Chapter 1
Numerals and their kin
Author
Affiliation
In this paper, I focus on the meaning of numerals, i.e., lexical items that express
a relationship with numbers. I discuss different functions that cardinal numerals
can have including their quantifying and arithmetical use and compare various attempts
to capture the relationship between those functions. Furthermore, I examine
several types of complex numerical expressions such as ordinals, collective and
taxonomic numerals and multipliers in Slavic in comparison to other languages.
Finally, I bring novel data concerning Slavic label numerals, i.e., expressions used
to identify entities via association with a number and propose a morpho-semantic
analysis of such expressions. The core of the proposal is that the arithmetical meaning
is the numerals’ underlying semantic core from which both the quantifying and
the label meaning are derived.
Keywords: numerals, cardinals, numerical expressions, labels
1 Introduction
Numerals are funny words. On the one hand, it seems they share some core component
that justifies grouping them together as if they formed a single category,
a practice common in both descriptive and theoretic approaches to language.1
On the other hand, though, morphologically and syntactically they form a very
heterogeneous class of expressions with different items often exhibiting distinct
properties. This is the case not only when one compares various subclasses of
numerals, e.g., cardinals, ordinals and multiplicatives, but also often within the
subclass of cardinals different items have adjectival, nominal, or both nominal
1This intuition dates back at least to the Neo-Latin grammarian tradition, which distinguished
between three subclasses within the category nomen, i.e., nomen substantivum, nomen adjectivum
and nomen numerale.
Author. Numerals and their kin. To appear in: Change Volume Editor & in localcommands.tex Change
volume title in localcommands.tex Berlin: Language Science Press. [preliminary page numbering]
Author
and adjectival features while in some languages they seem to behave as verbs
(Donohue 2005; Ionin & Matushansky 2018).
In generative linguistics, a lot of work has been focused on establishing the
syntactic status of numerals with different approaches differing in whether they
are lexical or functional categories or whether they are heads or maximal projections.
These questions have been studied thoroughly with respect to Slavic
numerals with their well-known idiosyncratic properties (e.g., Babby 1987; Pesetsky
1982; Franks 1994; Rutkowski 2002; Klockmann 2012).
On the other hand, since the early days of analytic philosophy and formal semantics
a lot of attention has been dedicated to the meaning of numerals. At
least from Frege (1884), it has been recognized that capturing the semantics of
cardinals is far from trivial. Intuitively, it seems straightforward that they are
linguistic expressions that somehow describe a numeric quantity. However, beyond
this vague characterization it is much less obvious how to state exactly
what they actually are.
In this paper, I will discuss the main formal approaches to the meaning of cardinal
numerals. In doing so, I will also focus on examining two issues that only
recently have attracted due attention in the semantic literature. Specifically, I will
investigate different uses of cardinals as well as various derivationally complex
quantifying expressions in Slavic. The two sets of data indicate that cardinals
are typally flexibile and that numerals in general are semantically complex expressions.
I will argue that a proper approach to the meaning of numerals should
describe a compositional mechanism deriving different semantic flavors from the
underlying arithmetical meaning.
The paper is outlined as follows. In §2, I will discuss various functions of cardinal
numerals. §3 will provide an overview of different semantic approaches to
the quantifying function of cardinals, as in five cats. Next, §4 will explore the relationship
between the quantifying meaning and the ability of cardinals to refer to
abstract arithmetical concepts. In §5, I will review the literature on the morphosemantics
of complex numerical expressions in Slavic which presents evidence
calling for a compositional treatment. §6 is a case study in which I will explore
the so-far neglected label use of numerals. Finally, §7 concludes the paper.
2 Cardinals and their various flavors
Let us start with cardinal numerals. One of the challenges in accounting for their
meaning is that they can be used in very different ways (Bultinck 2005; Geurts
2006). Consider, for instance, (1). It turns out that a simple word such as English
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five is in fact quite tricky. Of course, it can be used to quantify over entities
denoted by the modified NP when it appears prenominally as in (1a) but that is
surely not the only function it can have. For instance, when it combines with
a measure word as in (1b), it designates a portion of a substance rather than a
plurality of objects. It can also appear in a predicative context such as (1c) or as
part of a measure phrase like in (1d). And that is not it since it can also be used to
refer to an abstract arithmetical concept, see (1e), or to label an entity as in (1f).
(1) a. Five cats meowed.
b. Five liters of milk got spilled.
c. These are five cats.
d. That cat is five years old.
e. Five is a Fibonacci number.
f. Player number five twisted her ankle.
Below, I will briefly discuss how the measure, predicative and arithmetical function
of cardinal numerals relate to their most deeply studied quantifying use.2 A
thorough investigation into properties of the label function will be undertaken
in §6.
2.1 Quantifying vs. measure use
For many years, the mainstream research has been primarily focused on the quantifying
use of cardinals exemplified in (1a). This seems justified since counting
objects is perhaps the first thing that comes to mind when we think about expressions
such as five. However, when a numeral appears in a measure phrase
such as (1b), it seems to be involved in measuring within a certain dimension,
e.g., volume, rather than in counting objects.
Intuitively, the difference between counting and measuring is that the former
is about specifying how many discrete objects of a certain kind there are, whereas
the latter determines some quantity in relevant measure units. Admittedly, some
proposals attempt to reduce measuring to a particular type of counting based
on the individuation in terms of quantity (Lyons 1977; Matushansky & Zwarts
2017). Consequently, measuring would simply be counting units determined by
measure words. An opposing approach treats counting as a form of measuring,
i.e., measuring a quantity of a plural individual in terms of natural or object units
2For an exhaustive discussion of numeral NPs interpreted as measure phrases in Slavic, see
Matushansky & Ionin (this volume).
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(Krifka 1989; 1995). Yet, there is also the third account which views counting and
measuring as two independent operations (Rothstein 2017).
The contrast between the two functions becomes clear when one considers
classifier constructions involving container nouns, see (2). Such expressions are
ambiguous between a measure (content) and counting (container) interpretation
(e.g., Landman 2004; Grimshaw 2007; Partee & Borschev 2012). On the measure
reading of (2a), Tomek flushed wine to the quantity of two bottles. On the counting
reading, on the other hand, he did something quite spectacular since two
actual bottles containing wine went down. Moreover, derivational morphology
seems to be sensitive to the distinction since the suffix -ful selects only for the
measure sense of a container noun, compare (2b)–(2c) (Rothstein 2011).
(2) a. Tomek flushed two bottles of wine through the toilet.
b. Romek added two { glasses / glassfuls } of wine to the soup.
c. Marek brought two { glasses / #glassfuls } of wine for our guests.
Another argument for distinguishing counting and measuring as two distinct operations
comes from distinct restrictions on the domain of quantification (Wągiel
2018). Though both measuring and counting quantify over entities that need to
be disjoint (Landman 2016), only the latter requires objects individuated as coherent
integrated wholes. To realize the contrast, imagine someone has spilled some
wine on the table in such a way that there are two separate blobs a and b whose
volume is exactly one and a half milliliters each. In such a scenario, (3a) is true
despite the fact that one of the three milliliters must be split between a portion of
a and a portion of b. On the other hand, (3b) is simply false. This contrast shows
that measuring, unlike counting, ignores individuation in terms of integrity.
(3) Scenario: There are exactly two 1.5 ml blobs of wine on the table.
a. There are three milliliters of wine on the table. true
b. There are three objects on the table. false
Though monotonic systems of measurement track certain part-whole relations
(Schwarzschild 2002), they do not seem to be sensitive to the spatial arrangement
of parts making up a whole. This suggests that despite sharing a common core
counting and measuring are two distinct things.
2.2 Quantifying vs. predicative use
It is well known that cardinals used as quantifiers give rise to scalar implicatures.
For instance, five cats in (4a) allows for an at least reading, as witnessed by the
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compatibility with the in fact clause. Similarly, (4b) is typically interpreted in a
way that Tomek must take at least three cards.
(4) a. Five cats live in the barn; in fact, six cats live there.
b. Tomek must take three cards.
However, numeral NPs in predicate position systematically lack a lower-bounded
interpretation (Partee 1986; Landman 2003; Geurts 2006). For instance, (5a) can
only get the exactly reading. This is further corroborated by the infelicity of sentences
such as (5b).
(5) a. These are five cats.
b. # The guests are three filmmakers; in fact, they are four filmmakers.
Furthermore, it has been observed that also bare cardinals can appear as predicates
in predicate position, see (6a)–(6b) (Rothstein 2017). This resembles the
behavior of adjectives rather than determiners.
(6) a. The apostles are twelve.
b. The planets are eight.
Similarly to numeral NPs, bare cardinals in predicate position lack an at least
reading. Thus both (6a) and (6b) are true if there are exactly twelve apostles and
exactly eight planets, respectively. Furthermore, sentences such as (7) are infelic-
itous.
(7) # My reasons for saying this are four; in fact they are five.
The contrasts discussed above indicate that meaning of the predicative use of
cardinals differs from their semantics in the quantifying function.
2.3 Quantifying vs. arithmetical use
It is not always the case that cardinals designate a plurality or measure. In (1e),
five refers to an abstract number concept rather than to a collection of entities.
In this use, cardinals have different properties than in their quantifying function
(Bultinck 2005; Rothstein 2017; Wągiel & Caha 2020). Specifically, number
concepts can have special properties such as being prime, see (8a), and they can
occur in mathematical statements such as (8b) and dedicated grammatical construction
as in (8c). When compared, the dimension of comparison is based on
their relative ordering, see (8d).
