Overt Distributivity in
Algebraic Event Semantics∗
Lucas Champollion
July 22, 2015
Abstract
This is the second in a pair of papers that aim to provide a comprehensive
analysis of the semantic phenomenon of distributivity in natural language. This
paper provides a unified analysis of observable cross-linguistic differences in overt
distributivity operators in the framework of Neo-Davidsonian algebraic event
semantics, drawing on previous work by Zimmermann (2002b,a). The previous
paper postulated two covert distributivity operators, D and Part, in grammar, even
though the semantic effects of D can be subsumed under the workings of Part. This
paper motivates this split by arguing that D and Part are lexicalized as adverbial
and adnominal distributivity operators in individual languages. For example,
English each in its various forms lexicalizes D while the German distributive
operator jeweils lexicalizes Part. For this reason, jeweils occurs in a wider range of
distributive environments, including distribution over contextually given salient
occasions. The proposed analysis explains why those distributive elements that can
also used as distributive determiners, such as English each, never allow distribution
over occasions.
Keywords: distance-distributivity, crosslinguistic semantics, algebraic semantics,
adnominal each, adverbial each, quantifier float, covers
1 Introduction
This paper presents a surface-compositional theory of distance distributivity that
relates adnominal and adverbial distributive elements, atomic and cover-based distri∗For
helpful discussions and comments, I am grateful to Chris Barker, Benjamin Bruening, Dylan
Bumford, Seth Cable, Robert Henderson, Chris LaTerza, Anna Szabolcsi, Linmin Zhang, Malte Zimmermann
and to the audiences of the Stuttgart 2011 workshop on distributivity, the Amsterdam Colloquium 2011,
at the University of Potsdam, and to members of my 2013 NYU seminar on algebraic semantics, where
earlier versions of this paper and its counterpart, Champollion (2014a), were presented. I thank Manuel Križ
and Linmin Zhang for helping me with the surveys reported in this paper. For native speaker judgments
on other languages, I thank my consultants Meike Baumann, Isaac Bleaman, Heather Burnett, Ivano
Ciardelli, Tuğba Çolak-Champollion, Liz Coppock, Hana Filip, Daði Hafþór Helgason, Hildur Hrólfsdóttir,
Gianina Iordăchioaia, Zack Jaggers, Songhee Kim, Sonia Kasyanenko, Sverrir Kristinsson, Jeremy Kuhn,
Chigusa Kurumada, Yohei Oseki, Roumyana Pancheva, Floris Roelofsen, Kjell Johan Sæbø, and Gunnar Ingi
Valdimarsson. This paper and Champollion (2014a) build in part on some earlier work of mine, for which I
am indebted to many people and institutions as indicated there (Champollion, 2012, 2013).
1
butivity, and distributive determiners to each other. This is one of two self-contained
papers that can be read individually but that form a coherent whole. This paper focuses
on overt distributivity. Its counterpart, Champollion (2014a), focuses on covert
distributivity. While this paper builds on results obtained in Champollion (2014a), the
two papers can be read in either order.
Overt and covert distributivity are illustrated in the following examples:
(1) a. The girls each wore a black dress.
b. The girls wore a black dress.
In sentence (1a), the adverbial distributive element each distributes the predicate wear
a black dress over the individual girls and leads to the entailment that each of the girls
in question wears a black dress. Sentence (1b) is interpreted in the same way, even
though there is no each. The ability of verb phrases to distribute in the absence of
an overt distributive element has been attributed to what is traditionally called the D
operator, a silent counterpart of adverbial each (Link, 1987; Roberts, 1987).
The purpose of this paper and of Champollion (2014a) is to bring together several
strands of research on phenomena related to the semantics and pragmatics of
distributivity in natural language. One of these strands deals with overt distributivity,
which is crosslinguistically often expressed via adverbials and adnominals, such as
English each and German jeweils. Such elements differ with respect to whether they are
restricted to distribution over individuals mentioned in the same sentence, or whether
they can also distribute over pragmatically salient occasions that need not have been
explicitly mentioned. This strand is motivated by the properties of adverbial each
and its adnominal and determiner counterparts both in English and other languages
(Zimmermann, 2002b). As we will see, the meanings of these elements varies in ways
that sometimes require them to distribute over individuals, such as in the case of each,
and in other cases allow them to distribute over salient parts of spacetime, such as in
the case of jeweils, as illustrated in the following example. The focus of this paper is
on this strand.
(2) Hans
Hans
hat
has
jeweils
Dist
zwei
two
Affen
monkeys
gesehen.
seen.
(German)
‘Hans has seen two monkeys on each occasion.’
The other strand concerns the properties of silent distributivity operators such as
the one arguably present in (1b). Covert phrasal distributivity has been at the center
of a long debate as to whether it always involves distribution over atoms – singular
individuals – or whether it can also involve distribution over nonatomic entities
(Lasersohn, 1989; Gillon, 1990). This is the focus of Champollion (2014a). I reserve the
term D operator for distributivity operators that always distribute over atoms. As for
the nonatomic version of the operator, whose meaning may be paraphrased as each
salient part of, I will refer to it as the Part operator, following Schwarzschild (1996).
As this sketch already suggests, overt and covert distributivity share many similarities.
In both cases, some elements can only distribute to atoms (each, D) while
others can distribute to salient nonatomic entities (jeweils, Part). And as we will see,
in both cases, the former elements can only distribute over pluralities that have been
2
explicitly mentioned while the latter elements can also distribute over salient domains
that have not been explicitly mentioned, such as temporal occasions. These similarities
give rise to analogous questions in the overt and in the covert case. Can a given
distributive element (be it a covert operator or an overt lexical item) only distribute
down to singular entities or also to plural entities? Do these entities need to be of a
certain size or “granularity”, and can this size vary from element to element? Must
these entities have been overtly mentioned in the sentence and thereby contributed by
semantic means, or can they also be supplied by the context via pragmatic means?
A unified semantic analysis of distributivity should make it apparent which aspects
of the meanings of various distributivity operators are always the same, and along
which dimensions these meanings can differ. The theory should capture the semantic
variation across distributivity-related elements should be captured . The resulting
system should be fully formalized and explicit.
This paper, together with Champollion (2014a), contributes towards these goals. By
combining ideas from algebraic semantics and event semantics, the two papers provide
a framework in which the split in overt distance-distributive elements can be related
to the debate in the literature on covert distributivity. In this framework, the various
uses of each in English are all lexically related to the distributivity operator, either in
its semantic, atomic form as defined by Link (1987) or in its pragmatic, salience-related
form as defined by Schwarzschild (1996). As we will see, these various uses of each
and these silent operators share some part of their meanings with each other and
with their counterparts across languages. This fact is captured by deriving them from
related distributivity operators which differ only in possible settings of two parameters
and the ranges of values they allow for them. One parameter indicates the dimension
in which distributivity takes place. This can be a thematic role in some semantic
instances of distributivity, or a spatial or temporal dimension in other instances. The
other parameter indicates the size of the entities over which distributivity takes place,
such as atoms or salient amounts of space or time. These parameters interact with
each other against the background of assumptions about the metaphysics of natural
language. For example, time is assumed to be either nonatomic or in any case to not
make its atoms available to the semantics of natural language. As a result, when the
first parameter is set to time, the second cannot be set to anything involving atoms,
because time does not provide any atoms to distribute over. This simple idea turns
out to explain and connect a range of superficially unrelated facts observed in various
places in the literature. It is situated within a broader framework that connects it to
aspect and measurement under the name of strata theory (Champollion, 2010b).
In this paper, I provide a view on the seemingly noncompositional behavior of
distance-distributive elements on which they look much more well-behaved than
might be expected. The analysis is surface-compositional, reuses many independently
motivated assumptions, and avoids unusual semantic mechanisms such as index-driven
and crosswise lambda abstraction (Zimmermann, 2002b). I also avoid nonstandard and
otherwise unmotivated concepts such as distributive polarity items (Oh, 2001, 2006).
The analysis is placed in the context of algebraic event semantics. This allows us to
formally model the relations between distribution over individuals and over events, as
well as those between distribution over atoms and over nonatomic parts. The paper
provides a formal framework for algebraic event semantics and makes an explicit
3
proposal for how the compositional process can be modeled.
When we move to algebraic event semantics, the theory of distributivity operators
developed by Link (1987) and extended by Schwarzschild (1996) turns out to require
adjustments for a number of reasons, as discussed in Champollion (2014a). As shown
by the following examples, the Neo-Davidsonian event semantic setting gives us
the ability to think of the D and Part operators as being coindexable with different
thematic roles. This allows us to capture through a simple change in coindexation the
kinds of configurations that have otherwise been taken to require type-shifting-based
reformulations of these operators (Lasersohn, 1998):
(3) a. The first-year students D(took an exam). Target: agent
b. John D(gave a pumpkin pie) to two girls. Target: recipient
The reformulation of the distributivity operators in Champollion (2014a) provides the
groundwork on which I build in this paper in order to formally relate this ambiguity
to the one observable in examples like the following:
(4) The boys told the girls two stories each. Target: agent
(two stories per boy)
(5) The boys told the girls two stories each. Target: recipient
(two stories per girl)
To capture this and other parallels between covert and overt distributivity, I will
propose that distance-distributive elements across languages are in essence overt
versions of the D and Part operators.
The theoretical picture that has been sketched so far, and that is developed below
and in Champollion (2014a), provides us with a way to formulate commonalities and
differences across instances of distributivity in natural language. Individual elements
can be analyzed as being hard-wired for certain parameter values, so that, for example,
the difference between Link’s and Schwarzschild’s operators, as well as that between
each and jeweils, can be described in terms of whether the value of the granularity
parameter is prespecified to Atom or can be filled in by context. In this way, overt
and covert instances of distributivity fit together and into distributivity theory more
generally.
To develop this picture, Section 2 starts by describing relevant facts and generalizations
about overt instances of distributivity across languages, drawing largely on
the crosslinguistic discussion in Zimmermann (2002b). The two strands – overt and
covert distributivity – are brought together in Section 3, which develops a surfacecompositional
account of overt distance distributivity. Section 4 compares the present
analysis with Zimmermann (2002b). Section 5 extends the analysis to the determiners
each and every, and shows that it accounts for cumulative readings of the latter.
The way in which meanings of overt distributive elements vary across languages is
explained in Section 6. Section 7 deals with more complicated syntactic configurations,
some of which previously lacked a surface-compositional analysis. Section 8 concludes.
4
2 Overt Distributivity Across Languages
Distributive elements have different syntactic uses and different meanings across
languages. In English, the distributive quantifier each can be used in at least three
ways, which I will refer to as adnominal, adverbial, and determiner each respectively:
(6) a. Adnominal each: Two men have carried three suitcases each.
b. Adverbial each: Two men have each carried three suitcases.
c. Determiner each: Each man has carried three suitcases.
There are many terms for these three uses. Adnominal each has also been called shifted
(Postal, 1974), an anti-quantifier (Choe, 1987), binominal (Safir and Stowell, 1988), or
ditransitive (Roberts, 1987). Adverbial each has also been called floated (Choe, 1987).
Determiner each is also called prenominal (Safir and Stowell, 1988). For the purpose of
this paper, I set aside the use of each in other constructions, notably the reciprocal each
other and the partitive each of the men. On the connection between reciprocals and
each, see for example LaTerza (2014b). (I have not been able to access LaTerza (2014a)
in the preparation of this paper, but I understand from its author that it is likely to be
relevant to this discussion.) I will refer to the noun phrase two men in (6a) and (6b)
as the antecedent of each, and to the noun phrase three suitcases in (6a) as the host of
adnominal each.
I will refer to adnominal and adverbial each and to similar elements across languages
as distance-distributive elements. That term is taken from Zimmermann (2002b).
There is a slight difference in terminology: Zimmermann reserves the term distance
distributivity for adnominal elements while I use it both for adnominal and for adverbial
elements. This seems appropriate because adverbial each can be separated from its
antecedent, for example by an auxiliary as shown in (6b).
Adnominal each can be shown by movement tests to form a constituent with its
host noun phrase (Burzio, 1986; Safir and Stowell, 1988). Distance-distributive elements
like it are sometimes thought of as a challenge for compositional semantics, because
their interpretations are similar to those of distributive determiners even though their
surface syntactic structure appears to be fundamentally different (Oh, 2001, 2006). For
example, adnominal each in the object of sentence (6b) is contained in the constituent
over which it seems at first sight to distribute, namely the verb phrase carried three
suitcases each. This is of course similar to the challenge represented by quantifiers in
object position (carried every suitcase), and the standard solutions to that challenge
are available in both cases. For example, one can lift the type of the quantifier or the
verb in order to give the quantifier scope over the verb phrase (Hendriks, 1993; Barker,
2002). I will follow the same general strategy in the formal analysis below. In fact,
I will assume that the scope of adnominal each is even more restricted than that of
object quantifiers: it takes scope only over its host noun phrase (Dotlačil, 2011, 2012;
LaTerza, 2014b). So there is no special challenge for compositionality.
