Numeric quantification in copredication ∗
Matthew Gotham
Abstract
Copredication is the apparent attribution of incompatible properties to a single object, in a sentence
that is nevertheless acceptable and can readily be interpreted. Various attempts have been made
to devise theories that predict truth conditions for copredication sentences that avoid this attribution of incompatible properties. In this paper I examine three kinds of such theories: those based
on merological accounts of what nouns supporting copredication denote, accounts based on type
manipulation and the operation of repair rules in the process of semantic composition, and ambiguity accounts supplemented by pragmatically-driven predicate meaning transfer. I concentrate on
the nature of the problems that these attempts face in correctly predicting the truth conditions of
copredication sentences involving numeric quantifiers, and evaluate some possible ways of overcoming those problems. I conclude that native speaker judgements about copredication sentences
involving numeric quantification do not perfectly reflect the counting and individuation criteria that
the existing accounts predict, because copredication presents us with divergent criteria for identifying objects falling under the denotation of the noun, and hence potentially divergent criteria for
individuating and counting those objects.
Keywords: copredication, quantification, internalism, externalism, semantics
1 Introduction
According to Kennedy and Stanley (2009, p. 584),
Semantic theory [. . . ] can tell us what the costs would be of denying the existence
of certain kinds of entities [. . . ] If a straightforward semantic theory for arithmetic
is true, then a sentence such as ‘There is a prime number between two and five’
entails the existence of numbers. As a result, a nominalist who rejects the existence
of numbers is committed either to rejecting the simple semantics, or to rejecting the
truth of ‘There is a prime number between two and five’.
It has been argued by Chomsky (2000), Pietroski (2005b) and Collins (2009), among others,
that this view of the nature of semantic theories is undermined by sentences like (1) (Asher,
2011, p. 11).
(1)
Lunch was delicious but took forever.
‘Delicious’ is a predicate we expect only to be able to apply to food, while ‘took forever’ is
a predicate that should only be applicable to events. By Kennedy and Stanley’s logic, if a
straightforward semantic theory for meals is true, then (1) entails the existence of things that
can both be delicious and take forever, i.e. be both food and an event. If indeed nothing can
be both food and an event, as we would have thought, then deliciousness and taking forever
are properties both of which no single object can have. This would mean that a straightforward
semantic theory for meals would predict (1) to be always false. However, there are situations
in which competent speakers of English will judge (1) to be true. This phenomenon—an interpretable and possibly true sentence appearing to include the attribution of a combination of
∗
I am grateful to Nathan Klinedinst for very helpful discussion and comments on several previous drafts of
this paper. This research is funded by an AHRC Doctoral Studentship.
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properties to an object such that no object could bear all of them—is the problem of copredication (Pustejovsky, 1995, p. 236).
In (1), the two apparently conflicting predicates are joined by the conjuction ‘but’. This
invites the supposition that copredication in this case is to be explained by coordination reduction (Haspelmath, 2007, pp. 38–39). The argument would be that the sentence has an underlying structure like that shown in (10 ), and that ‘lunch’ is resolved differently in each coordinand,
i.e. that ‘lunch1 ’ denotes food and ‘lunch2 ’ denotes an event.
(10 )
Lunch1 was delicious but lunch2 took forever.
However, whether or not (10 ) is the correct analysis of (1), this does not suffice as an
explanation, as copredication is not limited to a particular syntactic structure like coordination,
as (2)–(7) show.
(2)
The delicious lunch took forever.
(3)
Bob stopped by during the delicious lunch.
(4)
The bank called in Bob’s debt and was vandalised.
(5)
The book is heavy but easy to understand.
(6)
The heavy book is easy to understand.
(7)
The book, which is heavy, is easy to understand.
In this paper I will not directly address the question of whether or not the moral to be
drawn here is that Kennedy and Stanley’s view should be abandoned, except to note that just
how sentences like (1)–(7) are interpreted will have to be explained in any theory. Instead, I will
suppose for the sake of argument that Kennedy and Stanley are right and that our remaining
option is to reject the ‘straightforward semantic theory’ for e.g. (1) according to which it would
be assigned an interpretation expressing something like ‘∃x(lunch(x) ∧ (delicious(x) ∧ tookforever(x)))’. Taking this idea seriously, a paraphrase of what speakers are assenting to when
they judge (1) true might be:
(8)
There was some food that was delicious, and there was an event that lasted a long time.
We call both of these ‘lunch’ and they are linked inasmuch as the food in question was
eaten during the event in question.
A possible way of proceeding, then, would be to develop a theory according to which (1)
is assigned an interpretation that looks more like (8), which I will henceforth refer to as an
‘ontologically respectable paraphrase’ (Pietroski, 2005a, p. 277). Similar points can be made
of the other examples. For instance, in (4) an institution called in a debt and a building was
vandalised, while the book qua physical object in (5)–(7) is heavy and its informational content
is easy to understand.
Various attempts have been made to develop theories that predict such ontologically respectable paraphrases. In this paper I will evaluate three kinds of such attempts: those based
on merological accounts of nouns (such as ‘lunch’) supporting copredication, accounts based
on type manipulation and the operation of repair rules in the process of semantic composition,
and ambiguity accounts supplemented by pragmatically-driven predicate meaning transfer. I
will show that they all face problems in correctly predicting the truth conditions of numerically quantified sentences involving copredication. I will then evaluate some possible ways of
overcoming those problems.
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2 Overview of some approaches to copredication
2.1 Mereological accounts
Faced with the apparent clash of properties in (1), Cooper (2007, p. 4) suggests that the food
(which is delicious) and the event (which took forever) are both parts of the lunch that is being
talked about.
I would prefer [. . . ] to say that the lunch is delicious in virtue of the food which is
part of the lunch being delicious. It is common in natural language for us to make
predications of objects in terms of predications that hold of some of their parts,
though not all of them. Consider [‘The house is locked’] in contrast to [‘The door
is locked’]. [. . . ] We say that the house (which for simplicity we will assume has
exactly one external door) is locked in virtue of its door being locked. The door,
arguably, is locked in virtue of its lock being locked. We apply the predicate locked
to the house, without requiring that all of its parts be locked.
In this spirit, Cooper analyses (1) as being true in virtue of the food part of the lunch
being delicious even though the event part isn’t (and could not be), and the event part of the
lunch as taking forever even though the food part doesn’t. On this approach, both predicates
apply to different parts of a lunch. Presumably, (5) can be true for similar reasons. If a book is
viewed as a composite of a physical object with an informational object, then the physical part
can be heavy and the informational part easy to understand.
