Cambridge Handbook of Semantics
Maria Aloni and Paul Dekker
This guide was compiled using
cambridge7A.cls 2010/09/09, v2.10
The latest version can be downloaded from:
https://authornet.cambridge.org/information/productionguide/
LaTeX files/cambridge7A.zip
1
Plurality
Rick Nouwen
1.1 Introduction
It is easy to indicate what plurality is from a morphological point of view:
Plurality is an instance of the inflectional category of number. From a
semantic point of view, however, the concept of plurality is much more
diffuse. To start with, the general folk linguistic intuition that being
plural expresses being more than one is inaccurate. For example, while
(1-a) suggests that there are multiple stains on the carpet, (1-b) does
not suggest that only the carrying of multiple guns is illegal, nor does
(1-c) suggest it might be the case that there has been a solitary alien
walking the earth.
(1) a. There are stains on the carpet.
b. Carrying guns is illegal in Illinois.
c. No aliens have ever walked the earth.
Furthermore, plurality may emerge from sentences that lack any plural
morphology, such as the first sentence in (2). Despite the morphologically
singular a picture, this sentence expresses that a multitude of pictures
was drawn, each by one of the boys. This emergent plurality becomes
apparent in the second sentence in which a plural pronoun refers to the
pictures that the boys drew.
(2) Each boy drew a picture. They are hanging on the wall.
It is sometimes thought that, in languages like English (and indeed IndoEuropean
languages, generally), plurality is an essentially nominal phenomenon.
Corbett 2000, for instance, uses (3) to point out that the
plural verbal form must be an uninterpreted agreement reflex triggered
2 Plurality
by the (unmarked) plurality of the noun. We cannot for instance use (3)
to express that a single sheep drinks from the stream more than once.
It has to mean that there were multiple sheep.
(3) The sheep drink from the stream. Corbett (2000)
In other languages verbal number marking does have a semantic substance.
For instance, (4), an example from Hausa from Souˇckov´a (2011),
is incompatible with there being a single kick to the table, and typically
expresses a quick repetition of kicks. This interpretation is triggered by
partial reduplication of the verb stem.
(4) Yaa
3sg.masc
sh`us-sh`uuri
redup-kick
teeb`uˆr
table
‘He kicked the table repeatedly’
Plurality involving the multiplicity of events and some kind of verbal
morphological marking is often referred to as pluractionality (Newman,
1980). One difficult question is how (or whether, Corbett 2000) pluractionality
differs from aspectual marking or Aktionsart.a In this chapter,
I will ignore the phenomena of pluractionality and focus on nominal
plurality as found in English.b
The observation that, in English, plural marking has obvious semantic
effects in the nominal, but not in the verbal domain, should however not
be mistaken for a deeper semantic conclusion, namely that there is nothing
to study beyond plural reference. In fact, it turns out that the most
puzzling data involving plurality concern exactly the interaction between
plural arguments and verbs. This can be illustrated by the following observation:
predicates differ with respect to the entailment patterns they
display for plural arguments. Take the distributivity entailment pattern
in (5).
(5) A predicate “VP” is distributive if and only if “X and Y VP”
entails “X VP” and “Y VP”
Being wounded is an example of a predicate that clearly supports the
entailment scheme in (5), for (6-a) entails both (6-b) and (6-c).
(6) a. Bob and Carl are wounded.
a See Wood (2007) for discussion.
b See Beck and von Stechow (2007) for a discussion of English adverbials that can
be analysed as having apluractional function.
1.1 Introduction 3
b. Bob is wounded.
c. Carl is wounded.
For other predicates, the pattern in (5) does not hold. Consider, the
following examples.
(7) a. Bob and Carl carried a piano up the stairs.
b. Bob carried a piano up the stairs.
c. Carl carried a piano up the stairs.
In a situation in which both (7-b) and (7-c) are true, (7-a) is true too.
However, (7-a) is also true in a situation in which Bob and Carl jointly
carried the piano up. In that case, it would be false to say (7-b) or
(7-c). It turns out then that whilst to carry a piano up the stairs is not
distributive, it does have the inverse property:
(8) A predicate “VP” is cumulative if and only if “X VP” and “Y
VP” entails “X and Y VP”
Note that being wounded, besides begin distributive, is cumulative, too.
It follows from (6-b) and (6-c) that (6-a). Yet other predicates are neither
distributive nor cumulative:
(9) a. Bob and Carl are a couple.
b. *Bob is a couple.
c. *Carl is a couple.
In sum, the kind of entailment patterns we can observe for plural arguments
depends on the predicate (among other things such as the presence
of certain adverbs, floated quantifiers etc., some of which will be
discussed below). Plurality will therefore need to be studied in a compositional
context. It is this interaction between plural arguments and
predicates that is the focus of this chapter. The structure is as follows.
In section 1.2, I will introduce the most commonly assumed complication
that plurality brings to the logical language, namely the inclusion
of individual constants that are not atomic but sums. Using such plural
individuals, in section 1.3 I turn to distributive entailments, as well as to
distributive readings of sentences. Section 1.4 turns to a special class of
interpretations for sentences with plurality, namely cumulative readings,
and the relation this bears to distributivity. Section 1.5 looks at dependency
phenomena involving plurals. Finally, in section 1.6, I conclude
by briefly touching upon some other issues in the semantics of plurality.
4 Plurality
1.2 Sums
Let us consider the simple statement in (10) again.
(10) Bob and Carl are wounded.
Given the observed distributivity entailment of being wounded, it is
tempting to analyse (10) as (11). Not only does (11) yield the intuitively
correct truth-conditions, it does so without complicating the predicatelogical
language. That is, plurality in natural language is distributed over
several conjuncts of essentially singular predications.
(11) wounded(b) ∧ wounded(c)
The discussion on entailment patterns above, however, shows that this
kind of analysis is not very promising, for it would fail on predicates that
lack distributive entailments. For instance, analysing (12-a) as (12-b)
incorrectly predicts that (12-a) is false on a scenario where Bob and
Carl collectively carry a piano up the stairs.