As we have seen in the pair of examples in (4) and (5) above, adnominal each can
target different antecedents. This dependency is generally regarded as a case of ambiguity
rather than underspecification, and I will follow this view. There are syntactic
constraints on the distribution of adnominal each with respect to its antecedent, such as
5
c-command requirements and clausemate conditions. Accordingly, the dependency has
been variously argued to be similar to that of reflexive pronouns with respect to their
antecedents (Burzio, 1986; Safir and Stowell, 1988) or to that of traces of noun phrases
that undergo raising with respect to these noun phrases (Sportiche, 1988). Similarly,
adverbial each has been variously claimed to be related to its antecedent by movement,
in the sense that it modifies the trace of its antecedent, or to be base-generated, in
which case its relation to its antecedent can be taken to be anaphoric. For an overview
of these conflicting claims and their implications, see Bobaljik (2001).
I will not have anything to add to the discussion on these syntactic constraints.
Since the nature of the dependency between adnominal and adverbial each and their
antecedents is not the focus of this paper, I will not take a strong position on it. In the
formal theory to be developed below, I will represent it via coindexing of thematic
relations, which is more in line with anaphora-based accounts than with movementbased
ones. In this, I follow previous semantic analyses that interpret adnominal each
without any movement, such as Zimmermann (2002b). My coindexing-based analysis
is compatible with syntactic accounts because it treats each as an anaphoric element. I
assume that coindexing of thematic relations is similar to other dependencies that are
commonly formalized by coindexation, for example those between reflexive pronouns
and their antecedents. These dependencies are subject to binding theory constraints,
which can vary from language to language (Chomsky, 1981; Büring, 2005). Should the
movement-based view turn out to be the correct one, the syntax-semantics interface
may need to be modified accordingly, for example by incorporating elements from the
theory of Cable (2014).
Turning now to other languages, adnominal and adverbial each are translated in
German by one word, jeweils (Moltmann, 1997; Zimmermann, 2002b). Determiner each,
however, is translated by another word, jed-. I gloss distance-distributive elements as
Dist since, as we will see, they have a wider range of readings than each. Example
(7a) is adnominal, example (7b) is adverbial, and example (7c) contains a determiner.
Though adverbial and adnominal jeweils take the same surface position in (7a) and
(7b), they can be teased apart syntactically, as discussed in Zimmermann (2002b).
(7) a. Die
The
Jungen
boys
haben
have
[jeweils
Dist
[drei
three
Koffer]]
suitcases
getragen.
carried.
b. Die
The
Jungen
boys
haben
have
[jeweils
Dist
[drei
three
Koffer
suitcases
getragen]].
carried.
c. Jeder/*Jeweils
Each.sg.m/Dist
Junge
boy
hat
has
drei
three
Koffer
suitcases
getragen.
carried.
As we will see, each and jeweils generalize to two classes of distance-distributive
elements across languages. Each-type distance-distributive elements can also be used
as determiners. Jeweils-type distance-distributive elements cannot double as determiners.
Some languages have distance-distributive elements which can also function
as distributive determiners, as in English, and others are like German in that they
have no such elements (Zimmermann, 2002b). Across these languages, Zimmermann
observes that distance-distributive elements which can also be used as determiners
(e.g. each) always distribute over individuals, as determiners do. In contrast, many of
6
those distance-distributive elements which are formally distinct from determiners (e.g.
jeweils) can also distribute over salient occasions, that is, over chunks of time or space.
Let me illustrate this observation by using German jeweils, a distance-distributive
element which cannot double as a distributive determiner. Jeweils can distribute over
individuals like English each, but also over spatial or temporal occasions, as long as
context provides a salient set of such occasions. I call this the occasion reading. It
corresponds to what is also called the spatial key reading and the temporal key reading
(Balusu, 2005; Balusu and Jayaseelan, 2013). I leave open the question of whether or
not the spatial and temporal cases should be distinguished as two separate readings.
The framework I will present can accommodate both possibilities. Another, less theoryneutral
term for the occasion reading is event-distributive reading (Oh, 2001, 2006).
Zimmermann (2002b) uses the term adverbial reading for it. This term is potentially
misleading, because it suggests that only the adverbial use of jeweils can give rise to
this reading. But adnominal jeweils can give rise to it as well (Zimmermann, 2002b, ch.
5). For example, in (8), jeweils is part of the subject noun phrase (we know this because
German as a V2 language allows only one constituent before the tensed verb standen)
and is therefore adnominal. However, as shown by the paraphrase, this instance of
jeweils distributes over occasions, not over individuals.
(8) Jeweils
Dist
zwei
two
Jungen
boys
standen
stood
Wache.
watch.
‘Each time, two boys kept watch.’ (German)
The following examples illustrate the occasion reading. Sentence (9) is ambiguous
between a reading that distributes over individuals – the ones of which their plural
subject consists, (9a) – and one that distributes over occasions (9b).
(9) Die
The
Jungen
boys
haben
have
jeweils
Dist
zwei
two
Affen
monkeys
gesehen.
seen.
a. ‘Each of the boys has seen two monkeys.’
b. ‘The boys have seen two monkeys each time.’ (German)
While the former reading is always available, the latter requires a supporting context.
That is, when (9) is uttered out of the blue, it only has the reading (9a). The reading
(9b), by contrast, is only available in contexts where there is a previously mentioned
or otherwise salient set of occasions, such as contexts in which the boys have been to
the zoo on several previous occasions.
Unlike each, jeweils can also occur with a singular subject, as in (10), repeated here
from (2), which only has an occasion reading.
(10) Hans
Hans
hat
has
jeweils
Dist
zwei
two
Affen
monkeys
gesehen.
seen.
‘Hans has seen two monkeys on each occasion.’ (German)
This sentence is odd out of the blue, and it requires supporting context in the same way
as reading (9b) does. Its other potential reading would involve vacuous distribution
over only one individual, Hans. This is presumably blocked through the Gricean maxim
7
of manner “Be brief” or a non-vacuity presupposition or implicature or whatever else
prevents vacuous distributivity (Roberts, 1987, p. 219). For more discussion of this point,
and for a fuller discussion of the kinds of noun phrases that can license adnominal
each in English, see Champollion (2014b).
In the presence of contextual cues, jeweils is also able to distribute over nonatomic
pluralities. Since this data point will be critical in the following development, I confirmed
it through a large-scale web survey (300 participants) run in June 2015 on the
online platform Ibex Farm (Drummond, 2013). Participants were recruited using the
German online labor market platform Clickworker (http://www.clickworker.de). The
survey contained the following target item and question:
(11) Dreißig
Thirty
Jungen
boys
wurden
were
in
in
Dreiergruppen
triplet.groups
aufgeteilt.
divided.
Sie
They
bekamen
received
jeweils
Dist
einen
one
Ball.
ball.
‘Thirty boys were divided into groups of three. Each group (or: each boy)
received one ball.’
(12) Wie
How
viele
many
Bälle
balls
wurden
were
insgesamt
altogether
verteilt?
distributed?
‘How many balls were distributed in total?’
Participants were asked to type their answer to this question into a text field. A filler
task and demographic questions were used to check whether subjects paid enough
attention and whether they satisfied the criteria for the survey. The filler task consisted
of a simple question:
(13) In
In
wie
how
viele
many
Gruppen
groups
wurden
were
die
the
Jungen
boys
aufgeteilt?
divided?
The target and filler tasks were displayed in random order, followed by the demographic
questions. 30 participants responded to the filler with an answer distinct from ?10?
(mostly ?3?) and were removed from the results. In addition, 36 participants were
removed because they were nonnative speakers of German or did not spend all of their
childhood in Germany. The remaining 234 data points are summarized in Table 1.
Choice Raw frequency Percentage
(i) 10 153 65%
(ii) 30 78 33%
(iii) 3 3 1%
(iv) other 0 0%
Total 234 100%
Table 1: Results of the survey on nonatomic readings with jeweils
These results suggest that about two thirds of the participants interpret jeweils as
meaning “per group”, while one third interprets it as meaning “per boy”. For reasons
8
discussed in Champollion (2014a), I model these pluralities of boys as salient nonatomic
entities, rather than as “group atoms” in the sense of Landman (1989).
The previous example suggests that a contextually salient distribution over pluralities
of individuals is sufficient for nonatomic interpretations of jeweils. In fact, it is a
prerequisite. Out of the blue, that is in the absence of supporting context, jeweils is not
able to distribute over nonatomic entities. This can be seen in the following sentence
suggested by a reviewer, which is based on an example in Gillon (1987) (for discussion,
see Champollion (2014a):
(14) Rodgers,
Rodgers,
Hammerstein
Hammerstein
und
and
Hart
Hart
haben
have
jeweils
Dist
ein
a
Musical
musical
geschrieben.
written.
Intended: ‘There is a cover over Rodgers, Hammerstein and Hart such that
each part of that cover wrote a musical.’
Out of the blue, (14) is false in the actual world, even though Rodgers and Hammerstein
wrote a musical together and Rodgers and Hart wrote another musical together. The
mere existence of a nonatomic cover is not sufficient to make a jeweils sentence true.
Rather, its cells – the two nonatomic entities that each wrote a musical together – need
to be made salient. Since (14) is uttered out of the blue, the relevant cover is not salient,
and so the putative reading on which the sentence would be true does not arise.
While jeweils allows distribution both over individuals and over salient nonatomic
entities, this is not the case for all distance-distributive elements (Zimmermann, 2002b).
Across languages, many adnominal distance-distributive elements can only distribute
over individuals. For example, English adnominal each lacks the occasion reading:
(15) The boys saw two monkeys each.
a. Available: ‘Each of the boys saw two monkeys.’
b. Unavailable: ‘The boys saw two monkeys on each occasion.’
When adnominal each is used in a sentence whose subject is singular, distribution over
individuals is not possible, again presumably for pragmatic reasons:
(16) *John saw two monkeys each.
Unlike (10), this sentence lacks an occasion reading, even with supporting context. To
make the occasion reading surface, one needs to add an overt noun like time:
(17) John saw two monkeys each time.
We have seen that English each also differs from German jeweils in that only the former
can also be used as a determiner. Coming back to what I mentioned at the beginning of
this section, Zimmermann reports the following generalization (Zimmermann, 2002b):
(18) Zimmermann’s generalization: All each-type distance-distributive elements
(i.e. those that can also be used as determiners) can only distribute over individuals.
This contrasts with jeweils-type distance-distributive elements, many
of which can also distribute over salient spatial or temporal occasions.
This generalization is based in part on the following examples, which show that
9
distance-distributive elements in Dutch, French, Italian, Icelandic, Japanese, Norwegian,
and Russian all behave like English each in two ways: They can also be used as
distributive determiners, and they lack the occasion reading except when an extra
noun time is added. Many of the following examples are taken from Zimmermann
(2002b).
(19) De
the
jongens
boys
hebben
have
elk
Dist
twee
two
boeken
books
gelezen.
read
‘The boys have read two books each.’ (Dutch)1
(20) Elk
Dist
jonge
boy
heeft
has
twee
two
boeken
books
gelezen.
read
‘Each boy has read two books’ (Dutch)2
(21) Hans
Hans
heeft
has
elke
Dist
*(keer)
time
twee
two
boeken
books
gelezen..
read.
‘Hans has read two books each time.’ (Dutch)3
(22) Les
the
professeurs
professors
ont
have
lu
read
deux
two
livres
books
chacun/chaque.
Dist
‘The professors have read two books each.’ (French)4
(23) Chaque
Dist
professeur
professor
a
has
lu
read
deux
two
livres.
books
‘Each professor has read two books.’ (French)5
(24) Pierre
Pierre
a
has
lu
read
deux
two
livres
books
chaque
Dist
*(fois)
time
/ *chac-un(e)(fois).
‘Pierre read two books each time.’ (French)6
(25) Strákarnir
the.boys
keyptu
bought
tvær
two
pylsur
sausages
hver.
Dist
‘The boys bought two sausages each.’ (Icelandic)7
(26) Hver
Dist
strákur
boy
keypti
bought
tvær
two
pylsur.
sausages.
‘Each boy bought two sausages.’ (Icelandic)8
(27) Pétur
Peter
keypti
bought
tvær
two
pylsur
sausages
hvert
Dist
*(sinn).
time.
1Zimmermann (2002b, p. 40)
2Zimmermann (2002b, p. 44)
3Floris Roelofsen, p.c. to the author
4Tellier and Valois (1993, p. 574, ex. 1a) quoted in Zimmermann (2002b, p. 41). Chaque is colloquial as
an adnominal. While French adnominal chacun and determiner chaque are not exactly identical, they are
historically related and can still be considered formally identical (Grevisse, 1980; Junker, 1995; Zimmermann,
2002b, p. 44, fn. 30).
5Zimmermann (2002b, p. 44)
6Author’s judgment, adapted from Zimmermann (2002b, p. 47)
7Meike Baumann, Daði Hafþór Helgason, Hildur Hrólfsdóttir, Sverrir Kristinsson, Gunnar Ingi Valdimarsson
(p.c.)