2.2 Type manipulation
In the words of Asher and Pustejovsky, the right account of the interpretation of a sentence such
as (1) is that the two predicates involved are predicated of the lunch object considered under
different ‘aspects’. Thus the relation between the food and the event associated with the word
‘lunch’ in (1) is not understood as these being distinct parts of one complex object, but rather
as these being the same object conceptualised in different ways.
Predication, the application of a property to an object, may sometimes be restricted
to a particular aspect of an object, something known in scholastic philosophy as
qua predication, where philosophers speak of an X qua Y as having the property P.
(Asher and Pustejovsky, 2007, p. 14)
Tropes [aspects] are thus not part of objects; rather it’s the other way around—the
object is a constituent of a trope or aspect. The sum of the tropes of an object is
not identical to that object, since each trope contains the object together with some
property that it has. (Asher, 2008, p. 165) (emphasis added)
Given the way I have defined aspects, the sum of an object’s aspects cannot be
identical to the object itself (since each aspect contains the object together with
some property that it has). A lunch object is wholly an event (under one aspect) and
wholly food (under another aspect). (Asher, 2011, pp. 149–150)
The way that the idea of aspects is used as a solution to the problem of copredication
is by first introducing a hierarchy of types into the compositional system such that there are
subtypes of e, for instance the type p of physical entities or the type i of informational entities.
Predicates are then thought to be more semantically selective than is taken to be the case in the
simply-typed lambda calculus: so for example we might have the predicate ‘is heavy’ as type
p → t, i.e. selecting for physical objects.1 Therefore, applying the predicate ‘is heavy’ to an
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informational object would result in a type clash and hence not be semantically well-formed.
However, there are certain type clashes that can be repaired by rules that introduce additional
information into logical form. These rules are defined so as to apply particularly in cases of
copredication.
Formally, those lexical items that support copredication are assigned a complex type
called a ‘dot type’, α • β. α and β are the types of the individual aspects under which the object
can be considered (so in the case of ‘lunch’ the type would be FOOD • EVENT).2 Then, the
various aspects under which an object can be considered can be exploited in the composition
logic by rules that add information to the interpretation given certain conditions regarding the
assignment of types to variables.
For example, take (1), repeated below.
(1)
Lunch was delicious but took forever.
Given the compositional semantics that Asher (2011) develops, the interpretation of the
VP ‘was delicious but took forever’ is as follows:3
delicious : FOOD ∗ ARGtook forever :
λΦλπΦ(π ∗ ARGwas
1
1
(λxλπ1 (λcλπ7 was delicious(c, λπ7 )(π1 )(x)
∧ λbλπ8 took forever(b, π8 )(π1 )(x)))
(9)
EVENT )
Φ is a variable that ranges over DP denotations, and π is a variable for type context. The
type requirements are formulated as presuppositions and stored in this extra parameter—π—in
order to ensure that the repair rules that will be crucial for dealing with copredication and other
type conflicts apply at the right time. In (9) it states that the first argument of was delicious
(i.e c) is of type FOOD and that the first argument of took forever (i.e b) is of type EVENT.
These type requirements all end up applying to the variable x and are inconsistent, since the
type hierarchy is formulated as a semilattice and the meet of the dot type FOOD • EVENT and
the simple type EVENT is ⊥. The repair rules that are defined for this kind of a type clash are
such that the metalanguage interpretation that we end up with for (1) is the one shown in (10).
λπ∃x(lunch(x, π) ∧ ((∃z(was delicious(z, π) ∧ o-elab(z, x, π)))
∧ ∃y(took forever(y, π) ∧ o-elab(y, x, π))))
(10)
π : x : FOOD • EVENT, z : FOOD and y : EVENT
‘O-elab’ (short for ‘object elaboration’) is the relation between objects and their aspects
such that o-elab(x, y) says that x is an aspect of y. It is the relation that makes individual
aspects available for predication , and the parts of the interpretation containing it are those that
have been introduced by the repair rules.
Of course, o-elab(x, y) only makes sense when y is of type α • β and x is either of type
α or type β. (10) says that there is a lunch such that, conceived of as food, it was delicious
1
This is something of a simplification, as can be seen below.
In Asher (2011), on which the following discussion is based, the type of ‘lunch’ is actually LUNCH, since
each distinct lexical entry has a distinct type. L UNCH is a subtype of FOOD • EVENT and hence can be used in a
context where a term of type FOOD • EVENT is required. FOOD • EVENT in turn is a subtype of E, FOOD • EVENT
v E. To make this work Asher has to formulate the type theory he uses in category theory in order to guarantee
that if X v E then the functional type X→T v E→T, which is not the case in type theory as it is usually understood
according to a set-theoretic formulation.
3
I will assume that the lexical entries for ‘and’ and ‘but’ are identical and that the difference is at a pragmatic
level not discussed here. I do not think this that this affects the discussion at this stage.
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and, conceived of as an event, it took forever (hyperbolically speaking). The system is set up
in such a way that this captures the content of (8) and hence, Asher contends, it is equivalent to
the understood meaning of (1).
2.3 Pragmatic accounts involving lexical ambiguity
The approach that we have just looked at in section 2.2 seeks to derive truth conditions equivalent to an ontologically respectable paraphrase for sentences like (2)–(7) by means of an increase in complexity of the semantic composition rules—rules that will guarantee that, in those
sentences, both predicates do not apply to the object denoted by their grammatical argument,
but rather to something that stands in some kind of defined relation to that object, an ‘aspect’
of it.
However, it has been argued on the basis of apparently quite different cases that a predicate can shift meaning in something like this way, and that the triggers for such shifts are pragmatic in nature rather than determined by compositional processes.4 The predicate/argument
relation is not what it appears to be on the surface. Particularly relevant in this respect is the
theory of Geoffrey Nunberg relating to examples like (11) (Nunberg, 2004, p. 346).
(11) I’m parked out back.
(11) can be true if uttered by someone who is in fact inside, provided that he or she is the
driver of a car or other parkable vehicle which is parked out back. These and similar examples
have prompted Nunberg to develop a theory according to which a predicate can ‘transfer’ its
application in context.
Nunberg points out that, although it is initially appealing, the view that it is the denotation
of ‘I’ that has changed here (from the speaker to the speaker’s car) is actually false, since (12)
is acceptable while (13) is not (Nunberg, 2004, p. 347).