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(8) a. Five is prime.
b. Ten divided by five equals two.
c. Hanna can count up to five.
d. Five is bigger than four.
This behavior contrasts with cardinals in their quantifying function, which lack
the properties mentioned above. (9a) is odd since being prime is not a property
that can be attributed to a collection of things. Similarly, expressions denoting a
plurality of entities are illicit in constructions calling for a numeric value such as
(9b)–(9c). Finally, (9d) has different truth conditions than (8d), e.g., it would not
be true if one compared five cherries with four watermelons.
(9) a. # Five things are prime.
b. # Ten things divided by five things equals two things.
c. # Hanna can count up to five things.
d. # Five things are bigger than four things.
Furthermore, cardinals referring to number concepts are incompatible with numeral
modifiers such as more than and at least, see (10a). Finally, the arithmetical
function does not give rise to scalar implicatures and always get an exactly reading,
as witnessed in (10b).
(10) a. # More than five is prime.
b. # Two multiplied by five equals ten, if not more.
As we have seen, cardinal numerals can be used in various ways each of which
has different semantic characteristics. In the next two sections, I will review major
approaches to the meaning of cardinals and compare how they account for
at least some aspects of the flexibility discussed above.
3 Theories of cardinals
Given the variety of uses discussed in the previous section, any quest for ‘the’
meaning of cardinal numerals is probably a misunderstanding. Rather, a theory
of cardinals should be able to capture the relationship between meanings of cardinals
in their various functions. And yet, the mainstream research has been mainly
focused on the quantifying use often ignoring how it relates to other uses (with
Bylinina & Nouwen’s 2020 systematic inquiry being a notable exception).
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3.1 Cardinals as determiners
Let us start with the earliest and most prominent formal account of the meaning
of cardinal numerals, namely the standard Generalized Quantifiers (GQ) approach
(Barwise & Cooper 1981; though it can be traced back via Montague to
Frege). The intuition behind this analysis is that a cardinal such as English five is
in fact a determiner similar to some, most or all, as suggested by its occurrence
in prenominal position as in (11).
(11) { Some / Most / All / Five } cats meowed.
In the GQ theory, a determiner expresses a particular relation holding between
two sets (type ⟨⟨e,t⟩, ⟨⟨e,t⟩,t⟩⟩), i.e., a set denoted by the NP and a set denoted
by the VP. For instance, in (11) some yields the truth value True if the intersection
between the set of cats and the set of entities that meowed is non-empty, see (12a).
Similarly, on the GQ analysis five returns True if the cardinality of the set of cats
and the set of entities that meowed equals 5, see (12b).
(12) a. some = λPλQ[P ∩ Q ]
b. five = λPλQ[|P ∩ Q| = 5]
A nice thing about the GQ approach is that it aims to develop a unified semantics
of all quantified DPs. Furthermore, it can be enriched with a type-shifting mechanism
which allows for systematic mapping between different semantic types
(Partee 1986). Consequently, the theory can account for predicative uses of numeral
NPs such as (1c) by lowering the type of a general quantifier (⟨⟨e,t⟩,t⟩) to
the type of a predicate (⟨e,t⟩).
However, for some time it has been realized that the GQ approach is most probably
not a good way of capturing what cardinals are (e.g., Krifka 1999; Landman
2003). In the next sections, I will briefly discuss problematic data and alternative
approaches.
3.2 Cardinals as predicates
There are several problems with the GQ approach to cardinals. First of all, there
is a well-known asymmetry between DPs with numerals and DPs with regular
determiners in predicate position. For instance, DPs headed by every cannot be
used predicatively, see (13a). Similarly, examples such as (13b) are infelicitous on
a non-partitive interpretation. Given the felicity of sentences such as (1c) and
others discussed in §2.2, this fact is puzzling (e.g., Landman 2003).
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(13) a. # Hanna is every filmmaker at the party.
b. # The guests are { most / some } filmmakers.
Furthermore, as witnessed in (14a) bare determiners also cannot be used predicatively.
This contrasts with examples such as (6), in which cardinals pattern with
adjectives, see (14b). Since there is no standard type-shifting rule allowing for
mapping the type of determiners (⟨⟨e,t⟩, ⟨⟨e,t⟩,t⟩⟩) onto the type of predicates
(⟨e,t⟩), the predicative use of bare cardinals is unaccounted for. But even if such a
rule were postulated, it would still fail to explain the contrast between cardinals
and determiners such most and all.
(14) a. # My reasons for saying this are { most / all / every }.
b. My reasons for saying this are { serious / personal / five }.
The last problematic data point to be discussed here concerns the fact that cardinals
can appear along with bona fide determiners within a single DP (e.g., Rothstein
2017). From the GQ perspective, the compatibility of five with the definite
article in (15a) and with every in (15b) is unexpected and raises serious questions
regarding the semantic contribution of each of the elements.
(15) a. The five cats that I saw meowed.
b. Every two students got to share a hotel room.
The evidence discussed above motivated developing an alternative approach,
which treats cardinals as predicates (Landman 2003; 2004). Within such a framework,
cardinals get an interpretation very similar to that of intersective adjectives,
specifically they express a cardinality property. For this approach to work,
it is required to distinguish between two types of entities within the domain of individuals,
namely atoms, i.e., singular entities, and pluralities (Link 1983). While
atoms are minimal elements of a nominal denotation, pluralities are formed from
atoms via a pluralization operation * which closes the predicate under sum, i.e.,
takes a set of atomic entities and returns that set extended with all the pluralities
that can be formed by summing the atoms. In addition, the two types of individuals
are associated with each other via a part-whole relation defined in terms of
mereological parthood, e.g., an atomic entity a is part of a plurality a ⊕ b.3
According to the semantics provided in (16a), five denotes a set of pluralities
each of which contains 5 atomic entities. Similarly to intersective adjectives, cardinals
combine with NPs they modify via Predicate Modification (Heim & Kratzer
3For an introduction to mereological theories of plurality, see, e.g., Champollion & Krifka (2016).
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1998). Therefore, when five is composed with cats, see (16b), the resulting phrase
denotes the intersection of a set of pluralities of cats and a set of plural individuals
that consist of five atomic entities, i.e., a set of sums of five cats.
(16) a. five = λx[#(x) = 5]
b. five cats = λx[*cat(x) ∧ #(x) = 5]
The approach discussed above gives a straightforward explanation for the contrasts
between cardinals and determiners such every and most in predicate position
as well as the well-formedness and felicity of constructions such as those in
(15). However, it faces some challenges as well.
3.3 Cardinals as predicate modifiers
A problem with treating cardinals as cardinal properties is that cross-linguistically
the use of bare cardinals in predicate position is very restricted (Ionin & Matushansky
2018: 33–34). For instance, in Russian (17) is ungrammatical. Similarly,
Dutch (18) can only mean that the children are two years old and not that they
are two in number. And even in English the predicative use of bare cardinals is
heavily restrained, as witnessed by the fact that examples such as (19) are highly
degraded. If cardinals are assigned type ⟨e,t⟩, then this is rather startling.
(17) * Deti
children
byli
were
{ dva
two
/ dvoe }.
two.gndr
Intended: ‘The children were two in number.’ (Russian)
(18) # De
the
kinderen
children
zijn
are
twee.
two
Intended: ‘The children are two in number.’ (Dutch)
(19) ⁇ The chairs in this room are twelve.
Another issue concerns the fact that what counts as ‘one’ seems to be highly dependent
on the meaning of the NP the cardinal combines with. Arguably, atoms
should be defined relative to a particular property rather than in absolute terms.
To see why, let us consider partitive constructions such as those in (20) (see Chierchia
2010; Wągiel 2018). Example (20a) is weird since the numeral NP designates
triples of body parts and cannot be understood as referring to three pluralities of
boys. Yet, in (20b) quantification over subgroups of boys is possible. Importantly,
neither parts of the boys nor parts of the group denote entities that are atomic in
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any absolute sense. The first is true of portions of matter whose parts are also
parts of the boys. Similarly, the latter designates subgroups which themselves
can also consist of subgroups of boys. This suggests that the cardinal determines
what counts as ‘one’ relative to the denotation of the modified NP (and possibly
some contextual hints, see Rothstein 2010) and it is not obvious how this fact
could be captured if cardinals were interpreted simply as intersective predicates.
(20) a. # Three parts of the boys were sleeping.
b. Three parts of the group were sleeping.
A prominent alternative to treating cardinals as cardinal properties is to analyze
them as predicate modifiers (type ⟨⟨e,t⟩, ⟨e,t⟩⟩) equipped with a pluralization
operation * and a measure function #(P) (Krifka 1989; 1995). In (21a), * closes
the predicate under sum, i.e., takes a set of atomic entities and returns that set
extended with all the pluralities one can form by summing atoms. On the other
hand, the operation # returns for a property P a measure function that yields
pluralities of five individuals having that property. In other words, five maps
pluralities of entities onto the natural number 5.4 When the cardinal combines
with an NP, after the predicate slot is saturated we obtain a set of relevant plural
individuals (type ⟨e,t⟩). For instance, five cats denotes a set of pluralities of cats
each of which consists of five cats, see (21b). Thus, what counts as ‘one’ in this
system is relativized to the denotation of the NP.
(21) a. five = λP⟨e,t ⟩λxe [*P(x) ∧ #(P)(x) = 5]
b. five cats = λxe [*cat(x) ∧ #(cat)(x) = 5]
Another possibility is to interpret cardinals as predicate modifiers providing the
cardinality of a partition of a plural individual (Ionin & Matushansky 2006; 2018).
