5.1. Copredication and individuation: an introduction

To appreciate the nature of the issues, consider again (5.1). In this example, the predicates “pick up” and “master” require physical and informational objects as their respective arguments. Formally, their domains are PHY and INFO, the types of physical and informational objects, respectively. However, they manage to be applied in coordination to the argument “the book”. What goes on in this example is that “book” is used in its physical sense with respect to the predicate “picked up” and in its informational sense with respect to the predicate “mastered”.

The term copredication was first coined by Pustejovsky (1995), according to whom lexical items like “book” in (5.1) are complex and have more than one sense, to be chosen appropriately according to the context. The topic has since been studied by many researchers and different attempts to account for copredication have been proposed in various semantic frameworks. Besides Pustejovsky’s initial proposal (Pustejovsky 1995), these include, to mention a few among many: Asher (2012) using type composition logic, Gotham (2014, 2017) using a mereological theory, Bassac et al. (2010) using the second-order λ-calculus and Cooper (2011) using type theory with records (TTR).²

Most of the above approaches have been studied in the Montagovian setting where, in particular, CNs are interpreted as predicates. This has raised a challenge to the researchers because a formal modeling of copredication in such a setting meets with a problem, pointed out by the second author in Luo (2009c, 2012b): the CNs-as-predicates paradigm is incompatible with the subtyping relations used in an intended analysis. Let us use a simple example to illustrate. Consider how to interpret the phrase “heavy book”, where “book” is assumed to be of a complex type having two aspects, one informational and other physical. Then, the adjective “heavy” is a predicate, a function from physical objects to truth values, and “book” is a predicate over some dot-type PHY • INFO, where PHY and INFO are the subtypes of e of physical and informational objects, respectively.³ In symbols, we have:

(5.2) heavy : PHY → t and book : PHY • INFO → t

With these assumptions in line, the account will not work. Here is why: ideally, what we want to get is a predicate of the following form:

(5.3)

However, this predicate (5.3) is not well-typed on the assumption that the dot-type is a subtype of its constituent types, in our case PHY • INFO ≤ PHY. To the contrary, in order for typing to work in the above example, and given the CNs as predicates approach, what we need is the subtyping relation PHY ≤ PHY • INFO. Of course, an assumption like this is totally counterintuitive and formally untenable. Assuming a higher order type for the adjective “heavy” will not work either, but we leave the details of this to the interested reader to work out (or one can find out the details in Luo (2012b)).

MTT-semantics, different from the Montague semantics, adopts the CNs-as-types paradigm and this makes a real difference in formalizing dot-types for a treatment of copredication: one does not get the problem of incompatibility with subtyping anymore. Recognizing this, the second author has proposed the introduction of dot-types in MTTs for the semantic study of copredication (Luo 2009c, 2012b). The idea is that, the phenomenon of the single occurrence of “book” being able to be used in both physical and informational senses can be captured by stipulating that the type Book, that interprets “book”, be a subtype of the dot-type of those of physical and informational objects:

The copredication phenomenon as exhibited in (5.1) can then be dealt with straightforwardly: the predicates “pick up” and “master” can be applied in coordination to the argument “the book” because they are both of type

Book → Prop under the usual subtyping laws (see section 5.2 for a more detailed description of dot-types and their use for copredication).

In order to give an adequate account of copredication, we need to take care of a more advanced issue as well – the involvement of quantification. In (5.4), taken from Asher (2012), copredication and quantification interact:

(5.4) John picked up and mastered three books.

The above sentence involves a number of different things that need to be taken care of in order to get a proper semantic analysis: individuation and counting with numerical quantifiers in copredication. An adequate account of copredication must take into consideration all these parameters and make sure that when combined, they produce the correct semantic interpretations. The crucial question is then, what is needed for such an adequate account? In order to answer this question, let us first start by considering the following simpler sentences that do not involve copredication:

(5.5) John picked up three books.