(12) I am parked out back and have been waiting for 15 minutes.
(13) # I am parked out back and may not start.
If it were ‘I’ that had undergone meaning transfer, then (13) would be acceptable, since the
speaker’s car could bear both of the properties attributed to it. Therefore it must be ‘parked out
back’ that has undergone meaning transfer, explaining the acceptability of (12). It no longer
means parked out back, but rather the driver of a vehicle that is parked out back. Formally,
if the context makes salient a functional relationship h between cars and their drivers then the
adjusted meaning of ‘parked out back’ is calculated as follows (Nunberg, 2004, p. 348):
λP.λy.∃x: x ∈ domain(h) ((h(x) = y) ∧ P (x))[λz.parked-out-back(z)]
=λy.∃x: x is a car ((driver-of(x) = y) ∧ parked-out-back(x))
(14)
In fact, although Nunberg does not address the issue of copredication as such, (12) is in
some ways similar to the copredication sentences that we have been looking at. For instance,
there is no way to change the meaning of ‘lunch’ in (1) such that the adjusted meaning could
bear both the properties of being delicious and taking forever; however, changing the meaning of one or other of the predicates could solve the problem. Concretely, we could end up
with truth conditions equivalent to an ontologically respectable paraphrase either by applying
4
Just a few examples are Carston (2002, chapter 5), Nunberg (2004), Recanati (2004, chapter 5), Sperber
and Wilson (1998), Wilson and Carston (2007).
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volume 1
I-Language
volume 2
I-Language
volume 3
Gospel and Kingdom
Gospel and Wisdom
Gospel in Revelation
Figure 1: Two copies of one book, and a trilogy
a transfer function to ‘took forever’ to transfer its meaning from took forever to participant in
an event that took forever (if ‘lunch’ denotes food) or by applying a transfer function to ‘was
delicious’ to transfer its meaning from was delicious to included food that was delicious (if
‘lunch’ denotes an event).
3 Issues of individuation and counting
Copredication poses particular problems when it comes to the truth conditions of numerically
quantified sentences involving copredication: sentences like (15) and (16).
(15) Five books are heavy but easy to understand.
(16) John picked up and mastered three books. (Asher, 2011, p. 175)
The basic problem will be the issue of how books (for example) are individuated and counted
based on what the word ‘book’ is taken to denote. On the view involving pragmatically-licensed
predicate transfer, the word ‘book’ is ambiguous between denoting physical objects and denoting informational objects, while in Asher’s Type Composition Logic ‘book’ is univocal but is
subject to two distinct criteria of individuation depending on the aspect under which it is considered. Meanwhile, the natural reading of Cooper’s statements is that books are composite
objects made up of a physical part and an informational part, and should be individuated as
such. I will show below that these theoretical assumptions cause each of the accounts to make
predictions that come apart from speaker judgements about sentences like (15) and (16) in some
situations.
3.1 Problems with mereological accounts
It seems that there are real differences between the parthood relation as applied to the relationship between doors and houses and as putatively applied to the relationship between food and
lunches or physical objects and books. However, I am going to focus on the issue of individuating these composite objects. There is a problem lurking here that can most clearly be seen in
the case of ‘book’, as Asher (2011, pp. 146-7) points out. Suppose that we have two copies of
the same informational book and one trilogy (three information books in a single physical volume) and, further, that each volume is heavy and each informational book easy to understand.
This the situation shown in figure 1. Now consider (15).
If ‘book’ denotes a composite object made of a part that is a physical object and a part that
is information, then in the interpretation of DPs involving numerals, we must be counting these
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volume 1 + I-Language
volume 2 + I-Language
two copies
volume 3 + Gospel and Kingdom
volume 3 + Gospel and Wisdom
trilogy
volume 3 + Gospel in Revelation
Figure 2: A mereological view of the situation shown in figure 1
composite objects. Presumably, then, we are to understand that (15) is true iff there are at least
five such composite objects, each having a heavy part and a part that is easy to understand. If
books really are composite objects consisting of a physical part and an informational part, then,
by the axioms of mereology, two books are distinct whenever either of those parts is distinct.
And so, in the situation shown in figure 1, we have the physical/informational book composites
shown in figure 2.
So in fact there are five distinct physical/informational book composites, each having a
heavy part and a part that is easy to understand. But no-one would judge (15) true in the situation described above. This seems to indicate that the mereological analysis of the semantics of
‘book’ is on the wrong track.
The alternative is to say that, although ‘book’ denotes a composite object, we can only
ever count books relative to either a physical or an informational criterion of individuation (so
in fact we never count physical-informational composites). This (without the parthood analysis
as such) is actually the counting principle that Asher (2011) adopts, and it is to this analysis
that we now turn.
3.2 Problems with Asher’s Type Composition Logic
How are we to understand Asher’s talk of ‘aspects’, or the claim that ‘a lunch object is wholly
an event [. . . ] and wholly food’? The way that this works in the semantics of predication is that
dot objects (e.g. lunches) are underspecified with respect to their ‘criterion of individuation’.
Establishing the relevant criterion of individuation is a matter of hearer judgement in any case
of quantification over dot objects themselves, as opposed to quantification over their aspects.
There are situations in which distinct criteria of individuation result in distinct truth conditions.5
The repair rules mentioned in section 2.2 are such that under certain conditions, e.g.
when applying a physical-object-selecting predicate to ‘book’ or a food-selecting-predicate to
‘lunch’, they introduce into the interpretation an ‘object-elaboration’ of the dot object that is of
the right type (e.g. physical object or food respectively in these cases) . In copredication sentences, where we have e.g. both physical-object-selecting and informational-object-selecting
predicates applying to the dot object, we will therefore have more than one object elaboration.
5
As Asher says regarding an example involving books, which have contrasting physical and informational
criteria of individuation:
‘we have quantification over P • I objects [. . . ] We now have to choose the criterion of counting and individuation
[. . . ] if we choose the i [informational] criterion [. . . ] On the other hand, we could also take the criterion of
counting and individuation provided by physical objects’ (Asher, 2011, p. 174).
