1. Lecture 7 Natural Language Determiners Ling 442

2. exercises 1. (a) is ambiguous. Explain the two interpretations. (a)Bill might have been killed. 2. Do you think (b) can receive a deontic reading? Regardless of your answer, provide a context that is/would be appropriate for the (alleged) deontic meaning of (b). (b) Mary must have been elected president.

3. Exercises 3. Translate each English sentence into Predicate Logic. (c) Everyone who has a ticket is welcome. (d) Mary met every candidate. (e) Sue received no letters (today).

4. most Suppose that you wish to extend Predicate Logic in such a way that an (unrestricted) quantifier MOST is introduced into it: most x [ … x …] = true iff most objects  in the universe is such that … x … = true when x is understood to denote . With this semantics of most, one can never obtain the right translation of sentences like (1). (2a, b) are both wrong translations. So what should we do to correct the problem? (1)Most birds fly. (2)a.most x [bird (x)  fly (x)] b.most x [bird (x) & fly (x)]

5. What about every, some, and no? Essentially, we were lucky that we could cover these determiners using the syntax/semantics of predicate logic. English and Predicate Logic are very different languages syntactically, as well as semantically.

6. Determiner Quantifiers Generally, determiners denote relations between sets. This is Frege’s observation. Kearns calls this “Generalized Quantifier Theory”, but it is not a standard usage. The term “generalized quantifier” is usually used to refer to the denotation of a DP (a set of sets of individuals) in a more compositional theory.

7. Examples Every A B = true iff A  B Some A B = true iff A  B ≠  No A B = true iff A  B =  Most A B = true iff |A  B| > ½ |A| The semantics of most cannot be captured by any straightforward extension of Predicate Logic.

8. Different types of determiners Asymmetric (what Kearns calls Group 1) — proportional, strong (Milsark’s term) most, all, (proportional) few Symmetric (what Kearns calls Group 2) — cardinal, weak (Milsark’s term) no, a(n), some, four, (cardinal) many, several, (cardinal) few, a few

9. restricted quantifier notation We need a new notational system (logical language) in order to translate English sentences. (1) translates into (2) in this notation. (1)Most birds fly. (2)[most x: bird(x)] fly (x) Syntax: [Det x: CN (x)] VP (x) Semantics: DET x such that x is a CN VPs

10. Restricted quantifier notation All other determiner quantifiers are translated in a similar fashion. This enables us to indicate scope ambiguity as well. (1)Every girl likes a boy. (2)[every x: girl (x)][a y: boy (y)] likes (x, y) (3)[a y: boy (y)][every x: girl (x)] likes (x, y)

11. Relative clauses (1)Every student who wrote a paper was happy. (2)[every x: student (x) & [[a y: paper (y)] wrote (x, y)]] happy (x) Structure: every: restriction student (x) & [a y: paper (y)] wrote (x, y) scope happy (x) a: restriction paper (y) scope wrote (x, y)

12. Compositionality To express the semantics of determiners in terms of relations “between” (asymmetric between) sets of individuals is accurate except that it is not compositional. A better account would allow a DP to have a denotation. This is what is normally referred to as Generalized Quantifier Theory. In this theory, a DP denotes a set of sets of individuals.