IND.
The next section will be dedicated to the discussion of how the
proposed analysis could be applied to yet another type of expression.
7.7.1 Italian partitives with irregular plurals
The last case of a possible extension of the developed system to be
discussed here concerns count explicit partitives involving irregular
plurals in Italian such as those in (7.80). In Chapter 2, a considerable
amount of attention was dedicated to the intriguing effect these expressions
give rise to. We have seen that at least a subset of Italian
irregular plurals does not merely encode plurality but also implies a
certain spatial configuration that the individual entities making up
a sum remain in. Arguably, this kind of behavior can be captured in
terms of the topological notion of connection. The Italian data served
as a crucial source of evidence for the role of integrity in quantification
over parts. In particular, the evidence strongly suggests that
entities are countable as long as they are conceptualized as spatially
continuous despite whether they are singularities or pluralities.
(7.80) Italian partitive with irregular plurals
due
two
parti
parts
delle
of-the
mura
wallcoll
‘two parts of the walled complex’
Given the arguments in favor of the connected nature of pluralities
denoted by the relevant class of Italian irregular plurals, I postulate
that it is plausible to model them in terms of clusters (Grimm 2012b,
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7. Mereotopological account for subatomic quantification
p. 144). A cluster is a special type of individual defined in terms
of the tc relation formulated in (6.32) and repeated here as (7.81).
Specifically, the definition in (7.82) states that an entity is a cluster if
it involves a plurality of individuals that are transitively connected,
i.e., form a connected sequence. Intuitively, clusters are entities that
are conceptualized as bundled pluralities whose minimal units are
cognitively not salient enough or insignificant to be considered objects
in their own rights. For instance, a bundle of rice or a pile of gravel
constitute clusters since each minimal unit, i.e., a grain or a pebble, in
a bundle or a pile is connected to another minimal unit. Since tc is
relativized to a connection, clusters can involve bundled pluralities
of entities that are externally or even proximately connected.32 In the
case of Italian irregular plurals, the first would correspond, e.g., to the
referents of mura ‘walls (in a complex)’, whereas the latter might be
useful to model the denotation of, e.g., ossa ‘bones (in a skeleton)’.
(7.81) Transitively connected
tc(x, y, P, C, Z)
def
= ∀z ∈ Z[P(z) ∧ (x = z1 ∧ y = zn) ∧
Cz1z2 ∧ Cz2z3 · · · ∧ Czn−1zn]
where Z = {z1, z2, . . . zn}
(Entities x and y are transitively connected relative to a property
P, a connection relation C, and a set of entities Z, when
all members of Z satisfy P and x and y are connected through
the sequence of zis in Z.)
(7.82) Cluster
cluster(x, P, C)
def
= ∃Z[x = Z ∧ ∀z∀z ∈ Z∃Y[tc(z, z , P, C, Y)]]
(An entity x is a cluster relative to a property P and a connection
relation C iff x is a sum of entities falling under the same
property which are all transitively connected relative to some
set Y under the same property and connection relation.)
If the proposed analysis of Italian irregular plurals in terms of clusters
is correct, one would probably need to elaborate on the meaning of
cardinal numerals. Notice that clusters can consist of parts that are
mssc entities, e.g., individual bones, as well as sequences of individuals,
32. Those connection can be formally captures by the notions of ec and pc introduced
in (6.29) and (6.33), respectively.
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7. Mereotopological account for subatomic quantification
e.g., pluralities of transitively connected bones. This might require
to develop a more general devise able to capture more fine-grained
shades of integrity than the current formulation of the individuation
presupposition. Though such an enterprise lies beyond the scope of
this study and I leave it for future research, I believe that the account
proposed here provides an inspiring background for such an attempt.
7.8 Summary
In this chapter, I have presented a formal account for subatomic quantification
in natural language grounded in the conceptual framework
described in Chapter 5. I rejected atomicity as a useful concept for
that purpose and instead proposed an analysis based on notions developed
within mereotopology, i.e., the theory of wholes extending
standard mereology with topological distinctions, and introduced
in Chapter 6. The mereotopological account for nominal semantics
allows us to get rid of atomicity as an unintuitive and undesirable
notion as well as to distinguish between singularities and pluralities
in terms of the distinction between part-whole structures that either
involve or do not involve the topological notion of connectedness. In
particular, I adopted a view on which singular count nouns denote
sets of integrated objects modeled in terms of mssc individuals that
can be pluralized in order to yield sets of pluralities of such entities.
