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LIN241 Intro to Semantics Winter 2019 Tutorial 6 Answers Learning objectives Review the semantics of Generalized Quantifiers Use Generalized Quantifiers to think about monotonicity Exercise 1 For each sentence: 1. translate the sentence in Generalized Quantifier Logic 2. give the truth-conditions of the translation with respect to a model M and assignment g 3. draw a Venn Diagram that illustrates scenarios in which they are true. (1) Some students are happy. SOMEX[STUDENT(x)][HAPPY(x)] ⟦SOMESOMEX[STUDENT(x)][HAPPY(x)] ⟧M,g = 1 iff {x: x ∊ I(STUDENT )} ⋂ {x: x ∊ I(HAPPY)} ≠ ∅ or: ⟦SOMESOMEX[STUDENT(x)][HAPPY(x)] ⟧M,g = 1 iff |{x: x ∊ I(STUDENT )} ⋂ {x: x ∊ I(HAPPY)}| ≠ 0 Note: this analysis assumes that the inference that you draw from (1) that there are more than one happy students is an implicature. Evidence: in the following discourse, the implicature is cancelled: A: Are some students happy? B: Yes, Jenny is. (2) Most students are happy. MOSTX[STUDENT(x)][HAPPY(x)] ⟦SOMEMOSTX[STUDENT(x)][HAPPY(x)] ⟧M,g = 1 iff |{x: x ∊ I(STUDENT )} ⋂ {x: x ∊ I(HAPPY)}| / |{x: x ∊ I(STUDENT )}| > 1/2 or: ⟦SOMEMOSTX[STUDENT(x)][HAPPY(x)] ⟧M,g = 1 iff |{x: x ∊ I(STUDENT )} ⋂ {x: x ∊ I(HAPPY)}| > |{x: x ∊ I(STUDENT )}|/2 (3) Every student is happy. EVERYX[STUDENT(x)][HAPPY(x)] ⟦SOMEEVERYX[STUDENT(x)][HAPPY(x)] ⟧M,g = 1 iff {x: x ∊ I(STUDENT )} ⊆ {x: x ∊ I(HAPPY)} (4) Less than 36 students watched Creed. LESS-THAN-36X[STUDENT(x)][WATCHED(x,c)] ⟦SOMELESS-THAN-36X[STUDENT(x)][WATCHED(x,c)] ⟧M,g = 1 iff |{x: x ∊ I(STUDENT )} ⋂ {x: <x,I(c)> ∊ I(WATCHED)}| < 36 Exercise 2 Translate the following sentences in GQL and give truth-conditions of the translation with respect to a model M and assignment g (5) A student read every book. [direct scope] SOMEX[STUDENT(x)][EVERYY[BOOK(y)][READ(x,y)]] ⟦SOMESOMEX[STUDENT(x)][EVERYY[BOOK(y)][READ(x,y)]]⟧M,g = 1 iff {x: x ∊ I(STUDENT )} ⋂ {x: {y: y ∊ I(BOOK)} ⊆ {y: <x,y> ∊ I(READ)}} ≠ ∅ hint: it helps to break it down in two steps: {x: x ∊ I(STUDENT )} ⋂ {x: …. } ≠ ∅ {y: y ∊ I(BOOK)} ⊆ {y: <x,y> ∊ I(READ)} (6) A student read every book. [inverse scope] EVERYY[BOOK(y)][SOMEX[STUDENT(x)][READ(x,y)]] ⟦SOMEEVERYY[BOOK(y)][SOMEX[STUDENT(x)][READ(x,y)]]⟧M,g = 1 iff {y: y ∊ I(BOOK)} ⊆ {y: {x: x ∊ I(STUDENT )} ⋂ {x: <x,y> ∊ I(READ)} ≠ ∅ } breaking it down in two steps: {y: y ∊ I(BOOK)} ⊆ {y: … } {x: x ∊ I(STUDENT )} ⋂ {x: <x,y> ∊ I(READ)} ≠ ∅ Exercise 3 Classify the following quantifiers as left or right DE or UE. Justify your answer by providing entailments that illustrate the relevant properties. (1) No NP VP Left-DE: No student is happy ⊨ No linguistics student is happy. Right-DE: No student smokes ⊨ No student smokes cigars. (2) Most NP VP Neither left DE not left UE: Most students are happy ⊭ Most linguistics students are happy. Most linguistics students are happy ⊭ Most students are happy. Right-UE : Most students smoke cigars ⊨ Most students smoke. (3) Few NP VP Left-DE: Few students are happy ⊨ Few linguistic students are happy. Right-DE: Few students smoke ⊨ Few students smoke cigars. Note: we are assuming a weak ‘cardinal’ interpretation of ‘few’ (4) A few NP VP Left-UE: A few linguistics students are happy ⊨ A few students are happy. Right-UE: A few students smoke cigars ⊨ A few students smoke.