(5.6) John mastered three books.

(5.5) is true in case John picked up three distinct physical objects. Thus, it is compatible with a situation where John picked up three copies of a book that are informationally identical as long as three distinct physical copies are picked up. Similarly, example (5.6) is true in case three distinct informational objects are mastered, but it does not impose any restrictions on whether these three informational objects should be different physical objects or not.⁴ A more explicit way of explaining this is that, intuitively, the following entailments should be the case:

(5.7) John picked up three books ⇒ John picked up three physical objects.

(5.8) John mastered three books ⇒ John mastered three informational objects.

At first appearance, the above examples may be puzzling. The books John picked up and the books he mastered must have different individuation criteria – being physically identical is different from being informationally identical. Are they the same books? Actually, they are not. The collection of books with being physically the same as its IC (call it =_p) is different from the collection of books with being informationally the same as its IC (call it =_i). In other words, ICs play a crucial role in fixing collections of objects represented by CNs, although they are not made explicit in NL sentences. “Books” in (5.5) and (5.7) refer to those with IC =_p, while “books” in (5.6) and (5.8) refer to those with IC =_i. Note that it is the predicates (“pick up” and “master”) that determine which IC should be: in (5.5), the domain of “picked up” is the type PHY of physical objects, and the IC for books is =_p (being physically the same); and in (5.6), the domain of “master” is the type INFO of informational objects, and the IC for books is =_i (being informationally the same).

Now, a further complication arises in cases like (5.4), where copredication and quantification interact and, therefore, in its semantic analysis, both ICs have to be used, i.e. we have to consider both collections of books: one with being physically the same as its IC and the other with being informationally the same as its IC. Giving semantics properly, we should be able to get the following entailment:

(5.9) John picked up and mastered three books ⇒ John picked up three physical objects and mastered three informational objects.

Here, there is some sort of double distinctness that should be accounted for, involving both ICs. Any adequate theory of copredication should not only account for (5.7) and (5.8), but also for (5.9), dealing with counting and individuation correctly. In order to provide such an account, we further develop the idea on ICs for CNs, initially studied for MTT-semantics by the second author in (Luo 2012a). We propose that, in general, CNs are interpreted as types associated with their ICs, i.e. they are setoids. A setoid is a pair (A, =), where A is a type (domain of the setoid) and = is an equivalence relation on A (equality of the setoid). If a setoid interprets a CN, its domain interprets that of the CN and its equality gives the IC for the CN. For example, “books” in (5.7) refer to those in the collection of books whose IC is =_p (being physically the same), while “books” in (5.8) refer to those in the collection of books whose IC is =_i (being informationally the same). They are books with different ICs; or put in a better way, they belong to different collections of books whose ICs are different. In the CNs-as-setoids setting, it is shown that dot-types offer a proper treatment of copredication, including the sophisticated cases involving both copredication and quantification.

At this point, one might want to ask: why do people working in MTT-semantics usually say that, in MTT-semantics, CNs are interpreted as types and do not pay attention to their ICs? The answer is rather simple. This is because most of the cases are not as sophisticated as those cases where copredication interacts with quantification and as such, in these less sophisticated cases, the related CNs usually have essentially the “same” IC. For example, type Man inherits its IC from type Human: two men are the same if and only if they are the same as humans. This explains why we usually consider the entailment below induced by the subtyping relationship Man ≤ Human as “straightforward” without the need to mention, or explicitly flesh out, their:

(5.10) Three men talk ⇒ Three humans talk

In other words, when interpreting a CN, its IC can usually be safely ignored. Of course, this is not always the case: “books” in (5.5) and (5.6) are referring to different collections of books that have different ICs and, in (5.4) where both copredication and quantification are involved, more than one ICs will be involved in constructing appropriate semantics. In other words, in general, CNs are setoids but, in the usual cases, their IC can be safely ignored and they are just types.

