Taking Scope: The Natural Semantics of Quantifiers

Chapter 7 Quantification and Pronominal Anaphora

Chapter 7 Quantification and Pronominal Anaphora A minimum of two other individuals at this meeting reviewed each question. —The web Type raising, which we saw in the last chapter in its morphosyntactic aspect as a rule of the CCG lexicon, is also well known as the operation that Montague (1973) used in semantics to treat quantification in natural language and capture the phenomena of scope illustrated in example (1) of chapter 2, Somebody loves everybody, with which we began discussing quantifier-scope. It is standard in this tradition to translate expressions like “every farmer” and “some donkey” into “generalized quantifiers ” 7.1 Generalized Quantifiers In terms of the notation and assumptions of CCG, the natural way to incorporate generalized quantifiers into the semantics of CG determiners, given the assumption that type raising is an operation of the lexicon, is to assign the following category schema to determiners like every. (1) every, each := NP↑ 3SG/ ⋄N3SG : λpλqλ ...∀x[px → qx...] The category makes them functions from nouns to type-raised nounphrases, schematizing over them using the NP↑ abbreviation, where the schematized types are simply the syntactic types corresponding to a (polymorphic )generalized quantifier and . . . stands for any further arguments of the predicate q:1 Under the definition of environments developed below, the two applications of this category to a noun property p and a predicate q will contribute the newly quantified variable x to their environments. 1. The use of polymorphic generalized quantifiers avoids the complication of further raising the type of transitive verbs over monomorphic GQs, as in Montague 1973. The ⋄ modality is required on all English determiners to prevent crossed composition into them analogous to that permitted for verbs (see note 11 in the previous chapter and Baldridge 2002). . 110 Chapter 7 Category (1) schematizes over a number of different raised types, via the expression NP↑, which ranges over the (in English, finite) set of all lexical and derived function category types over NP. The dots “...” in the schematized logical form again represent the fact that q may bind more variables than x, and that these variables get passed in to q (under a wrapping ordering convention discussed in chapter 6 and inSS&I) (cf. Partee and Rooth 1983). Thus, (1) schematizes over the following categories, among others:2 (2) a. every, each := (S/(S\NP3SG))/ ⋄N3SG : λpλq.∀x[px → qx] b. every, each := (S\(S/NP3SG))/ ⋄N3SG : λpλq.∀x[px → qx] c. every, each := ((S\NP)\((S\NP)/NP3SG))/ ⋄N3SG : λpλq.λy.∀x[px → qxy] d. every, each := (((S\NP)/NP)\(((S\NP)/NP)/NP3SG))/ ⋄N3SG : λpλq.λyλz.∀x[px → qxyz] e. etc. In what follows, for example in derivations like (17) and (18) below, the interpretations will usually be spelled out as the relevant specific instance. On occasion, where the instance is obvious, the syntactic type of raised NPs will be abbreviated as NP↑ to save space and avoid visual clutter in derivations. There is a strong relation between the semantic interpretation of the universal determiners schematized in (1) and its instances (2) and the linguists’ notion of “covert quantifier movement to specifier of CP” (Steedman 2005, 2006). Both give the universal quantifier scope over the entire clause. The difference is that categorial grammars of all kinds achieve this effect statically, via lexicalization. As in Montague Grammar, such categories distribute correctly over conjunction and disjunction (see example (4) in chapter 2). Thus: (3) a. Every man walks and talks. b. ∀x[man′x → (walks′x∧talks′x)] (4) a. Every man walks or talks. b. ∀x[man′x → (walks′x∨talks′x)] 2. Polarity is ignored in this chapter, although in chapter 11 it will become apparent that the restrictor N (though not the corresponding antecedent in the logical form) is marked for polarity N−. Quantification and Pronominal Anaphora 111 7.2 Skolem Terms In contrast to the universal determiners, the interpretations assigned to the inde finite determiners are not generalized quantifiers at all. Rather, they are determiners of Skolem terms that differ from the generalized Skolem terms we have seen so far in being initially unspecified as to their bound variables, if any.3 Thus, the underspecified translation of a donkey in the standard donkey sentence (1) of chapter 4 is written skolem′ ndonkey′. (The subscript n is again a number unique...