On the other hand, I postulated a compositional semantics for numerical
expressions such as cardinals and multipliers that involves
numeral roots treated here as names of numbers as well as classifiers
that turn expressions of type n into predicate modifiers. In particular, I
proposed that those classifier elements employ measure functions, i.e.,
operations that relate entities with integers. While cardinals utilize a
measure function that simply counts mssc entities, multipliers involve
quantification over essential parts of an individual. Compatibility of
the numerical expressions in question is ensured by the individuation
presupposition that requires the nodes cardinals and multipliers
combine with to be predicates of mssc objects.
Another component of the analysis regarded the semantics of partitive
words. Given the advantages of mereotopology, it was possible
to model subatomic quantification without postulating sorted domain
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7. Mereotopological account for subatomic quantification
with different parthood relations defined over them. Instead, particular
partitive words were treated as expressions employing the unified
parthood relation. Specifically, what they do is that they select an
entity and return a set of its parts. Different types of partitive words
differ with respect to the nature of those parts. For instance, part-words
denote sets of any proper parts, whereas half -words refer to parts that
constitute approximately 50% of the cardinality of a plurality or the
volume of an mssc individual or a scattered entity such as substance.
On the other hand, topology-neutral partitive words denote all kinds
of overlapping continuous as well as discontinuous parts, and thus
are compatible with singular count nouns, mass terms, and plurals,
whereas denotations of topology-sensitive partitive words are more
restricted. For instance, the Polish half -word pół introduces the integrated
individual presupposition which imposes a constraint that it
can only select mssc entities. This fact explains why it does not combine
with mass and plural complements. On the other hand, morphologically
complex połówka is a compositional expression involving the
suffix -k- that I argue is a formal exponent of the individuating element
that partitions the set of parts so that overlapping halves are excluded
as well as removes those parts that are discontinuous. As a result, the
extension of połówka involves only a set of disjoint integrated halves
that could serve as an argument for cardinals. Though the individuating
element is formally expressed in the morphology of some Polish
partitive words, it can also be silent. I argued that that is the case in, e.g.,
German count explicit partitives since given the counting principles
postulated in Chapter 5, one can only count non-overlapping continuous
parts. The chapter was concluded by considerations how the
proposed system could be extended to some of the other expressions
discussed in this study including Italian partitives involving irregular
plurals that could be analyzed as clusters, i.e., bundled pluralities of
transitively connected entities. However, the precise implementation
of the suggested ideas is left for future research.
284
8 Conclusion
The research on different aspects of quantification in natural language
is abundant and since the early days of formal semantics has continually
led to better understanding of certain properties of the language
faculty. Nevertheless, one perspective seems to be somewhat neglected.
The main aim of this thesis was to contribute to our understanding of
quantification in natural language by exploring the so far neglected
domain of subatomic quantification, i.e., quantification over parts of
building blocks of the denotations of singular count nouns. I have
provided compelling evidence for the relevance of this phenomenon
for natural language semantics. In particular, I have explored various
aspects of meaning of a broad range of linguistic expressions such
as different types of partitive constructions, certain types of adjectives
as well as multipliers from a cross-linguistic perspective. One
of the key findings is the significance of the topological notion of
integrity with respect to part-whole structures encoded in nominal
semantics. The examination of properties of different types of investigated
expressions suggests that there is one unified parthood relation
for various types of entities. At the same time, different part-whole
structures result from distinct topological relations holding between
particular elements. Entities conceptualized as integrated wholes, i.e.,
objects whose parts stick together, differ from those whose parts are
not bound by any topological constraints or, alternatively, are arranged
in some other type of spatial configuration. This concerns both wholes,
e.g., the prototypical referents of singular count nouns and regular
plurals, and parts, e.g., spatially continuous pieces as opposed to
arbitrary discontinuous portions of matter. The major claim of the
thesis is that subatomic quantification is subject to the very same constraints
as quantification over wholes. Specifically, only entities that
are conceptualized as non-overlapping integrated contiguous objects
can be put in one-to-one correspondence with numbers be it either
wholes or parts. In order to capture this generalization, I propose a
universal mechanism which allows for counting of both entire objects
and their parts. Its formal implementation is based on the theory of
wholes called mereotopology which extends standard mereological
parthood with topological notions such as connectedness. Building
285
8. Conclusion
on this framework, I develop a system which derives the predicted
semantics of representative constructions discussed in the empirical
part of the thesis. Different aspects of their meaning arise as a result of
the interaction between topology, partitivity, and numerical quantification.
