Semantics I, Rutgers University Week 7 Yimei Xiang October 15, 2018 eterminers and generalized quantifiers 1. The A -phrase denotes an referential individual of type e. The definite determiner is of type et, e : it combines with a / of type to return an referential individual of type e. (1) a. w = λp e,t.ιx e [P(x)] (The function that applies to a predicate P and returns unique entity x s.t. P(x) holds) b. cat w = ιx e [cat w (x)] (The unique entity x such that x is a cat) c. The cat snores. S t snore w (ιx e [cat w (x)]) P e ιx e [cat w (x)] λx e.snore w (x) et, t λp e,t.ιx e [P(x)] λx e.cat w (x) λx e.cat(x) cat λx e.snore w (x) snores The definite determiner presupposes uniqueness. (2) a. [Pointing at one cat], cat snores. b. [Pointing at two cats], # cat snores. This presupposition is modeled as introduced by presupposition of ι-operator: (3) ιx e [P(x)] is defined iff re exists exactly one x such that P(x) = 1. Formally: a. = λp e,t : 1!x[P(x)].ιx e [P(x)] b. = λp e,t : x[p(x) y[p(y) y x]].ιx e [P(x) y[p(y) y x]] NB: efinition (b) is more preferable since it also extends to plural definite descriptions like cats. 1
Adding relative clauses () iscussion: Which of following (simplified) trees correctly describes structure of definite description? [In or words, does relative clause who invited Andy modify or?] Why? (4) a. P b. P P Exercise: Compose following sentence: (5) The left. S t left w (ιx e [ w (x) invite w (x, a)]) P e ιx e [ w (x) invite w (x, a)] λx e.left w (x) et, t λp e,t.ιx[p(x)] λx e [ w (x) invite w (x, a)] λx e. w (x) λx e.invite w (x, a) λx e.left w (x) left λx e. w (x) 2
2. Generalized quantifiers and quantificational determiners Recall: We define quantificational determiners like some and every as relations between two sets of entities. (6) a. Some cat meows. b. Every cat meows. c. No cat meows. 2.1. Generalized quantifiers Quantificational Ps (e.g. everything, something, nothing, every cat, some cat, no cat) are not individuals (of type e): (cf. proper names like John, definite Ps like cat), nor individual sets (of type ) (cf. common nouns like cat). They are not individuals. Compare with e-type s: (7) Law of Contradiction a. Mary is coming and Mary is not coming. (Contradiction) b. Someone is coming and someone is not coming. (Not contradiction) (8) Law of Excluded middle a. Mary is coming or Mary is not coming. (Tautology) b. Every is coming or everyone is not coming. (Not tautology) (9) Only an e-type can normally license a singular discourse pronoun. a. John / man/ a man walked in. He looked tired. b. Every man /no man/ more than one man walked in. *He looked tired. They are also not sets/predicates. * * N-words: It is hard to think of nobody as a set. The best thing we can do is to treat it as an empty set. But n nobody and no linguist would be semantically equivalent, contra fact. Numeral modified quantifiers: It s unclear what set at least one question and at most three questions refer to. iscussion: Can you think of more differences between generalized quantifiers and e-type s or common nouns? 3
We treat quantificational Ps as second-order functions of type et, t, called generalized quantifiers. In (10), moews is an argument of every cat. (10) Every cat meows. S t P et,t e,t every meows e,t cat e,t (11) a. every cat w = λp e,t. x[cat w (x) P(x)] b. every cat meows w = every cat w ( meows w ) = (λp e,t. x[cat w (x) P(x)])(λy e.meows w (y)) = x[cat w (x) meows w (x)]) (12) a. some cat w = λp e,t. x[cat w (x) P(x)] b. some cat meows w = x[cat w (x) meows w (x)] (13) a. no cat w = λp e,t. x[cat w (x) P(x)] b. no cat meows w = x[cat w (x) meows w (x)] 2.2. Type-shifters Individuals (of type e) can also be shifted into generalized quantifiers via type-lifting. (14) LIFT = λa α λp α,t.p(a) (15) a. Kitty w = k b. LIFT( Kitty w ) = λp e,t.p(k) c. (LIFT( Kitty w ))( meows w ) = (λp e,t.p(k))(λx e.meows w (x)) = (λx e.meows w (x))(k) = meows w (k) We can extract quantification domain of an -quantifier via BE-shifter (Partee 1986): (16) BE = λpλz[p(λy.y = z)] (17) BE( some cat w ) = λz[(λ f e,t. x[cat w (x) f (x)])(λy.y = z)] = λz. x[cat w (x) x = z] = {z cat w (z)} iscussion: (i) What do we get by applying BE to every cat w and LIFT( John w )? (ii) What do we get by applying LIFT to every cat w and LIFT( John w )? 4
2.3. Quantificational determiners The determiner every combines with a common noun of type to return a generalized quantifier of type et, t. Therefore, its type is quite complex: et, et, t. (18) a. every w = λq e,t λp e,t. x[q(x) P(x)] b. some w = λq e,t λp e,t. x[q(x) P(x)] c. no w = λq e,t λp e,t. x[q(x) P(x)] A quantificational determiner takes two arguments (both of which are of type ). The first argument is its restrictor, and second argument is its scope. (19) S every P (RESTRICTOR) cat (SCOPE) meows iscussion: Identify restrictor and scope of every in following sentences. (20) a. Every student who read chapter 5 passed exam. b. Everyone passed exam. c. John read every chapter. 5