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The precise nature of these rules requires some further comment, since the manner in
which type conflicts are resolved make certain predictions with respect to the possible meanings
of sentences involving both dot objects and quantifiers. By default, in a structure [[D N] VP], if
the N is dual-aspect noun like ‘book’, then the dot-typed variable (introduced by the noun) will
be existentially bound and the expression as a whole will inherit the purely physical or purely
informational typing of the VP (Asher, 2011, pp. 113–114). This means that if the determiner is
quantificational, then the quantifier ends up ranging over aspects: in the case of books, physical
aspects if the VP is physically-typed and informational aspects if it is informationally-typed.
To take an example, the interpretation of the sentence ‘A student carried off every book’
in Asher’s theory is that shown in (17) (Asher, 2011, p. 175).6
λπ 0 ∃y(student(y, π 0 ) ∧ ∀v(∃w(book(w, π 0 ) ∧ o-elab(v, w, π 0 )) → carry off(y, v, π 0 )))
π 0 : y : HUMAN, v : PHYS, w : PHYS • INFO
(17)
Note that in (17) the dot-typed variable has been existentially bound by the repair rules
and that the universal quantifier ranges over physical aspects (object elaborations) of books
(v)—that is, books individuated physically. (17) therefore means that the student in question
carried off all the physical copies of books. This is a welcome result, as ‘carried off every book’
surely does mean carried off every physical copy of every book.
However, in a case of copredication like (16) the default rule cannot apply, as here the type
conflict occurs within the VP, and so existentially binding the dot-typed variable introduced by
the noun ‘book’ will not resolve it. Therefore, in this case it is the variables that are introduced
by the conflicting predicates ‘picked up’ and ‘mastered’ that are existentially bound, and the
expression as a whole inherits the dot typing of ‘book’. The result is as shown in (18) (Asher,
2011, p. 179):
λπ∃u(u = j(π) ∧ ∃3 w(book(w, π) ∧ ∃z(pick-up(u, z, π)
∧ o-elab(z, w, π)) ∧ ∃z 0 (master(u, z 0 , π) ∧ o-elab(z 0 , w, π))))
(18)
π : w : PHYS • INFO, z : PHYS, z 0 : INFO
In this case the numeric quantifier (Asher’s ‘∃3 ’) quantifies over dot objects themselves.
This is necessary in terms of Asher’s repair rules for the reasons given above, and it means
that (18) is underspecified with respect to criterion of individuation for ‘book’.7 What (18)
says, then, is that there are three books such that John picked up their physical aspect(s) and
mastered their informational aspect(s). However, whether that means three books individuated
physically or informationally is undetermined by (18).
If the hearer judges (18) relative to a physical criterion of individuation then ∃3 w(book(w)∧
. . .) says that the three books are physically distinct, and if the hearer judges (18) relative to an
informational criterion of individuation then ∃3 w(book(w) ∧ . . . ) says that the three books are
informationally distinct (see figure 4). Taking a physical criterion of individuation therefore
means that (18) would be true in a situation in which John picked up three copies of the same
(informational) book and mastered that book (once). On the other hand, taking an informational physical criterion of individuation means that (18) would be true in a situation in which
6
Or rather, the interpretation that gives ‘a student’ wide scope. This is a simplification of the example that
Asher gives: ‘a student carried off every book in the library’, where ‘library’ itself is understood as being of type
PHYS • LOCATION .
7
In this particular case it would be possible for Asher to define his repair rules such that either physical or
informational individuation is stipulated. This would not help since, as noted below, (16) can only be true if the
three books are both physically and informationally distinct.
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volume 1
I-Language
volume 2
I-Language
volume 3
I-Language
Figure 3: Three copies of I-Language
John picked up a single copy of a trilogy (three informational books) and mastered the contents.
On neither reading does it entail both that John picked up three (physical) books, and that he
mastered three (informational) books. I submit that this is a bad result, and that (16) can only
be interpreted as meaning that John picked up three distinct physical books and mastered the
distinct informational contents of each.
Now, it might be argued that this impression is not a reflection of the truth conditions of
(16). The argument would be that (16) really can be true in a situation in which John picked up
a single copy of a trilogy and mastered the contents, or a situation in which John picked up three
copies of the same informational book and mastered the contents, and that our judgements to
the contrary are the result of a process of pragmatic strengthening of the basic meaning of (16)
that is derived. The extra aspect of meaning contributed by strengthening should be cancellable
in that case, however. (19)–(22) show that attempts to cancel this purportedly pragmaticallycontributed meaning result in contradictions.
(19) John picked up and mastered three books, but he didn’t pick up three books.
(20) John picked up and mastered three books; in fact, he picked up exactly one book.
(21) John picked up and mastered three books, but he didn’t master three books.
(22) John picked up and mastered three books; in fact, he mastered exactly one book.
I conclude that (18) does not capture the truth conditions of (16), since on either criterion of
individuation the truth conditions of (18) are weaker than the truth conditions of (16). For (16)
to be true the three books being talked about must be doubly distinct, i.e. both physically and
informationally distinct.
4 Addressing the counting problems
4.1 Criteria of individuation in TCL
Take a situation in which John (physically) picked up the volumes and mastered the information
shown in figure 3. According to Asher’s theory of predication, (18) can only be judged true or
false relative to a ‘criterion of individuation’ for ‘book’, as dot objects can only ever be counted
together with one or other of their aspects. On the informational criterion of individuation, (18)
is false, because on that criterion each ‘book’ has one and only one informational aspect and
each informational aspect is the informational aspect of at most one book. In the situation
shown in figure 3 there are not three distinct informational book aspects and hence the sentence
is false.
However, on the physical criterion of individuation, the sentence is true, and there is no
principle guiding how to choose between the criteria of individuation in this case. It should also
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be noted that, in a situation in which there is one copy of three easily comprehensible books
in one heavy volume then (18) would be true on the informational criterion of individuation
and false on the physical one. I submit that, once again, (16) should not be judged true in this
situation.
It seems, then, that the obvious way to begin to attempt to get the right counting conditions is to go further than Asher does and say that, in TCL, a sentence in the metalanguage is
true only if it is true under both (or all) criteria of individuation for its variables of dot type,
rather than being merely true relative to a criterion of individuation as outlined in section 2.2
above. However, on Asher’s own terms this is problematic, because it removes some amibiguity in an unwelcome way. For example, take (23), the metalanguage interpretation of which8 is
shown in (24) (Asher, 2011, p. 174).
(23) A student read every book in the library.