An important advantage of the proposed approach is that it does
not define countability in terms of atomicity since devising counting
as quantification over mereological atoms, i.e., entities that have no
proper parts, is very problematic for subatomic quantification.
This chapter concludes the thesis by summarizing its empirical
insights as well as theoretical contributions to the study of quantification
in natural language. First, I will provide an overall summary,
then I will discuss some of the open questions and possible topics for
future investigations.
8.1 Overall summary
As outlined in Chapter 1, this thesis was intended to contribute to
the field of formal semantics by exploring subatomic quantification.
Throughout its pages, I have provided a broad range of novel evidence
indicating its significance in natural language and developed
a novel approach accounting for the observed phenomena. The thesis
is divided into two conceptual parts. In the first part comprising
Chapter 2, 3, and 4, I have provided a broad range of novel evidence
indicating that natural language involves elaborate means to express
subatomic quantification. The second part involving Chapter 5, 6, and
7 was dedicated to developing a conceptual framework that would
provide means to model quantification over both wholes and parts and
a formal approach based on it that would account for a representative
subset of the observed phenomena.
The data discussed in Chapter 2 provide cross-linguistic evidence
that the same partitive word, e.g., part, half, and most, can commonly
appear both in entity and set partitives, i.e., constructions that refer
to parts of singularities and pluralities and typically combine with
singular and plural DPs, respectively. This fact indicates that such
expressions are able to operate both at the superatomic and subatomic
level depending on what structure is provided by the embedded DP
(cf. Moltmann 1997, 1998). On the other hand, explicit partitives mod-
286
8. Conclusion
ified by the cardinal numeral can only quantify over material parts of
singular individuals irrespective of whether the complement DP is
singular or plural, see Table 8.1. This might suggest that since partitive
words have distinct properties in different structures, they differ
significantly after all (Schwarzschild 1996).
singulars plurals
bare count bare count
subatomic quantification *
quantification over wholes * * *
Table 8.1: Properties of partitive words
Nonetheless, the crucial piece of data coming from Italian partitives
involving irregular plurals, i.e., expression that infer not only plurality
but also cohesion or integrity of parts (Ojeda 1995, Acquaviva 2008),
provides evidence to the contrary. In particular, when such a partitive
is modified by the cardinal numeral, the whole phrase can get either
a part-of-a-singularity reading or a part-of-a-plurality interpretation
as long as the parts making up the plurality constitute an integrated
entity. Therefore, countability results from the interplay between the
meaning of a particular partitive word and the semantic properties of
a singular or plural DP it combines with. If a plural expression denotes
a plural entity constituted by a cohesive, i.e., spatially related, sum of
parts, quantification over continuous parts of such a sum is possible,
see Table 8.2.
singulars regular pl irregular pl
bare count bare count bare count
subatomic quantification *
quantification over wholes * * *
Table 8.2: Properties of Italian parte ‘part’
Furthermore, I have shown that at least in some languages the
parallelism between entity and set partitives cannot be explained
in terms of semantic ambiguity of partitive words. Instead, I have
argued that what allows for the cross-linguistically widely attested
287
8. Conclusion
distribution of those expressions is the fact that singulars and plurals
actually involve part-whole structures that are based on the same
unified parthood relation. Crucially, what distinguishes the two is an
additional notion that is responsible for how parts are topologically
arranged with respect to each other. In other words, I have posited
that the intuitive distinction between integrated wholes and arbitrary
sums of parts is reflected in grammar by the syntactic distinction
between count singulars and regular plurals, respectively. In addition,
the evidence from Italian irregular plurals shows that there are natural
language expressions designating entities that are similar to plurals in
that they comprise a number integrated objects but at the same time
their sum is arranged in such a way that it constitutes a cluster.