5.2. Dot-types for copredication: a brief introduction

Dot-types in MTT-semantics were introduced by the second author (Luo 2009c, 2012b) to model copredication. In what follows, we shall first present the basic ideas behind this approach and then explain the rules for dot-types. Consider the copredication example (5.1), repeated below:

(5.11) John picked up and mastered the book.

Let PHY and INFO represent the types of physical and informational objects, respectively, and further assume that the interpretations of “pick up” and “master” have the following types:

In (5.11), “pick up” and “master” are applied in coordination to “the book”. In order for such coordination to happen, the two verbs must be of the same type.⁵ Our question is: given the above typing, can both of them be of the same type? The introduction of dot-types make this happen naturally. Let us informally start with the dot-type PHY • INFO. PHY•INFO is the dot-type that satisfies the following property: it is a subtype of both PHY and INFO. So far, so good. Now, we need to account for the assumption that books have both a physical and an informational aspect. This can be easily captured using subtyping, i.e. by assuming that Book is a subtype of PHY • INFO:

Given that subtyping is contravariant for function types, i.e. if A ≤ B, then B → Type ≤ A → Type (informally, subtyping goes the other way around), the following holds:

In other words, pick up and master can be both considered of type Human → Book → Prop given subtyping and, therefore, the coordination in (5.11) and its interpretation can proceed straightforwardly as intended.

With this taken care of, let us dig a little bit deeper on what dot-types are and how they are introduced in MTT-semantics. In order to get there, we need first to mention that one of the basic features of dot-types: their formation. To form a dot-type A • B, its constituent types A and B should not share common parts. This is what Pustejovsky calls inherent polysemy, as witnessed by the passage below:

Dot objects have a property that I will refer to as inherent polysemy. This is the ability to appear in selectional contexts that are contradictory in type specification. (Pustejovsky 2005)

Thus, a dot-type A • B can only be formed if the types A and B do not share any components. Consider the following:

• – PHY • PHY should not be a dot-type because its constituent types are the of same type PHY.
• – PHY • (PHY • INFO) should not be a dot-type because its constituent types PHY and PHY • INFO share the component PHY.

To rephrase what we have just said, given types A and B, we can form the dot-type A • B just in case that A and B do not share common parts (formally called components). For instance, PHY • INFO is a legitimate dot-type, while PHY • (PHY • INFO) is not, because in the latter, the constituent types PHY and PHY •INFO share the common part PHY. Furthermore, as already mentioned, a dot-type A • B is a subtype of both of its constituent types A and B. Note also that the whole dot-type formation is based on the notion of component. Its definition is as follows:

DEFINITION 5.1 (components).– Let T : Type be a type in the empty context. Then, C(T), the set of components of T , is defined as:

where SUP(T) = {T′ | T ≤ T′}. □

Note that because of the non-sharing requirement for components and the subtyping relationships with its component types, a dot-type is not an ordinary type (like an inductive type) already available in MTTs, but we still specify them by means of formation, introduction, elimination and computation rules, with suitable conditions and some extra coercive subtyping rules to stipulate that the elimination/projection operators are coercions. The rules for dot-types can be found in Appendix 6. Here, we will have a look at some of them four of these rules, starting with the formation rule of dot-types, which basically is the rule that takes care of the formation of dot-types based on what we just been discussing:

What this rule intuitively says is that if types A and B do not share any components, the dot-type A • B can be formed. The important point here that we need to stress again is the non-sharing of components. Thus, given the types INFO and PHY, we can form the dot-type PHY • INFO, but not the dot-type PHY • PHY since the two types are identical (PHY and PHY) and thus do not obey the non-sharing components restriction. The next rule is the introduction rule:

Intuitively, the introduction rule says that given an element a of type A, b of type B and the dot-type A • B, the pair a, b is of type A. The next rule is the elimination rule:

This rule gives us a way to access the components of a dot-type via its two projections. Given a : A • B, we can use projections p₁ and p₂ to access the two components of c of type A and B, respectively. The last rule we want to have a look at is the projections as coercions rule:

These two rules simply say that in case we have a dot-type A • B : B, its projections p₁ and p₂ are declared as coercions, i.e. the dot-type is a subtype of its constituent types. For example, PHY • INFO (the dot-type) is a subtype of its constituent types PHY and INFO, i.e. PHY • INFO ≤ PHY, INFO.