A partition of a plurality x is a cover of x, i.e., a set of entities such that the sum
of all those entities forms x; in addition, it is a cover whose cells do not overlap,
i.e., do not share any parts (see Gillon 1987; Schwarzschild 1996). In (22a), S is
a partition Π of an individual x and the cardinality of S equals 5. The cardinal
combines with the NP via standard Function Application and the result is of
type ⟨e,t⟩. For instance, five cats gets the meaning in (22b), i.e., it denotes a set of
pluralities divisible into 5 non-overlapping entities each of which has a property
4This is a slight simplification since in fact Krifka (1989) postulates a special operation nu for
measuring pluralities in terms of ‘natural units’ they consist of, whereas Krifka (1995) proposes
ou for measuring the number of ‘object units’ realizing a particular kind.
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of being a cat.5
(22) a. five = λP⟨e,t ⟩λxe ∃S⟨e,t ⟩[Π(S)(x) ∧ |S| = 5 ∧ ∀s ∈ S[P(s)]]
b. five cats = λxe ∃S⟨e,t ⟩[Π(S)(x) ∧ |S| = 5 ∧ ∀s ∈ S[ cat (s)]]
Notice, however, that both approaches discussed above require the denotation
of NPs the cardinal combines with to be singular, i.e., to consist only of atoms. It
is postulated that the source of plurality is the numeral and the plural marking
on the noun is, e.g., due to agreement with no semantic interpretation. Supporting
evidence comes from expressions such as those in (23) where the plural is
not associated with a plurality (Krifka 1989; Bylinina & Nouwen 2018) as well as
from languages such as Finnish and Turkish which display the singular/plural
distinction but in which unlike in, say, English cardinals systematically require
singular NPs, see (24).
(23) a. zero students
b. 1.0 students
(24) üç
three
{ çocuk
child
/ *çocuklar }
children
‘three children’ (Turkish)
Finally, analyzing cardinals as predicate modifiers makes it easier to account for
restrictions on bare cardinals in predicate position, see (18)–(19), by appealing to
the well-described peculiarities of the copula which would allow for a type-shift
only under particular circumstances (Ionin & Matushansky 2018: 33–34).
3.4 Cardinals as degree quantifiers
The last approach to be discussed in this sections builds on two observations regarding
the behavior of cardinals. The first observation is that sentences with
existential modals such as those in (25) are ambiguous (Kennedy 2013; 2015). On
the strong reading, (25a) means that Hanna is allowed to eat five cookies and she
is not allowed to eat more. The strong reading is probably the most natural interpretation
of (25a). There is, however, also a weak reading which merely states
that eating five cookies by Hanna is allowed without saying anything about eating
other quantities of cookies. This interpretation becomes more prominent in
questions, see (25b).
5The system is devised this way in order to provide a unified analysis of both simple and complex
numerals such as five hundred. For a more detailed discussion of the approach, see Matushansky
& Ionin (this volume).
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(25) a. Hanna is allowed to eat five cookies.
b. Is Hanna allowed to eat five cookies?
The second observation is that numerals can take scope independently of the rest
of the NP. The evidence comes from decimals in sentences such as (26a) (Kennedy
& Stanley 2009). Crucially, it does not entail the existence of families with 2.1 cats.
This contrasts with (26b) which is infelicitous due to such an entailment.
(26) a. The average Polish family has 2.1 cats.
b. # A normal Polish family has 2.1 cats.
The data discussed above suggest that cardinal numerals are in fact quantifiers.
We have already seen that the analysis of five as a determiner is most probably
incorrect and obviously treating it as a generalized quantifier over individuals
(type ⟨⟨e,t⟩,t⟩) would not make much sense. However, an interesting idea is to
interpret cardinals as quantifiers over degrees, i.e., expressions of type ⟨⟨d,t⟩,t⟩
(Kennedy 2013; 2015). Degrees (type d) are objects similar to individuals (type e)
with the crucial difference that the domain of degrees Dd , unlike the domain of
individuals De , is ordered.6 Hence, on the degree quantifier analysis five denotes
a set of degree properties whose maximal value equals 5, see (27a). A sentence
with a numeral NP would be then interpreted as in (27b), e.g., the maximal number
of entities that Hanna had and that are cats is 5.
(27) a. five = λD⟨d,t ⟩[MAX(D) = 5]
b. Hanna had five cats =
= MAX{n | ∃x[had(x)(hanna) ∧ *cat(x) ∧ #(x) = n)} = 5
The analysis of cardinals as degree quantifiers provides a promising perspective
to explain interactions between modals and both unmodified and modified cardinals
such as more than five and at least five (Nouwen 2010). In the next section,
I will discuss the arithmetical function of cardinals and how it relates to the theories
described so far.
4 Relating cardinalities and number concepts
The theories of cardinal numerals discussed so far focus on the quantifying function.
However, as we have already seen in (1e) and §2.3, cardinals can also be
used to refer to abstract numeric values.
6For more discussion on degrees and an introduction to degree semantics, see, e.g., Morzycki
(2016: Ch. 3).
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4.1 Cardinals as names of numbers
On the arithmetical function, cardinals seem to behave as proper names of number
concepts (type n) or, alternatively, degrees (type d).7 From this perspective,
the arithmetical meaning of five is simply the number 5, see (28a). Expressions
such as prime would then denote properties of numbers (type ⟨n,t⟩). For instance
the extension of (28b) would be the set {2, 3, 5, 7, 11, 13, . . . }. On the other hand,
expressions used to formulate mathematical statements such as plus, times, divided
by and equals seem to describe relations between number concepts. For
instance, (28c) is interpreted in terms of addition of two numeric values.
(28) a. five = 5
b. prime = λnn[prime(n)]
c. plus = λnnλmn[n + m]
The key question is what the relationship between names of number concepts,
i.e., the arithmetical use, and meanings of cardinals used in the quantifying function
is. In the next sections, I will discuss two possible approaches to how to
relate the two meanings.
4.2 Deriving number concepts from cardinalities
The mainstream view is to take the quantifying meaning as basic, be it the predicate,
predicate modifier or degree quantifier analysis, and to derive the arithmetical
meaning from it. Let us discuss three variants of that view.
As we have already seen in §3.2, on the predicate analysis cardinals are interpreted
as denoting a cardinal property, see (29a). Rothstein (2017) assumes this
semantics to be basic.8 The arithmetical meaning is then derived via a special
type-shifting operation ∩ which when applied to a cardinal property yields a
proper name, see (29b).
(29) a. five ⟨e,t ⟩ = λxe [#(x) = 5]
b. five n = ∩ five ⟨e,t ⟩ = 5
7Throughout the paper, I will assume there is no difference between the two notions and following
the widespread convention in the literature on the arithmetical use of cardinals I will
simply use the label n.
8In fact, this is a slight simplification since the theory distinguishes typally between lower numerals
and ‘lexical powers’, i.e., multiplicands such as hundred in five hundred.
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The proposal is seemingly motivated by an analogy with adjectival and nominal
uses of expressions such as white. It builds on Chierchia’s (1985) theory of
predication according to which every property has an entity correlate of type π
derived by ∩. For instance, the predicate white can be turned into the name of
the property of being white. Similarly, ∩ turns five into the name of the property
of being five in number.
However, as argued convincingly by Ionin & Matushansky (2018: 31–33) this
analogy does not hold since arithmetical environments such as those discussed in
§2.3 require the number 5 itself as an argument rather than the property of being
five in number. Consequently, in order to derive the arithmetical function Ionin
& Matushansky (2018: 26–27) propose a null suffix NOMNUM which is a nominalizer
that when applied to a predicate modifier, turns it into a numeric singular term.
More specifically, for any cardinal numeral it yields the corresponding cardinality.
An argument for such an account comes from the fact that cardinal numerals
as names of numbers display full morpho-syntactic uniformity suggesting they
are derived expressions. On the other hand, cardinals in the quantifying function
often show variation with respect to φ-features, e.g., grammatical gender.
Another mode of relating the arithmetical and quantifying meaning is proposed
by Kennedy (2015). The idea is to derive the first from the latter by the
standard type-shifting operations BE and IOTA defined in (30a) and (30b), respectively
(Partee 1986). BE takes a quantifier and returns a property. On the other
hand, IOTA yields the unique entity of which the relevant property is true.
(30) a. BE = λP⟨⟨e,t, ⟩,t ⟩λxe [P(λye [y = x])]
b. IOTA = λP⟨e,t ⟩ιx[P(x)]
Generalizing the type-shifts defined above so that they can also apply to quantifiers
over degrees and properties of degrees (or numbers), respectively, allows
to derive the arithmetical meaning by applying BE and IOTA consecutively to the
degree quantifier semantics. Specifically, when BE applies to a set of degree properties
whose maximal value equals 5, see (27a), the result in (31a) is the singleton
set {5}. The subsequent application of IOTA yields the number 5, as shown in (31b).
(31) a. BE( five ) = λnn[n = 5]
b. IOTA(BE( five )) = IOTA(λnn[n = 5]) = 5
A nice feature of such an approach is that it allows to relate the quantifying
and arithmetical functions using standard independently motivated type-shifting
machinery. In the next section, I will discuss accounts that derive cardinalities
from the arithmetical meaning.
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4.3 Deriving cardinalities from number concepts
Though the view that the quantifying function of cardinals is the basic one seems
to have become mainstream, there are a number of approaches that build on a
contrary intuition, namely that underlyingly cardinal numerals are simply names
of number concepts (e.g., Scha 1981; Krifka 1989; Scontras 2014; Wągiel 2018). According
to that alternative view, cardinals in prenominal position are in fact syntactically
and semantically complex expressions derived from the arithmetical
meaning via various shifts.