8Meike Baumann, Daði Hafþór Helgason, Hildur Hrólfsdóttir, Sverrir Kristinsson, Gunnar Ingi Valdimarsson
(p.c.)
10
‘Peter bought two sausages each time.’ (Icelandic)9
(28) I
the
ragazzi
boys
comprarono
bought
un
a
libro
book
ciascuno.
Dist
‘The boys bought one book each.’ (Italian)10
(29) Ciascun
Dist
ragazzo
boy
ha
has
comprato
bought
due
two
salsicce.
sausages
‘Each boy has bought two sausages.’ (Italian)11
(30) *Peter
Peter
ha
has
comprato
bought
due
two
salsicce
sausages
ciascun/-o/-e.
Dist
Intended: ‘Peter has bought two sausages.’ (Italian)12
(31) Otoko=tati-ga
men=pl-nom
sorezore
Dist
huta=ri-no
two=cl-gen
zyosei-o
women-acc
aisi
love-asp
teiru
fact
koto.
‘The men love two women each.’ (Japanese)13
(32) Sorezore-no
Dist-gen
gakusei-ga
student-nom
iti=dai-no
one=cl-gen
piano-o
piano-acc
motiage-ta.
lift-past
‘Each student lifted one piano.’ (Japanese)14
(33) Taroo-wa
Taroo-top
sorezore-?(de)
Dist(-loc)
iti-dai-no
one-cl-gen
piano-o
piano-acc
motiage-ta.
lift-past
‘Taroo lifted one piano on each occasion.’ (Japanese)15
(34) Guttene
boys-the
har
have
kjøpt
bought
to
two
pølser
sausages
hver.
Dist
‘The boys bought two sausages each.’ (Norwegian)16
(35) Hver
Dist
gutt
boy
har
has
kjøpt
bought
to
two
pølser.
sausages
‘Each boy has bought two sausages.’ (Norwegian)17
(36) Jon
Jon
har
have
kjøpt
bought
to
two
pølser
sausages
hver
Dist
*(gang).
Intended: ‘Jon has bought two sausages each time.’ (Norwegian)18
(37) Mal’chiki
boys.nom
kupili
bought
(po) dve
two
sosiski
sausage.gen.sg
kazhdyj.
Dist
‘The boys bought two sausages each.’ (Russian)19
9Meike Baumann, Daði Hafþór Helgason, Hildur Hrólfsdóttir, Sverrir Kristinsson, Gunnar Ingi Valdimarsson
(p.c.)
10Burzio (1986, p. 198, ex. 50b) quoted in Zimmermann (2002b, p. 41)
11Zimmermann (2002b, p. 44)
12Ivano Ciardelli, p.c. to the author
13Sakaguchi (1998, p. 115, ex. 1) quoted in Zimmermann (2002b, p. 41)
14Sakaguchi (1998, p. 4, ex. 7)
15Chigusa Kurumada, p.c. to the author. Kurumada comments that the sentence without de feels like an
elliptical version of the sentence with de.
16Øystein Vangsnes, p.c. to Zimmermann (2002b, p. 40)
17Zimmermann (2002b, p. 44)
18Kjell Johan Sæbø, p.c. to the author
19Olga Borik, p.c. to Zimmermann (2002b, p. 41)
11
(38) Kazhdyj
Dist
mal’chik
boy
kupil
bought
dve
two
sosiski.
sausages
‘Each boy bought two sausages’ (Russian)20
(39) Petja
Petja
pokupal
buy.perf
dve
two
sosiski
sausage
kazhdyj
Dist
*(raz).
(time)
‘Peter bought two sausages each time.’ (Russian)21
Zimmermann’s generalization states that every distance-distributive element that
can be used as a determiner lacks the occasion reading. The opposite is not the case,
however. For example, the Romanian distance-distributive element cîte lacks the
occasion reading, but it cannot be used as a determiner. This is illustrated in the
following:
(40) Doi
two
oameni
men
au
have
cărat
carried
cîte
Dist
trei
three
valize.
suitcases
‘Two men have carried three suitcases each.’ (Romanian)22
(41) *Cît(e)
Dist
student
student
a
has
plecat.
left.
Intended: ‘Each student has left.’ (Romanian)23
(42) *Petru
Peter
a
has
cîs, tigat
won
cîtă/cîte
Dist.f/Dist.adv
(dată).
time.
Intended: ‘Peter won each time.’ (Romanian)24
Zimmermann considers Japanese as an example of the same phenomenon that I have
illustrated with Romanian, but whether this is correct is not very clear. Zimmermann
bases his view on the fact that the Japanese distance-distributive item sorezore differs
formally from what he calls the Japanese distributive determiner-quantifier wh...+ mo,
which is illustrated in (43).
(43) Dono
which
gakusei-mo
student-mo
sooseezi-o
sausage-acc
hutatu
two-cl
katta.
bought
‘Every student bought two sausages.’ (Japanese)25
However, sorezore can also be used in the position of a determiner, as example (32)
above shows. The syntactic status of sorezore in this example, and therefore the import
of Japanese on Zimmermann’s generalization, is debatable since Japanese is usually
assumed to lack overt determiners.
While Romanian and possibly Japanese are counterexamples to the inverse of
Zimmermann’s generalization, that inverse direction still describes a tendency. That
is, in many languages where adnominal distance-distributive elements have occasion
readings, they cannot be used as determiners. In addition to German jeweils, adnominal
20Olga Borik, p.c. to Zimmermann (2002b, p. 44)
21Sonia Kasyanenko, p.c. to the author
22Gil (1982, p. 19, ex. 1f), Zimmermann (2002b, p. 41)
23Brasoveanu and Farkas (2011, p. 10), Gianina Iordăchioaia, p.c.
24Gianina Iordăchioaia, p.c. There is a related expression cîte-o-dată that means from time to time.
25Satoshi Tomioka, p.c. to Zimmermann (2002b, p. 45)
12
distance-distributive elements in Bulgarian, Czech, and Korean have occasion readings
cannot be used as determiners, as is shown below. Most of these observations are due
to Zimmermann (2002b). The Korean case is also discussed in depth by Choe (1987)
and by Oh (2001, 2006).
(44) John
John
i
and
Mary
Mary
kupiha
bought
po
Dist
edna
one
tetradka.
notebook
‘John and Mary bought one notebook each.’ (Bulgarian)26
(45) Vsjako/*Po
Dist
momche
boy
kupi
bought
dve
two
jabulki.
sausages
‘Each boy bought two sausages.’ (Bulgarian)27
(46) Mary
Mary
byaga
runs
po
Dist
5
5
mili
miles
predi
before
zakuska.
breakfast
‘Mary runs 5 miles before breakfast (every morning).’ (Bulgarian)28
(47) Chlapci
boys
koupili
bought
po
Dist
dvou
two
párcích/párkách.
sausages.loc
‘The boys bought two sausages each.’ (Czech)29
(48) Každý/*Po
Dist
chlapec
boy
koupil
bought
dva
two
párky.
sausages
‘Each boy bought two sausages.’ (Czech)30
(49) Po
Dist
třech
three.loc
ženách
women.loc
vstupovalo
entered.3sg
do
into
místnosti.
room
‘Three women entered the room [i.e., one triplet after another]’ (Czech)31
(50) ai-tul-i
child-pl-nom
phwungsen-hana-ssik-ul
balloon-one-Dist-acc
sa-ess-ta.
bought
‘The children bought one balloon each.’ (Korean)32
(51) Sonyen-mata
boy-Dist
chayk-ul
book-acc
twu
two
kwen-ssik
cl-dist
sa-(a)t-ta.
buy-past-dec
‘Every boy bought two books.’ (Korean)33
(52) na-nun
I-top
[phwungsen-
balloon
hana-ssik-ul]
one-Dist-acc
sa-ess-ta.
bought
‘I bought a balloon (each time / each day / at each store).’ (Korean)34
Many languages express adnominal distance distributivity by a bound morpheme
(either an affix or a reduplicative morpheme) that attaches to a numeral (Gil, 1982).
In this category, on the one hand, we find cases where this process does not give rise
26Petrova (2000) quoted in Zimmermann (2002b, p. 41)
27Milena Petrova, p.c. to Zimmermann (2002b, p. 45); Roumyana Pancheva, p.c. to the author
28Petrova (2000, ex. 3b) quoted in Zimmermann (2002b, p. 41)
29Hana Filip, p.c. to Zimmermann (2002b, p. 41)
30Hana Filip, p.c. to Zimmermann (2002b, p. 45) and to the author
31Hana Filip, p.c. to Zimmermann (2002b, p. 47) and to the author
32Choe (1987, p. 49, ex. 13) quoted in Zimmermann (2002b, p. 41)
33Kim, p.c. to Zimmermann (2002b, p. 45)
34Choe (1987, p. 49, ex. 13) quoted in Zimmermann (2002b, p. 47)
13
to occasion readings, such as Hungarian (Farkas, 1997; Szabolcsi, 2010) and Turkish
(Tuğba Çolak-Champollion, p.c.). On the other hand, we find cases where it does,
such as Telugu (Balusu, 2005; Balusu and Jayaseelan, 2013) and Tlingit (Cable, 2014).
All these cases are illustrated below. The import of these facts on Zimmermann’s
generalization is unclear, since bound morphemes are not expected to be able to act as
determiners. I mention them here for completeness and because the compositional
analysis given below extends to them.
(53) pilla-lu
kid-pl
renDu
two
renDu
two
kootu-lu-ni
monkey-pl-acc
cuus-ee-ru.
see-past-3pl
a. ‘The kids each saw two monkeys.’
b. ‘The kids saw two monkeys each time.’
c. ‘The kids saw monkeys in groups of two.’ (Telugu)35
(54) A
The
gyerekek
children
két-két
two-two
majmot
monkey.acc
láttak.
saw.3pl
a. Available: ‘Each of the children saw two monkeys.’
b. Unavailable: ‘The children saw two monkeys each time.’ (Hungarian)36
(55) Çocuklar
The children
iki-ş-er
two.Dist
sosis
sausage
aldı.
bought.
‘The children bought two sausages each.’ (Turkish)37
(56) *Can
Can
iki-ş-er
two.Dist
sosis
sausage
aldı.
bought.
Intended: ‘Can bought two sausages each time.’ (Turkish)38
(57) Nás’gigáa
three.Dist
xáat
fish
has aawasháat.
3pls.3o.caught
a. ‘They caught three fish each.’
b. ‘They caught three fish each time.’ (Tlingit)39
The facts discussed so far suggest the following requirements for a semantic analysis of
distance-distributivity. First, the synonymy of the determiner, adnominal and adverbial
uses of each in English should be captured, ideally by essentially identical lexical
entries. Second, the fact that distance-distributive elements across languages share
some part of their meanings (namely their individual-distributive readings) should be
represented, as well as the fact that some of them can also have occasion readings. Third,
the analysis should clarify the connections between distance-distributive elements
and distributivity theory more generally, and the semantic variation across distancedistributive
elements should be captured. Finally, an explanation for Zimmermann’s
generalization should be readily available. The rest of the paper develops an analysis
35Balusu and Jayaseelan (2013, p. 67, ex. 15a)
36Szabolcsi (2010, p. 138, ex. 99)
37Tuğba Çolak-Champollion, p.c. to the author.
38Tuğba Çolak-Champollion, p.c. to the author. This sentence is unacceptable on the intended reading.
According to my consultant, it may be thatăit is acceptable under the reading Can bought two sausages of
each kind.
39Cable (2014, ex. 3b)
14
that fulfills these requirements.
The semantic analysis I will propose is semantic and not syntactic in nature. As
such, it does not aim to explain every crosslinguistic difference there is. For example,
I will not be able to shine additional light on why English-type languages are more
reluctant to allow for the inverse distribution of subject over object denotations than
some German-type languages are.
(58) *One journalist each interviewed the politicians.
(59) Jeweils
Dist
ein
one
Journalist
journalist
interviewte
interviewed
die
the
Politiker.
policitians.
‘The politicians were interviewed by one journalist each.
See Zimmermann (2002b, Sect. 5.4.2 and 5.4.3) for extensive discussion and a syntactic
account.
3 Relating Overt and Covert Distributivity
The connection between the D operator from Link (1987) and adverbial each that was
illustrated in (1) has been noted many times. I take adverbial and adnominal each and
related distance-distributive elements in Dutch, French, Italian, Japanese, Norwegian,
and Russian to be essentially D operators. These are the languages mentioned in Section
2 as being English-type. As for jeweils and its relatives in German-type languages like
Bulgarian, Czech and Korean, we have seen that they can distribute over spatial and
temporal intervals – arguably nonatomic entities. Link’s D operator always distributes
down to individual atoms and can therefore not be extended to these cases. So I will
connect them to the nonatomic distributivity operator Part from Schwarzschild (1996).