Let us use again example (5.11) repeated below as (5.12), but this time with more explanations given and referral to the relevant rules used when needed:

(5.12) John picked up and mastered the book.

Let us repeat the types involved:

The first thing to note is that PHY • INFO is a well-formed dot-type, since PHY and INFO do not share any components. This means that we can find elements of that type, e.g. the_book : Book. Now in order to compose picked up, on the one hand, and master, on the other hand, with the_book, we need to use the projections as coercions rule. In the first instance, we use the p₁ to extract the PHY component of the_book, while in the second case p₂ to extract the INFO component. Because of the subtyping declarations Book ≤ PHY • INFO ≤ PHY and Book ≤ PHY• INFO ≤ INFO, as well as the contravariance of subtyping for function types, i.e. PHY → Prop ≤ PHY • INFO → Prop ≤ Book → Prop and INFO → Prop ≤ PHY • INFO → Prop ≤ Book → Prop, we are able to do the composition. This concerns the individual cases of putting together the individual predicates with the argument the_book. We are in a coordination environment however. This is how we coordinate: we take the A argument needed by the typing of and to be Book → Prop. Both the typings of the two predicates are subtypes of that type, thus they can be coordinated and together applied to the argument the_book. We can also access the individual components of the dot-type. For example, we can access the informational component of the the_book in mastered(John, the_book) using the second elimination rule.

With this brief introduction to dot-types in place, there is a remark that should be made concerning these types. As already extensively discussed in this book, in MTT-semantics, CNs are interpreted as types (rather than predicates): for instance, the CNs “table” and “man” can be interpreted as types Table and Man, respectively. However, there is something that needs to be stressed: even though every CN corresponds to a type, the opposite relation does not hold, i.e. not every type represents a CN. Dot-types are a good example of this: dot-types such as Table • Man do not represent any CNs in natural language, at least to the extent that we know. The way we see it is that dot-types are useful type theoretical constructions that help us in getting from natural language to logical form in a way that corresponds to the intuitions with regard to copredication.

5.3. Identity criteria: individuation and CNs as setoids

As already mentioned in the introduction, individuation is the process by which objects in a particular collection are distinguished from one another. The discussion on individuation goes back to at least Aristotle (Cohen 1984) and has been the subject of enquiry of great philosophers in both the continental as well as the analytical tradition. However, it is outside the scope of the present chapter to present an overview of individuation from a philosophical point of view. Our point of departure will be individuation as discussed in the philosophy of language and the linguistic semantics tradition.

In linguistic semantics, individuation is very much related to the idea that a CN may have its own IC for individuation, an idea already discussed in (Geach 1962). In mathematical terms, the idea amounts to the association of an equivalence relation (the IC) to each CN. In fact, in the tradition of constructive mathematics, a set or a type is indeed a collection of objects together with an equivalence relation that serves as an IC of that collection.⁶ For CNs in MTT-semantics, this is the same as saying that a CN should in general be interpreted as a setoid – a type associated with an equivalence relation over the type.

In our discussions of MTT-semantics, the author of this book have been extensively discussing and endorsing the view that CNs are better treated as types rather than predicates. In doing so, we have been skipping some detail for the sake of simplicity.

These details will now become significant, as well as handy, when discussing the issue of individuation concerning CNs. Put it in another way, these details we have been skipping are now crucial in giving an account of individuation and correct predictions in complex cases involving dot-types and counting like (5.7), (5.8) and dot-types, counting and copredication in cases like (5.9), where individuation, and the derived issue of counting, becomes significant and cannot be ignored anymore.

