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```LIN241 Intro to Semantics
Winter 2019
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)
[direct scope]
{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)
[inverse scope]
{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.
```