An early theory of cardinals postulating that they are derived from number
concepts was developed by Scha (1981). This system distinguishes between numbers,
numerals and numerical determiners. The first simply denote number concepts.
On the other hand, the numeral head shifts an integer (typen) to a cardinal
predicate while the numerical D head transforms such a cardinal predicate into
a determiner with a GQ-style semantics, see (32).9
(32) [ d⟨⟨e,t ⟩, ⟨⟨e,t ⟩,t ⟩⟩ [ numeral⟨e,t ⟩ [ numbern five ] ] ]
A related idea is the core of the system developed by Hackl (2000) who makes
use of a covert quantificational operator MANY in order to turn a number concept
into a determiner, see (33a). On this approach, the cardinal on its quantifying use
is accompanied with an additional null element that ensures the shift, as depicted
in (33b). The result combines with the denotation of the NP to form a generalized
quantifier.
(33) a. MANY = λnnλP⟨e,t ⟩λQ⟨e,t ⟩∃xe [#(x) = n ∧ P(x) ∧ Q(x)]
b. five cats = [ [ 5 MANY ] cats ]
Another proposal on how to derive the quantifying meaning from a name of a
number concept dates back to Krifka (1995). On this view, all cardinals are assumed
to underlyingly designate number concepts but they also involve an additional
classifier element that depending on a language can be either overt or
covert. There are several variants of this approach but what they all share is that
such a classifier element takes an integer and returns a counting device equipped
with an appropriate measure function, see (34a) (Scontras 2014; Wągiel 2018). For
instance, in the derivation of the phrase five cats the name of a number concept
9The exact denotation of the numerical determiner depends on whether the whole sentence
gets a collective, distributive or cover reading. In other words, there are three distinct Ds each
of which deriving a different interpretation.
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is first turned into a quantifying predicate modifier which then combines with
the noun to yield the meaning in (34b).
(34) a. CL = λnnλP⟨e,t ⟩λxe [*P(x) ∧ #(P)(x) = n]
b. five cats = [ [ 5 CL ] cats ] = λxe [*cat(x) ∧ #(x) = 5]
In yet another variant of the discussed approach, the classifier semantics turns an
arithmetical concept into a predicate (Sudo 2016). The classifier takes a numeric
value and yields a cardinal property (type ⟨e,t⟩) rather than a predicate modifier.
In addition, it can introduce a special presupposition concerning the nature of
referents of the modified noun. The resulting expression then combines with the
NP via Predicate Modification.
From the theoretical perspective both types of the relationship between the
arithmetical and quantifying function, i.e., the derivation from the quantifying
meaning to the arithmetical meaning and vice versa, seem plausible. Thus, the
question which one is correct appears to be an empirical issue. In §4.2, we have
already seen that certain aspects of cardinals’ morphology have been argued to
support the quantifying-function-is-basic view. In the next section, I will discuss
a different type of evidence that can be useful in determining the direction of the
derivation.
4.4 Morphological evidence
It has been observed that many languages distinguish formally between so-called
counting, i.e., arithmetical, and attributive, i.e., quantifying, numerals (Hurford
1998; 2001). For instance, in Mandarin the form èr ‘two’ is used in arithmetical environments
whereas liǎng ‘two’ is an attributive cardinal appearing in the quantifying
use. Similarly, the Maltese numeral tnejn ‘two’ indicates the arithmetical
function while żewġ ‘two’ has the quantifying meaning. Though the distinction in
question is typically restricted to a subset of numerals in a language and there are
a number of patterns of how the two forms can differ, it is a relatively widespread
phenomenon across languages.
Interestingly, the two families of theories discussed above make different predictions
regarding the morphological make-up of cardinal numerals cross-linguistically.
On the assumption that morphology expresses meaning, if the arithmetical
function is derived from the quantifying function, see §4.2, then we
would expect that across languages a pattern with arithmetical numerals being
morphologically more complex than quantifying cardinals should be relatively
widespread. This is because an additional affix is expected to introduce a shift
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from a counting device to a number concept. On the other hand, if the quantifying
function is derived from the arithmetical one, see §4.3, then we would expect
an inverse pattern to be common. Namely, quantifying cardinals should include
an additional morpheme encoding a shift from number concepts to quantifying
modifiers.
It turns out that the cross-linguistically widespread asymmetry attested, e.g.,
in many obligatory classifier languages, is of the latter type (Wągiel to appear;
Wągiel & Caha 2020). For instance, bare cardinals in Japanese cannot modify
NPs and require an additional element, traditionally referred to as a classifier, see
(35a).10 However, such an element is illicit in an arithmetical environment like
(35b) despite the fact that ko being a general classifier could be used to indicate
any type of inanimate entity.
(35) a. { *go-no
five-gen
/ go-ko-no }
five-clf-gen
ringo
apple
‘five apples’
b. juu
ten
waru
divided
{ go-wa
five-top
/ #go-ko-wa }
five-clf-top
ni-da.
two-cop
‘Ten divided by five equals two.’ (Japanese)
Therefore, it seems that the approaches that derive the arithmetical semantics
from the quantifying one make incorrect predictions with respect to meaning/
form correspondences in cardinals cross-linguistically. However, there is a twist
here since a pattern with more complex arithmetical numerals is also attested,
though scarce. For instance, in German the arithmetical form for 1 consists of an
additional morpheme compared to its quantifying equivalent, see (36).
(36) a. { ein
one
/ *ein-s }
one-nbr
Mädchen
girl
‘one girl’
b. Zehn
ten
geteilt
divided
durch
by
{ *ein
one
/ ein-s }
one-nbr
ist
is
gleich
equal
zehn.
ten
‘Ten divided by one equals ten.’ (German)
The pattern attested in German is puzzling and seemingly indicates that both
types of relationship between the two functions in question are possible. Never-
10Typically, classifiers also introduce certain restrictions on the referents of the NP modified by
the cardinal, e.g., being a round object, a flat object, a plant, a big animal etc. This aspect of their
meaning can be captured by accommodating various presuppositions concerning the nature
of the denoted individuals (Sudo 2016).
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theless, it can be accounted for by postulating a more complex morpho-semantic
structure of numerals along with a spell-out mechanism based on late insertion
(Wągiel & Caha 2020). On this analysis, though the arithmetical meaning is more
basic, it is still derived from an even more primitive concept of a number scale.
The quantifying meaning is then obtained by turning a number concept into a
quantificational modifier. The pattern in (36) can be then explained by postulating
that ein denotes the concept of a number scale, the suffix -s forges the name
of the numerical value and that there is yet another phonologically null morpheme
- that shifts the number scale into a counting device. Crucially, not only
does such an account capture cross-linguistic variation but also it explains why
the German pattern is rare. On the other hand, no alternative explanation of the
typological facts that would build on the quantifying meaning as the primitive
component has been proposed so far.
To conclude, the patterns discussed above suggest that the arithmetical meaning
of cardinals is the basic one and that other uses can be derived from it. In the
next section, I will discuss complex numerical expressions in Slavic and suggest
how this type of evidence can provide further hints on which of the two ways of
relating number concepts and cardinalities discussed so far is on the right track.
5 Complex numerical expressions
While the mainstream semantic research focuses mostly on cardinal numerals,
there is a growing body of formal literature acknowledging that cardinals are
in fact only a subset of various numerical expressions. This fact has been well
known in Slavic descriptive traditions. Since Slavic languages have a rich morphology,
it is not surprising that one can find abundant formal complexity also in
numerals. In this section, I will briefly review recent developments in the study
of different types of derivationally and semantically complex numerical expressions.
Though the focus of the section is on Slavic data, some parallels with other
languages will be drawn.
5.1 Ordinals
The cross-linguistically most widespread type of derived numerical expressions
are ordinal numerals, i.e., forms such as English fifth and Russian pjatyj ‘fifth’.
Intuitively, ordinals represent position or rank in a sequential order. If a language
distinguishes morphologically a class of ordinals, they are always derived by an
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additional affix (Stolz & Veselinova 2013).11
Perhaps somewhat surprisingly, the semantics of ordinals has been typically
linked to superlatives. Cross-linguistically, the two types of expressions often
share identical or related morphemes (Hurford 1987; Veselinova 1998)12 and have
the same syntax (Barbiers 2007). Furthermore, the acquistion of ordinals seems
to be influenced by superlative morphology (Meyer et al. 2018). Even more importantly,
ordinals, just like superlatives, give rise to an ambiguity between an
absolute and comparative interpretation (Bhatt 2006; Sharvit 2010). On the absolute
reading of (37a), John gave Mary a telescope that is older than all other
telescopes. On a comparative reading, however, he might have given her a relatively
new telescope than happens to be older than the telescopes other people
gave her, i.e., John’s gift is compared to other gifts rather that to telescopes in
general. Similarly, (37b) can mean either that Mary got a telescope that was older
than other telescopes, or that John’s telescope was the first Mary has received.
(37) a. John gave Mary the oldest telescope.
b. John gave Mary the first telescope.
However, it has been observed that ordinals, unlike superlatives, do not give
rise to so-called upstairs de dicto readings in sentences with intensional operators
such as want (Bylinina et al. 2015). For instance, (38a) can have a particular
comparative interpretation which makes it true in the scenario below. Yet, that
reading is unavailable for (38b).