(12) a. Bob and Carl carried a piano up the stairs.
b. [[carried a piano up the stairs]] (b) ∧
[[carried a piano up the stairs]] (c)
The common solution is to assume that the phrase Bob and Carl refers
to a plural individual and that this individual is, as a single collective
individual, the argument of the predicate. It would not do, however,
to just assume that the conjunction Bob and Carl is a name for some
plural individual. There must be some connection between the plural
individual that Bob and Carl refers to and the individual entities Bob
and Carl. This is because, ultimately, we want an account of how certain
predicates allow for distributivity and/or cumulativity entailments that
link statements involving plural arguments to statements involving the
singular parts. Here we can just assume that pluralities are to singularities
what sets are to their members. That is, if De is the set of singular
entities, then the set of plural individuals will be the powerset of this
domain with the empty set removed: ℘(De)\∅. We can now say that Bob
and Carl refers to the smallest set that has Bob and Carl as members.
So, for (12-a) we get (13).
(13) [[carried a piano up the stairs]] ({b, c})
1.2 Sums 5
While there exist analyses along this line (Hoeksema (1983), Gillon
(1987), Schwarzschild (1996), Winter (2002)), a more common alternative
is to use an approach inspired by the work of Link. (See especially
Link (1983). See Krifka (1989) and Landman (1989), Landman (1996),
Landman (2000) for foundational works building in part on Link.) In
Link’s approach there is no type-theoretic difference between plural and
singular individuals. So, plural individuals are e-type entities just like
singular individuals are. However, the domain of entities is structured
exactly like the powerset of the set of atomic individuals.
We can set this up as follows. Let De be the set of all (i.e. both singular
and plural) entities and let be a binary operation on elements of De,
called summation. Summation has the following properties:
(14) a. (α β) γ = α (β γ) associative
b. α β = β α commutative
c. α α = α idempotent
Given summation, we now define a partial order ≤ on De:
(15) α ≤ β if and only if α β = β
The idea is that just like {b, c} is the smallest set that contains b and
c as elements, so is b c – i.e. the sum of b and c – the smallest entity
that has b and c as its parts. This means that is a supremum or join
operator. The desired structure is a particular kind of lattice, namely a
complete atomic join semi-lattice (see chapter 6 of Link (1998) as well
as Landman (1991) for details). Complete since we want the domain of
entities to be closed under ; atomic since we also want that all atomic
parts of sums in the domain are part of the domain themselves. (If we
can talk about Bob and Carl, then we can also talk about Bob.) Finally,
it is a semi-lattice, since we are only interested in joins, not in meets.
An example of a complete atomic join semi-lattice for three atoms a, b
and c is given in figure 1.1.
In (16), I give two handy definitions inspired by Link’s work. First of
all (16-a) says that something is an atomic entity if and only if it only
has itself as a part. The operation in (16-b) restricts a set of individuals
to the ones that are atomic.
(16) a. Atom(α) if and only ∀β ≤ α[α = β]
b. Atoms(A) = λα.α ∈ A ∧ Atom(α)
6 Plurality
Figure 1.1 A depiction of the complete atomic join semi-lattice with
a, b and c as the atomic elements. The arcs represent the ≤ relation.
Given these definitions, we can formally express the correspondence between
the lattice structure assumed by Link and the powerset structure
we were aspiring to:
(17) De, is isomorphic to ℘(Atoms(De)) \ ∅, ∪
The operator most borrowed from Link’s work is the sum closure operator
‘*’ for sets or one-place predicates:
(18) *X is the smallest set such that:
*X ⊇ X and
∀x, y ∈ *X : x y ∈ *X
Link reasoned that while singular nouns referred to sets of atoms (i.e. elements
in ℘(Atoms(De))), plural nouns refer to sets that include plural
individuals. (We will ignore for now the hairy issue of whether or not
such plural nouns also contain atomic entities in their extension. See
below for discussion.)
(19) [[boys]] = * [[boy]]
So, if [[boy]] is the set {b, c, d}, then [[boys]] is the set {b, c, d, b c, c d, b
d, b c d}. The example (20-a) is now predicted to be true, by virtue of
the fact that the plural individual b c is in the pluralised – i.e. * -ed –
extension of the noun boy, which we could express in a predicate logical
language as in (20-b).
(20) a. Bob and Carl are boys.
b. *boy(b c)
1.3 Distributivity 7
The question is now how to account for predicates with different entailment
patterns. For this, we need to have a closer look at distributivity.
1.3 Distributivity
The sentence in (21) is true in a number of quite different scenarios. In
a collective understanding of the sentence, Bob, Carl and Dirk jointly
carry a piano up the stairs. Conversely on a distributive understanding
Bob carried a piano up the stairs by himself, and so did Carl, and so did
Dirk.
(21) Bob, Carl and Dirk carried a piano up the stairs.
The distributive understanding can be isolated by adding a floating distributive
quantifier like each: that is, Bob, Carl and Dirk each carried a
piano up the stairs only has the distributive reading. With this in mind,
predicates with distributive entailments are always understood distributively,
since for (22-a) the adding of each in (22-b) seems vacuous.
(22) a. Bob, Carl and Dirk are wounded.
b. Bob, Carl and Dirk are each wounded.
In summary, we get the following overview:
(23) entailments understandings
distr cumul - distr collec
being wounded yes yes - yes no
to carry a piano up x no yes - yes yes
The stable factor in this table is that all examples allow for cumulative
inferences and all examples have distributive understandings. Note
that Link’s *-operation amounts to enforcing cumulative entailments.
That is, if predicates with plural arguments are pluralised using *, then
cumulativity follows, due to the following fact:
(24) P(α) ∧ P(β) ⇒ *P(α β)
The inverse entailment does not follow, for *P(α β) could be the case
without P(α) being the case. Just take a P such that P(α β) is true
but not P(α).