Chapter 3 was dedicated to the systematic exploration of the role
the topological notion of integrity plays with respect to part-whole
structures and subatomic quantification in particular. I have provided
novel evidence that shows that the distinction between topologyneutral
and topology-sensitive partitive words is lexicalized in Polish.
Specifically, Polish has three morphologically distinct half -words, i.e.,
połowa, pół, and połówka all ‘half’. When applied to a DP denoting an
entity, they all yield its part constituting approximately 50% of the
whole. However, where they differ is the type of entity they select for
as well as the type of part they return. Given different distributional
and referential properties of each expression in question, I have interpreted
the alternation in terms of topological sensitivity. While the
half -word połowa is topology-neutral, i.e., it simply measures halves
of any type of entity and yields a topologically indeterminate portion
of a whole, pół and połówka are semantically more complex. Similar
to its topology-neutral counterpart, in the case of pół the outcome of
quantification is either a continuous or discontinuous half. However,
unlike połowa it selects only for integrated wholes, i.e., individuated
entities that come in one piece. On the other hand, the meaning of
morphologically most complex połówka even more restricted. While it
shares selectional restrictions with pół, in addition it imposes topological
constraints on the extension of the resulting partitive construction.
As a result, the whole partitive can only be true of a continuous integrated
part constituting approximately 50% of the volume of an
individuated object, see Table 8.3.
288
8. Conclusion
continuous part discontinous part
połowa
pół
połówka *
Table 8.3: Denotations of Polish half -words
Similar alternations are attested in different classes of other partitive
expressions in Polish, e.g., piece-words and quarter-words. In
each of the examined cases, the notion of integrity played a crucial
role in predicting possible denotations of distinct types of partitives.
Even more interestingly, the cross-linguistic investigation revealed
that the phenomenon observed is not a Polish idiosyncrasy. Instead,
it appears in several other languages. Though the means particular
languages employ to express certain spatial configurations differ, e.g.,
English utilizes determiner whereas Mandarin classifier semantics,
the resulting effect is similar and demonstrates the relevance of the
topological notion of integrity in subatomic quantification.
In addition, I have discussed the contrast between two types of
Polish whole-adjectives (cf. Moltmann 1997, Morzycki 2002), i.e., cały
‘whole’ and kompletny ‘complete, which emphasizes two distinct aspects
of wholeness, namely maximality and integrity. While the first
involves both, the latter does imply that no part is missing but does
not infer that they remain in a particular topological configuration.
The significance of the presented data lays primarily in revealing the
relevance of topological relations holding between parts of individuals
forcing us to recognize that natural language semantics is sensitive to
whether parts come in one piece or constitute discontinuous entities.
maximality integrity
cały
kompletny *
Table 8.4: Interpretations of Polish whole-adjectives
The final piece of data has been discussed in Chapter 4. It concerns
a neglected class of numerical expressions, namely multipliers such
289
8. Conclusion
as English double, that display non-trivial quantificational properties.
Specifically, I have provided novel evidence demonstrating that multipliers
are specialized for subatomic quantification, i.e., they count
cognitively salient parts that are essential for a whole to be considered
having a certain property. In many cases, those parts are self-sufficient,
i.e., have themselves a property comparable to that of the whole, but
sometimes they are just intuitively the most salient elements. Though
the distribution of multipliers involves also abstract entities denoted
by event nominals as well as role nouns, I have argued that the semantic
behavior observed in combination with singular count nouns
constitutes is the basic quantificational mechanism that can be further
extended to other types of entities. Interestingly, when the multiplier
combines with the mass term, it forces a portion, i.e., count interpretation
of the modified noun.