The crucial detail that has been mostly ignored so far in this book is embodied in the proposal put forward by the second author in Luo (2012a), according to which the interpretation of a CN is not just a type, but rather a type associated with an IC for that specific CN. In other words, a CN is in general interpreted as a setoid, i.e. a pair

(5.13) (A, =), where A is a type and = : A → A → Prop is an equivalence relation over A.

The notion of setoids is not new in type theory and in constructive mathematics, and reflects the view that a type is basically a setoid, meaning that it is composed of a type plus an equivalence relation on this type. We want to pursue the same idea in linguistic semantics and assume that CNs are more nuanced with respect to ICs: two CNs may have the same base type, but their IC can be different and, if so, they are different CNs.

To see how this proposal works, let us first consider some simple examples for illustration purposes. In a simplified view, one would only interpret a CN as a type: for instance, “human” would be interpreted as a type Human, as in (5.14). However, in the elaborate view, CNs are interpreted as setoids, i.e. pairs of the form (5.13) and, therefore the CN “human” is now interpreted as in 5.15:

(5.14) [human] = Human : Type (CNs-as-types view)

(5.15) [human] = (Human, =_h), where =_h : Human → Human → Prop is the equivalence relation that represents the identity criterion for humans (CNs-as-setoids view)

Interpreting CNs as setoids makes explicit the individuation criteria (or their ICs) of CNs. This idea seems to be on the correct track in order to deal with copredication properly, i.e. in order to take care of the more fine-grained cases where there is interaction of copredication with numerical quantification. Indeed, this is what we are going to show, i.e. the dot-type as introduced in the previous section in conjunction with the CNs-as-setoids view, is adequate to deal with cases of copredication that also involve numerical quantification (see section 5.3.3). Let us now have a look at simple cases of individuation that do not involve copredication and the ICs of related CNs are essentially the same – inherited from supertypes.

5.4. Concluding remarks and related work

The introduction of dot-types in MTTs offers an adequate formal account of copredication in general. In combination with the formal treatment of CNs as setoids, dot-types provide a proper treatment of the more sophisticated cases in copredication where its interaction with quantification is involved. In particular, in this chapter we have shown that such notoriously problematic double distinctness cases are also correctly captured within our account, once CNs are interpreted as setoids, rather than just types.

In this section, we shall give a review of some of the related work on copredication and individuation, especially about identity criteria for CNs. As far as we know (see section 3.2.1), the origin of the notion of identity criteria can at least be traced back to Frege (1884) and it was Geach (1962) who first connected ICs with CNs in linguistic semantics. Subsequent studies include Gupta (1980), Baker (2003), Barker (2008) and Luo (2012a), among others. As mentioned in footnote 1 on p. 100, studying the issues of copredication with quantification in (Chatzikyriakidis and Luo 2015), we failed to properly acknowledge the necessity of considering ICs for these cases, although it was already recognized in Luo (2012a). The issue has been taken up again in (Chatzikyriakidis and Luo 2018) and expanded here, providing an account, which is capable of dealing with ICs under copredication with quantification. The same problem, under the name of individuation, was studied by Asher (2008, 2012) and Gotham (2017), among others.¹¹ In particular, Gotham (2017), starting from his PhD thesis (Gotham 2014), gives a first thorough account of the individuation problem discussed in section 5.3 in a mereological setting, as related to Link (1983) among others. In the following, besides giving a more detailed summary of Gotham’s work, we shall also discuss a recent very interesting paper by Liebesman and Magidor (2017) that studies copredication from a philosophical point of view.