λπ∃y(student(y) ∧ ∀v(∃u∃x∃z(library(x, π) ∧ in(u, z, π) ∧ o-elab(z, x, π)
∧ book(v, π) ∧ o-elab(u, v, π) → read(y, v, π)))
(24)
On the informational criterion of individuation, (24) is true if a student read every informational book of which there is a physical copy in the library. On the physical criterion of
individuation, however, (24) can only be true if the student read every physical book in the library, which normally would require reading some informational books several times. It seems
that this tracks a genuine ambiguity in the English sentence. However, if we require (24) to be
true on both criteria of individuation for the variable v, then (24) can only be true if some student read every physical copy of every book in the library, which is actually the less favourable
of the two readings.
Perhaps, though, all this shows is that the semantics of ‘read’ needs to be tweaked somehow in order to get the right result.9 Even if this is the case, however, it should be noted that
this proposed augmenation of Asher’s method predicts different results in some situations when
compared to the accounts proposed in sections 4.2 and 5 below. Consider again the situation
shown in figure 1. (18) is true on both physical and informational criteria of individuation in
this situation; as figure 4 shows, on both criteria of individuation there are at least three books
that meet the following criteria: there is an object elaboration of them that John picked up and
there is an object elaboration of them that John mastered.
So which of the prospective accounts is to be preferred in this situation depends on
speaker judgements about the truth value of sentence (16) in the situation shown in figure
1. All else being equal, if sentence (16) is true then the account outlined in this section is to
be preferred, whereas if it is false then either the account developed in section 4.2 or the one
developed in section 5 is to be preferred. It is my judgement that (16) is false in the situation
shown in figure 1, or at least somehow defective, and so the account in this section is to be
avoided for that reason.
8
9
Or rather, the interpretation that gives ‘a student’ wide scope (again).
‘Read’ is actually very unusual in Asher’s system in taking objects of dot type.
11
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I-Language
O
7
Kingdom
f
Wisdom
O
book 1
book 2
book 3
volume 1
volume 2
volume 3
Revelation
8
Physical individuation:
3 books
Informational individuation:
4 books
I-Language
O
Kingdom
O
Wisdom
O
Revelation
O
book 1
book 2
book 3
book 4
w
volume 1
volume 2
& w
volume 3
a
/
b : o-elab(b, a)
Figure 4: Figure 1 according to physical and informational criteria of individuation
4.2 Plural predication in mereological accounts
The mereological analysis of (15) says that there are five things that are books, have a heavy
physical part and have an easy-to-understand informational part. This logical form can be
expressed as shown in (26).10
|{x :(book(x) ∧ ∃y(part-of(y, x) ∧ heavy(y))
∧ ∃z(part-of(z, x) ∧ easy-to-understand(z)))}| ≥ 5
(26)
The problem with the mereological analysis presented in (26) is that the formula can be
true when you have five ‘books’ (again, conceived of as composites of physical objects and
informational objects) that all have the same informational part that is easy to understand, provided that their (heavy) physical parts are distinct.11 So how might this problem be addressed,
while retaining the idea that a book is a physical/informational composite?
For the reconstruction of (25) given in (26), I have assumed a denotation of ‘heavy’ as
shown in (27), which I will henceforth refer to as a ‘weakened’ denotation of ‘heavy’. This is
10
Cooper’s formalism is actually that shown in (25); however, as discussed in section 2.1, underlying the
formalisation in (25) is the supposition that ‘book’ denotes a complex of a single physical object with a single
informational object. In order to bring this out more explicitly (and express it in terms that are more familiar
to more readers) I have expressed it in the formalism shown in (26), which I believe is a fair reflection of these
assumptions.
(25) 



x : Ind



λr : c : phys(x) c :
3
c
:
f
ive
1




c2 : info (x)
11
Or vice versa


x : Ind
c1 : phys(x) 

book(r.x)
, λr0 : 
c2 : info(x) 
c3 : book(x)
c4
c5

!

: heavy(r0 .x) 


: e t u(r0 .x) 

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a function that characterises the set of things that have a heavy part.12
(27) λx.∃y(part-of(y, x) ∧ heavy(y))
What is needed in order to avoid the problems described above is some kind of requirement
that the set that this weakened predicate characterises have no two members with the same
heavy part. However, adding this kind of requirement to (27) is not straightforward, because
the set-specification conditions shown in (28) are contradictory:
(28) It is the set such that:
everything that has a physical part that is heavy is a member, and
no two of its members have the same heavy part in common.
Perhaps, instead, what we want is the maximum subset of the set of things with a heavy
part such that no two members of this subset have the same heavy part. That would give us the
set-specification conditions shown in (29).
(29)
It is the set such that:
1. it is a subset of the set of things that have a heavy part,
2. no two of its members have the same heavy part in common, and
3. it is a proper subset of no other set meeting conditions 1 and 2.
The problem with (29) is not that the conditions are contradictory, but rather that in most cases
they do not determine a unique set. For instance, in the situation graphically outlined in figures
1 and 2, the six sets shown in (30) all satisfy conditions 1–3 of (29).
(30) {volume 1 + I-Language, volume 3 + Gospel and Kingdom},
{volume 1 + I-Language, volume 3 + Gospel and Wisdom},
{volume 1 + I-Language, volume 3 + Gospel in Revelation},
{volume 2 + I-Language, volume 3 + Gospel and Kingdom},
{volume 2 + I-Language, volume 3 + Gospel and Wisdom}, and
{volume 2 + I-Language, volume 3 + Gospel in Revelation}.
However, what if we were to treat the denotation of the predicate as being the characteristic function of the set of all the sets listed in (30)? That would be the approach shown in (31),
where we have the function that, in the situation shown in figure (2), returns 1 when applied to
(the characteristic function of) any of the sets listed in (30) and 0 otherwise.
(31) λP.∀x(P (x) → ∃y(part-of(y, x) ∧ heavy(y) ∧ ¬∃z(P (z) ∧ z 6= x ∧ part-of(y, z))))
Given that this is a monadic predicate of sets rather than individuals (i.e. of type (e →
t) → t rather than e → t), it is not the usual type of intransitive verb such as was used to
derive the interpretation shown in (26). However, the noun phrase in (15) is in plural form—
this is what causes the counting problem in the first place—and in fact it has been argued
elsewhere (Schwarzschild (1996), Winter (2001)) that at least some plural nominals are of type
type (e → t) → t rather than e → t. In general, on these accounts the denotation of the plural
form of a noun is (the characteristic function of) the set of non-empty subsets of the denotation
12
Admittedly, this definition ignores the requirement that the part y of x that is heavy must be the maximal
physical part of x. I take this as understood in what follows, not forgetting that this kind of requirement would
impose further constraints on the semantic theory at later points.