I now present a formally explicit, surface-compositional account of the overt distributive
elements described in Section 2 in terms of the covert distributive operators D
and Part discussed in Champollion (2014a) and in the literature on covert distributivity
as summarized there. For the sake of brevity, I will only execute the analysis for English
each and German jeweils, but it should be clear how to extend it to other distributive
elements depending on which one of these two they pattern with. I will only show
how to model the individual-distributive and the temporal occasion readings. The
extension from the temporal to the spatial occasion reading is straightforward.
The guiding idea of the analysis is that overt distributive elements include two
versions of the distributivity operator. Each includes the atomic distributivity operator
D, which can only distribute over count domains because only those domains have
atoms. Jeweils includes the nonatomic distributivity operator Part. I argue in Champollion
(2014a) that the latter operator can also distribute over noncount domains
like time. I adopt the strata-theoretic perspective from Champollion (2010b). According
to this theory, distributivity is a property with two parameters: dimension and
granularity. I will suggest that each, just like the D operator, comes prespecified for
“granularity=atom”. This blocks the setting “dimension=time”, so distributivity over
occasions is unavailable. By contrast, jeweils does not come prespecified for anything
but is anaphoric on context, so it can distribute over salient covers, or salient stretches
15
of time, just like the Part operator.
In claiming that each is an overt form of the D operator, I loosely follow proposals
made for German jeweils and its short form je by Link (1998b, 1987) and for English
each by Roberts (1987). While Link and Roberts did not give explicit compositional
implementations and did not or not fully consider the crosslinguistic picture, this paper
can be seen, in a way, as an update to these early ideas which benefits from later work on
algebraic semantics, nonatomic distributivity, and compositional implementations. In
particular, Link and Roberts did not have access to the theory that has been built around
the commonalities and differences between the D and Part operators (Champollion,
2014a, and references therein).
Arriving at the meaning of each from the meaning of distributivity operators is
somewhat of the reverse of the process by which Schwarzschild arrived at his Part
operator, which “was based on a generalization of Dowty and Brodie’s (1984) account
of floated quantifiers as verb phrase modifiers” (Schwarzschild, 1996, p. 137).
Schwarzschild himself notes that the history of Part should not be taken for an endorsement
that floated quantifiers are related to it, and argues that floated quantifiers
should be distinguished from Part operators because he takes reciprocals to be licensed
by distributivity operators, but not by adverbial each. He gives the following examples
to support this claim. These kinds of examples, as well as the idea that reciprocals are
licensed by distributivity operators of some kind or other, go back to Heim, Lasnik,
and May (1991).
(60) a. Theyj Parti [saw each otherj,i].
b. *Theyj eachi saw each otherj,i.
While reciprocals are not the topic of this paper, let me briefly note that Schwarzschild’s
argument rests crucially on the assumption that reciprocals are licensed by VP-level
distributivity operators. This has been argued to make wrong predictions, since not
all VPs with reciprocals in them are interpreted distributively (see Dotlačil (2013) and
references therein):
(61) a. John and Mary wrote to each other on two cold days.
?? under the reading ‘John wrote to Mary on two cold days and Mary
wrote to John on two other cold days’ (Moltmann, 1992)
b. The doctors gave each other a new nose.
?? under the reading ‘each doctor gave the other doctor a different new
nose’ (Williams, 1991)
c. The two children gave each other a Christmas present.
?? under the reading ‘each child giving a different present’ (Williams,
1991)
I conclude that neither overt and covert adverbial distributive operators seem to be
able to license reciprocals, and therefore there is in principle no reason to avoid giving
them a unified analysis.
I use the following typing conventions: t for propositions, e for ordinary objects,
and v for events. I use the symbols x, y, z, x , y , z and so on for variables that range
over ordinary objects, and e, e , e for events. I use P for predicates of type et , V
16
for predicates of type vt , θ and Θ for functions of type ve . I assume that ordinary
objects and events are each closed under mereological sum formation (Link, 1998a).
Intuitively, this means that these categories include plural entities. The lowercase
variables just mentioned should therefore be taken to range over both singular and
plural entities. In the literature on plurals, the distinction between singular and plural
entities is often indicated by lowercase and uppercase variables. Since almost all
the variables in my representations range over potentially plural entities, I do not
follow this convention. I will adopt the framework of Champollion (2014a), including
the following assumptions: atomicity of singular individuals, thematic uniqueness,
cumulativity of thematic roles, verbs as pluralized event predicates.
I assume that noun phrases are interpreted in situ, because I do not consider
quantifier raising in this paper. Silent theta role heads, which denote thematic roles of
type ve (event to individual), are located between noun phrases and verbal projections.
The one that denotes the agent role can be either thought of as a silent case-marker-like
part of the noun phrase, or as little v or Voice, depending on whether it first combines
with the subject noun phrase or with the verb phrase. The heads that denote the theme
and recipient roles bear some conceptual similarity with applicative heads (Pylkkänen,
2008). I will occasionally omit or abbreviate these heads in my LFs but they should
always be assumed to be there. The precise nature of the compositional process is not
essential, but it affects the types of the lexical entries of distance-distributive elements
so let me make it concrete. I assume that the following type shifters apply first to
the theta role head, then to the noun phrase, and finally to the verbal projection. I
will introduce a third type shifter in Section 7 below in order to accommodate the
alternative theoretical assumption that the theta role head combines first with the
verbal projection and then with the noun phrase it belongs to, as opposed to the other
way round. The choice between these assumptions is immaterial for the theories in
this paper, but it matters to theories of the syntax-semantics interface more generally.
(62) a. Type shifter for indefinites: λθveλPetλe.P(θ(e))
b. Type shifter for definites: λθveλxλe.θ(e) = x
Each of these type shifters combines a noun phrase with its theta role head to build an
event predicate of type vt which can combine with other predicates of the same type
via intersection. For example, after the noun phrases the boys (definite) and two monkeys
(indefinite) combine with the theta role heads [agent] and [theme] respectively, their
denotations are as follows.
(63) [[[[agent] the boys]]] = λe[∗
agent(e) = boy]
(64) [[[[theme] two monkeys]]] = λe[|∗
theme(e)| = 2 ∧ ∗
monkey(∗
theme(e))]
After the verb has combined with all its arguments, the event variable is existentially
bound if the sentence is uttered out of the blue. If the sentence is understood as
referring to a specific event, the event variable is instead resolved to that event. If the
noun phrases combine directly with the verb, we get a scopeless reading as in (65). Here
and below, I write 2M as a shorthand for λe[|∗
theme(e)| = 2 ∧ ∗
monkey(∗
theme(e))].
(65) [[The boys saw two monkeys]] = ∃e[∗
agent(e) = boy ∧ ∗
see(e) ∧ 2M(e)]
17
To generate distributive readings, we use Link’s D operator, reformulated in the
counterpart of this paper and repeated here as (66). As in Champollion (2014a), I
assume that the D operator is coindexed with the thematic role of its target.
(66) Definition: Event-based D operator
[[Dθ]]
def
= λV vt λe[e ∈ ∗
λe (V (e ) ∧ Atom(θ(e )))]
As an example, the distributive reading of (65) is derived like this:
(67) [[[[agent] The boys] [Dagent [saw [[theme] two monkeys]]]]]]
= ∃e[∗
agent(e) = boy ∧ e ∈ [[[Dagent]](λe [∗
see(e ) ∧ 2M(e )])]]
= ∃e[∗
agent(e) = boy ∧ e ∈ ∗
λe [∗
see(e ) ∧ 2M(e ) ∧ Atom(agent(e ))]]
This formula is true just in case there is an event e whose agent is the boys, and which
consists of seeing-two-monkeys events whose agents are atomic. As discussed in
Champollion (2014a), the background assumptions of algebraic semantics ensure that
the seeing-two-monkeys events have boys as agents even though the formula does
not explicitly state this. I come back to this point at the end of this section.
Here are the entries for adverbial, adnominal, and determiner each. An explanation
follows below. An illustration of the derivation of a basic sentence like The boys saw
two monkeys each is shown in Figure 1. Adverbial and determiner each work similarly.
Adverbial each is simply synonymous with the D operator. I come back to determiner
each in Section 5.
(68) [[eachθ]]adverbial = [[Dθ]] = (66)
(69) [[eachθ]]adnominal = λPetλΘveλe.e ∈ [[Dθ]](λe .P(Θ(e )))
I assume that adverbial each, as shown in (68), is a verb phrase modifier just like
the D operator, and can therefore be given the same entry as that operator. I adopt for
concreteness the assumption that adverbial each is an adverb adjoined to VP. This is
similar what has been argued for floating quantifiers in general by Dowty and Brodie
(1984), Bobaljik (1995) and Doetjes (1997). For another view that analyzes floating
quantifiers as the remaining part of a noun phrase the rest of which has moved away
from it, see for example Safir and Stowell (1988) and Sportiche (1988). The movement
view makes a formal link between each and its antecedent available for independent
reasons since there is a movement relation between them, while the adverbial view
requires one to assume that the two are coindexed. Therefore the movement view,
which I do not adopt here, would be likely to be at least as compatible with my approach
than the adverbial view.
Adnominal each, as shown in (69), needs to be type-shifted, but like adverbial each
it is defined in terms of the D operator. This captures the fact that they are essentially
synonymous. As shown in (69), adnominal each carries an index. I assume that it
is coindexed with the theta role θ of its antecedent. It first combines with its host
predicate P (e.g. two monkeys), and then with the theta head Θ of the host (which
is not to be confused with the theta role of its antecedent). Afterwards, it combines
intersectively with the verbal projection to which its host attaches (e.g. the verb see).
This means that adnominal each does not take scope over the verbal projection but
18
CP
∃e.
∗
agent(e) = boy
∧ ∗
see(e) ∧ e ∈ [[[Dagent]](λe .
two-monkeys(∗
theme(e )))]
[closure]
λV ∃e.V (e)
IP
λe.
∗
agent(e) = boy
∧ ∗
see(e) ∧ e ∈ [[[Dagent]](λe .
two-monkeys(∗
theme(e )))]
DP
λe.
∗
agent(e) = boy
The boys [agent]
VP
λe.∗
see(e) ∧
e ∈ [[[Dagent]](λe .
two-monkeys(∗
theme(e )))]
saw
λe.∗
see(e)
DP
λe.
e ∈ [[Dagent]](λe
[two-monkeys(∗
theme(e ))])
[theme]
∗
theme
λΘ ve λe.
e ∈ [[Dagent]](λe [
two-monkeys(Θ(e ))])
NP
λx.two-monkeys(x)
two monkeys
eachagent
λP et λΘ ve λe.
e ∈ [[Dagent]]
(λe [P(Θ(e ))])
Figure 1: Deriving The boys saw two monkeys each.
19
only over its own complement (Dotlačil, 2012). By contrast, adverbial each takes scope
over the verbal projection. In this, I follow LaTerza (2014b), who evidence the following
minimal pair as evidence for this contrast:
(70) a. John and Bill served [four meals each] to (exactly) three judges.
b. John and Bill each [served four meals to (exactly) three judges].
As LaTerza reports, speakers judge (70a) true of situations where there are at most
three judges, while (70b) is true in situations which allow up to six different judges.
As he notes, this is predicted by giving adnominal and adverbial each different scopes,
as indicated by the square brackets in these examples. For example, sentence (70b) can
be derived as in Figure 2.
My entry for adnominal each combines with its host in two steps, in order to give
it access to both the predicate and the theta head. This is not essential, but it allows
us to ensure that the type of the predicate is et . I do so to provide a hook on which
to build future accounts of the “counting quantifier requirement” that prevents such
phrases as *most men each (Safir and Stowell, 1988; Sutton, 1993; Szabolcsi, 2010, §10.5).
The theory in this paper does not to provide an account of this requirement, since it
will not rule out bare plurals as in *They saw monkeys each, as pointed out in Cable
(2014). If an independent account of these kinds of mismatches can be provided that
does not need to have access to the host predicate and its theta role separately, then
it may not be necessary to place each between the host predicate and the theta head
after all. The choice is immaterial for the rest of this paper.
Let us now turn to jeweils. My reformulation of the Part operator in Champollion
(2014a), repeated here as (71), is also the lexical entry of adverbial jeweils, as shown in
(72). The same type shift as in (69) brings us from (72) to adnominal jeweils, as shown
in (73).
(71) Definition: Event-based Part operator
[[Partθ,C]]
def
= λP vt λe[ e, θ(e) ∈ ∗
λe (P(e ) ∧ C(θ(e )))] (Takes an event
predicate P and returns a predicate that holds of any event e which can be
divided into events that are in P and whose θs satisfy the contextually salient
predicate C.)
(72) [[jeweilsθ,C]]adverbial = [[Partθ,C]] = (71)
(73) [[jeweilsθ,C]]adnominal = λPλΘλe[[[Partθ,C]](λe [P(Θ(e ))])(e)]
As in the case of the Part operator, the granularity parameter C of jeweils can be set to
Atom so long as its dimension parameter θ is set to a function into a count domain,
such as agent. In that case, Part distributes over individuals and is equivalent to the D
operator, as explained in Champollion (2014a). This accounts for the fact that when
jeweils distributes over individuals, it is equivalent to each. The following example
illustrates this with sentence (7a); sentence (7b) is equivalent.