Landman (1996) proposes to account for the difference between predicates
like being wounded and a predicate like to carry a piano up the
8 Plurality
stairs by just assuming two things. Firstly, predicates with plural arguments
are plural in the sense that they are pluralised using the *operation.
Second, predicates differ with respect to the kind of entities
they have in their (singular) extension. For being wounded it suffices to
assume that its extension only contains atoms. This captures the intuition
that it does not make sense for the property of being wounded to
be shared among multiple entities. We can now analyse (22-a) as (25).
(25) *wounded(b c d)
The cumulativity entailment follows from (25). The inverse distributivity
entailment follows from the fact that the only way a plurality can end
up in the pluralised extension of the predicate is if its atomic parts
were in the singular extension, for wounded(α) is only true for atomic
α’s. Another way of saying this is that there cannot be any collective
understandings. All plural arguments are the result of the *-operation.
Adopting a similar strategy for (21) results in (26).
(26) * [[carried a piano up the stairs]] (b c d)
In Landman’s framework, the key to accounting for the properties of
(26) is to assume that the extension of predicates like to carry a piano
up the stairs might or might not include atoms and it might or might not
include sums; that this fully depends on the state of affairs. In a distributive
scenario, b, c and d occur in the extension as singular individuals.
Given the effect of pluralisation, (26) is then true in that scenario. In
the collective scenario the extension just contains b c d as a plurality,
and hence (26-a) is also true. There is no distributivity entailment, since
we cannot know which scenario made (26-a) true.
Contrary to what my presentation above suggests, it is sometimes
assumed that what I have vaguely called understandings are in fact different
readings of an ambiguity.c On such accounts, (21) is ambiguous
between a reading in which the distributivity entailment holds (the distributive
reading) and one in which it does not (the collective one). The
latter reading is more basic, while the distributive one is derived using
the insertion of a distributivity operator. There exist two main variations
on such approaches, depending on whether the operator in question is
applied to the verb’s argument or to the verb itself. Take, for instance,
the following two options.
c As far as I can see, no consensus on this topic has been reached. In fact, this
topic is surprisingly absent from the recent literature.
1.3 Distributivity 9
(27) a. DIST = λα.λP.∀β ≤ α[Atom(β) → P(β)]
b. DIST = λP.λα.∀β ≤ α[Atom(β) → P(β)]
In a case like (21), insertion of DIST will result in a reading that carries
a distributivity entailment: ∀x ≤ b c d[Atom(x) → [[carried a piano
up the stairs]](x)]. Hence, this reading is exactly what one would get by
adding a floating each to (21). For cases like (21) the choice between
(27-a) and (27-b) is immaterial. It is easy to find examples, however,
where these two operators differ in predictions. Following Lasersohn
(1995), a sentence like (28-a) illustrates this. Here, it is possible for
one part of the conjoined verb phrase to get a collective reading whilst
the other part gets a distributive reading. Using (27-b), this is easy to
capture, as in (28-b). However, no easy analysis seems to be available
using (27-a).
(28) a. Bob and Carl carried a piano up the stairs and then drank
a glass of water.
b.
λx. [[carried a piano up the stairs]] (x) ∧
DIST( [[drank a glass of water]] )(x)
(b c)
Assuming we should abandon the option in (27-a), we now have two competing
accounts of distributivity: one in the form of a freely insertable
operator DIST, and one as a semantic repercussion of pluralisation using
*, as in Landman (1996, 2000). These accounts are closely related.
In fact, it is easy to see that the following correspondence holds:
(29) DIST(A) = *(Atoms(A))
There exists some evidence against thinking of distributive understandings
as separate interpretations that arise through augmenting a logical
form with an additional operator. Schwarzschild (1996) uses one of the
classic ambiguity tests from Zwicky and Sadock (1975). Whenever an
ambiguous term is elided, it has to be resolved to the same interpretation
as its antecedent. As an illustration, take the elaborated version of
the famous Groucho Marx joke in (30-a). This sentence can mean that
the speaker shot an elephant that was wearing pyjamas and that the
speaker’s sister shot a similar animal or it means that the speaker shot
an elephant whilst wearing pyjamas and that his or her sister shot an
elephant in similar attire. Crucially, it cannot mean that the speaker
shot an elephant that was wearing pyjamas, whilst the sister shot an
elephant while she was wearing pyjamas. Schwarzschild now observes
10 Plurality
that for sentences like (30-b) such a mixed interpretation is perfectly
possible. For instance, (30-b) is intuitively true if each of the computers
were paid in two instalments and the entire collection of diskettes
were collectively paid in two instalments. (See Schwarzschild for a more
elaborate scenario that makes this understanding likely.)
(30) a. I shot an elephant wearing pyjamas and my sister did too.
b. The computers were paid in two instalments and the diskettes
were too.
While Schwarzschild’s observation strongly suggests that we should not
really be talking about readings, there is some data that seems to indicate
that plural sentences at least involve some covert operator (as,
incidentally, is the case in Schwarzschild’s own account). One reason to
believe that this is the case comes from Heim et al. (1991). Consider
(31).
(31) Bob and Carl think they will win e1000.
This example has multiple readings depending on how the pronoun they
is resolved and whether the verb phrases are construed distributively or
collectively. Three of those are in (32).
(32) a. Bob and Carl argue that Bob and Carl, as a group, will win
e1000.
b. Bob argues that Bob will win e1000 and Carl argues that
Carl will win e1000.
c. Bob argues that Bob and Carl, as a group, will win e1000
and Carl argues the same.
In (32-a), the plural pronoun they is understood as co-referring with the
matrix subject Bob and Carl. This is a fully expected reading. However,
the reading in (32-b) is not so trivial to account for. In this case, the pronoun
is interpreted as a bound variable. But that means that there has
to be some quantificational operator that is responsible for distribution
over the atoms in the subject.