Gotham (2014, 2017). The first account to deal in depth with the problems discussed in section 5.3 is given in (Gotham 2014, 2017). Gotham has managed to provide a compositional account of copredication that derives the correct ICs in the cases where double distinctness is needed. Gotham also provides a comprehensive critique and discussion of a number of the previous accounts, most importantly Asher (2012), Cooper (2011) and Chatzikyriakidis and Luo (2015), showing their inefficiency to derive the correct individuation criteria in quantified copredication cases. He then moves on to propose his account that tackles the issues correctly.

Gotham manages to provide a compositional account of co-predication that derives the correct ICs in the cases where double distinctness is needed. The main components of the account are as follows:

1) Both complex and plural objects exist

• – complex objects are denoted with the + operator while plural objects with the ⊕ operator (assumes a join semilattice ála Link (1983));
• – any property that holds of p holds of p + i, and likewise any property that holds of i also holds of p + i.

2) Lexical entries for CI and verbs are more complex than what is traditionally assumed in Montague Grammar. They do not only decide extensions, but they further specify a distinctness criterion.

3) Different objects have different distinctness criteria. Gotham calls these individuation relations. They are two-place predicates of the following form:

• a) PHY = λx, y : e.phys-equiv(x)(y)
• b) INFO = λx, y : e.equiv(x)(y)

4) A non-compressibility statement denoting that no two members of a plurality stand to a relation R. This basically is the way to ensure that members of the plurality do not bear any relation to each other (see Gotham (2014) for the formal definition of compressibility).

5) A further assumption that verbs also somehow point to the individuation criteria that have to be used.

6) Finally, operator, called Ω, is used that helps in choosing the individuation criteria for each of the argument.¹²

Putting all these assumptions together, Gotham manages to get the correct ICs for the different cases interested. But before we present you with an example, some notes are presented on the Ω operator. First of all, it must be already clear to the reader that the type of CNs according to Gotham departs from the classic conception of CNs as being predicates (or sets if you wish). The first proposal involves the type e → (t × R), where (t × R) is a product type with t a truth value and R the type of relations, basically an abbreviation for e → e → t. The second and final proposal takes CNs like “book” to involve the type e → (t × ((e → R) → t)), which Gotham abbreviates to e → T. The second projection p₂ is now a set of functions that map the type e argument, say x, to a relation of type e → R, where R R_cn, and where R_cn is taken to be the individuation relation given by the noun or the predicate. For example, the entry for book is now as follows (where ∗Pd means that the predicate P is a set of pluralities):

(5.65)

Assuming such an entry for a CN, the function Ω computes the least upper bound of the set of relations R. In the case of books, this is (PHYS INFO), for master it is (INFO), and so on.¹³ The end result of the semantics of the sentence “John picked up and mastered three books” is shown below:

(5.65)

This gives the correct ICs for the double distinctness cases. To put it in words: the formula talks of a plurality of three objects that are books, that are all picked up and mastered and that are neither informationally nor physically compressible (thus distinct). The second projection just gives the ICs for the arguments, PHYS INFO for the object and ANI for the subject. There are a number of similarities between Gotham’s account and the one proposed in this chapter. First of all, both of them make the claim that the definition of CNs has to be extended with a notion of ICs. For Gotham, this amounts to extending the traditional notion of predicates as sets to a considerably more complex type. For our account, the idea is that CNs are not just types anymore, but rather setoids, with their second component being an equivalence relation on the type, i.e. its IC. Gotham also assumes individuation relations. With regard deciding the ICs for verbs or adjectives, Gotham uses the Ω function. For our account, the ICs criteria for the arguments of the verb are decided by the type of the argument and whether it is a complex object or not. Furthermore, Gotham uses a mereological account to capture the individuation criteria, but makes an additional crucial assumption that, as far as we know, is not found in standard mereological studies. This is the introduction of the + operator and the introduction of the following axiom:

(5.65)

for any predicate P. This is crucial for the account to work and pretty much plays the role of the definition we have for setoids of dot-types in section 5.3.3. We can question its naturalness, as Gotham (2017) seems to be doing in footnote 2, where he says:

One can imagine other ways of implementing this mereological approach to copredication, both in terms of what composite objects there are and in terms of what their properties are. That in turn would require a revision to the definition of compressibility in Section 3.1.1. The approach adopted in this article is adequate for criteria of individuation to be determined compositionally, and can be adapted if revisions of this kind turn out to be necessary (Gotham 2017, footnote 2).