(74) Die
The
Jungen
boys
haben
have
jeweilsagent,Atom
Dist
zwei
two
Affen
monkeys
gesehen.
seen.
“The boys have each seen two monkeys.”
20
CP
∃e.∗
serve(e) ∧
∗
agent(e) = j ⊕ m
∧ three-judges(∗
goal(e))
∧ e ∈ [[[Dagent]](λe .
four-meals(∗
theme(e )))]
[closure]
λV ∃e.V (e)
IP
λe.∗
serve(e) ∧
∗
agent(e) = j ⊕ m
∧ three-judges(∗
goal(e))
∧ e ∈ [[[Dagent]](λe .
∧ four-meals(∗
theme(e )))]
DP
λe.
∗
agent(e) = j ⊕ m
John and Mary [agent]
VP
V’
λe.∗
serve(e) ∧
e ∈ [[[Dagent]](λe .
four-meals(∗
theme(e )))]
V
served
λe.∗
serve(e)
DP
λe.
e ∈ [[Dagent]](λe
[four-meals(∗
theme(e ))])
[theme]
∗
theme
λΘ ve λe.
e ∈ [[Dagent]](λe [
four-meals(Θ(e ))])
NP
λx.four-meals(x)
four meals
eachagent
λP et λΘ ve λe.
e ∈ [[Dagent]]
(λe [P(Θ(e ))])
PP
λe.
three-judges(∗
goal(e))
to exactly three judges
Figure 2: Deriving John and Mary served four meals each to exactly three judges.
21
If – and only if – there is a supporting context, the anaphoric predicate C can be set
to a salient antecedent other than Atom, and in that case θ is free to adopt values
such as τ (runtime). This leads to occasion readings. Suppose for example that it is in
the common ground that the boys have been to the zoo to see animals last Monday,
last Wednesday and last Friday, and that (7a) is uttered with reference to that state of
affairs, or sum event. It is interpreted as follows.
(75) [[Die Jungen haben jeweilsτ,zoovisit zwei Affen gesehen.]] =
∗
agent(e0) = boy ∧ ∗
see(e0) ∧ e0 ∈ ∗
λe λt[|∗
theme(e )| = 2 ∧
∗
monkey(∗
theme(e )) ∧ zoovisit(τ(e ))]
“The boys have seen two monkeys on each occasion.”
Since the sentence refers specifically to the sum e0 of the three events in question,
the event variable in (75) is resolved to e0 rather than being existentially bound. The
predicate that is true of any time interval at which a zoo visit takes place, call it zoovisit,
is also salient in this context. So C can be resolved to zoovisit rather than to Atom.
Since there are no atoms in time, it is only now that θ can be set to τ, rather than
to agent as in (74). What (75) asserts is that e0 has the boys as its agents; that it
can be divided into subevents, each of whose runtimes is the time of a zoo visit; and
that each of these subevents is an event whose theme are two monkeys. That these
subevents are seeing events is entailed by the fact that see is lexically distributive on
its theme argument, which in turn is formally represented as a meaning postulate, as
discussed in Champollion (2014a). I assume that runtime is closed under sum just like
other thematic roles (τ = ∗
τ), or in other words, it is a sum homomorphism. This
means that any way of dividing e0 must result in parts whose runtimes sum up to
τ(e0). Assuming that τ(e0) is the (discontinuous) sum of the times of the three zoo
visits in question, this entails that each of these zoo visits is the runtime of one of the
seeing-two-monkeys events. This is the occasion reading.
4 Previous Work: Zimmermann (2002b)
The most detailed semantic account of jeweils and each is offered in Zimmermann
(2002b). I summarize and critically review it here; other discussions of this account are
found in a number of places (Zimmermann, 2002a; Blaheta, 2003; Dotlačil, 2012).
Zimmermann takes adnominal each and jeweils to be prepositional phrases that
are only partially pronounced, but this aspect does not really influence the semantic
composition. The meaning of each, or more precisely of the prepositional phrase
that is supposed to contain it, is as follows (Zimmermann, 2002b, p. 210). While the
relevant discussion is actually about adnominal jeweils, it carries over to adnominal
each without changes, so I present it in terms of each for clarity.
(76) [[each]] (Zimmermann) = λP.∀z[z ∈ Zi → ∃x[P(x) ∧ ∗
Rj(z, x)]]
This meaning is a property of predicates that holds of a given predicate P iff every
member of a certain plurality Zi stands in the algebraic closure ∗
Rj of a certain relation
Rj to some entity of which P holds. In this definition, Zi and Rj are free variables
22
that are assumed to be coindexed, respectively, with the antecedent of each and with
the relation that holds between the two, typically denoted by the verb. Take sentence
(15), repeated here as (77) with the coindexation added. Here P is the denotation of
two monkeys, Z is coindexed with the boys, and R is coindexed with saw.
(77) [The boys]i sawj two monkeys eachi,j.
Here is how this sentence would be represented by Zimmermann (2002b). First, the
entry for each is applied to two monkeys, which is taken to denote a predicate of sum
individuals that I will represent here by the shorthand two-monkeys. This results in an
open proposition with two free variables:
(78) [[two monkeys eachi,j]] = ∀z[z ∈ Zi → ∃x[two-monkeys(x) ∧ ∗
Rj(z, x)]]
The next steps involve lambda-abstracting over the free variables, via a rule that
Zimmermann calls “index-triggered lambda abstraction”, a variant of a rule which is
taken to apply when a type mismatch makes function application impossible (Bittner,
1994, p. 69).
(79) Index-triggered λ-abstraction (Zimmermann, 2002b, p. 217):
If the semantic types of a proposition-denoting expression α and its syntactic
sister β do not match, and if [[α]] contains a free variable ui that shares an
index ‘i’ with β, λ-abstraction in [[α]] over index ‘i’ is licensed, and λui.[[α]] is
a value for α.
This rule allows a constituent with a free variable in it to combine with another
constituent that is coindexed with that variable. For example, in (77), the constituent
two monkeys eachi,j has the free variable j in it, which carries the same index as the
constituent sawj. Since the two constituents are sisters, index-triggered λ-abstraction
applies, with the result as shown in (80), as discussed in Zimmermann (2002b, p. 226).
(80) λRj.∀z[z ∈ Zi → ∃x[two-monkeys(x) ∧ ∗
Rj(z, x)]]
Zimmermann takes the classical Davidsonian view on verb meaning, under which
n-ary verbs denote n + 1-ary relations between arguments and events. He tentatively
proposes that the event argument can “at least sometimes” (p. 226) be saturated inside
the verb phrase by existential closure. This means that the verb saw can have the right
type to combine with (80), as shown below in the derivation taken from Zimmermann
(2002b, p. 227):
(81) [[sawj]] = λyλx.∃e[see(x, y, e)]
(82) [[(80)]]([[(81)]]) = ∀z[z ∈ Zi → ∃x[two-monkeys(x) ∧ ∃e[∗
see(z, x, e)]]]
In (82), the existential quantifier over events has ended up (somewhat mysteriously to
me) outside the scope of the star operator, even though the star takes scope over R in
(81). The result of the computation, in (82), is another open proposition. The last step
in the derivation is to combine this with the antecedent, the boysi, in another instance
of index-triggered λ-abstraction. The result is as follows:
23
(83) ∀z[(z ∈ boy) → ∃x[two-monkeys(x) ∧ ∃e[∗
see(z, x, e)]]]
This formula says that for every boy there exists a sum of two monkeys such that
the boy saw the monkeys. This is an accurate rendering of the truth conditions of
sentence (77).
In Zimmermann’s system, the denotation of the host phrase of adnominal each,
given in (78), is of type t. This means that the only way it can combine with other
constituents is via index-triggered lambda abstraction. The only two indices that can
trigger this operation are the ones on Z and R. The values for these two variables will
therefore always be provided by the two constituents which are closest to the host
phrase. Put another way, the only thing that can intervene between the host phrase
of each and its antecedent is the constituent that denotes R. Zimmermann’s system
requires the host phrase of adnominal each to be adjacent either to its antecedent or to
the constituent that denotes the relation between the two.
As a point in favor of this adjacency requirement, Zimmermann (2002b, p. 240f.)
notes that jeweils cannot distribute over individual-denoting noun phrases in a higher
clause:
(84) *Die
the
Verkäuferi
store.clerks
sagen,
say,
dass
that
Peter
Peter
jeweilsi
Dist
einen
a
Ballon
balloon
gekauft
bought
hat.
has.
*‘The store clerks said that Peter had bought a balloon each.’ (Zimmermann,
2002b, p. 241)
As mentioned earlier, this clausemate condition can also be explained by syntactic
means, such as binding theory (e.g. Burzio, 1986) or LF movement (e.g. Safir and
Stowell, 1988). As long as the syntactic locality requirement is sufficiently stringent,
there is no need for an additional semantic adjacency requirement on top of it.
Zimmermann’s adjacency requirement is not only redundant in those cases where
it agrees with these syntactic constraints, it is also too strong in those cases where it
goes beyond it. Establishing this requires us to think about the relationship between
Z and R. The universal quantifier over parts of Z in (78) has scope over R. Therefore,
if R contains a scope-taking element such as a numeral, this element cannot stand
in a scopeless relation with Z, such as a cumulative relation (Scha, 1981; Beck and
Sauerland, 2000). Either it should take scope below the universal quantifier, or else
take above it due to scope displacement. Therefore if Zimmermann’s account is right,
the antecedent of adnominal each cannot stand in a cumulative relation with a noun
phrase that intervenes between its antecedent and its host noun phrase. For example,
if the indirect object in a ditransitive construction stands in a cumulative relation with
the subject and if the direct object contains adnominal each, then only the indirect
object can be its antecedent, the subject cannot:
(85) Subjecti Ditransitive-Verb Indirect-Objectj Direct-Object eachj/∗i.
To test this prediction, I conducted a large-scale web survey using TurkTools (Erlewine
and Kotek, to appear), which relies on the online labor market platform Amazon
Mechanical Turk (http://www.mturk.com). The survey took place in June 2015 and
contained the following target task:
24
(86) Imagine there is an election with two candidates. The day after, you read in the
newspaper: “100 million voters gave two candidates one vote each.” According
to the newspaper, how many votes were cast in total?
Participants were asked to choose one of four answers: (i) 100 million, (ii) 200 million,
(iii) 100 million or 200 million — it could be either one; (iv) none of the above. They
were instructed to choose only one answer from the four choices. A filler task and
demographic questions were used to check whether subjects paid enough attention
and whether they satisfied the criteria for the survey. The filler task consisted of a
simple question (“Imagine I have five coins in my pocket. Then I put ten more coins
into my pocket. How many coins are in my pocket now?”) and four answers: (i)
ten; (ii) fifteen; (iii) ten or fifteen – it could be either one; (iv) none of the above. 200
participants were given the survey, with the target task preceding the filler task in one
half of the cases and following it in the other half. Each participant was located in the
United States and had an MTurk approval rate of at least 95%. Each was paid $0.02.
Data were rejected from participants who worked on more than one group or did not
finish the survey (10 rejected data points), who gave their native language as anything
other than American English (9 rejected data points), and who answered the filler task
with anything other than fifteen (6 data points). The remaining 175 data points are
summarized in Table 2.
Choice Raw frequency Percentage
(i) 100 million 63 36%
(ii) 200 million 77 44%
(iii) 100 million or 200 million 30 17%
(iv) none of the above 5 3%
Total 175 100%
Table 2: Results of the survey on cumulative readings
Of interest are the 93 participants (53%) who answered (i) or (iii). It is likely that
these participants understood the sentence to being able to describe a situation in which
every one of the 100 million voters voted for only one of the candidates, with some of
these votes going to the first candidates and the rest going to the second. In this reading,
the word each can be paraphrased as per voter but not as per candidate, so its antecedent
is the subject. The reading also involves a scopeless relation between the two noun
phrases 100 million voters and two candidates, since neither one distributes over the
other. Since every voter is related by a giving event to one of the two candidates but
not to the other one, that scopeless relation is cumulative. This is exactly the pattern
that is unexpected on Zimmermann’s account, which predicts that all participants
should answer either (ii) or (iv).
As for those participants that were able to access the “200 million” reading, they
presumably took the word each as taking the indirect object two candidates as its
antecedent, and interpreted the vP gave two candidates one vote each as being in the
scope of a D operator that distributes it over the 100 million voters. If the D operator
25
is absent but each still targets the indirect object, the resulting reading should entail
that a total of two votes were cast. Indeed, a small number of participants commented
that the answer could also be “2 votes”.
In Zimmermann’s account, the universal quantifier introduced by each takes scope
over whatever constituent provides the value for R, and that constituent must include
everything between the host phrase of each and its antecedent. When his account
is applied to (86), that constituent must be give two candidates, since it includes everything
between the subject and the direct object. Zimmermann’s entry for each
then distributes this predicate over the voters. This prevents the candidates from
cumulating with the voters. Therefore Zimmermann’s account does not generate the
relevant reading of (86).