While these data are usually thought to indicate the presence of a
proper distributivity operator akin to the natural language floating each
in distributive readings, the observations are not incompatible with a
theory in which a single operator accounts for a range of understandings,
including Landman’s use of *. Also, Schwarzschild 1996 offers an analysis
that uses a distributivity operator, without predicting that there are
1.3 Distributivity 11
multiple readings. In a nutshell, and simplifying drastically, the idea is
that pluralities can be partitioned, i.e. divided up into parts and that
the distributivity operator quantifies over such parts.d A definition of
minimal sum covers as well as some examples are given in (33), while
(34) defines the distributivity operator.
(33) C minimally covers α iff α ∈ *C and ¬∃C ⊂ C[α ∈ *C ]
a. {b, c d} minimally covers b c d
b. {b, c, d} minimally covers b c d
c. {b c d} minimally covers b c d
d. {b c, c d} minimally covers b c d
(34) DISTC(P) = λx.∀y ∈ Cx[P(y)]
where Cx is some pragmatically determined minimal cover of x
This one definition covers a range of readings, depending on what kind
of cover is pragmatically given. For instance, (35-a) will be interpreted
as (35-b).
(35) a. These six eggs costs e2.
b. DIST(λx.cost(x, e2))(the-6-eggs)
On its most salient reading, the relevant cover would be a cover that does
not divide the eggs up, but one which just offers a single cell containing
the six eggs. This results in a collective understanding. The distributive
reading (where these are particularly expensive eggs) results from taking
a cover that divides the six eggs up into the individual eggs such that
(35-b) ends up expressing that each egg costs two euros. There are other
theoretical possibilities, which are however not so salient for the example
in (35-a). For an example like (36), however, the relevant cover would
be one in which the shoes are divided up in pairs, since, pragmatically,
that is how shoes are normally valuated. (See Gillon (1990) for the first
indepth discussion of this type of reading.)
(36) The shoes cost $75. Lasersohn (2006)
So far, I have discussed distributive and collective understandings of
plural predication, as well as intermediate ones as in (36). I now turn to
yet another understanding, cumulativity.
d See Dotlacil (2010) for a more advanced integration of Schwarzschild’s idea in a
sum-based approach to plurality. The idea of using covers to account for different
understandings of plural statements originates in Gillon (1987).
12 Plurality
1.4 Cumulativity
In the discussion of distributivity above, we saw that one approach (basically
that of Landman 1996, 2000) involves reducing distributivity to
pluralisation of e, t predicates, using Link’s *-operation. Notice that in
the examples so far we have always *-ed the VP denotation. This is not
the only theoretical option. Consider (37).
(37) Two boys carried three pianos up the stairs.
We could imagine quantifier raising the object and thereby creating a
derived predicate ‘λx.two boys carried x up the stairs’.e Using the *operation,
we would then expect there to exist a distributive reading
for the object: there are three pianos such that each of these pianos
was carried up the stairs by a potentially different pair of boys. This
reading is available to some, but certainly not all speakers of English.
While there may be linguistic reasons to limit such object distributivity,
it shows that distributivity is tightly linked to scope.
One way to make the relevant scopal relations more visible is by counting
the number of entities that are involved in a verifying scenario. To
do this reliably, let me change the example to (38). (A crucial difference
to (37) is that whilst you can carry the same piano multiple times, a
book cannot be written more than once.)
(38) Two Dutch authors wrote three gothic novels.
A collective understanding of (38) is verified by a group of two Dutch
authors who collectively wrote three gothic novels. On a distributive
reading the verifying situations involve six gothic novels, three for each
Dutch author. It is the dependency of the number of books on the number
of authors that shows that the subject has scope over the object.
Using the number of entities involved as a diagnostic of scope, we can
show that there exists another kind of understanding, on top of the
distributive and the collective one. This is the so-called cumulative reading
(Scha (1981)). For instance, (39) (a variation on an example from
Landman 2000), is true in a situation in which there were (in total)
six ladybird-swallowing frogs and there were (in total) twelve ladybirds
being swallowed by frogs.f
e As would be standard in, say, the textbook account of Heim & Kratzer 1998.
f Landman 2000 uses a similar example to argue against the reduction of
cumulative readings to collective reading, e.g. Roberts (1987).
1.4 Cumulativity 13
(39) Six frogs swallowed twelve ladybirds.
The reported reading cannot be a collective reading, since swallow is a
predicate with distributive entailments. (It makes no sense to swallow a
single entity in a joint effort.) It also clearly is not a distributive reading,
since the number of ladybirds is twelve, and not seventy-two.g
Similarly, (38) has a cumulative reading in which there were two Dutch
authors writing gothic novels and there were three gothic novels written
by Dutch authors. Again, this is not the collective reading. Take for
instance, a verifying situation like one in which one Dutch author writes
a gothic novel by himself and then co-writes two more gothic novels with
another Dutch author.
To introduce the popular account of cumulativity, let us go through
an elaborate illustration of (38). Assume the following semantics for the
subject and the object respectively.
(40) a. λP.∃x[#x = 2 ∧ *dutch.author(x) ∧ P(x)]
b. λP.∃y[#y = 3 ∧ *gothic.novel(y) ∧ P(y)]
Here I use # for the ‘cardinality’ of a sum, as given by the following
definition.
(41) #α := |Atoms(λβ.β ≤ α)|
The (subject) distributive reading is now given by pluralising the verb
phrase:
(42) ∃x[#x = 2∧*dutch.author(x)∧*λx .∃y[#y = 3∧*gothic.novel(y)∧
wrote(x , y)](x)]
This is not entirely accurate, since this assumes that the object is interpreted
collectively. (It is not entirely clear how books can be collectively
written.) So, a probably more accurate semantics is a doubly distributive
reading, as in (43):
(43) ∃x[#x = 2∧*dutch.author(x)∧*λx .∃y[#y = 3∧*gothic.novel(y)∧
*λy .wrote(x , y )(y)](x)]
The cumulative reading results not from pluralising a derived predicate,
but from directly pluralising the binary relation write (Krifka 1989,
g See Winter 2000 for an attempt at reducing cumulativity to distributivity and
dependency and Beck and Sauerland for a criticism of that account.