(38) Scenario: There are many trains throughout the day. John wants to take
a train. Any of the trains between 3 pm and 4 pm is fine for him.
Similarly, Bill and Steve want to take a train, and they are fine as far as
the departure time is between 5 pm and 6 pm and between 7 pm and 8
pm, respectively. These people do not know anything about one another.
a. John wants to take the earliest train. true
b. John wants to take the first train. false
As a result, Bylinina et al. (2015) argue that superlatives and ordinals differ with
respect to where they are interpreted. According to that proposal, while superlatives
involve movement, ordinals are always interpreted in situ.
11Unless an ordinal is suppletive, e.g., one ∼ first in English.
12Notice that this is also the case in some Slavic languages. For instance, Polish pierw-szy and
Ukrainian per-šyj, both ‘first’, contain the comparative markers -szy and -šyj which also appear
in superlatives, e.g., naj-star-szy and naj-star-šyj, both ‘the oldest’, respectively.
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5.2 Collective numerals
One of the key topics discussed in the literature on plurality regards distributive
and collective interpretations of sentences including plural DPs such as (39)
(e.g., Scha 1981; Link 1983; Landman 1989; 2000; Schwarzschild 1996). For instance,
(39b) is ambiguous between a reading on which a total of three letters have been
written, i.e., a distributive interpretation, and a reading on which only one letter
has been written, i.e., a collective interpretation. The source of the ambiguity has
often been located inside the VP (e.g., Hoeksema 1983; Link 1984; Schwarzschild
1991; see also Lasersohn 1995: Ch. 7 for an overview).
(39) a. The boys wrote a letter.
b. Three boys wrote a letter.
Interestingly, Slavic languages have special numerical expressions which force a
collective interpretation of the whole sentence. For instance, consider the Czech
sentences in (40) (Dočekal 2012; 2013).13 While (40a) patterns with (39b) in that
it is ambiguous between a distributive and a collective reading, the sentence in
(40b) rules out the distributive interpretation, and thus the sole possible reading
is that only one letter has been written.
(40) a. Tři
three
chlapci
boys
napsali
wrote
dopis.
letter.acc
✓coll, ✓distr
‘Three boys wrote a letter.’
b. Trojice
three.coll
chlapců
boys.gen
napsala
wrote
dopis.
letter.acc
✓coll, # distr
‘A group of three boys wrote a letter.’ (Czech)
Since the difference between the two sentences boils down to the alternation
between the basic cardinal numeral tři ‘three’ and the morphologically complex
derived form trojice ‘group of three’, the suffix -ice has been proposed to be a
morphological reflex of a semantic operation forcing a collective interpretation,
specifically the group-forming operator ↑ proposed by Landman (1989).14
Importantly, collective numerals cannot be used to refer to number concepts,
as witnessed in (41). This fact is surprising if the arithmetic meaning were derived
13In the literature, expressions such as trojice ‘group of three’ have been referred to as collective
or group numerals as well as denumeral group nouns (due to their nominal-like behavior).
14The affix -oj- appears only in morphologically complex numerical expressions derived from
the roots for 2 and 3 and is usually assumed to have a purely structural function, i.e., it marks
non-cardinal stems.
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Numerals and their kin
from the quantifying one, see §4.2. Despite the collective meaning component
described above, NPs involving collective numerals denote pluralities and one
would expect the same shift that is posited to turn the quantificational meaning
of a cardinal into the name of a number concept should be applicable also in this
case, contrary to facts.
(41) a. # Trojice
three.coll
je
is
prvočíslo.
prime.number
Intended: ‘Three is prime.’
b. # Šestice
six.coll
děleno
divided
trojicí
three.coll.ins
je
is
dvojice.
two.coll
Intended: ‘Six divided by three is two.’ (Czech)
Collective numerals have been identified also in other Slavic languages. In particular,
Polish trójka ‘group of three’ patterns with (40b) (Wągiel 2015) and similar
examples include Slovak trojica, Bosnian/Croatian/Montenegrin/Serbian (BCMS)
trojka and Russian trojka, all ‘group of three’.15 The existence of the discussed
phenomenon in Slavic is unexpected since it is typically postulated to hold crosslinguistically
that if a language has a scopal quantifier, it is a quantifier forcing a
distributive interpretation (Gil 1992).16 Hence, the Slavic data discussed above
compel to revise that generalization. In any case, a systematic theory-driven
cross-linguistic research of collective numerals has not been carried out so far.
5.3 Both
An interesting feature of many languages including Slavic is that alongside a regular
cardinal numeral corresponding to English two there is also another form
corresponding to English both. In an early GQ account, both has been analyzed
as a quantificational determiner with the same meaning as the two (Barwise &
Cooper 1981). However, it has been observed that the two expressions are not
equivalent (Ladusaw 1982). For instance, both is incompatible with collective
predicates, see (42a), and cannot appear inside partitives, see (42b). As witnessed
in (43), the same behavior is also attested in Slavic (Łazorczyk & Pancheva 2009).
(42) a. { The two / #Both } students are a happy couple.
15Russian shows a strong tendency to lexicalize collective numerals, e.g., trojka is also a word
for a carriage drawn by a team of three horses abreast.
16In Hebrew, there are derived expressions that at first blush resemble Slavic collective numerals,
e.g., šlišiya ‘threesome of’. However, they have been reported to differ in that they do not force
a collective interpretation (Gil 1992).
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b. One of { the two / #both } children sneezed.
(43) a. # Obe
both
ženščiny
woman.gen
prišli
came
vmeste.
together
Intended: ‘Both women came together.’
b. Kto
who
iz
from
etix
these.gen
{ dvoix
two.gen
/ *oboix }
both.gen
sdelal
did
to,
this.acc
čto
what
xotel
wanted
otec?
father
‘Which of these two did what his father wanted?’ (Russian)
The data presented above led to proposals postulating that both, unlike cardinals,
involves an additional distributive component (e.g., Ladusaw 1982; Landman
1989).17 Interestingly, however, there is yet another respect in which the
two differ. Specifically, both and its counterparts in other languages cannot be
used to refer to a number concept in an arithmetical environment, see (44).
(44) a. # Oba
both
to
that
liczba
number
parzysta.
even
Intended: ‘Both is an even number.’
b. # Dziesięć
ten
dzielone
divided
przez
by
oba
both.acc
równa
equals
się
refl
pięć.
five
Intended: ‘Ten divided by both equals five.’ (Polish)
Similarly to collective numerals, the behavior of both and its Slavic equivalents
presented in (44) poses a serious challenge for approaches deriving the arithmetical
function of numerals from the allegedly basic quantifying meaning.
5.4 Taxonomic numerals
A unique property of some Slavic languages is that they have taxonomic numerals.
Likewise collective numerals, they are derived by a special suffix which
triggers a non-trivial semantic effect. Unlike cardinals, such expressions do not
quantify over object-level entities, i.e., tokens of a type, but rather over subkinds
of a particular kind corresponding to the meaning of the modified noun.18 For
instance, let us consider the taxonomic numeral dvojí ‘two kinds of’ in Czech
17For the recent research on bona fide distributive numerals, see, e.g., Gil (2013), Cable (2014)
and Wohlmuth (2019).
18For an introduction to the kind/object distinction, see, e.g., Krifka et al. (1995).
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Numerals and their kin
(Dočekal 2012; 2013; Grimm & Dočekal to appear). The phrase in (45a) does not
refer to two musical instruments but rather to two types of violin, e.g., a set of
classical violins and a set of electric violins. Likewise, (45b) denotes two kinds of
the beverage, e.g., white and red wine, irrespective of its amount.
(45) a. dvojí
two.tax
housle
violin
‘two kinds of violin’
b. dvojí
two.tax
víno
wine
‘two kinds of wine’ (Czech)
Unlike cardinals, taxonomic numerals specify the domain of quantification to
be within the realm of kinds exclusively. Therefore, the suffix -í has been proposed
to introduce a quantificational operation dedicated to counting subkinds,
specifically the ku (for ‘kind unit’) operation proposed by Krifka (1995).
Though taxonomic numerals seem to be best preserved in Czech, Polish dwojaki
and dwoisty, Bulgarian dvojak and Russian dvojakij, all ‘twofold’, are arguably
expressions of a similar type. However, there appear to be certain differences
with respect to their distribution and behavior that have not been described
sufficiently so far. Crucially, none of the taxonomic numerals can be used
to refer to an abstract number concept.
5.5 Aggregate numerals
Another intriguing type of Slavic derivationally complex numerical expressions
is sometimes referred to as aggregate numerals, e.g., forms such as Czech dvoje
‘two collections’ (Dočekal 2012; 2013) as well as BCMS dvoji (Lučić 2015). Likewise
collective numerals, they are derived by a special suffix which triggers a
non-trivial semantic effect and lack the arithmetical meaning. While cardinals
simply count atomic individuals in the denotation of the modified NP, aggregate
numerals are used to quantify over collections of entities that typically form a
particular spatial configuration. For instance, (46a) refers to two aggregates of
cards, e.g., two decks of cards. Similarly, (46b) denotes two pairs of shoes.
(46) a. dvoje
two.aggr
karty
cards
‘two sets of cards’
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Author
b. dvoje
two.aggr
boty
shoes
‘two pairs of shoes’ (Czech)
Aggregate numerals are not limited to Slavic. For instance, similar expressions
seem to have existed in Latin, e.g., trēs ‘three’ ∼ trīnī ‘three collections’ (Ojeda
1997), and Estonian uses pluralized numerals for this purpose, e.g., kaks ‘two’ ∼
kahed ‘two collections’ (Norris 2018). However, it is the Slavic data what has been
taken as evidence for the relevance of mereotopology in natural language, specifically
the significance of topological relations holding between atomic members
of a plurality within a part-whole structure.19 In particular, the suffix -e in phrases
such as (46) is sensitive to how parts making up a whole are arranged, and thus it
has been proposed to introduce a semantic operation dedicated to counting clusters,
i.e., topologically structured spatial groupings of entities (Grimm & Dočekal
to appear).