On the account presented here, representing that reading is straightforward because
adnominal each takes semantic scope only over its host argument (the noun phrase
that contains it), and not over the verbal predicate. As discussed earlier, this insight
is derived from Dotlačil (2012) and LaTerza (2014b). Zimmermann’s account does
not capture it because the relation between the voters and the votes corresponds
to the variable Rj, which is in the scope of the universal quantifier contributed by
adnominal each. On my account, the relationship of adnominal each with its antecedent
is only subject to whatever (perhaps binding-theoretic) locality restrictions need to be
imposed on theta-indexing. As long as these restrictions allow for theta-indexing of
coarguments, the dependency in (86) can be modeled. Sentence (86) can be analyzed
as in Figure 3, which gives the following result:
(87) ∃e.[∗
give(e)
∧ 120-million-voters(∗
agent(e))
∧ two-candidates(∗
beneficiary(e))
∧ e ∈ ∗
λe [vote(∗
theme(e )) ∧ Atom(agent(e ))]]
This representation states that there is a sum of giving events whose agents sum up to
100 million voters, whose beneficiaries sum up to two candidates, and which consist
of subevents with atomic agents and individual votes as themes. The background
assumptions of algebraic semantics ensure that this entails that there were 100 million
such subevents.
A few comments on the derivation in Figure 3. The current consensus is that
the syntax of ditransitives differs somewhat from their surface structure. Although
nothing here hinges on this, I have chosen to make my analysis harmonize with the
syntax proposed in Marantz (1993) and recently defended by Bruening (2010). For this
purpose, I have assumed that the Voice and Appl heads introduce the thematic roles
agent and beneficiary respectively, and I have type-shifted these heads accordingly.
For reasons of clarity, I show the meanings of these heads only in their type-shifted
form, but really they should be thought of as having the same meanings as above, that
is, ∗
agent and ∗
beneficiary. The type-shifter I have used here is the following:
(88) λθveλVvtλPetλe.V (e) ∧ P(∗
θ(e))
An alternative to this approach, if one prefers additional compositional operations to
type-shifters, would be to use the event identification operation from Kratzer (1996).
26
For details on how this would work, see Harley (2012).
Another problem for Zimmermann’s account, which is similar to the one discussed
above, was noted by Blaheta (2003):
(89) Alex and Sasha lifted a piano with two jacks each.
In (89), it is possible for the piano-lifting event to be collective. As Blaheta puts it,
the phrase with two jacks each “needs to distribute itself in some fashion over each
member of the subject, without making the verb phrase itself distribute!” Indeed it
is not obvious how to represent this kind of configuration in Zimmermann’s system.
Blaheta leaves (89) as an open problem for his own account, which is closely related to
Zimmermann’s, but notes he suspects that event semantics may hold the key to the
solution. Indeed it does. The compositional derivation of (89) is similar to the one of
(86). Each of the three arguments is intersected with the event predicate. This is the
result:
(90) ∃e.[∗
lift(e)
∧ ∗
agent(e) = alex ⊕ sasha
∧ piano(∗
theme(e))
∧ e ∈ ∗
λe [two-jacks(∗
instrument(e )) ∧ Atom(agent(e ))]]
This formula entails that Alex and Sasha together lifted a piano, and that each of them
was the agent of a part of the lifting event which had two jacks as its instrument. I
does not entail that the parts of the lifting events need to be lifting events themselves.
This is as it should be, because lift is not distributive on its agent position.
Schwarzschild (2014) points out a potential problem for my line of analysis. Suppose
that a group of artists build a wall of books on the sidewalk, with each artist putting
one book down next to or on top of other books until a wall is built. In this scenario,
each artist did something to one book. If we assume that all these events sum up to a
building event, sentence (91) should be acceptable and true.
(91) #The artists built one book each.
The deviant status of (91) can be explained by assuming that the building event is not
in fact the sum of the individual events in which artists put down books. For discussion
of an analogous problem involving the collective planting of a rosebush, see Kratzer
(2007), Williams (2009) and Champollion (2010b). An alternative line of analysis would
be to assume that in some cases including (91), the scope of adnominal each includes
the verb phrase after all. This would raise the question how to delineate the cases in
which adnominal each does and does not take scope over the verb phrase. I have not
adopted this analysis because I do not see an easy way to answer this question.
While the analysis so far has focused on adnominal and adverbial each and their
counterparts across languages, it is possible to assimilate distributive determiners such
as each and every to these items. I turn to them now.
27
CP
∃e.∗
give(e)
∧ 120-million-voters(∗
agent(e))
∧ two-candidates(∗
beneficiary(e))
∧ e ∈ ∗
λe [vote(∗
theme(e ))
∧Atom(agent(e ))]
[closure]
λV ∃e.V (e)
VoiceP
DP
λx.
120-million-voters(x)
100 million voters
Voice’
[Voice]
λVvtλPetλe.
V (e) ∧ P(∗
agent(e))
VP
DP
λx.two-candidates(x)
two candidates
Appl’
[Appl]
λVvtλPetλe.
V (e)∧
P(∗
beneficiary(e))
VP
λe.∗
give(e) ∧
e ∈ ∗
λe [vote(∗
theme(e ))
∧Atom(agent(e ))]
V
gave
λe.∗
give(e)
DP
λV vt λe.
e ∈ ∗
λe [vote(∗
theme(e ))
∧Atom(agent(e ))]
[theme]
∗
theme
λΘ ve λe.
e ∈ ∗
λe [vote(Θ(e ))
∧Atom(agent(e ))]
NP
λx.vote(x)
one vote
eachagent
λP et λΘ ve λe.
e ∈ ∗
λe [P(Θ(e ))
∧Atom(agent(e ))]
Figure 3: Deriving 100 million voters gave two candidates one vote each.
28
5 Distributive Determiners
As we have seen, English each along with some of its crosslinguistic relatives can be
used adnominally, adverbially, and as a determiner. I have suggested that the synonymy
of these uses should be captured, ideally by essentially identical lexical entries. Another
distributive determiner in English is every. As shown by their incompatibility with
collective predicates, both every and each are distributive (92).
(92) a. #Every/#Each soldier surrounded the castle.
b. The soldiers surrounded the castle.
Traditionally, the determiners every and each are analyzed in terms of universal quantification
(e.g. Montague, 1973):
(93) [[every boy]] (traditional)
= λPet∀x[boy(x) → P(x)]
This style of analysis is especially useful when one is interested in comparing them
with other determiners from the perspective of generalized quantifier theory (e.g.
Barwise and Cooper, 1981). This paper, however, focuses on the parallels between
determiner each and its adnominal and adverbial counterparts. Therefore, instead of
the traditional approach I will assimilate it, as far as it goes, to the analyses of adverbial
and adnominal each that we have already encountered. Since the differences between
each and every are not a core concern of this paper, I will adopt the same analysis
for both determiners. This is not to deny that there are differences between every
and determiner each. Determiner each has a strong preference for taking wide scope
over its environment, more so than every (e.g. Ioup, 1975; Beghelli and Stowell, 1997).
Moreover, each also appears to impose a condition of spatial or temporal separateness
on its subevents, as shown in (94) (e.g. Tunstall, 1998; Brasoveanu and Dotlačil, 2015).
There may be other differences as well.
(94) Jake photographed every/#each student in the class, but not separately.
The analysis I will adopt is couched in terms of the D operator (95). The determiner
combines first with a nominal (of type et ) and then with a theta head. Unlike its
adnominal and adverbial counterparts, determiner each is not coindexed with anything
because it is not a distance-distributive element. The thematic relation is contributed
by the theta head. The result is a phrase of VP modifier type vt, vt , ready to combine
with the verb phrase or other verbal projection V . A sample noun phrase denotation
is shown in (96), and a sample sentence in (97).
(95) [[each]]determiner = [[every]] = λPetλθveλVvtλe [θ(e) = P ∧ [[Dθ]](V )(e)]
= λPetλθveλVvtλe [θ(e) = P ∧ e ∈ ∗
λe [V (e ) ∧ Atom(θ(e ))]]
(96) [[[every boy [agent]]]]
= λVvtλe[∗
agent(e) = dog ∧ e ∈ ∗
λe [V (e ) ∧ Atom(agent(e ))]]
(Takes an event predicate V and returns a predicate that holds of any event e
whose agent is all the boys and which consists entirely of events that are in V
and whose agents are individual boys.)
29
(97) [[[every boy [agent] carried three suitcases]]]
= ∃e[∗
agent(e) = boy∧e ∈ ∗
λe [∗
carry(e )∧Atom(agent(e ))∧|∗
theme(e )| =
3 ∧ ∗
suitcase(∗
theme(e ))]]
This says that there is an event e whose agent is all the boys and which consists
entirely of carrying events whose agents are individual boys and whose themes are
sums of three suitcases. From this and from the assumption that the agent role is a
sum homomorphism, we can conclude that every boy carried three suitcases.
The entry in (95) can be refined in various ways. For example, the underlined subformula
could be replaced by a contextually supplied variable that specifies the domain
of quantification. This variable can be dependent on another universal quantifier in
the sentence (see e.g. Stanley and Szabó, 2000):
(98) Every child ate every apple. (Farkas, 1997)
In order to keep the system simple, I will refrain from adding these refinements here.
Treating universally quantified noun phrases as involving distributivity operators
in the present event semantic framework avoids leakage, as discussed in Champollion
(2011) in connection with examples like (99). Briefly, the event modifier unharmoniously
needs access to the sum of the events whose agents are the individual students
quantified over by every (Schein, 1993). The traditional account of every in terms of
generalized quantifiers does not provide us with access to this sum event.
(99) Unharmoniously, every organ student sustained a note on the Wurlitzer.
(Schein, 1993)
(100) [[(99)]] = ∃e[∗
agent(e) = organ.student
∧e ∈ ∗
λe [sustain(e ) ∧ note(theme(e ) ∧ Atom(agent(e ))]]
(There is an unharmonious event e whose agent is all the students and
which consists entirely of note-sustaining events whose agents are individual
students.)
Another advantage relates to cumulative readings of every such as the ones available
in (101a) and (101b), from Schein (1993) and Kratzer (2000) respectively.
(101) a. Three video games taught every quarterback two new plays.
b. Three copy editors caught every mistake in the manuscript.
Such configurations cause problems for the traditional analysis of every in (93). Just
like the adverbial modifier unharmoniously needs access to the sum of all the individual
events in (99), so does the subject noun phrase in (101a) and in (101b). For example, the
cumulative reading of (101b) can be paraphrased roughly as “There is a sum of mistakecatching
events, whose agents sum up to three copy editors, and every mistake was
caught in at least one of these events” (Schein, 1993; Kratzer, 2000; Champollion, 2010a).
In this reading, the relationship between the two verbal arguments is cumulative and
symmetric. There is no entailment that any mistake was caught by more than one
copy editor, as would be expected if every mistake took scope either above or below
three copy editors. My analysis of this reading is as follows.
30
(102) [[(101b)]]
= ∃e[|∗
agent(e)| = 3∧∗
copy-editor(∗
agent(e))∧∗
theme(e) = mistake∧
e ∈ ∗
λe [∗
catch(e ) ∧ Atom(theme(e ))]]
This formula says that there is an event whose agents sum up to three copy editors,
whose themes sum up to all the mistakes, and which consists of catching events with
atomic themes. That these themes are individual mistakes follows from cumulativity
of thematic roles.
It has been suggested that every can never enter a cumulative relation with arguments
in its syntactic scope (Champollion, 2010a, cf. Kratzer, 2000). For example,
Kratzer (2000) notes that (103) does not have a cumulative reading, in contrast to (101b).
Bayer (1997) judges (104a) to be “clearly bizarre” because scripts cannot be written
more than once, but reports that (104b) has a reading where every screenwriter in
Hollywood contributed to the writing of the movie. Assuming that Gone with the
Wind denotes a sum entity, we can represent (104b) as a cumulative reading. Similarly,
Zweig (2008) reports that (105a) entails that each game was won by both teams at
once, but (105b) has a cumulative reading, in which either team won games and every
game was won by only one of the teams.
(103) Every copy editor caught 500 mistakes.
(104) a. Every screenwriter in Hollywood wrote Gone with the Wind.
b. Gone with the Wind was written by every screenwriter in Hollywood.
(105) a. Every game was won by the Fijians and the Peruvians.
b. The Fijians and the Peruvians won every game.
These facts are in line with what we would expect, since every distributes over its
syntactic scope but makes the large event available for arguments or adverbs further
up the tree. In (104a) and (105a), the syntactic scope of the argument headed by every
is the entire verb phrase. The verb phrase includes the other argument, which is then
related as a whole to each of the individual screenwriters or games. As a result, the
every-phrase takes scope over its coargument and a cumulative reading is ruled out.