14 Plurality
1992, Landman 1996, 2000). That is, we need to generalise the *-operator
to many-placed predicates. Let X be a set of pairs:h
(44) α, β γ, δ := α γ, β δ
Now the *-operator can be generalised to binary relations.
(45) *X is the smallest set such that *X ⊆ X and ∀x, x ∈ *X :
x x ∈ *X
The cumulative reading of (38) is now simply (46).
(46) ∃x[#x = 2∧*dutch.author(x)∧∃y[#y = 3∧*gothic.novel(y)∧
*wrote(x, y)]]
For instance, if the extension of wrote is { b, gn1 , b c, gn2 , b
c, gn3 }, then *wrote will contain b c, gn1 gn2 gn3 , verifying
(38).
There are two well-studied problems with this account. First of all, because
the cumulative reading is scopeless, the above account only works
for examples with commutative pairs of quantifiers, such as the two existentials
in (46). For instance, (47) also has a cumulative reading, but
on a standard generalised quantifier approach to modified numerals, as
in (48), the quantifiers are not commutative, since they count atoms and
thereby distribute over their scope.
(47) In 2011, more than 100 Dutch authors wrote more than 200
gothic novels.
(48) [[more than n]] (P)(Q) = true iff the number of atoms that have
both property P and Q exceeds n
Krifka’s solution is to analyse numerals as existential quantifiers (as in
(40)) and their modifiers as propositional operators on alternative numerical
values. So, the interpretation of (47) yields the set of cumulative
propositions in (49). The role of the numeral modifiers is now to assert
that there is a factual proposition in this set for n > 100 and m > 200.i
h More general, for n-ary tuples f, g: f g := that n-ary tuple h, such that
∀1 ≤ m ≤ n : h(m) = f(m) g(m).
i An alternative account is to treat numeral modification as a degree phenomenon.
See, for instance, Hackl (2000); Nouwen (2010).
1.5 Distributivity and dependency 15
(49)



∃x[#x = n ∧ *dutch.author(x)∧
∃y[#y = m ∧ *gothic.novel(y)∧ n, m ∈ N
*wrote(x, y)]]



A second problem with the particular pluralisation account of cumulativity
comes from examples where a cumulative reading is combined
with a distributive one. Such examples were divised by Schein (1993).
The standard example is (50). Crucially, there is cumulativity with respect
to the three cash machines and the two new members, but the
passwords are distributed over these two.
(50) Three cash machines gave two new members each exactly two
passwords.
Since each new member was given two passwords, two new members
has to have scope over two passwords. But that scoping should be independent
of the scopeless relation between the cash machines and the
members. Schein uses examples like (50) as part of an elaborate argument
that a plural semantics should be part of a neo-Davidsonian event
semantics, where events have a similar part-whole structure as entities.
Partly due to Schein’s argument, events play a major role in the dominating
approaches to plurality.j See, for instance, Kratzer (2007) for
recent discussion and Champollion (2010) for a criticism of some of the
arguments for the neo-Davidsonian approach. In the interest of brevity,
I refrain from discussing events in this chapter.
1.5 Distributivity and dependency
Since, unlike cumulative readings, distributive readings are scopal, transitive
distributive sentences may introduce a dependency between the
arguments of the verb. For instance, on the distributive reading, (51)
expresses a relation between three boys and (at least) three essays.
(51) Three boys wrote an essay.
This dependency becomes grammatically relevant in at least two different
ways. First of all, in a number of languages, indefinites that are
dependent in the way an essay is in the distributive reading of (51) have
to be marked as such. (For instance, see Farkas (1997) for Hungarian,
j See also Parsons (1990); Lasersohn (1995).
16 Plurality
Farkas (2002) for Romanian, Yanovich (2005) for Russian, and Henderson
(2011) for the Mayan language Kaqchikel). Dubbed dependent
indefinites by Farkas, such indefinites come with a constraint that they
have to co-vary with some other variable.
(52) A
the
gyerekek
children
hoztak
brought
egy-egy
a-a
k¨onyvet
book.acc
‘The children brought a book each.’
Another form of dependency involves anaphora.
(53) a. Three students wrote an article.
They sent it to L&P. Krifka (1996)
b. John bought a gift for every girl in his class and asked their
desk mates to wrap them. after Brasoveanu (2008)
The example in (53-a) has many readings. The most salient one is the
one in which both sentences are distributive. On that reading, the first
sentence introduces a dependency between students and articles, which
is then accessed by the pronouns in the second sentence. That is, the
second sentence can only mean that each student sent his or her paper
to L&P. Similarly, there is a salient reading for (53-b) in which for each
girl, John asked the desk mate of that girl to wrap the gift he bought
for her.
Such data are hard to account for, especially given some other complicating
features of anaphora. For instance, dependent indefinites are
antecedents of plural, not singular, pronouns, which have maximal reference.
In an example like (54), the singular pronoun is infelicitous and the
plural pronoun is interpreted as referring to the set of all essay written
by a student.
(54) Several students wrote an essay. I will need to grade them / *it
before tomorrow.
In contrast to (54), singular anaphoric reference to a dependent indefinite
is possible once the pronoun is in the same kind of dependency
environment, as shown by the examples in (53).
A powerful account of these facts is offered by the dynamic mechanisms
proposed in van den Berg (1996) and subsequent related work
1.5 Distributivity and dependency 17
Nouwen (2003, 2007); Brasoveanu (2006, 2008).k To get the idea behind
these proposals, first consider the simple example in (55).
(55) Two boys were wounded. They are in hospital.
Under a dynamic semantic approach to anaphora), anaphoric relations
are captured by assigning antecedent and pronoun the same variable
name. (The relevant classic literature is Kamp (1981); Groenendijk and
Stokhof (1991); Kamp and Reyle (1993), but see also chapter 4 of this
volume for an overview of discourse semantics that includes a discussion
of dynamics.) For instance, the first sentence in (55) could correspond
to ∃x[#x = 2 ∧ *boy(x) ∧ *wounded(x)]. This sentence is verified in
a model in which, say, b and c were wounded, by assigning b c to
x. Let us write f[x := α] to indicate that the assignment function f
assigns α to x, and so the relevant variable assignment for the first
sentence in (55) are functions of the form f[x := b c]. The second
sentence may now be represented by *inhospital(x). By interpreting this
open proposition with respect to an assignment function that verifies the
antecedent sentence, the pronoun is automatically understood to refer
to whichever two boys were referred to in the first sentence.