5.6 Gender-sensitive numerals
All Slavic languages mark grammatical gender on low numerals whereas some
of them, e.g., Bulgarian, Polish and Slovak, distinguish between virile (personal
masculine) and non-virile forms also in higher numerals. (Cinque & Krapova
2007; Pancheva 2018; Wągiel to appear). However, what is probably even more
interesting is the existence of complex expressions which I will refer here to as
gender-sensitive numerals.20 Such expressions introduce non-trivial restrictions
with respect to the natural gender of individuals making up a plurality.
Let us consider Polish gender-sensitive numerals such as troje ‘three (at least
one male and at least one female)’ (Wągiel 2014; 2015). Unlike virile and non-virile
cardinals in (47a) and (47b), respectively, they do not show grammatical gender
agreement with modified NPs. Instead, they are neuter, as demonstrated in (47c).
And yet they introduce a gender-related effect. While (47a) denotes a plurality of
either males or students whose natural gender is not known to the speaker and
(47b) refers to three females, (47a) is interpreted as designating a mixed-gender
group.
(47) a. tych
these.acc.v
trzech
two.v
studentów
students.gen.v
‘these three students (male or indeterminate)’
19For an introduction to mereotopology, see Grimm (2012: Ch. 4), Wągiel (2018: Ch. 6).
20In traditional Slavic linguistics such forms are typically termed ‘collective numerals’. In order
to avoid confusion with expressions discussed in §5.2, I will not use that term.
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b. te
these.acc.nv
trzy
three.nv
studentki
students.nom.nv
‘these three female students’
c. to
this.nom.n
troje
three.gndr.n
studentów
students.gen.v
‘these three students (at least one male and at least one female)’
(Polish)
Based on the evidence discussed above, numerals such as Polish troje have been
proposed to introduce a presupposition that a plurality of individuals referred to
by the whole phrase is heterogeneous and consists of both male and female individuals.
Though similar expressions seem to be only attested in BCMS, e.g., dvoje,
‘two (one male and one female)’, there are also other types of gender-sensitive numerals
in Slavic. For instance, BCMS dvojica and Russian dvoe, both ‘two (male)’,
presuppose a homogeneous group consisting of male individuals only (Kim 2009;
Lučić 2015; Khrizman 2020). Interestingly, none of the gender-sensitive numerals
can be used to refer to a number concept.
5.7 Indefinite numerals
Another intriguing class of numerical expressions concerns indefinite numerals
such as English several (Kayne 2007). Similar quantifiers appear in all branches
of Slavic, e.g., compare Czech několik, Russian neskol’ko, Bulgarian nyakolko and
BCMS nekoliko, all ‘several’. Particularly interesting data come from Polish which
distinguishes between two different expressions of this type, specifically kilka
‘several, a few’ and ileś (tam) ‘some, some amount’. While the former is only
compatible with count NPs and designates a cardinality between 3 and 9, the
latter also combines with mass NPs and can indicate any quantity (Wągiel 2020b).
In terms of morphosyntax and many aspects of semantics, several and its Slavic
equivalents pattern with cardinals. However, one crucial respect in which the two
differ is that indefinite numerals do not fit contexts calling for number concepts,
see (48a) and (49) (Schwarzschild 2002; Wągiel 2020b). Despite the fact that one
can easily imagine the intended meaning and paraphrase it using the existential
quantifier, as in (48b), this is not what one gets in (48a) and (49).
(48) a. # Four plus several is less than ten.
b. There is a number n such that four plus n is less than ten.
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(49) # Cztery
four
plus
plus
{ kilka
several
/ ileś (tam) }
some
to
this
mniej
less
niż
than
dziesięć.
ten
Intended: ‘Four plus several/some is less than ten.’ (Polish)
Polish indefinite numerals have been analyzed as complex expressions with an
incorporated measure function and choice function. The difference between kilka
and ileś (tam) is due to a different type of set the choice function selects a number
from and a different kind of built-in measure function (Wągiel 2020b).
5.8 Multipliers
In many languages there is a heavily understudied class of numerical expressions
sometimes referred to as multipliers, e.g., English double. Though Germanic and
Romance languages have borrowed their multipliers from Latin, in Slavic they
are morphologically transparent and one can easily notice that they are derived
from numerical roots by various affixes, e.g., compare Polish podwójny, Russian
dvojnoj and Czech dvojitý, all ‘double’.
Interestingly, multipliers display non-trivial quantificational behavior. Specifically,
they do not quantify over whole entities or events, as cardinals would, but
rather over parts thereof (Wągiel 2018; 2020a). For instance, the phrase in (50a)
denotes a set of singular objects having a complex internal structure. Each of
those objects is a crown but crucially it also has two parts that could be considered
as having the property of being a crown themselves, e.g., entities such as
the ancient Egyptian Pschent or the papal tiara. Similarly, (50b) refers to a murdering
event with two victims. But this means that there were two parts of that
event each of which could be described as a murder in its own right.
(50) a. dvostruka
two.mult.f
kruna
crown.n
‘double crown’
b. dvostruko
two.mult.n
ubistvo
murder.n
‘double murder’ (BCMS)
The behavior of multipliers has been argued to stem from the phenomenon of
subatomic quantification (Wągiel 2018). According to the proposal, natural language
is sensitive to the arrangement of parts within a singular entity. Certain expressions,
e.g., proportional partitives and multipliers, can access such subatomic
part-whole structures and quantify over parts conceptualized as contiguous objects
within a whole.
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Numerals and their kin
5.9 Event numerals
Yet another class of cross-linguistically frequent though surprisingly understudied
numerical expressions consists of event numerals also known as multiplicatives
(but see Landman 2006; Donazzan 2013; Kayne 2015). They include adverbial
expressions such as English twice and two times which are primarily used to
quantify over events but can be also employed in comparative and equative constructions.
Event numerals are attested in all Slavic languages with BCMS dvaput
and Czech dvakrát, both ‘two times, twice’, being representative examples.
Intuitively, event numerals seem to be a type of a broader class of adverbs of
quantification including frequency adverbs such English often and French souvent
‘often’ (Abeillé et al. 2004; Doetjes 2007). However, close examination reveals
interesting differences between event numerals and frequency adverbs one
of which regards degree modification (Dočekal & Wągiel 2018). For instance, the
sentence with the event numeral in (51a) is ambiguous between the quantifiedevent
reading and the quantified-degree reading. On the former interpretation
there were two events of increasing the demand and the total value of the increase
is unknown whereas on the latter interpretation there was a twofold increase
in the demand irrespective of the number of events on which it increased.
Interestingly, (51b) lacks the quantified-degree reading.
(51) a. Poptávka
demand
po
after
dotacích
subsidies.loc
vzrostla
increased.pfv
dvakrát.
twice
✓event, ✓deg
‘The demand for subsidies increased (by) two times.’
b. Poptávka
demand
po
after
dotacích
subsidies.loc
rostla
increased.ipfv
často.
often
✓event, #deg
‘The demand for subsidies increased often.’ (Czech)
In addition, while event numerals are perfectly fine in comparatives and equatives,
frequency adjectives are infelicitous in such constructions, as witnessed by
the contrast in (52).
(52) Petr
Petr
je
is
{ dvakrát
twice
/ #často
often
} vyšší
taller
než
than
Marie.
Marie
‘Petr is two times taller than Marie.’ (Czech)
Unlike all other types of numerical expressions discussed in this section, event
numerals can be used in arithmetical environments provided that they express
multiplication. However, it is puzzling that not all event numerals can do that. For
instance, Polish has two sets of event numerals but only one can be felicitously
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Author
used in a context such as (53) which suggests that the element razy ‘times’ might
be a homonymous form for a multiplicative morpheme used to quantify over
events and an expression of an arithmetical relation.
(53) { Dwa
two
razy
times
/ #Dwukrotnie
twice
} trzy
three
równa
equals
się
refl
sześć.
six
‘Two times three equals six.’ (Polish)
Though event numerals seem key to understanding quantification in the domains
of events and degrees, it is surprising that so far they have not received nearly
as much attention as cardinals.
5.10 Frequency numerals
Finally, in many languages there are expressions I will refer to as frequency numerals,
i.e., multiplicative adjectives equivalent to English two-time. Slavic is no
exception here and Polish dwukrotny, Czech dvojnásobný and Russian dvukratnyj,
all ‘two-time’, are representative examples of the class in question.
At first sight, it seems that frequency numerals such as two-time are expressions
of the same type as frequency adjectives like occasional and frequent (Zimmermann
2003; Schäfer 2007; Gehrke & McNally 2015). Notice, however, that
while occasional can have the adverbial reading, expressions such as two-time
pattern with frequent in that they lack such an interpretation. For instance, while
(54a) means that occasionally a sailor strolled by, both (54b) and (54c) cannot be
understood that way.
(54) a. An occasional sailor strolled by. ✓adverbial
b. A frequent sailor strolled by. #adverbial
c. A two-time senator strolled by. #adverbial
Furthermore, frequency numerals fail to combine with sortal nouns and seem to
require role nouns denoting socially salient capacities such as positions with a
term and award recipients that can be acquired repetitively. That is because frequency
numerals quantify over conventionalized events of acquiring a property,
e.g., via a ceremony, compare (55a) and (55b).