13
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of the singular form of the noun. For instance, this would given rise to the alternation shown in
(32):
[[book]] = λx.book(x)
[[books]] = λP.(∀x(P (x) → book(x)) ∧ ∃x0 (P (x0 )))
(32)
In this view, a set that is in the extension of [[books]] would be of the right type to be an
argument of the function shown in (31). Assuming for the sake of elucidation that the semantic
contribution of the simple numeral is to introduce existential quantification over sets with five
or more members, and giving a treatment to ‘easy to understand’ analogous to that given to
‘heavy’ shown in (31), (15) would be interpreted as shown in (33).
∃X |X| ≥ 5 ∧ ∀y(X(y) → book(y)) ∧ ∃y 0 (X(y 0 ))
∧ ∀x X(x) → ∃v(part-of(v, x) ∧ heavy(v)
∧ ¬∃z(X(z) ∧ z 6= x ∧ part-of(v, z)))
(33)
∧ ∀x0 X(x0 ) → ∃v 0 (part-of(v 0 , x0 ) ∧ easy-to-understand(v 0 )
∧ ¬∃z 0 (X(z 0 ) ∧ z 0 6= x0 ∧ part-of(v 0 , z 0 )))
This states that there is at least one five-membered set of books (i.e. composites of a single
informational book with a single physical book) X such that each of its members has a heavy
part that is a part of no other member of X and each of its members has an easily comprehensible part that is a part of no other member of X . Unlike (26), (33) is not true in the situation
shown in figure 1, even given the mereological understanding shown in figure 2.13
Furthermore, if we extend this treatment to transitive verbs then we end up with the
interpretation of (16) shown in (34).
∃X |X| ≥ 3 ∧ ∀y(X(y) → book(y)) ∧ ∃y 0 (X(y 0 ))
∧ ∀x X(x) → ∃v(part-of(v, x) ∧ picked-up(John, v)
∧ ¬∃z(X(z) ∧ z 6= x ∧ part-of(v, z)))
(34)
∧ ∀x0 X(x0 ) → ∃v 0 (part-of(v 0 , x0 ) ∧ mastered(John, v 0 )
∧ ¬∃z 0 (X(z 0 ) ∧ z 0 6= x0 ∧ part-of(v 0 , z 0 )))
Unlike (18) under the additional constraint mentioned at the end of section 4.1, (34) is not true
in the situation shown in figure 2; and, as discussed at the end of section 4.1, this seems to be a
welcome result.
However, these welcome results have come about because of the interpretations that have
been assigned to the predicates of pluralities, as shown for example in (31). Importantly, no
difference has been made to the way in which books themselves are counted—we still have
five books in the situation shown in figure 2. This might not be a problem if we could be sure
that in any sentence in which a plurality of books is referred to, something is predicated of
that plurality that would avoid these unwelcome results. But is does not seem that this can be
guaranteed. Take the very simple sentence shown in (35).
13
The plural predicate meaning shown in (31) dervies distributive meanings, and hence the metalanguage
formula in (33) is a distributive reading. I have chosen to work this way around because the distributive reading of
(15) is much more readily available than the collective one, and I will not be going into the question of how these
tentative steps at a resolution to the counting problem deal with the distributive/collective alternation, although I
am well aware that in the end such an account will have to be given.
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(35) There are five books.
Unless we implausibly say that ‘there are’ cannot apply to books as such, but rather only
physical or informational parts of books, then on the current approach we are forced to say that
(35) is true in the situation shown in figure 2. And this is an unwelcome result. So what we
need, in fact, is a system that forces us only to count books either physically or informationally,
but which also does not fall into the problems of Asher’s account.
There most obvious way to guarantee that books will be counted either physically or informationally is to say that the word ‘book’ is ambiguous, with a meaning that denotes physical
objects and a meaning that denotes informational objects. This approach leaves us facing the
original problem of copredication, of course. However, it is possible to develop an account
of copredication that nevertheless retains this ambiguity idea. We have already seen such an
account.
5 Counting and the semantics/pragmatics interface
5.1 Prospects for resolving the counting issues
Nunberg’s account relies on the pragmatic notions of salience and noteworthiness as constraints
on the process by which certain predicates can undergo meaning transfer. For example, on his
account (11) is acceptable because the context makes salient the relationship between cars and
their drivers and because, in the context, being the driver of a car that is parked out back makes
someone noteworthy.
It certainly seems that some kind of pragmatic story needs to be told about the acceptability or otherwise of some sentences that look like copredication, since the contrast between
(4) (repeated below) and (36) shows that it is not simply the case that the word ‘bank’ licenses
the coordination of any predicate appropriate of financial institutions with any predicate appropriate of buildings that those institutions operate.
(4)
The bank called in Bob’s debt and was vandalised.
(36) # The bank is FTSE-100 listed and used to be a school.
Some of the pragmatic effects on the acceptability of copredication have been investigated by Regine Brandtner in her dissertation (Brandtner, 2011), and an analysis developed
on Nunbergian lines. The analysis that Brandtner offers is that in a sentence containing two
conflicting predicates applying to a single argument, the second predicate can undergo meaning transfer in such a way as to be compatible with the first, provided that certain discourse
conditions are met.
The subject of Brandtner’s dissertation is exclusively deverbal -ung nominals in German,
but it is nevertheless worth examining how well this approach would transfer to other nouns
supporting copredication, since we can ask whether or not all copredication could be accounted
for on a meaning-transfer approach like this. As noted at the end of section 2.3, the natural way
to apply predicate meaning transfer as a way to resolve the problem of copredication is to
postulate a lexical ambiguity between e.g. physical books and informational books, and to say
that—in sentences where informational and physical predicates are both applied to a ‘book’—
one or other of these predicates has undergone meaning transfer in order to make it compatible
with the other. In this view, in any given sentence ‘book’ denotes either physical books or
informational books, but not both.
Suppose that ‘book’ denotes physical books, and that the meaning of ‘easy to understand’
changes in context along the lines proposed by Brandtner for predicates of deverbal nominals
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in German. The general idea is that (5) would be interpreted as shown in (37), where I have
adopted the notation for predicate meaning transfer that Brandtner (2011) uses to show how the
interpretation of the informational predicate has changed in context in order that it be applicable
to physical objects.
