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# section_4.11_answers - PHIL12A Section answers 11 April...

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PHIL12A Section answers, 11 April 2011 Julian Jonker 1 How much do you know? 1. Translate the following pairs of sentences, and show by a chain of equivalences that they are equivalent: (a) i. It is not the case that all P’s are Q’s. ¬∀ x(P(x) Q(x)) ii. Some P’s are not Q’s. x(P(x) ∧ ¬ Q(x)) Proof of equivalence: ¬∀ x(P(x) Q(x)) ¬∀ x ¬ (P(x) ∧ ¬ Q(x)) equivalence for x ¬¬ (P(x) ∧ ¬ Q(x)) De Morgan x(P(x) ∧ ¬ Q(x)) Double negation (b) i. It is not the case that some P’s are Q’s. ¬∃ x(P(x) Q(x)) ii. No P’s are Q’s. x(P(x) → ¬ Q(x)) Proof of equivalence: ¬∃ x(P(x) Q(x)) x ¬ (P(x) Q(x)) De Morgan x( ¬ P(x) ∨ ¬ Q(x)) De Morgan x(P(x) → ¬ Q(x)) equivalence for 2. Which of the following sentences are logical truths? For each sentence, you should be able to explain why it is a logical truth, or produce a counterexample that shows that it is not. (a) (Ex 10.24) ( xCube(x) ∨ ∀ xDodec(x)) ↔ ∀ x(Cube(x) Dodec(x)) The direction of implication doesn’t hold. Suppose half of the objects in the world are cubes and half

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section_4.11_answers - PHIL12A Section answers 11 April...

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