The innovation brought in van den Berg’s work is to assume that the
assignment of pluralities to variables happens in a much more structured
way. So, rather than using the assignment in (56-a), a plurality assigned
to a variable, van den Berg uses (56-b), a plurality (set) of assignments
to the same variable.l
(56) a. f[x := b c]
b. {f1[x := b], f2[x := c]}
The main reason for doing this is that it opens up a way to represent
dependencies created by distributivity. For instance, say that Bob is
dating Estelle and Carl is dating Ann. This makes (57) true. In a van
den Bergian approach to plural dynamic semantics, the sentence will now
introduce the variable assignment in (57-b), rather than the simplistic
(57-a).
k See Krifka (1996) and Kamp and Reyle (1993) for alternative approaches,
involving parametrised sums and a representational rule for recovering
antecedents respectively. Nouwen (2003) contains a comparison of Krifka’s
proposal and distributed assignment. See Nouwen (2007) for a (technical)
discussion and critique of Kamp & Reyle’s approach.
l I see no reason why sets are needed. The same idea could in principle be worked
out in a lattice theoretic framework with sum functions. However, I follow here
the standard move to use sets of functions.
18 Plurality
(57) Bob and Carl are dating a girl.
a. f[x := b c; y = e a]
b. C = {f1[x := b; y := e], f2[x := c; y := a]}
The plurality of variable assignment in (57-b) has stored the dependency
between boys and girls. In a subsequent sentence, like (58-a), a
distributive reading can have access to that dependency. Van den Berg
therefore proposes that distributivity is not universal quantification over
atoms in plural individuals, but rather over atomic assignment functions
in a contextual set of functions, such as C above.m
(58) a. They each brought her a flower.
b. ∀f ∈ C[f(x) brought f(y) a flower]
The same framework also has an elegant way of capturing the constraint
on Farkas’ dependent indefinites, such as the Hungarian reduplication
indefinite in (52). Brasoveanu (2010) proposes that such indefinites are
necessarily evaluation plural. If C is a set of assignments, let C(v) be the
sum given by *λα.∃f ∈ C[f(v) = α], i.e. the projection of all values for v
in C into a single sum entity. If a dependent indefinite like a Hungarian
egy-egy noun phrase is associated to a variable x, it presupposes that
C(x) is not atomic. For instance, an example like (52), repeated here in
(59), is felicitous in the context in (59-a), but not in (59-b).
(59) A
the
gyerekek
children
hoztak
brought
[egy-egy
a-a
k¨onyvet]y
book.acc
a. C = {f1[x := child1; y := book1], f2[x := child2; y :=
book2], f3[x := child3; y := book3]}
b. C = {f1[x := child1; y := book1], f2[x := child2; y :=
book1], f3[x := child3; y := book1]}
1.6 To conclude
Space limitations have enforced a somewhat narrow focus on plurality
in this chapter. There are a number of key issues I have not been able to
pay attention to. In the conclusion of this chapter, let me mention two
of these.
m For the sake of brevity, I am simplifying severely. See Nouwen (2003) for
discussion of some complicating factors and for various ways of implementing
this idea.
1.6 To conclude 19
So far, I have mentioned two sources of distributivity. First of all, lexically,
some predicates are only compatible with atomic entities in their
(singular) extension (being wounded was the running example). Second,
(in at least some approaches) there is additionally a distributivity operator
that quantifies over atoms in a sum. A third source is the presence
of an inherently distributive quantifier. That is, some quantifiers are
distributive in the sense that they quantify over atoms only.
(60) A quantifier Q of type e, t , t is distributive if and only if for
any P of type e, t : Q(P) ⇔ Q(λx.P(x) ∧ Atom(x)).
An indefinite quantifier like three girls or the definite the three girls is
not distributive. An example like (61) allows for collective as well as
distributive construals.
(61) Three girls carried a piano up the stairs.
Things are different for quantifiers like every girl, which are distributive.n
In (62), any carrying of pianos that involves joint action is irrelevant.
Only if each girl carried a piano by herself is (62) true.
(62) Every girl carried a piano up the stairs.
Similarly distributive is most girls:
(63) Most girls carried a piano up the stairs.
If (63) is true, then it is true of most individual girls that they carried
a piano by themselves. If the majority of girls jointly carried a piano up
the stairs, whilst a minority stood by and did nothing, (63) is intuitively
false.
Quantifiers that are especially interesting from the perspective of plurality
are floating quantifiers like each and all. Whilst each enforces a
distributive reading in a sentence, as in (64-a), the presence of all allows
for both distributive and collective understandings. Yet, all is incompatible
with certain collective predicates, as illustrated in (65).
(64) a. The boys each carried a piano up the stairs.
b. The boys all carried a piano up the stairs.
n One complication is that every does have non-distributive readings, as in:
(i) It took every/each boy to lift the piano. Beghelli and Stowell (1996)
20 Plurality
(65) *The boys are all a good team.
See Brisson (2003) for an analysis of all using Schwarzschild’s 1996 cover
analysis of distributivity and collectivity.
Whilst the semantics of each is obviously closely related to (atomlevel)
distributivity, the main puzzle posed by each is of a compositional
nature. This is especially the case for so-called binominal each (Safir and
Stowell (1988)). For instance, in (66) each distributes over the atoms in
the subject plurality, while it appears to be compositionally related to
the object. See Zimmermann (2002) for extensive discussion.
(66) These three Dutch authors wrote five gothic novels each.
Another major issue I have not discussed in this chapter is the interpretation
of plural morphology. As I observed at the start, plurals do not
generally express more than one, for (67) suggests that not even a single
alien walked the earth.