One of the interesting connections of Gotham’s work with ours concerns with double distinctness. In Gotham’s work, this is reflected in its basic assumption that (5.67) is true for every predicate, which can be found in the first paragraph of section 2 of Gotham (2017). Although the two formal frameworks are completely different, this is informally related to our notion of pre-setoids for semantic interpretations of dot-types. However, there seems to be a potential problem with the formalization in Gotham (2017): consider the following predicate P :

(5.68) P (p) = true, if p = a + x;

P (p) = false, otherwise.

Then, according to (5.67), P(a) is false and P(a + x) is true, and thus P does not satisfy (5.67). We do not know how serious of a flaw this is and whether a straightforward fixing can be obtained for an amendment.

Liebesman and Magidor (2017). Here, we want to discuss an interesting account of copredication from a more philosophical point of view, that of Liebesman and Magidor (2017). In this paper, the authors claim that copredication does not pose any problems for traditional referential semantics and no extra work needs to be done in formal semantics or in terms of metaphysics to account for them. According to Liebesman and Magidor (2017), “book” involves one sense that designates both informational and physical books, instead of two senses. One of the examples they use to prove this is as follows: there is a bookshelf that has two copies of the French translation of “Naming and Necessity”, and one copy of the Hebrew translation. Given this situation, consider the following sentences:

(5.69) One book is on the shelf.

(5.70) Two books are on the shelf.

(5.71) Three books are on the shelf.

They expand the true readings for each of the sentences:

(5.72) One book is on the shelf: Naming and Necessity.

(5.73) Two books are on the shelf: the Hebrew translation of Naming and Necessity and the French translation of Naming and Necessity.

(5.74) Three books are on the shelf: the one on the left, the one in the middle, and the one on the right.

Liebesman and Magidor (2017) argue that in the dot-type approach, we will have to stipulate at least three senses for “book”, one for physical books, one for informational books that different translations do not count as different informational objects and one where different translations matter. We do not have to posit three senses, but rather to allow the informational sense to have different individuation criteria in different contexts. For example, for a given speaker or a given situation the IC for the informational sense might not distinguish between translations of the same book, and for other speakers or given contexts it can. This is something that is afforded by the theory proposed here by making use of local coercions already discussed in section 3.2.2.

Consider another example by Liebesman and Magidor (2017):

(5.75) Each week, you will be assigned a single book, on which you will be required to submit a report: on odd weeks, your book will be a theoretical book on the history of book making, while on even weeks, our special collections librarian will assign you a volume from our historical collection to examine.

Liebesman and Magidor (2017) state with respect to that example:

“Book” ranges over both physical and informational books: the books the students are mastering on odd weeks are informational books, while the ones they are mastering on even weeks are physical books. (Liebesman and Magidor 2017, p. 136).

However, this is hardly a problem in the account we have presented so far. Volume is a dot-type. In the case of “assign”, its PHY sense is used, and in the case of “examine” its INFO sense is used. On this assumption, the problem vanishes. In general, the problem that is dubbed Univocity by Liebesman and Magidor (2017) does not seem to be a real problem, at least for our account.

Another problem that the authors identify with respect to accounts that posit different senses for elements like “book” is what they name inheritance. In the authors’ words:

In particular, the key challenge raised by copredication arises from the supposition that the properties ascribed in copredicational sentences cannot be instantiated by both. For instance, physical books can be heavy but not informative and informational books can be informative but not heavy. (Liebesman and Magidor 2017, p. 137).