(55) a. Jan
Jan
to
this
dwukrotny
two-time
{ mistrz
champion
/ #człowiek
man
}.
‘Jan is a two-time champion.’
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Numerals and their kin
b. Jan
Jan
został
became
{ mistrzem
champion.ins
/ #człowiekiem
man.ins
} dwa
two
razy.
times
‘Jan became a champion twice.’ (Polish)
Having examined various complex numerical expressions, let us now summarize
what the discussed data show us.
5.11 Summary
In this section, I reviewed various types of numerical expressions in Slavic and
beyond. In particular, I discussed a wide range of morphologically complex forms
and examined how they correspond to certain non-trivial semantic effects. The
reviewed dataset is far larger and more diverse than typically analyzed in mainstream
semantic theories of numerals, see §3. And yet, there is an intuition that
a general theory of numerals should cover all of the examined cases. Therefore,
I believe the relevance of the above-discussed Slavic data for such a theory is
twofold.
First, morphological transparency of the discussed forms clearly shows that
all complex numerical expressions share a common component, i.e., a numerical
root designating a certain number.21 That number is somehow employed in
counting but what exactly is counted may vary depending on a type of numeral,
e.g., whole objects in the case of cardinals, parts in the case of multipliers and subkinds
in the case of taxonomic numerals. Therefore, a theory of numerals should
aim at providing a general mechanism that will allow us to capture not only the
behavior of cardinals but also other numerical expressions discussed here.
Second, despite sharing an element designating a number none of the complex
numerical expressions can be used to refer to abstract arithmetical concepts, i.e.,
the arithmetical function is never available except for basic cardinals.22 This fact
turns out to be problematic for the approaches deriving the arithmetical meaning
from the quantifying one, see §4.2. The reason is that they shift a counting device
semantics of a cardinal into a proper name of a number concept. But if so, why
cannot that shift be also applied in the case of other types of counting expressions,
e.g., a taxonomic, an aggregate or an indefinite numeral, or both? Semantically,
these expressions do not seem to be radically different from cardinals (they all
count something) to justify why shifting their meaning to a name of a number
concept is impossible. If, on the other hand, the arithmetical meaning is basic,
one expects to derive various quantificational flavors from it. Therefore, it would
21Unless, of course, suppletion is involved.
22And perhaps event numerals in contexts expressing multiplication.
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Author
not be surprising that only basic cardinals can have the arithmetical function and
other types of numerical expressions would lack it.
In the next section, I will attempt to show how the two conclusions reached
above can guide formulating a particular proposal of yet another type of numerical
expressions, namely label numerals.
6 Case study: Numerals as labels
In this section, I will return to the label use of numerals, as already introduced
in (1f) in §2, repeated here as (56).
(56) Player number five twisted her ankle.
So far, the label meaning has received little attention in the literature (see Hurford
1987: 167–168; Wiese 2003: 37–42; Bultinck 2005: 119–129). I will demonstrate that
not only is this function conceptually different from other functions discussed in
§2 but also that the distinction is reflected in grammar. I will argue that the label
function is derived from the arithmetical meaning and I will propose a morphosemantic
account that allows for deriving both the form and the meaning of
dedicated label numerals in Slavic. The proposed analysis will combine standard
compositional semantics with a Nanosyntax model of morphology.
6.1 Label uses are distinct
Intuitively, label numerals provide means for identification of an object via its
association with a unique number, which allows for the recognition of that object
among other entities. This mechanism is sometimes referred to as nominal
number assignment (Wiese 2003: 37–38). A set of numbers would be consistent
labels as long as the numerals corresponding to distinct numbers in that set are
uniquely used for each distinct object. This can be achieved by a one-to-one mapping
between a set of labeled entities and a given set of numbers. Importantly, the
actual values of the numbers represented by label numerals are irrelevant since
they do not indicate quantity, order or any other type of measurement. Furthermore,
the set of labels can change over time, e.g., it can grow as new entities are
labeled. The only thing that matters is that at a given time every labeled entity
is associated with a unique number.
It is clear that from a conceptual point of view the label function differs from
the quantifying and the arithmetical meaning. Importantly, however, the distinction
is also encoded in grammar. First, numerals used as labels do not allow for
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Numerals and their kin
modification by numeral modifiers and prequantifiers, see (57a). Moreover, label
constructions do not give rise to scalar implicatures. For instance, in the context
of (57b), taking bus number six would not guarantee getting to the university.
(57) a. # Tram { more than / at least / all } number five has left.
b. You must take tram number five to get to the university. #at least
The data in (57) differ from the quantifying function and pattern with the properties
of the arithmetical use of numerals discussed in §2.3. However, these two
functions can be distinguished linguistically as well. For instance, English constructions
with numerals used as labels differ from other count NPs in that they
are ungrammatical with the indefinite article, see (58a). On the other hand, names
of number concepts cannot appear without the definite article, as demonstrated
in (58b) (for a more detailed discussion on the distribution of English phrases
such as number five, see Bultinck 2005: 124–129).
(58) a. Kim took (*a) tram number five.
b. * (The) number five is odd.
Above, I argued that the label meaning of numerals is both conceptually and
linguistically different from the quantifying and the arithmetical function. In the
following two sections, I will discuss further evidence for the relevance of the
label function as well as its relationship with the other two discussed meanings.
6.2 Evidence from co-lexicalizations
In many languages, an additional expression can be optionally used alongside
the numeral in order to indicate its quantifying, arithmetical or label use. Sometimes,
the same lexical item can introduce all of the functions above. For instance,
English employs a single noun, namely number, to indicate cardinality, to designate
a mathematical object and to signal that an entity is identified via association
with a relevant integer, see (59).
(59) a. The cats are five in number.
b. The number five is odd.
c. Tram number five left.
While English shows full co-lexicalization, i.e., the noun number can be used to
indicate all three functions, more often lexicons develop a different word for at
least one of the functions in question. For instance, Bulgarian exhibits a pattern
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with no co-lexication, i.e., each of the discussed functions is signalled by a morphologically
distinct expression, specifically broj is for quantification, čislo is for
arithmetic and nomer is for the label function, all ‘number’.
Though ways in which meanings and forms correspond to each other vary, it
seems that they do not vary in an unrestricted manner. The co-lexicalization patterns
found in Slavic are summarized in Table 1. Interestingly, the pattern that is
not attested in the sample is a case of non-contiguous co-lexicalization where the
quantifying and label function are expressed with the same form while the arithmetical
function is expressed by a different lexical item. Perhaps surprisingly,
this result resembles the well-known *ABA principle (Bobaljik 2012).
Table 1: Co-lexicalization of ‘number’
language qantifying arithmetical label
BCMS A broj A broj A broj
Polish A liczba A liczba B numer
Slovak A počet B číslo B číslo
Bulgarian A broj B čislo C nomer
unattested A B A
Examining permissible patterns of co-lexicalization is rarely considered evidence
in formal semantics. However, I believe that looking at how certain parts
of lexicon are structured can be instructive. Arguably, it can reveal that some
meanings have more in common than others and in some cases that certain categories,
though theoretically possible, are empirically unrealized.23 Consequently,
one can gain valuable hints regarding what those meanings are.
On the assumption that co-lexicalization does in fact reflect the relationship
between concepts, the data in Table 1 suggest that while the quantifying and the
label meaning are unrelated, they both seem to stem from the arithmetical function.
The reason is that if both the quantifying and the label meaning are derived
from the arithmetical meaning, the attested co-lexicalization patterns are unsurprising
since the unrelated derived functions are not expected to be expressed
by a single term that is different from the one expressing the underlying meaning
(ABA). On the other hand, if the quantifying meaning were basic, then one
would expect to find the ABA pattern instead of the Czech and Slovak ABB.
23This way of viewing data has been successful in lexicology, e.g., in generalizations about kinship
terms (see, e.g., Hage 1997).
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Numerals and their kin
Whether a language which has a term for the quantifying and the label function
to the exclusion of the arithmetical function can be found is an empirical
issue. So far, the Slavic co-lexicalization data suggest a strong relationship between
the label and the arithmetical meaning and no relationship between the
label and the quantifying meaning. In the next section, I will discuss a class of
complex numerical expressions for labeling in Slavic.
6.3 Derived label numerals
An interesting property of Slavic languages is that they have specialized label
numerals. Such dedicated expressions are derived with the suffixes -ka and -ica.
Table 2 gives an overview of the label forms for the numerals forresponding to 5
across Slavic.
Table 2: Cardinal and label ‘five’ in Slavic
language number cardinal label
Czech 5 pět pětka
Polish 5 pięć piątka
Russian 5 pjat’ pjaterka
Slovenian 5 pet petka/petica
BCMS 5 pet petica
Though there are a number of interesting differences in their productivity and
distribution, e.g., Polish derives label forms with -ka up to 999 while Russian has
them only for 2–10, 20 and 30, all Slavic label numerals are morphosyntactically
nominal expressions used to refer to objects that can be identified via nominal
number assignment. Though very often they appear bare, the noun specifying
what entity is labeled can optionally precede the label numeral, as in (60). In the
absence of the noun this information is typically provided by the context.
(60) (Tramvaj)
tram
pět-ka
five-lbl
vyjede
will.go.out
na
on
novou
new.acc
trať
track.acc
v
in
listopadu.