(106) [[(104a)]]
= ∃e[∗
agent(e) = screenwriter∧e ∈ ∗
λe [∗
write(e )∧Atom(agent(e ))∧
∗
theme(e) =[[Gone with the wind]]]]
(107) [[(105a)]]
= ∃e[∗
theme(e) = game ∧ e ∈ ∗
λe [∗
win(e ) ∧ Atom(theme(e )) ∧
∗
agent(e) =[[the fijians and the peruvians]]]
By contrast, in (104b) and (105b), the syntactic scope of the every-phrase only includes
the verb. For this reason, it does not distribute over the other argument, and a cumulative
reading is possible. Distributing over the verb does not amount to anything much
in (105b) since win is already distributive on its theme.
(108) [[(104b)]]
= ∃e[∗
theme(e) =[[Gone with the wind]]∧∗
agent(e) = screenwriter∧e ∈
∗
λe [∗
write(e ) ∧ Atom(agent(e ))]]
31
(109) [[(105b)]]
= ∃e[∗
agent(e) =[[the fijians and the peruvians]]∧∗
theme(e) = game∧e ∈
∗
λe [∗
win(e ) ∧ Atom(theme(e ))]]
Kratzer (2000) claims that the availability of cumulative readings depends on the
thematic role of the coargument of the every-phrase. According to her, cumulativity is
only possible when the coargument plays the agent role. Champollion (2010a) points
out that (104b), where the role of the coargument is theme, is a counterexample.
6 Zimmermann’s Generalization Explained
How can we capture the correlation expressed in Zimmermann’s generalization (18)?
That is to say, why does a distance-distributive element which can also be used as a
distributive determiner lack the occasion reading? Zimmermann himself proposes a
syntactic explanation: Determiners must agree with their complement; adnominal
or adverbial each also has a complement, a proform that must acquire its agreement
features from its antecedent, which is the antecedent of each; only overt antecedents
have agreement features; so adnominal/adverbial each cannot have a covert antecedent;
so it cannot refer to a contextually supplied but not overtly mentioned antecedent
such as a salient set of occasions.
This explanation is compatible with the present framework, and it makes the right
predictions given the assumption that covert antecedents cannot have agreement.
However, this assumption is problematic. To mention a simple example, German has
grammatical gender, so for example, the gender of Tisch ‘table’ is feminine. Knowing
this, a German speaker can point to a table and say with reference to it:
(110) Den hab ich gebraucht gekauft.
This.m have I used bought.
‘I have bought this used.’ (German)
But the same speaker cannot point to it and say:
(111) *Die hab ich gebraucht gekauft.
This.f. have I used bought.
Intended: ‘I have bought this used.’ (German)
As this example shows, a deictic pronoun in German has to agree in grammatical
gender with the gender of the noun phrase that would most aptly describe this
antecedent, even though the antecedent has not been mentioned explicitly.
A similar phenomenon was pointed out for English by Tasmowski-De Ryck and Verluyten
(1982). English pronouns agree with their antecedents based on syntactic rather
semantic grounds, as is shown by pluralia tantum such as pants and scissors which are
syntactically plural but semantically singular. Pronouns show syntactic agreement
with their antecedents even when these antecedents are not overtly mentioned:
(112) a. (John wants his pants that are on a chair and he says to Mary:)
Could you hand them/*it to me, please?
32
b. (Same situation but with a shirt:)
Could you hand *them/it to me, please?
For a recent discussion of these facts, and an explanation in terms of covert syntactic
antecedents that are included in the syntactic structure and c-command the entire
sentence in question, see Collins and Postal (2012, ch. 4). In the following I will remain
neutral on whether the covert syntactic antecedent should be thought of as being
included in the syntactic structure or not.
While Zimmermann’s account seems problematic, its difficulties can perhaps be
overcome, and it is by itself not incompatible with the present framework. But in
the context of the general theory adopted here, a more straightforward explanation
suggests itself. Distributive determiners like English each are only compatible with
count nouns, not with mass nouns. We know this since, for example, *each mud is
ungrammatical. Formally, this amounts to an atomicity requirement of the kind the
Link’s D operator provides, as discussed in Champollion (2014a). This requirement can
be seen as independent evidence of the atomic distributivity hard-coded in the entry
(95) via the D operator. In other words, the distance-distributive element inherits the
atomicity requirement of the determiner. This explanation is in line with the notion
of parameter settings imported from strata theory and described above. That is, in
English, adnominal, adverbial and determiner each have identical meanings up to
type-shifting. Determiner each is only compatible with count domains because its
granularity parameter is hardwired to the value “Atom”. Adnominal each is formally
identical to determiner each, so it inherits this property.
Zimmermann’s generalization is not about languages but about items. As such, it
can even be observed within one language. The German distributive determiner that
corresponds to each and every is jed-. The distance-distributive item jeweils cannot be
used in this position. This is illustrated in sentence (7c), repeated here:
(113) Jeder/*Jeweils Junge hat drei Koffer getragen.
Each.sg.m/Dist boy has three suitcases carried.
‘Every boy has carried three suitcases.’
This determiner can in turn also be used as an adverbial distance-distributive item.
Like English each, and unlike German jeweils, it can only distribute over individuals,
but not over salient occasions.
(114) Die
The
Jungen
boys
haben
have
jeder
each.sg.m
zweimal
sneezed.
geniest.
Available: ‘The boys have each twice sneezed.’
Unavailable: ‘The boys have sneezed twice on each occasion.’
(115) *Hans
Hans
hat
has
jeder
each.sg.m
zweimal
twice
geniest.
sneezed.
Intended: ‘Hans has sneezed twice on each occasion.’
As we see, the distance-distributive element jeder can also be used as a distributive
determiner, and it lacks the occasion reading. The distance-distributive element jeweils
cannot be used as a distributive determiner, and as we have seen before, it has the
33
occasion reading. All this is in line with Zimmermann’s generalization. As I have
suggested, we can account for it by assuming that jed-, like each, corresponds to the D
operator (its granularity parameter can only be Atom), while jeweils corresponds to
the Part operator (its granularity can be set to a nonatomic predicate if it is salient is
context). Concretely, I propose the following denotations for adverbial and determiner
jed-. They are identical with adverbial and determiner each respectively. As for jeweils,
we have already seen its entry in (72).
(116) [[jed-θ]]adverbial = [[Dθ]] = (66)
(117) [[jed-]]determiner = [[every]] = λPetλθveλVvtλe [θ(e) = P∧e ∈ ∗
λe [V (e )∧
Atom(θ(e ))]] = (95)
The derivation of (113) is exactly analogous to (97) above. Let me show a derivation of
(114). For convenience, and to avoid getting into the difficult question of how to count
events, I represent zweimal (“twice”) as an unanalyzed intersective predicate of sum
events.
(118) [[[[agent] Die Jungen]]] = λe[∗
agent(e) = boy] = (63)
(119) [[zweimal geniest]] = λe[∧twice(e) ∧ ∗
sneeze(e)]
(120) [[jederagent zweimal geniest]] = (117)((119)) = λe[e ∈ ∗
λe (twice(e )∧∗
sneeze(e ) ∧
Atom(agent(e )))]
(121) [[(114)]] = ∃e.e ∈[[(118)]]∩[[(120)]]
= ∃e[∗
agent(e) = boy∧e ∈ ∗
λe (twice(e )∧∗
sneeze(e ) ∧ Atom(agent(e )))]
This says that there is a sum event whose agents sum up to the boys, and which
consists of sneezing-twice events with atomic agents.
As a reviewer notes, jeweils and jed- are morphologically related. They are both
built around the distributive element je, which also occurs as a free morpheme; in that
case it is an each-type distance-distributive item (Link, 1998b). On the present account
that the underlying semantics of jed- and jeweils is related via the common core of
the D and Part operators, discussed in Champollion (2014a). An interesting question
is whether we can explain the two of them denote related different distributivity
operators. A starting point might be the observation that the morpheme weil in
jeweils is related to the noun Weile ‘timespan, while’. However, the reviewer notes
that the morpheme je is also found in words with quite distinct meanings, such as nie
‘never’, jeglich ‘any kind’, je ...je ‘the ...the’ and others. As we can see, a common
morphological core implies related but not necessarily identical meanings. On these
questions see also Zimmermann (2002b), who argues that weil is a proform; in terms of
the present account, it might be the part of jeweils that is anaphoric on the variable C.
7 Some More Complicated Cases
I will now go through a few configurations that are more complicated than those I
have discussed so far and demonstrate the versatility of my analysis of adnominal
distance-distributive items.
34
7.1 Each as a PP Modifier
Each can occur as the modifier of a prepositional phrase. Example (123) is a simple
case. Example (122) plays an important role in Schein (1993) and has not received a
compositional semantic analysis so far.
(122) 300 quilt patches covered two workbenches each with two bedspreads.
(Schein, 1993)
(123) Mary put the books each back on the bookshelf.
(Maling, 1976)
To analyze these sentences, I assume that each modifies the prepositional phrase to its
right, similarly to the adverb back in back at the farm, rather than the noun phrase to
its left. (As in (123) shows, these modifiers can be stacked.) My assumption is plausible
because adnominal each cannot modify definite plurals like the books. I assume for
concreteness that the syntactic structure of these sentences is [[V DP] PP] rather than
[V [DP PP]]; for discussion on the choice between these two analyses, see Janke and
Neeleman (2012).
Example (122) has a reading according to which there are a total of two workbenches
and a total of four bedspreads that cover them. The workbenches stand in a cumulative
relation with the 300 quilt patches. The following formula captures this reading:
(124) ∃e.∗
cover(e)
∧ ∗
quilt-patch(∗
agent(e)) ∧ |∗
agent(e)| = 300
∧ ∗
workbench(∗
theme(e)) ∧ |∗
theme(e)| = 2
∧ e ∈ ∗
λe [∗
bedspread(∗
instrument(e ))
∧|∗
instrument(e )| = 2 ∧ Atom(theme(e ))]
(There is a sum of covering events whose agents sum up to 300 quilt patches,
whose themes sum up to two workbenches, and which can be divided into
two smaller sum events, each of which involves two bedspreads and one of
the workbenches.)
Formula (124) is derived as follows. I have used shortcuts like 300-quilt-patches for
better readability. The derivation is straightforward and does not make use of any
new ingredients. The entry for each is the same as the adverbial one, even though
it modifies a prepositional phrase and not a verb phrase. This works because the
prepositional phrase is represented as an event predicate, in the same way as a verb
phrase.
(125) [[eachtheme]] = λVvtλe.e ∈ ∗
λe [V (e ) ∧ Atom(theme(e ))]
(126) [[with two bedspreads]] = λe.two-bedspreads(∗
instrument(e))
(127) [[each with two bedspreads]] = λe.e ∈ ∗
λe [two-bedspreads(∗
instrument(e ))∧
Atom(theme(e ))]
(128) [[covered two workbenches [theme]]]
= λe.∗
cover(e) ∧ two-workbenches(∗
theme(e))
(129) [[300 quilt patches [agent]]] = λe.300-quilt-patches(∗
agent(e))
35
(130) [[(122)]] = ∃e.e ∈ (129) ∩ (128) ∩ (127) = (124)
7.2 Jeweils Distributing over a Conjunction of Verbs
German jeweils can take a conjunction of verbs as its antecedent and distribute over the
two events described by the conjuncts, which results in a meaning for which English
uses the word respectively.
(131) Peter
Peter
kritisierte
criticized
und
and
lobte
praised
Maria
Mary
aus
for
jeweils
Dist
zwei
two
Gründen.
reasons.
‘Peter criticized and praised Mary for two reasons respectively.’
(Zimmermann, 2002b, p. 46)
(132) Der
The
Professor
professor
hat
has
jeweils
Dist
drei
three
Studenten
students
gelobt
praised
und
and
kritisiert.
criticized.
‘The professor praised three students and criticized three students.’
As for English each, it cannot be used for that purpose:
(133) *Peter criticized and praised Mary for two reasons each. (Zimmermann,
2002b, p. 134)
According to Zimmermann (2002b, p. 143f.), other languages that pattern with English
in this respect include Bulgarian, Czech, Dutch, French, Italian, Norwegian and Russian.
As we have seen in Section 2, this list includes many languages whose distancedistributive
items can also be used as determiners and lack the occasion reading. I
have suggested earlier that the occasion reading is only possible if the granularity
parameter can be set to a nonatomic value. Therefore, distributivity over conjunctsăis
predicted to be impossible as long as the events described by the two conjuncts are
nonatomic, contra Zimmermann (2002a). There is ample reason to assume that they
are indeed nonatomic. For example, in (131), the praising event could be imagined
to possibly consist of two subevents corresponding to the two reasons that caused it.
Sentence (132) entails that each of the six students was either praised or criticized,
which means that the two verbs are distributive on their themes. This in turn means
that the praising event in (132) consists of three praising subevents, and similarly for
the criticizing event. More generally, praise and criticize are atelic predicates, so any
praising event that goes on for five minutes will have parts that take less than five
minutes.
