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PHIL12A Section answers, 9 February 2011 Julian Jonker 1 How much do you know? 1. I have constructed a world in Tarski’s World using objects named a through f , but I’m not going to show it to you. Now consider the sentences below, and decide whether you can determine their truth value even though you can’t see my world. You should be able to explain why or why not. (a) d=d We know that we can use the =Intro to introduce an identity at any point. Of course there must exist such an object in the world, but I have told you that objects a through f exist. So d=d must be true. (b) BackOf(b,a) BackOf(a,b) You cannot determine the truth value of this sentence without looking at the world. After all, if a and b are in the same row, then neither BackOf(b,a) nor BackOf(a,b) will be true. (c) LeftOf(e,f) RightOf(f,e) LeftOf(e,f) and RightOf(f,e) are equivalent, and if e and f are in the same column, then these will not be true. (d) b=a a 6 = b b=a is the same as a=b , and a 6 = b is its negation, so this sentence says that either a and b are identical objects, or they are not. One of the must be true, so the whole sentence is true, regardless of the world. (e) Large(a) ∨¬ Large(a) Either a is large, or it is not. So this sentence must be true, regardless of the world. 2. (Based on Ex 3.14) Is ¬ (Small(a) Small(b)) a logical consequence of ¬ Small(a) Small(b) ? If it is, write an informal proof. If it is not, describe a counterexample. The following counterexample shows that ¬ (Small(a) Small(b)) is not a logical consequence of ¬ Small(a) Small(b) Let a be a large object, and b be a small object. Then a is not small, so ¬ Small(a) is true and therefore ¬ Small(a) Small(b) is true. 1

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However, ¬ (Small(a) Small(b)) is false, since (Small(a) Small(b)) is true, since Small(b) is true. 3. Is ¬ (Small(a) Small(b)) a logical consequence of ¬ Small(a) ∧¬ Small(b) ? If it is, write an informal proof. If it is not, describe a counterexample. Yes. Suppose ¬ Small(a) ∧¬ Small(b) is true. Then both conjuncts must be true, that is ¬ Small(a) is true and ¬ Small(b) is true. But then Small(a) is false and so is Small(b) . This just means that a is either medium or large, and b is either medium or large.
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