Furthermore, I have discussed Slavic multipliers that display morphological
complexity suggesting semantic compositionality. For instance,
the Russian multiplier dvojnoj ‘double’ consists of the numeral
root corresponding to the number 2 as well as some additional morphology
including a special multiplicative affix. This fact indicates
that multipliers share with cardinals reference to integers, but differ
in that they are devised to count entities of a different type, see Table
8.5. In particular, cardinals are semantically equipped to count
wholes. Though they can be used in subatomic quantification, e.g., in
count explicit partitives, it is only possible when they combine with
a partitive word. In such a case, however, entities designated by the
whole partitive phrase are treated as objects in their own right. Hence,
the source of subatomic quantification is the partitive and the cardinal
simply counts provided entities in their relative entirety. On the
other hand, multipliers always quantify over parts of objects referred
to by a modified nominal. Given the cross-linguistically widespread
appearance of multipliers, the examined data set shows that similar
to cardinals whose purpose is to count wholes natural language developed
expressions dedicated to numerical subatomic quantification.
In Chapter 5, I have provided a general conceptual framework
intended as a basis for developing a formal account for subatomic
quantification in natural language. I have suggested how the linguistic
evidence discussed in Chapter 2, 3, and 4 could be linked with insights
provided by cognitive psychology. Specifically, extensive research on
290
8. Conclusion
cardinals multipliers
subatomic quantification *
quantification over wholes *
Table 8.5: Properties of cardinals and multipliers
perception and cognitive development indicates that since early childhood
we conceptualize objects, i.e., integrated wholes, in a different
way than other types of entities. Furthermore, even young children
possess an ability to simultaneously perceive a whole as an object
in their own right as well as a collection of parts and this capacity
guides their linguistic development. Finally, human number sense
seems to be sensitive to the intuitive notion of integrity. Specifically,
young children always count each separate integrated physical entity
as one. Based on both linguistic and psychological evidence, I
have presented the three claims constituting the conceptual core of
the thesis. I have postulated that natural language is sensitive to the
topological notion of integrity which manifests itself in how nominal
lexicon is classified into different grammatical categories. For instance,
count singular nouns designate entities conceptualized as integrated
wholes. On the other hand, plural nominals presuppose parts of their
referents to be integrated wholes but impose no topological constraints
on them, i.e., denote arbitrary sums individuals. The second claim
concerns the general counting principles, i.e., a set of rules constraining
what can be counted. Specifically, the principle of non-overlap
ensures that enumerated entities are disjoint, i.e., each thing can be
counted once and once only. The principle of maximality demands
that counting involves entities in their mereological entirety, i.e., no
part is left out. Finally, the principle of integrity guarantees that an
entity that can be put in one-to-one correspondence with numbers
needs to be conceptualized as an object that comes in one piece. All
things considered, the general counting principles ensure that only
sets consisting exclusively of elements that are discrete integrated
object can be enumerated. Importantly, there is no reference to atomicity
in the postulated quantificational mechanism. Finally, the third
claim extends the general counting principles to subatomic quantification.
I posit that the set of constraints described above is a universal
291
8. Conclusion
mechanism applicable both at the level of wholes and at the level of
parts.
In Chapter 6, I have introduced a theory of wholes called mereotopology
which enhances standard mereology based on the parthood
relation with additional topological features (Casati and Varzi 1999; for
linguistic application see Grimm 2012b,a). Within this framework, the
primitive notion of connectedness enables us to derive complex notions
capturing distinct configurations of parts within a part-whole structure.
One of such notions is a property of being maximally strongly
self-connected (mssc) which allows us to capture an intuition on what
it means to be an integrated whole. Specifically, an mssc entity is an
object whose every interior part overlaps the whole and anything else
that has a relevant property and overlaps it is part of it. Such a perspective
seems to correspond neatly to the intuitive view on individuals
as configurations of parts arranged in a certain manner rather than
atomic elements with no proper parts, and thus constitutes an advantageous
alternative for accounts based on atomicity. This alternative
view turns out to be of significant value in modeling quantification
over both wholes and parts.
