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This document from ling 106 explores entailment relationships between sentences and monotonicity patterns of sentential operators. It includes exercises to test understanding of entailment and monotonicity concepts. The document also touches upon the difference between extensional and intensional semantics.
Typology: Exercises
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LING 106. Knowledge of Meaning Lecture 5- Yimei Xiang Feb 22, 2017
Exercise: Based on the semantics of propositional logic and predicate logic, identify whether each of the following claim is right or wrong.
(1) a. For any two propositions p and q: i. p ^ q entails p; ii. p ^ q entails p _ q; iii. p _ q entails pp ^ qq; iv. p entails p Ñ q.
b. For any two sets A and B: i. a P A entails a P A Y B; ii. a P A does not entail a P A X B. iii. @xrx P A Ñ x P Bs entails Dxrx P A ^ x P Bs. (^1) We are now working on extensional semantics, which interprets a sentence as truth value. Once you work on intensional semantics, which interprets a sentence as a set of possible worlds, be aware that some of the concepts shall be defined differently.
(2) φ entails ψ if and only if “φ but not ψ” is intuitively contradictory.
For example:
(3) a. John and Mary left, but Suzi didn’t leave. (Not contradictory. Hence “John and Mary left” œ “Suzi left”.) b. # John and Mary left, but John didn’t leave. (Contradictory. Hence “John and Mary left” ñ “John left”.)
Nevertheless, this test doesn’t always work:
(4) # John or Mary left, but both John and Mary left. (Sounds odd. But “John or Mary left” œ “John and Mary didn’t both leave”.)
Why this test fails? How do you think?
(5) a. Q: Did John and Mary leave? A: Yes. # Actually, John didn’t leave. b. Q: Did John or Mary leave? A: Yes. Actually, they both left.
Exercise: Use contradiction tests to identify whether each (a) sentence entails the paired (b) sentence.
(6) a. All of the students left. b. Some of the students left. (7) a. Some of the students left. b. Not all of the students left. (8) a. John’s daughter will come. b. John has a daughter.
(14) a. For any individual x, Chinese-studentpxq ñ studentpxq. Therefore, Chinese student ñ student. b. For any individual x, semanticistpxq ñ linguistpxq. Therefore, semanticist ñ linguist. c. ...
(16) Restriction of a quantifier: a. Some semanticist arrived ñ Some linguist arrived. some ( ) arrived is UE b. Every semanticist arrived ð Every linguist arrived. some ( ) arrived is DE c. No semanticist arrived ð No linguist arrived. no ( ) arrived is DE Exercise: Identify the monotonicity pattern of the following quantifiers: (17) a. exactly three students b. not every student c. Every participant who got an award
Exercise: For each of the following claims, identify whether it is right or wrong. (18) a. Negation is a DE operator. b. Every is a DE operator. c. Conditionals create DE environments. d. Any environment containing negation is DE.