(37) The (physical) book is heavy but {instantiates (an) informational book that is/are [easy
to understand]}
To make this idea formal, we can adopt Nunberg’s predicate transfer mechanism shown
in (14) in order to implement it. If h is the functional relationship, made salient by the context,
between informational books and the physical books that instantiate them, then the interpretation of the predicate after having undergone meaning transfer would be as shown in (38).
λP.λy.∃x: x ∈ domain(h) ((h(x) = y) ∧ P (x))[λz.easy-to-understand(z)]
=λy.∃x: x is an infobook ((physbook-instantiating(x) = y) ∧ easy-to-understand(x))
(38)
Actually, (38) does not get us the interpretation that we want. h is supposed to be a
function, and this means that for each argument in its domain it will return a single value.
Therefore, (38) has the unwelcome effect of requiring that x be something that is instantiated
by a single physical book. In general, this is certainly not the case when it comes to the
relationship between informational books and the physical books that instantiate them. In the
situation shown in figure 1, what is the value of ‘physbook-instantiating(I-Language)’? Volume
1 or volume 2?
Conversely, suppose that ‘book’ denotes informational books, and it is [[heavy]] that undergoes meaning transfer:
λP.λy.∃x: x ∈ domain(h) ((h(x) = y) ∧ P (x))[λz.heavy(z)]
=λy.∃x: x is a physbook ((infobook-instantiated-by(x) = y) ∧ heavy(x))
(39)
(39) is even stranger than (38). Not only is it possible for one physical book to instantiate more
than one informational book, making the functional relationship here dubious as in the case
above, but also, use of this predicate in the interpretation of (40) would have the bizarre result
that (40) would be true in a situation in which the informational book referred to is easy to
understand and is instantiated by some physical book that is heavy, even if that physical book
is not present in the situation.14
(40) This book is heavy but easy to understand.
So in this context h can be neither the function that returns the physical book instantiating
any given informational book (because there is no such function), nor the function that returns
the informational book instantiated by any given physical book. But perhaps it is a different
function: one that (say) pairs the informational book that is relevant in this context to the
physical book that is relevant in this context. In that case, if we take physical books to be basic
and the shifted predicate to be as shown in (41), then the interpretation of (16) would be as
shown in (42).
(41) λy.∃x: x is an infobook ((relevant-physbook-instantiating(x) = y) ∧ easy-to-understand(x))
|{y :(physbook(y) ∧ picked-up(John, y)
(42)
∧ ∃x: x is an infobook ((relevant-physbook-instantiating(x) = y)
∧ mastered(John, x)))}| ≥ 3
14
Thanks to Nathan Klinedinst for this observation.
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(42) manages to avoid the counting problems faced by the accounts described in sections
2.1 and 2.2. Unlike the interpretations that those accounts predict (e.g. (18)), (42) is not true
in a situation in which John picks up three copies of the same informational book and masters
their content. This is because in that situation, while there would be three things that John
picked up, they could not all be the unique physical book instantiating a single informational
book (since there are three of them, none of them would be unique).
Let us call the meaning of the noun to which the unshifted predicate applies in some
sentence its ‘primary’ meaning (so e.g. the primary meaning of ‘book’ in (5) according to (42)
is physical books), and the apparent other meaning of the noun, which in fact is only present in
the sentence contained within the shifted predicate, it’s ‘secondary’ meaning. If this treatment
generalises then,when we have numeric quantification in copredication, the number according
to the secondary meaning of the noun cannot be any lower than the number according to the
primary meaning: so in this case, we cannot be talking about fewer informational books than
physical books.
5.2 Objections
To get this result the transfer function has to be more or less exactly as shown in (14). This
being the case, is worth asking not only how well motivated (14) is in its precise formulation,
but also whether or not it covers all the data regarding predicate transfer equally well. For
instance, consider (43).
(43) John and Mary are parked outside.
(43) can be true in a situation in which John and Mary are both drivers of a shared car in which
they have both arrived and which is parked outside. However, if (43) is interpreted by means
of [[parked outside]] undergoing meaning transfer as shown in (14), then we end up with the
interpretation shown in (44).
(44)
∃x: x is a car ((driver-of(x) = John) ∧ parked-outside(x))
∧ ∃x0 : x0 is a car ((driver-of(x0 ) = Mary) ∧ parked-outside(x0 ))
(44) cannot be true if x = x0 , because John and Mary cannot both be values of the driver-of
function for the same argument. It seems that in the case of (43) we do not want to have the
same number entailments that we have in (16).
However, it might instead be the case that in (43) there really is a single value of the
function h, namely the group or plurality John⊕Mary. The problem in that case is how to
capture the inference shown in (45).
(45)
John and Mary are parked outside
∴ John is parked outside
It seems that (45) is valid; if John and Mary are parked outside then John is parked
outside. But in the naı̈ve formalisation of (45) shown in (46) the conclusion actually contradicts
the premise.
(46) #
∃x: x is a car ((driver-of(x) = John⊕Mary) ∧ parked-outside(x))
∴ ∃x: x is a car ((driver-of(x) = John) ∧ parked-outside(x))
It might be argued that in (45) some subtle change in the shifted meaning of ‘parked
outside’ has taken place between the premise and the conclusion of the inference. This idea
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can be captured by treating the inference shown in (45) as being that shown in (47).
(47) #
∃x: x is a car ((joint-drivers-of(x) = John⊕Mary) ∧ parked-outside(x))
∴ ∃x: x is a car ((one-driver-of(x) = John) ∧ parked-outside(x))
In the case of (47) the conclusion and the premise are at least consistent with each other,
but the former does not follow from the latter. If some kind of predicate meaning transfer
account is correct then, to the extent that native speakers judge (45) valid, either the logical
form of (45) is something like (47) and speakers are entertaining an additional premise along
the lines shown in (48), or its logical form is not like (47), in which case the transfer function
must be different from that shown in (14).15
(48) ∀x∀y∀z((atom(x) ∧ joint-drivers-of(z) = x ⊕ y) → one-driver-of(z) = x)
This treatment and its resulting introduction of (48) raises the additional worry that the actual
relations in view are becoming difficult to specify in anything but the most ad-hoc way. This
is also reflected in the replacement of ‘physbook-instantiating(x)’ with ‘relevant-physbookinstantiating(x)’ in (41) in order to avoid absurdity, i.e., the replacement of an intuitive relation
with one that is much harder to satisfactorily characterise.