Finally, in Chapter 7 I have developed a formal account for subatomic
quantification based on the conceptual background and mereotopological
notion of mssc discussed above. In particular, I have argued
that an approach building on mssc entities instead of atoms is more
auspicious in modeling quantification over parts. For instance, a significant
advantage of a mereotopological account is that it does not
require postulating sorted domain with different parthood relations
defined over them (Link 1983). The distinction between part-whole
structures associated with singular count nouns and regular plurals
can be recast in terms of connectedness. While the first encode that
the elements of their denotations are mssc entities, the latter require
parts of their referents to be such individuals but impose no topological
constraints on pluralities thereof. Furthermore, I have proposed
that numerical expressions including cardinals and multipliers are
complex semantic expressions derived from names of numbers via
different overt or covert classifiers specialized in counting distinct
types of objects, e.g., mssc entities and cognitively salient parts thereof,
respectively. The last piece of the puzzle concerned how to capture
different semantic interpretations of distinct types of topology-neutral
292
8. Conclusion
and topology-sensitive partitive words. For this purpose, I have proposed
a special individuating element that can be either expressed
formally, e.g., as a special suffix on Polish topology-sensitive partitive
words, or null. In any case, what this individuating element does is
that it partitions the set of entities denoted by a partitive and introduces
the mssc constraint relative to a given partition. This guarantees
that the extension of a whole partitive involves only non-overlapping
continuous parts. Such integrated parts are conceptualized as objects
in their own right within a part-whole structure of an individual, and
thus can be quantified over similar to other mssc entities. With these
components in place, I have provided step-by-step derivations of multiplier
phrases as well as different types of partitives that account for
distinct interpretations of constructions involving topology-neutral
and topology-sensitive partitive words. In the proposed system, distinct
semantic effects arise as a result of different interactions between
partitivity, topology, and quantification. Its great advantage is that
it provides explanation for the topological phenomena concerning
reference to continuous and discontinuous parts in different types of
partitives, an issue that previous approaches failed not only to account
for but even to acknowledge (e.g., Chierchia 2010).
8.2 Open questions and future research
The account developed in this thesis provides explanation for a number
of phenomena in natural language. I believe its contribution is
valuable both from a theoretical perspective and in terms of novel data
it covers. However, some questions remain unanswered and many
issues require further investigation. One of the most serious challenges
concerns the interpretation of count explicit partitives with the embedded
plural DP as discussed in Section 2.2.2. Unlike standard explicit
set partitives, such expressions only get a part-of-a-singularity reading.
I have argued that this effect is due to the general counting principles
which require that a counted entity constitutes an integrated whole.
Since pluralities fail to satisfy this condition, the only possibility how
a relevant phrase could be interpreted involves quantification over
material parts. Nonetheless, the issue concerning the interpretation
of the morpho-syntactic plural on the downstairs nominal remains
293
8. Conclusion
open. The question of what the exact semantic contribution of number
in such a structure is (if any) requires further investigation. It might
be the case that in such constructions the plural is semantically void.
However, there is an intuition that within a set of parts denoted by a
count explicit partitive each element is part of a different object. At
this point, I can only hypothesize that the mechanism deriving this
effect is pragmatic in nature. However, it might also be the case that
the plural is interpreted as a distributive operator similar to what has
been postulated to account for covariational uses of different (see, e.g.,
Beck 2000, Brasoveanu 2008, Dotlačil 2010). Before a detailed analysis
could be proposed, definitely more research is required.
As indicated in Section 7.7, the proposed account can be easily
extended to topology-sensitive partitives in other languages than Polish
and German as well as to other types of constructions involving
subatomic quantification. However, the question whether such an
extension is empirically appropriate will in each case require careful
consideration. After examining the cross-linguistic data presented in
Section 3.4, at first sight it seems plausible that, e.g., in English explicit
and proportional partitives it is the determiner that introduces the
individuating element. Similar, it appears that the same semantics
is associated with a special classifier that follows a partitive word
in Mandarin. However, more tests need to be applied to reveal the
real nature of the interaction between topology, partitivity, and the
determiner or classifier semantics, respectively.
A further issue concerns a detailed analysis of Italian irregular
plurals (Ojeda 1995, Acquaviva 2008). Though the data from count
explicit partitives involving such expressions were of great significance
for capturing the nature of the interaction between partitivity
and numerical quantification, the proper semantic treatment of such
constructions would require accommodating more complex topological
notions. Although this task is suit for another enterprise, I believe
this study provides a neat framework for pursuing it, see Section
7.7.1. Similar, a potentially interesting topic concerns investigation
into partitives in languages displaying the collective-singulative number
alternation (Grimm 2012a,b). Moreover, if the proposed analysis
is on the right track, effects similar to what we have observed in partitives
with irregular plurals in Italian should also be detectable in
Slavic partitives with topologically sensitive derived mass nouns that
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8. Conclusion
denote a plurality of discrete objects forming a cluster (Grimm and
Dočekal to appear) as well as swarm nouns (Henderson 2017). A pilot
study concerning Czech and Polish suggests that this is in fact the case,
yet more work needs to be done before a conclusion can be drawn.