(67) No aliens have ever walked the earth.
Krifka 1989 suggests that while plurals are semantically inclusive, that
is they consist of both non-atomic and atomic sums, singulars can be
taken to necessarily refer to atoms. The idea is now that, under normal
circumstances, use of a plural implicates that the more specific contribution
a singular would have made was inappropriate, and hence that
a non-atomic reference was intended. Cases like (67) are simply cases
in which the implicature is somehow suppressed, for instance, because
of a downward monotone environment. See Sauerland (2003); Sauerland
et al. (2005) and Spector (2007) for detailed analyses along these lines.
Such approaches often break with the intuition that the plural form is
marked, while the singular form is unmarked and so we would expect
that the plural but not the singular contributes some essential meaning
ingredient (Horn (1989)).o Farkas and de Swart (2010) formulate an account
of the semantics of number marking that is in the spirit of Krifka’s
suggestion but is at the same time faithful to formal markedness in the
singular/plural distinction.
Zweig (2008) offers an extensive discussion of the related issue of deo
Most natural languages mark the plural and leave the singular unmarked (e.g.
Greenberg (1966), cf. de Swart and Zwarts (2010) for discussion of exceptions).
1.6 To conclude 21
pendent bare plurals.p The most salient reading of (68) is one in which
bankers wear one suit at a time, despite the plural marking on the object.
(68) All bankers wear suits.
A typical analysis of (68) is to assume that such sentences are cumulative
(Bosveld-deSmet (1998), cf. de Swart (2006)) and that the plural
marking on suits triggers an interpretation of the object as a multiplicity
of suits. Contrary to this, one could assume that this is a distributive
reading and that the plural marking on the object is a result of the dependency
the object is part of. Here, the issue of the interpretation of
plural morphology meets the issues concerning distributive and cumulative
entailments that were central to this chapter. Zweig (2008), for
instance, argues that although the cumulative analysis of (68) might be
on the right track, this will not do for all examples involving dependent
bare plurals. This is because of examples like (69).
(69) Most students wrote essays.
There exists a dependent reading for essays in (69). But there does
not exist a cumulative reading for the whole sentence, since most is
a distributive quantifier. Such examples illustrate the complexity of the
semantics of the plural and stress that plurality is a compositional rather
than a referential phenomenon.
p Moreover, he discusses a complication that an event-based semantics for plurality
brings.
References
Beck, Sigrid, and von Stechow, Arnim. 2007. Pluractional Adverbials. Journal
of Semantics, 24(3), 215–254.
Beghelli, Filippo, and Stowell, Tim. 1996. Distributivity and Negation. In:
Szabolsci, Anna (ed), Ways of Scope Taking. Kluwer Academic Publish-
ers.
van den Berg, M.H. 1996. Some aspects of the internal structure of discourse:
the dynamics of nominal anaphora. Ph.D. thesis, ILLC, Universiteit van
Amsterdam.
Bosveld-deSmet, Leonie. 1998. On Mass and Plural Quantification. Ph.D.
thesis, Rijksuniversiteit Groningen.
Brasoveanu, Adrian. 2006. Structured Nominal and Modal Reference. Ph.D.
thesis, Rutgers, The State University of New Jersey.
Brasoveanu, Adrian. 2008. Donkey pluralities: plural information states versus
non-atomic individuals. Linguistics and philosophy, 31(2), 129209.
Brasoveanu, Adrian. 2010. Plural discourse reference. In: van Benthem, Johan,
and ter Meulen, Alice (eds), Handbook of logic and language, 2nd edition.
Elsevier.
Brisson, Christine. 2003. Plurals, All, and the Nonuniformity of Collective
Predication. Linguistics and Philosophy, 26(2), 129–184.
Champollion, Lucas. 2010. Cumulative readings of every do not provide evidence
for events and thematic roles. Pages 213–222 of: Aloni, Maria,
Bastiaanse, Harald, de Jager, T., and Schulz, K. (eds), Logic, Language
and Meaning. Lecture Notes in Computer Science. Springer.
Corbett, Greville. 2000. Number. Cambridge University Press.
de Swart, Henriette. 2006. Aspectual implications of the semantics of plural
indefinites. Pages 161–189 of: Vogeleer, Svetlana, and Tasmowski, Liliane
(eds), Non-definiteness and plurality. Linguistik Aktuell/Linguistics
Today. Benjamins.
de Swart, Henriette, and Zwarts, Joost. 2010. Optimization principles in the
typology of number and articles. In: Heine, Bernd, and Narrog, Heiko
(eds), Handbook of linguistic analysis. Oxford University Press.
Dotlacil, Jakub. 2010. Anaphora and distributivity: a study of same, different,
reciprocals and others. Ph.D. thesis, Utrecht University. LOT series 231.
References 23
Farkas, Donka. 1997. Dependent indefinites. Page 243267 of: Corblin, Francis,
Godard, Dani`ele, and Marandin, Jean-Marie (eds), Empirical issues in
formal syntax and semantics. Peter Lang Publishers.
Farkas, Donka. 2002. Extreme non-specificity in Romanian. Page 127151
of: Beyssade, Claire (ed), Romance languages and linguistic theory 2000.
John Benjamins.
Farkas, Donka, and de Swart, Henriette. 2010. The semantics and pragmatics
of plurals. ?Semantics & Pragmatics, 3(6), 154.
Gillon, Brendan. 1987. The Readings of Plural Noun Phrases. Linguistics and
Philosophy, 10(2), 199–220.
Gillon, Brendan. 1990. Plural noun phrases and their readings: A reply to
Lasersohn. Linguistics and Philosophy, 13, 477–485.
Greenberg, Joseph. 1966. Language universals. Mouton.
Groenendijk, J.A.G., and Stokhof, M.J.B. 1991. Dynamic Predicate Logic.
Linguistics and Philosophy, 14, 39–100.
Hackl, Martin. 2000. Comparative Quantifiers. Ph.D. thesis, Department of
Linguistics and Philosophy, Massachusetts Institute of Technology.
