Individuation Criteria, Dot-types and Copredication: A View from Modern Type Theories

In this paper we revisit the issue of copredication from the perspective of modern type theories. Specifically, we look at: a) the counting properties of dot-types, and b) the case of a complex dot-type that has remained unsolved in the literature, i.e. that of newspaper. As regards a), we show that the account proposed in (Luo, 2010) for dot-types makes the correct predictions as regards counting. In order to verify this, we implement the account in the Coq proof-assistant and check that the desired inferences follow. Then, we look at the case of b), the case of a dot-type which is both resource and context sensitive. We propose a further resource sensitive version of the dottype, in effect a linear dot-type. This along with local coercions can account for the behaviour attested. 1 Copredication: Dot Types and Individuation Criteria One of the issues that should be taken care of when giving an account of co-predication, concerns cases of coordination like the one shown below: (1) John picked up and mastered three books In the above sentence, the CN book is used in its physical sense (PHY) with respect to the predicate ∗ This work is partially supported by the research grants from Leverhulme, Royal Academy of Engineering, the CAS/SAFEA International Partnership Program for Creative Research Teams as well as the ANR ContInt Polymnie project in France. picked up, while for the predicate mastered it is rather used in its informational content sense (INFO). A theory of co-predication should be able to take care of these facts. This is true for the account by means of the dot-types proposed by (Luo, 2010; Luo, 2012b). However, besides capturing this behaviour of dot objects, there is an additional property that has to be captured. The account provided must also make the correct predictions as regards individuation and counting. Let us explain. Consider the following sentences: (2) John picked up three books (3) John mastered three books (4) John picked up and mastered three books The first example (2) is true in case John picked three distinct physical objects. Thus, it is compatible with a situation where John picked up three copies of the same book. (3) is true in case three distinct informational objects are mastered but does not impose any restrictions on whether these three informational objects should be different physical objects or not. To the contrary, (4) is only compatible with an interpretation where three distinct physical objects as well as three distinct informational objects is involved.1 Another issue pertaining to dot types concerns cases of what Retoré (2014) calls rigid and flexible coercions in co-predication cases. These cases in contrast to cases like Book where both senses can This is basically an issue of how complex objects, i.e. dottypes, are individuated and stems from the work of (Asher, 2008; Asher, 2011). be coordinated, involve examples where if one of the senses is used the other one cannot be used anymore: (5) Liverpool is spread out and voted (last Sunday). (6) # Liverpool voted and won (last Sunday). Perhaps a better example for such cases is Pustejovsky’s newspaper examples. The CN newspaper is associated with three senses: a) physical object, b) informational object and c) institution. It is a strange fact that whereas senses a) and b) can appear together in a coordinated structure, sense c) cannot appear with any of the other two (examples taken from (Antunes and Chaves, 2003)): (7) # That newspaper is owned by a trust and is covered with coffee. (8) # The newspaper fired the reporter and fell off the table. (9) # John sued and ripped the newspaper. Pustejovsky’s proposal (Pustejovsky, 1995) to treat newspaper as a composite dot object does not explain the above facts. Likewise, the proposal of using (ordinary) dot-types in (Luo, 2010) has a similar problem: one would consider newspaper to be a subtype of the dot-type INST • (PHY • INFO), which would not disallow the above bad examples. The picture gets complicated in the light of examples like the following, in which it seems that the institutional sense can be used together with one of the two other senses in some cases: (10) The newspaper you are reading is being sued by Mia. As far as we know, no satisfactory account has been provided to these questions yet. In this paper, following earlier work on dot-types in MTTs (Luo, 2010; Luo, 2012b; Xue and Luo, 2012) and coordination (Chatzikyriakidis and Luo, 2012), we take up the challenge of providing an account that correctly predicts the individuation criteria in cases of co-predication while it furthermore provides a first look at capturing the behaviour of problematic cases like newspaper. 2 Formal Semantics in Modern Type Theories: a Brief Introduction The term Modern Type Theories (MTTs) refers to type theories studied and developed within the tradition of Martin-Löf, which include predicative type theories such as Martin-Löf’s type theory (Nordström et al., 1990) and impredicative type theories such as CICp as implemented in Coq (The Coq team, 2007) and UTT (Luo, 1994). Formal semantics in Modern Type Theories (MTT-semantics for short) was first studied by Ranta in his pioneering work (Ranta, 1994).2 It has been further developed in the last several years, including the crucial employment of the theory of coercive subtyping (Luo, 1999; Luo, Soloviev and Xue, 2012) among other developments and made MTT-semantics a viable and full-blown alternative to the traditional Montagovian framework. In this paper, we use one such modern type theory, UTT with Coercive Subtyping (Luo, 1994; Luo, 1999), whose application to linguistic semantics was first discussed in (Luo, 2010). Two features of MTTs are worth being mentioned, both important for being a foundational language for linguistic semantics. The first is that an MTT has a consistent internal logic according to the propositions-as-types principle (Curry and Feys, 1958; Howard, 1980).3 For instance, the higherorder logic is embedded in UTT and it is essentially used when we employ UTT for linguistic semantics (just like how higher-order logic is used in Montague’s semantics.) The second feature of MTTs is that it has rich type structures, which have been recognised by many researchers as very useful in formal semantics. In this Potentially, even further back, with the work of Sundholm (Sundholm, 1986; Sundholm, 1989), but Ranta (Ranta, 1994) was the first systematic study of formal semantics in a modern type theory. Having such an internal logic is a basic requirement for a type theory to be employed for linguistic semantics and we need to be careful to keep the internal logic to be consistent when trying to extend an existing type theory to do linguistic semantics, for otherwise, we could be in a muddle situation if the basic requirement is violated. For instance, the framework of Type Theory with Records (TTR) (Cooper, 2011) is based on set theory and, as a consequence, TTR does not have such an internal logic based on the propositions-as-types principle (to see this, it suffices to note that TTR’s a : A is just the settheoretical membership relation a ∈ A and undecidable). section, we shall briefly discuss some of these distinctive features of MTTs, specifically the ones most relevant to this paper. 2.1 Type Many-sortedness and CNs as Types The domain of individuals in MTTs is multi-sorted and not single-sorted as in Church’s simple type theory (Church, 1940). Instead of using one coarsegrained domain of entities, like it is done in the Montague Semantics (MS) (Montague, 1974), MTTs contain many types that allow one to make finegrained distinctions between individuals and further use those different types to interpret subclasses of individuals. For example, one can find John : [[man]] and Mary : [[woman]], where [[man]] and [[woman]] are different types. A further difference closely related to type manysortedness concerns the interpretation of CNs. In MS, CNs are interpreted as predicates of type e → t, whereas in MTTs CNs are interpreted as types. Thus, in MTTs, CNs man, human, table and book are interpreted as types [[man]], [[human]], [[table]] and [[book]], respectively. (Such types may be defined by means of type constructors such as Σ etc – see below.) Then, individuals are interpreted as being of one of the types used to interpret CNs. Such interpretations of CNs as types would not work without a proper subtyping mechanism that extends MTTs – coercive subtyping provides us with such a framework.4