The authors provide the following example as a counterexample to approaches like the approach described in this chapter. Imagine a case where there are three copies of “War and Peace” on the shelf. Even in that case, the following seems to have a true reading the authors claim:

(5.76) One book is on the top shelf.

Since what is quantified here is a single entity, the authors conclude that this single entity cannot be a physical, but rather an informational object. The authors take this to be an argument that accounts like the one put forth in this example are wrong. But there is nothing problematic with these examples. Let us take the elaborated versions of the examples, given by the authors to further prove their point:

(5.77) One book is on the top shelf: War and Peace, and it is 587,287 words long.

(5.78) (#) One book is on the top shelf: War and Peace, and it is worn out.

Some typing:¹⁴

Now, the interpretations (5.77) and (5.78) are:

(5.79) ∃b : Book. OT S(b) ∧ WaP = b ∧ XW L(b)

(5.80) ∃b : Book. OT S(b) ∧ WaP = b ∧ WO(b)

We strongly disagree with Liebesman and Magidor (2017) that (5.78) is an infelicitous example. It is a perfectly good example given the context, and should be captured in giving a formal semantics account. Consider the following:

(5.81) One book is on the top shelf: War and Peace, and it is worn out. All three copies.

This example shows that there is nothing wrong with 5.78 in the first place. In sum, the problems raised in Liebesman and Magidor (2017) do not seem to apply to our account. To the contrary, the account proposed in Liebesman and Magidor (2017) cannot deal with the double distinctness cases, as the authors themselves note. They claim that they are not sure that double distinctness actually exists. We are not going to get into this discussion. What we are going to say is that the accounts as given in the formal semantics literature have the double distinctness cases at their center and are evaluated as to whether they can capture these cases or not. Our account does not really suffer from the problems that Liebesman and Magidor (2017) identify, and furthermore, it also captures the double distinctness cases. For us, this is a gain compared to what Liebesman and Magidor (2017) propose.

REMARK.– While finalizing the book, two relevant papers came to our attention: one by Mery et al. (2018) and another by Ortega-Andrés and Vicente (2019). In Mery et al. (2018), the authors, as they say, attempt to solve the quantification in copredication problem. The idea is to connect Retore’s Montagovian Generative Lexicon Retoré (2013) with discourse representation theory. In essence, the authors consider the problem of counting to be a discourse-related problem. The example they concentrate on is as follows:

(5.82) Oliver stole the books and read them.

In the above example, “the books” are individuated physically in the first instance and informationally in the second instance. The authors present a partial implementation of this sentence, going from the actual utterance (parsing) to logical form. What the authors do not seem to notice is that the counting problem is not an anaphoric problem, or it may not need be so. The most important issue in individuation in copredication cases, and the most difficult to tackle, concerns cases where a numerical quantifier is involved (like “three”), i.e. the double distinctness cases we have been discussing. These cases are not brought up by the authors, and as far as we can see, it is not clear how these cases are going to be handled with the system Mery et al. (2018) propose.

Ortega-Andrés and Vicente (2019) is a different line of work that concentrates on the question of why some groups of senses (the components of the dot-type in our case) allow for copredication and why some others do not. They present interesting cross-linguistic data that can be further elaborated in fully understanding the nature of copredication and explain why some cases like “newspaper” present restrictions of the type we have discussed. The argument is that the legit copredication cases are possible because the senses in question are accessible by the time the speaker/hearer has to retrieve them. They present and discuss interesting data from the psycholinguistics literature to support this claim. This discussion, albeit extremely interesting, has no effect on our account whatsoever (even though a complete account of copredication needs to be informed from these results). Our account is agnostic to the discussion, in the sense that it operates as soon as the legit and illicit copredication senses are decided. Simply put, formal semantics accounts like the account proposed here or any formal semantics account proposed in the literature do not have anything to say on why some senses can appear in copredication and some others do not. □