This explanation will work for most of the languages just mentioned, but not
for all of them. As we have seen in Section 2, in Bulgarian and Czech the distancedistributive
item po can be used to distribute over salient occasions and cannot be
used as a distributive determiner. We would therefore expect that this item allows
distribution over conjuncts, but it does not. Like Zimmermann, I have no explanation
for this fact.
To derive (131) compositionally, I assume that the dimension parameter θ of jeweils
is resolved to the identity function id rather than to a thematic role. I also assume that
the cover variable C is resolved to the pragmatically salient cover {ec, ep} where ec
36
is the salient criticizing-for-two-reasons event and ep is the salient praising-for-tworeasons
event.
(134) [[jeweilsθ,C]]adnominal
= λPλΘλe.e ∈ ∗
(P(Θ(e )) ∧ C(θ(e )))
(135) [[jeweilsid,{ec, ep} zwei Gründen]] = λΘλe.e ∈ ∗
λe .2-reasons(Θ(e )) ∧ e ∈
{ec, ep}
I assume that aus (in this context) denotes a function from events to their causes (or
whatever is the relation between a praising/criticizing event and its reason).
(136) [[aus]] = λe.∗
cause(e)
(137) [[aus jeweilsid,{ec, ep} zwei Gründen]] = λe.e ∈ ∗
λe .2-reasons(∗
cause(e )) ∧
e ∈ {ec, ep}
I represent the denotation of the verbal conjunction using sum formation as in (138).
I remain noncommittal about the compositional derivation of this conjunction. For
present purposes, we do not need to choose between a sum-based denotation of and,
as in (Lasersohn, 1995), and an intersective denotation that involves lifting of verb
phrases as in Winter (2001) and Champollion (2015, to appear).
(138) [[kritisierte und lobte]] = λe.∃e1∃e2[∗
criticize(e1)∧∗
praise(e2)∧e = e1 ⊕e2]
Once all these building blocks have been put together and conjoined with the agent
and theme, the result is as follows:
(139) [[(131)]] = ∃e.agent(e) = peter ∧ theme(e) = maria ∧
∃e1∃e2[∗
criticize(e1) ∧ ∗
praise(e2) ∧ e = e1 ⊕ e2∧
e ∈ ∗
λe .2-reasons(∗
cause(e )) ∧ e ∈ {ec, ep}]
This is true iff there is an event whose agent is Peter, whose theme is Maria, and
which consists of two events ec and ep such that one of them is a criticizing event, the
other one is a praising event, and each of them is caused by two reasons. By thematic
uniqueness, each of these two events has Peter as agent and Maria as theme.
7.3 Jeweils in Subject Position
German adnominal jeweils can occur as part of the subject of a clause (Zimmermann,
2002b, p. 27). When the subject is at the beginning of the clause, one may speak of
“backwards distributivity” since the antecedent of jeweils occurs to its right:
(140) Jeweils
Dist
zwei
two
Offiziere
officers
begleiteten
accompanied
die
the
Ballerinen
ballerinas
nach
to
Hause.
home.
‘The ballerinas were accompanied home by two officers each.’
It has been claimed that backwards distribution is not possible in English, perhaps
except for passives and unaccusatives (Burzio, 1986; Safir and Stowell, 1988):
(141) *One officer each accompanied the ballerinas home. (Zimmermann, 2002b)
37
(142) ?One interpreter each was assigned to the visitors. (Burzio, 1986, p. 200)
(143) Table 3 shows the dissertation topics for those holding earned doctorates.
[...] Three dissertations each dealt with assessment, transfer, trustees, and
technical education. Two dissertations each were on accreditation, counseling,
effectiveness, and mission. One dissertation each focused on economic
development, learning resources, performing arts, and strategic planning.40
(144) Indeed, Mr. Mitsotakis commanded only 144 seats [...] The Socialists won
125 seats [...] and one seat each went to a conservative independent and to
an ethnic Turk from Thrace, near the Turkish border.41
I do not have a semantic explanation for the restriction against backwards distributivity
in English. As in the case of locality restrictions, I assume that this restriction can be
dealt with by syntactic accounts such as the ones already proposed, for example by
Safir and Stowell (1988). I come back to this point in Section 8. For the German case,
where there is no restriction, my account can easily be used to derive the meaning of
(140) if we assume that the dimension paramter of jeweils is provided by the theta role
of die Ballerinen. Here are the core elements of the derivation:
(145) [[jeweilstheme,Atom]] (adnominal) = λPλΘλe.e ∈ ∗
λe .P(Θ(e )) ∧ Atom(theme(e ))
(146) [[jeweilstheme,Atom zwei Offiziere [agent]]] = λe.e ∈ ∗
λe .2-officers(∗
agent(e )) ∧
Atom(theme(e ))
(147) [[begleiten die Ballerinen [theme]]] = λe.∗
accompany(e)∧∗
theme(e) = ballerina
(148) [[jeweilstheme,Atom zwei Offiziere [agent] begleiten die Ballerinen [theme]]]
∃e.∗
accompany(e)∧∗
theme(e) = ballerina∧e ∈ ∗
λe .2-officers(∗
agent(e )) ∧
Atom(theme(e )))
What (148) says is that there is a sum of accompanying events whose themes sum up
to the ballerinas and which consists of parts e such that each e has an atomic theme
and two officers as its sum agent.
7.4 Reverse DP-internal distributivity
Finally, here is an example of how the analysis can be extended to a use of a distancedistributive
item which is halfway between the adverbial and adnominal case: backwards
distributivity cases within a noun phrase like the following. I illustrate this case
with a German example; a reviewer notes that parallel constructions are available with
Polish distributive po.
40Attested example. Keim and Murray (2008, p. 125f.)
41Attested example. The New York Times, Greek Conservative Is Seeking Coalition. June 20, 1989.
38
(149) Das
The
Parlament
parliament
hat
has
jeweils
Dist
zwei
two
Abgeordnete
representatives
aus
from
den
the
drei
three
baltischen
Baltic
Staaten
states
eingeladen.
invited.
‘From each of the three Baltic states, two representatives were invited by the
parliament.’
The analysis proceeds as follows. I write es ⊕ la ⊕ li for the sum of the three Baltic
states Estonia, Latvia and Lithuania.
(150) [[den drei baltischen Staaten]] = es ⊕ la ⊕ li
I assume that in this context, aus (“from”) denotes a function that maps individuals to
their origins:
(151) [[aus]] = λyλx.∗
origin(x) = y
The prepositional phrase then denotes the set of all plural individuals whose origins
are the three Baltic states:
(152) [[aus den drei baltischen Staaten]] = λx.∗
origin(x) = es ⊕ la ⊕ li
The complex noun phrase denotes a sum of six representatives consisting of three
pairs, with each pair coming from one of the three Baltic states. I will assume that
jeweils here has the same denotation as adverbial jeweils in (71) except that it ranges
over individuals instead of events. In the sentence at hand, its dimension parameter is
set to the origin function I used as the denotation of aus, and its granularity parameter
to Atom (since each pair of representatives comes from a single Baltic state:
(153) [[jeweilsorigin,Atom]] = λPetλx.x ∈ ∗
λy.P(y) ∧ Atom(origin(x))
(154) [[jeweilsorigin,Atom zwei Abgeordnete]] = λx.x ∈ ∗
λy.|y| = 2 ∧ ∗
rep(y) ∧
Atom(origin(y))
The denotation of the complex noun phrase is the intersection of (154) and (152). After
it combines with the theme theta head via the appropriate type shifter in (62a), the
result is this:
(155) [[[theme] jeweils zwei Abgeordnete aus den drei baltischen Staaten]]
= λe.∗
origin(∗
theme(e)) = es ⊕ la ⊕ li ∧ ∗
theme(e) ∈ ∗
λy.|y| = 2 ∧
∗
rep(y) ∧ Atom(origin(y))
After combining with the main verb eingeladen and with the subject Das Parlament,
the final result is as follows:
(156) [[(149)]]
= ∃e.∗
agent(e) = ιx[parliament(x)]∧mystarinvite(e)∧∗
origin(∗
theme(e)) =
es ⊕ la ⊕ li ∧ ∗
theme(e) ∈ ∗
λy.|y| = 2 ∧ ∗
rep(y) ∧ Atom(origin(y))
This says that there is an inviting event whose agent is the parliament, and whose
theme has the following properties: its origins sum up to the three Baltic states, and it
39
consists of sums of two representatives, each of which has an atomic origin.
8 Summary and Discussion
At the end of Section 2, I had suggested that the facts reviewed in that section suggest
the following requirements for a theory of distributivity. First, the synonymy of
the determiner, adnominal and adverbial uses of each in English should be captured,
ideally by essentially identical lexical entries. Second, the fact that distance-distributive
elements across languages share some part of their meanings (namely their individualdistributive
readings) should be represented, as well as the fact that some of them
can also have occasion readings. Third, the analysis should clarify the connections
between distance-distributive elements and distributivity theory more generally, and
the semantic variation across distance-distributive elements should be captured. Finally,
an explanation should be readily available for the crosslinguistic observation that
distance-distributive elements that can also be used as determiners can only distribute
over individuals (Zimmermann’s generalization).
Let me briefly summarize how the theory presented in this paper addresses these
issues. The synonymy of the determiner, adnominal and adverbial uses of each in
English is captured by the fact that they are all derived from Link’s D operator. I have
represent the fact that distance-distributive elements across languages share some
part of their meanings by derived them from related distributivity operators (Link’s or
Schwarzschild’s) which differ from each other only in their parameter settings. On the
theory presented here, distance-distributive items display the same parametric variation
as covert distributivity operators do, not only insofar as nonatomic distributivity
is concerned, but also insofar as the ability is concerned to target different thematic
roles or time. The semantic variation among distance-distributive elements is captured
by restriction on parameter settings. One type of element, exemplified by English each,
is hard-wired for distribution over atoms and the other one, exemplified by German
jeweils also allows distribution over nonatomic contexts. Zimmermann’s generalization
is explained by the natural assumption that distance-distributive elements are formally
identical to distributive determiners and therefore inherit their inability to distribute
over nonatomic domains, no matter if these domains are mass or temporal.
The theory in this paper, while surface-compositional, is semantic and pragmatic.
In this, it contrasts with some previous accounts discussed here. For example, Zimmermann
(2002b) is an integrated syntactic and semantic approach. I have not specified
the syntactic constraints that govern the theta-indexing of distributivity operators.
While it would not be in the scope of this paper to do so, clearly a theory of thetaindexing
eventually needs to supplement the proposed analysis in order to account
for the locality conditions on binominal each constructions. These conditions have
been studied in detail in the syntactic literature (e.g. Burzio, 1986; Safir and Stowell,
1988; Zimmermann, 2002b, ch. 3). Other examples of coindexation that are subject to
syntactic locality constraints are familiar from binding theory (Chomsky, 1981; Büring,
2005). As one reviewer suggests, one might expect θ-indexing to turn out to obey a
hierarchy comparable to the hierarchies of thematic roles that are sometimes claimed
to be at work in binding theory (Jackendoff (1972, p. 148), see also Büring (2005, p. 16)).
40
If correct, this may help explain why attested cases of inverse distribution in English,
such as the ones we have seen in (143) and (144), tend to involve nonagent subjects.
One possibility is that there are language specific constraints on θ-indexing. This
may seem necessary in the light of differences in distribution of distance-distributive
items in subject position that we have seen in (140). Another possibility is that θindexing
is subject to language-universal constraints, but that distance-distributive
items differ across languages in whether they require θ-indexing in order to distribute
over another element in the sentence. For example, one might speculate that jeweils in
(140) is not actually coindexed with the θ-role of the ballerinas. Rather, it distributes
over a set of occasions which stand in a one-to-one relation with the ballerinas and
which are made salient by the fact that the ballerinas are mentioned in the sentence.
Thus, on this option there is no formal link between jeweils and the ballerinas. This
makes an interesting prediction: The languages that allow distance-distributive items
in subject position might turn out to be just the ones that allow distribution over salient
entities that need not be overtly mentioned and need not be atomic. This prediction
indeed appears to be borne out, as discussed in Zimmermann (2002b, p. 48-50). Besides
German, Korean, Bulgarian and Czech have distance-distributive items that can occur
in subject position. We have seen in Section 2 that these languages are in the group
that allow distribution over salient entities.
Taken together, this paper and its counterpart, Champollion (2014a), suggest
the following general picture of distributivity. No matter whether distributivity is
introduced by an overt or by a covert element, it always involves a certain domain
that contains the individuals or the material to be distributed over, and a certain size
or granularity that specifies how finely the relevant predicates are distributed. When
the domain question is a count domain, for example when we distribute over people
or objects, then it is always possible to distribute over these objects one by one. When
the domain in question does not make such atomic units available, as in the case of
time or space, two things can happen. Either the element in question does not allow
distribution over such nonatomic domains, for example because it is incompatible
with non count domains to begin with, or else it looks for a salient cover or set in
the context, such as a salient set of temporal locations. Those distributive elements
that can do this in principle can also do this in count domains even though atoms are
available.
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