Heim, Irene, and Kratzer, Angelika. 1998. Semantics in Generative Grammar.
Heim, Irene, Lasnik, Howard, and May, Robert. 1991. Reciprocity and Plurality.
Linguistic Inquiry, 22(1), 63–101.
Henderson, Robert. 2011. Pluractional distributivity and dependence. Page
218235 of: Proceedings of SALT 21.
Hoeksema, Jack. 1983. Plurality and Conjunction. Pages 63–83 of: ter Meulen,
Alice (ed), Studies in modeltheoretic semantics. GRASS, vol. 1. Foris.
Horn, Laurence. 1989. A natural history of negation. Chicago: University of
Chicago Press.
Kamp, H. 1981. A Theory of Truth and Semantic Representation. In: Groenendijk,
J. A. G., Janssen, T. M. V., and Stokhof, M. J. B. (eds), Formal
Methods in the Study of Language. Amsterdam: Mathematical Centre.
Kamp, Hans, and Reyle, Uwe. 1993. From Discourse to Logic. Kluwer.
Kratzer, Angelika. 2007. On the Plurality of Verbs. Pages 269–300 of: D¨olling,
J., Heyde-Zybatow, T., and Sch¨afer, M. (eds), Event Structures in Linguistic
Form and Interpretation. Mouton de Gruyter.
Krifka, Manfred. 1989. Nominal reference, temporal constitution, and quantification
in event semantics. Pages 75–115 of: Bartsch, R., van Benthem,
J., and van Emde Boas, P. (eds), Semantics and Contextual Expressions.
Dordrecht: Foris.
Krifka, Manfred. 1996. Parametrized Sum individuals for plural reference and
partitive quantification. Linguistics and Philosophy, 19, 555–598.
Landman, Fred. 1989. Groups, I. Linguistics and Philosophy, 12(5), 559–605.
Landman, Fred. 1991. Structures for Semantics. Kluwer.
Landman, Fred. 1996. Plurality. Pages 425–457 of: Lappin, Shalom (ed), The
Handbook of Contemporary Semantic Theory. Blackwell.
Landman, Fred. 2000. Events and Plurality. Kluwer.
Lasersohn, Peter. 1995. Plurality, Conjunction, and Events. Kluwer.
Lasersohn, Peter. 2006. Plurality. Page 642645 of: Brown, K. (ed), Encyclopedia
of Language and Linguistics, 2nd edition, vol. 9. Elsevier.
24 References
Link, Godehard. 1983. The Logical Analysis of Plurals and Mass Terms: a
Lattice-theoretical Approach. Pages 302–323 of: Schwarze, C., Bauerle,
R., and von Stechow, A. (eds), Meaning, Use, and the Interpretation of
Language. deGruyter.
Link, Godehard. 1998. Algebraic Semantics in Language and Philosophy.
CSLI.
Newman, Paul. 1980. The Classification of Chadic within Afroasiatic. Leiden
Universitaire Pers.
Nouwen, Rick. 2003. Plural pronominal anaphora in context. Netherlands
Graduate School of Linguistics Dissertations, no. 84. Utrecht: LOT.
Nouwen, Rick. 2007. On Dependent Pronouns and Dynamic Semantics. Journal
of Philosophical Logic, 36(2), 123–154.
Nouwen, Rick. 2010. Two kinds of modified numerals. Semantics and Pragmatics,
3(3), 1–41.
Parsons, Terence. 1990. Events in the Semantics of English: A Study in Subatomic
Semantics. MIT Press.
Roberts, Craige. 1987. Modal Subordination, Anaphora, and Distributivity.
Garland.
Safir, Ken, and Stowell, Tim. 1988. Binominal each. In: Proceedings of NELS
18.
Sauerland, U., Anderssen, J., and Yatsushiro, K. 2005. The Plural is Semantically
Unmarked. In: Kesper, S., and Reis, M. (eds), Linguistic Evidence.
Mouton de Gruyter.
Sauerland, Uli. 2003. A new semantics for number. Page 258275 of: Young,
Rob, and Zou, Yuping (eds), SALT 13. CLC publications.
Scha, Remko. 1981. Distributive, Collective, and Cumulative Quantification.
In: Groenendijk, Jeroen, Janssen, Theo, and Stokhof, Martin (eds),
Truth, Interpretation and Information. GRASS, vol. 2. Foris.
Schein, Barry. 1993. Plurals and Events. Current Studies in Linguistics, vol.
23. The MIT Press.
Schwarzschild, Roger. 1996. Pluralities. Kluwer.
Souˇckov´a, Kateˇrina. 2011. Pluractionality in Hausa. Ph.D. thesis, Universiteit
Leiden.
Spector, Benjamin. 2007. Aspects of the Pragmatics of Plural Morphology:
On Higher- Order Implicatures. Pages 243–281 of: Sauerland, Uli, and
Stateva, Penka (eds), Presupposition and Implicature in Compositional
Semantics. Palgrave Macmillan.
Winter, Yoad. 2002. Atoms and Sets: a characterization of semantic number.
Linguistic Inquiry, 33, 493505.
Wood, Esther. 2007. The Semantic Typology of Pluractionality. Ph.D. thesis,
University of California, Berkeley.
Yanovich, Igor. 2005. Choice-functional series of indefinites and Hamblin Semantics.
Page 309326 of: Georgala, Effi, and Howell, Jonathan (eds),
Proceedings of SALT 15. CLC Publications.
Zimmermann, Malte. 2002. Boys buying two sausages each: on the syntax
and semantics of distance-distributivity. Ph.D. thesis, Universiteit van
Amsterdam.
References 25
Zweig, Eytan. 2008. Dependent Plurals and Plural Meaning. Ph.D. thesis,
New York University.
Zwicky, Arnold, and Sadock, Jerrold. 1975. Ambiguity Tests and How to Fail
Them. Page 136 of: Kimball (ed), Syntax and Semantics 4. Academic
Press.