November.loc
‘Tram number five will head out on the new track in November.’ (Czech)
Similarly to classifier constructions and other complex numerical expressions, recall
§4.4 and §5, Slavic label numerals exhibit uniform behavior in that they cannot
be felicitously used in mathematical statements such as (61), This, in turn, sug-
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gests that their semantics is more intricate compared to the arithmetical meaning
of cardinal numerals.
(61) a. Dva-krát
two-times
pět
five
se
refl
rovná
equals
deset.
ten
‘Two times five equals ten.’
b. # Dvoj-ka
five-lbl
krát
times
pět-ka
two-lbl
se rovná
equals
desít-ka.
ten-lbl
Intended: ‘Two times five equals ten.’ (Czech)
The data above show that Slavic label numerals are both morphologically and semantically
complex. With this in mind, let us now propose an account that will
capture their morpho-semantics as well as the relationship between the arithmetical
meaning, quantification and labeling.
6.4 Analysis
In order to account for the relationship between the three uses of cardinal numerals
discussed in this paper, i.e., their arithmetical, quantifying and label function,
I develop a morpho-semantic system combining a standard inventory of formal
semantics with the Nanosyntax model of morphology (see Wągiel & Caha 2020).
The core of the proposal is that the quantifying and the label function are both
derived from the arithmetical meaning which is taken to be basic.
I propose the ingredients in (62) as a set of morpho-semantic components that
numeral expressions conveying the relevant functions are formed of. Specifically,
I postulate the bottom-most NumPn (for ‘number’) layer along with two additional
heads, namely Cl (for ‘classifier’) and Lbl (for ‘label’).
(62) a. NumPn n = n
b. Cl ⟨n, ⟨⟨e,t ⟩, ⟨e,t ⟩⟩⟩ = λnλPλx[*P(x) ∧ #(P)(x) = n]
c. Lbl ⟨n, ⟨⟨e,t ⟩,e ⟩⟩ = λnλPιx[P(x) ∧ LABEL(n,x)]
The meaning of NumPn is simply an abstract number concept (type n), see (62a).
The subscript indicates that NumPn is lexically encoded. Though for the purpose
of this paper I assume that it is the basic structural component of each cardinal
numeral, I do not claim that it is primitive. It might very well be further decomposed
into a number of features.24 On the other hand, the components Cl and
Lbl are interpreted as shifts from n to a more complex type.
24This is indicated by the triangle in Figure 1 and 2. For why one would want to further decompose
NumPn, see Wągiel & Caha (2020).
xxxiv
Numerals and their kin
I will assume here that Cl is interpreted as a function from a number concept to
a predicate modifier devised for counting individuals (type ⟨n, ⟨⟨e,t⟩, ⟨e,t⟩⟩⟩), see
(62b).25 Cl takes a number and yields an expression with the built-in pluralization
operation ∗ and measure function #(P), see §3.3. In cases such as English five or
Czech pět ‘five’, Cl is already incorporated in the structure of a numeral but in
some other languages it is encoded in as additional morpheme, e.g., a classifier.
On the other hand, Lbl is a labeling device (type ⟨n, ⟨⟨e,t⟩,e⟩⟩) which selects
a number and returns a function from properties to unique individuals such that
they have the relevant property and are labeled with the corresponding number
by the LABEL operation, as provided in (62c). Lbl is typically introduced by a free
morpheme such as number or a specialized affix like the Czech suffix -ka, as
discussed in §6.2 and §6.3, respectively.
As a result of combining the components in (62) by syntax, we obtain the
structures in Figure 1 and 2. The tree in Figure 1 represents the structure and
meaning of the quantifying function of a numeral corresponding to 5 whereas
the tree in Figure 2 depicts the parallel label function.
ClP⟨⟨e,t ⟩, ⟨e,t ⟩⟩
λPλx[*P(x) ∧ #(P)(x) = 5]
Cl⟨n, ⟨⟨e,t ⟩, ⟨e,t ⟩⟩⟩
λnλPλx[*P(x) ∧ #(P)(x) = n]
NumP5 n
5
. . .
Figure 1: The quantifying function
LblP⟨⟨e,t ⟩,e ⟩
λPιx[P(x) ∧ LABEL(5, x)]
Lbl⟨n, ⟨⟨e,t ⟩,e ⟩⟩
λnλPιx[P(x) ∧ LABEL(n, x)]
NumP5 n
5
. . .
Figure 2: The label function
When a quantifying numeral, e.g., Czech pět ‘five’, and a label expression, e.g.,
Czech číslo pět ‘number five’, combine with an NP, what we obtain are the denotations
such as those in (63). In particular, (63a) denotes a set of pluralities of
players such that each plurality in that set consists of five players. On the other
hand, (63b) refers to the contextually unique player identified via their association
with the number 5.
(63) a. pět hráčů ⟨e,t ⟩ = λx[∗player(x) ∧ #(player)(x) = 5]
b. hráč číslo pět e = ιx[player(x) ∧ LABEL(5,x)]
25For the purpose of this paper, it is not important what the exact interpretation of Cl is. The
proposed system is compatible with other theories of quantifying numerals discussed in §3 as
long as it is possible to shift the primitive type n to a type for the quantifying function.
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Author
But how to account for the fact that a numeral such as Czech pět ‘five’ can be
used to express both the quantifying and the arithmetical meaning? This ambiguity
can be captured as a consequence of a core property of late insertion, namely
that lexical entries are not tailor-made for one specific use.26 In Nanosyntax, a lexical
entry pairs a well-formed syntactic structure with phonology. For instance,
let us assign to the cardinal pět the lexical entry in Figure 3. When syntax builds
a structure interpreted as the quantifying meaning, as in Figure 1, the entry in
Figure 3 can be used to lexicalize it. Consequently, the structure or, more precisely,
its top-most phrasal node is pronounced by /pjɛt/, as indicated by the big
circle in Figure 4.
ClP
Cl NumP5
. . .
⇔ /pjɛt/
Figure 3: Lexical entry
ClP
Cl NumP5
. . .
pět
Figure 4: Lexicalizations
Crucially, however, when syntax builds a structure interpreted as the arithmetical
meaning, i.e., the sole NumP5, it can also be pronounced by Figure 3, as
indicated by the small circle in Figure 4. This is due to the Superset Principle,
defined in (64) (Starke 2009), which states that a lexical entry can be used to pronounce
a particular structure iff it contains that structure as a sub-part (either
proper or improper). As a result, the Czech numeral pět ‘five’ is ambiguous between
the quantifying and the arithmetical meaning since both circled structures
in Figure 4 are contained in the tree in Figure 3.
(64) The Superset Principle (Starke 2009)
A lexically stored tree matches a syntactic node iff the lexically stored
tree contains the syntactic node.
Let us now consider how the label meaning arises. For this purpose, I will demonstrate
how specialized label numerals such as Czech pětka ‘number five’ are derived.
Since pět ‘five’ is stored as Figure 3, and thus cannot express the label
meaning, the Lbl component needs to be spelled out by a separate entry, e.g., as
26For an introduction to late insertion models such as Distributed Morphology and Nanosyntax,
see, e.g., Bobaljik (2017) and Baunaz & Lander (2018), respectively.
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Numerals and their kin
the suffix -ka in Figure 5. Notice that the semantics of Lbl is not compatible with
the quantifying meaning and requires type n as its input, recall (62c). Thanks to
the Superset Principle, see (64), pět can also pronounce the arithmetical meaning,
i.e., the sole NumP5, as indicated by the small circle in Figure 4. However,
in order for this structure to apply at the LblP node, NumP5 must move from
its base-position. In Figure 6, its trace, i.e., the lower copy of the displaced element,
is shaded.27 After the NumP5 is moved, Figure 5 matches the lower LblP,
as shown in Figure 7, where the label suffix -ka is inserted at the relevant node.
LblP
Lbl
⇔ ka
Figure 5: Czech label suffix
LblP
NumP5
. . .
LblP
Lbl NumP5
. . .
pět
Figure 6: Movement
LblP
NumP5
. . .
LblP
Lbl
pět ka
Figure 7: Spellout
The proposed analysis has several advantages. First of all, it provides a unified
approach that captures the relationship between the arithmetical, the quantifying
and the label meaning. Second, it explains when and how a numeral can be
ambiguous between different meanings. Finally, it gives a detailed account for
the meaning/form correspondences both in cardinal and in label numerals.
7 Conclusion
In this paper, I explored the semantics of numerals in Slavic and beyond. I discussed
various functions that cardinal numerals can have as well as different approaches
to capturing the relationships between these functions. Furthermore, I
27In Nanosyntax, the movement in Figure 6 is driven by the need to lexicalize the phrasal LblP
node by the lexical entry in Figure 5. For a detailed explanation of how precisely the movement
in Figure 6 is triggered, see Baunaz & Lander (2018) and Caha et al. (2019).
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Author
examined many types of complex numerical expressions that only recently attracted
attention in the literature. I also provided novel evidence in favor of the
view suggesting that the arithmetical meaning is basic while other functions are
derived from it. Finally, I explored the so-far understudied label function and proposed
a morpho-semantic analysis of Slavic derived label numerals that could be
further extended to other types of complex numerical expressions.
Abbreviations
acc accusative
aggr aggregate numeral
clf classifier
cop copula
f feminine
gen genitive
gndr gender-sensitive numeral
ins instrumental
ipfv imperfective
lbl label numeral
loc locative
mult multiplier
n neuter
nbr number-denoting numeral
nv non-virile
pfv perfective
refl reflexive pronoun
tax taxonomic numeral
top topic
v virile
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