It is also worth noting at this juncture that (43) can be generalised as shown in (49), which
is likewise true in the situation described.16
(49) Two people are parked outside.
If (14) is the right way to think about predicate meaning transfer (and predicate meaning transfer is the right way to think about (49)), then (49) can only be true in this situation given a
collective reading of ‘two people’. A plausible account of how this collective reading is derived is that ‘two people’ denotes the set of sets that have a non-empty intersection with the set
of two(or more)-membered pluralities of people. The meaning of (49) in context, then, would
be that at least one such plurality is in the extension of the (shifted) meaning of ‘parked outside’
(Link, 2002, p. 141).
In contrast, sentences like (16) cannot be given this kind of collective reading. If they
could, then (16) could be true if John picked up and mastered the book(s) in the situation shown
in figure 3, contrary to fact, since in that situation there is a plurality of three physical books that
collectively instantiates the informational book I-Language.17 Accordingly, the metalanguage
formula shown in (50)18 would actually be true in the situation shown in 3.
∃z picked-up(John, z)
∧ ∃y(infobook(y) ∧ relevant-physbook-instantiating(y) = z ∧ mastered(John, y))
∧ (book∗ (z) ∧ |{y : atom(y) ∧ z ≥i y}| ≥ 3)
(50)
Of course, the (non-)availability of collective readings for plurals is a large topic in itself
and there are almost certainly other relevant considerations here. But clearly there is some
difference that makes the counting conditions for cases like (16) different to those for cases
like (11).
15
Here and in what follows I will be relying on the treatment of pluralities given by Link (2002), according
to which ‘atom(x)’ means that x is an individual member of a plurality (and not a plurality itself).
16
That is, the situation in which John and Mary are both drivers of a shared car in which they have both
arrived and which is parked outside
17
And mutatis mutandis for a situation involving a trilogy, since in that situation there would be a plurality
of three informational books that is collectively instantiated by at least one physical book.
18
‘book∗ (z)’ means that z is either a book or a plurality of books, and z ≥i y means that y is an ‘individual
part’ of z (Link, 2002).
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5.3 Different transfer functions
Given these considerations, would we do better to alter the transfer function shown in (14) in
order to take account of (43)? If we have (51) instead of (14), then (43) will be interpreted as
shown in (52).
(51)
λP.λy.∃x: hx, 1i ∈ range(h) (h(y, x) ∧ P (x))[λz.parked-out-back(z)]
=λy.∃x: x is a car (drives(y, x) ∧ parked-out-back(x))
(52)
∃x: x is a car (drives(John, x) ∧ parked-out-back(x))
∧ ∃y : y is a car (drives(Mary, y) ∧ parked-out-back(y))
This means that the inference in (45) will be validated, as shown in (53).
(53)
∃x: x is a car (drives(John, x) ∧ parked-out-back(x))
∧ ∃y : y is a car (drives(Mary, y) ∧ parked-out-back(y))
∴ ∃x: x is a car (drives(John, x) ∧ parked-out-back(x))
The problem with altering the transfer function in this way is that by doing so we end up
with the same counting problem as faces the account of Asher (2011) when addressing cases
like e.g. (16). In (16) we must decide to interpret ‘book’ as denoting either physical books or
informational books and then apply meaning transfer to the appropriate predicates as per (14).
The two options are shown in (54)–(540 ) and (55)–(550 ) respectively.
(54) John picked up and {[mastered] the information instantiated by} three (physical) books.
(55) John {[picked up] the physical object(s) instantiating} and mastered three
(informational) books.
(540 )
|{y :(physbook(y) ∧ picked-up(John, y)
∧ ∃x: infobook(x) (instantiates(y, x) ∧ mastered(John, x)))}| ≥ 3
(550 )
|{y :(infobook(y) ∧ mastered(John, y)
∧ ∃x: physbook(x) (instantiated-by(y, x) ∧ picked-up(John, x)))}| ≥ 3
Again, 540 would be true in a situation in which John picked up three copies of the same
(informational) book and mastered that book (once), while 550 would be true in a situation in
which John picked up a single copy of a trilogy (three informational books) and mastered the
contents. This is unwelcome as neither of these predictions seems to be right; we need an
interpretation that is false in both the circumstances just described.
Of course, one could propose that sometimes the transfer function is similar to that shown
in (14), while sometimes it is similar to that shown in (51). But without further specification of
when (14) should be used over (51) (and vice versa) this seems to be merely a re-description
of the data—namely, that in some cases we have the number entailments and in others we do
not. If the idea is to say that copredication is simply a case of the same kind of pragmaticallylicensed predicate transfer that makes ‘I am parked out back’ acceptable then something explanatory needs to be said about why the number entailments are different for different sentences involving predicate transfer.
It seems that the moral of this extended look into variations on Nunberg’s formalism is
that our copredication examples are not simply instances of the general case of pragmaticallydriven predicate meaning transfer which that formalism was devised to capture. Whether or not
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a particular sentence has the kind of number entailments noted for (16) seems to depend on the
noun to which the coordinated predicates are applied. In any sentence involving informational
and physical predications being made of some number n of ‘books’, the sentence entails that
at least n physical books and at least n informational books are in view. However, this is not
the case when the mapping is between people and their cars rather than between informational
books and the physical books that instantiate them: as discussed above, (49) can be true when
there is only one vehicle parked outside.
6 Conclusion
Existing attempts to devise theories that predict ontologically respectable paraphrases as the
natural language interpretations of copredication sentences all face problems of greater or lesser
severity when it comes to extending those theories to cases where the copredication sentences
involve numeric quantification. The general issue is that copredication presents us with divergent criteria for identifying objects falling under the denotation of the noun, and hence potentially divergent criteria for individuating and counting those objects. Native speaker judgements about copredication sentences involving numeric quantification do not perfectly reflect
the counting and individuation criteria that the existing accounts predict.
The theory that most closely matches natural judgements for copredication sentences
involving numeric quantification is one based on pragmatically-licensed predicate meaning
transfer. This kind of account can be fixed so that these incorrect counting predications are
not made. However, the kind of fix that is needed seems to tell against simply identifying
copredication with the independently-observed phenomenon of predicate meaning transfer in
context. Exactly how and where these phenomena overlap and diverge is the subject of ongoing
research.
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