So far, whole-adjectives did not receive a lot of attention in the
semantic literature. In this thesis, I have argued that different types of
Polish expressions of this kind provide evidence for distinguishing
between maximality and integrity. However, a detailed analysis of
the meaning of whole-adjectives awaits for being developed. As a
starting point, one could consider an interaction between universal
quantification at the subatomic level and the individuating element
encoding topological integrity or some similar component. Though the
idea that whole-adjectives could be interpreted as universal quantifiers
over parts has already been proposed (Moltmann 1997). Nevertheless,
careful investigations into scope dependencies and related factors
are required in order to test whether such treatment is empirically
plausible (see Morzycki 2002 for potentially problematic data).
Another area where the proposed perspective could be applied concerns
adverbials quantifying over parts of a singularity such as mostly,
wholly, largely (cf. Morzycki 2002) as well as all wet (cf. Schwarzschild
1996, pp. 162–170). Such explorations would most probably require
extending the account advocated here with the insights of the prolific
research on scale structures (e.g., Kennedy and McNally 1999, 2005). If
both approaches turn out to be compatible, it potentially opens a new
perspective on the opposition between partial and total predicates,
e.g., dirty, wet, and touch as opposed to clean, dry, and eat (e.g., Yoon
1996, Rotstein and Winter 2004).
A separate subject concerns the interpretation of multipliers with
event nominals, role and abstract nouns as well as collectives. In Section
4.2, I have suggested that extending ontology with additional
types for events as well as social roles will allow us to extend the
basic mechanism proposed to account for the semantic behavior of
multiplier modifying concrete nouns to more abstract types of objects.
However, the exact implementation of the idea requires careful consideration
and rigorous study. It might turn out that a more general
notion is necessary or, alternatively, that several derived concepts will
prove more advantageous in tackling the distribution of multipliers
across the board. In any case, figuring this out seems to be an intrigu-
295
8. Conclusion
ing challenge that would almost certainly provide a novel perspective
on quantification in natural language.
If the approach I have argued for is correct, I expect that scrupulous
cross-linguistic investigation will reveal even more types of constructions
indicating the relevance of topological relations within partwhole
structures. Though a number of directions worth examining
has already been indicated, e.g., adjectival half -words in German and
Romance, I believe that there are even more parts of the lexicon that are
sensitive to topological relations at the subatomic level. One potential
example is a class of expressions I refer to as verbs of separation such
as separate, dismantle, and dissolve mentioned briefly in Section 2.3.1. At
first blush, their meaning seems to affect the topological component of
part-whole structures. Examining them from a perspective developed
here might be a promising enterprise that would shed light at subatomic
quantification in the verbal domain. It seems that an especially
auspicious area of research concerns Slavic verbal morphology with
its various prefixes and aspectual distinctions (e.g., Filip 1999).
Furthermore, I believe that rigorous experimental investigations
could also shed new light on subatomic quantification, as they did
in other areas of the field. For instance, what would be of significant
interest is the exploration of the relative strength of topological inferences
in different types of constructions cross-linguistically. It might
be informative to compare structures that encode the individuating
element formally such as Polish and German topology-sensitive proportional
partitives with constructions where a similar effect arises
pragmatically. Similar, I can imagine that testing in multiple contexts
to reveal could also provide interesting insights on different aspects
of quantification over parts.
To conclude, I hope that this study provides a valuable perspective
on previously neglected semantic phenomena in natural language.
Though I have presented a number of novel observations concerning
distinct types of relatively well-studied constructions as well as
genuine insights concerning expressions that so far have not received
attention in the study of meaning, I expect that many more obscure
regions await to be explored. I believe that a detailed map of the phenomena
related to subatomic quantification in natural language can
only emerge as a result of careful systematic typological as well as
296
8. Conclusion
experimental research. This is, however, where other journeys are to
begin.
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