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PHIL12A Section answers, 7 February 2011 Julian Jonker 1 How much do you know? 1. (Ex 2.22) Is the following argument valid? Sound? If it is valid, give an informal proof of it. If it is not, give an informal counterexample to it. All computer scientists are rich. Anyone who knows how to program a computer is a computer scientist. Bill Gates is rich. Therefore, Bill Gates knows how to program a computer. This is invalid, and therefore we know immediately that it cannot be sound. (Why?) Here is a counterexample showing that it is invalid: suppose the world contains exactly four people. All of them are rich. Exactly three of these people know how to program a computer, and they are all computer scientists. The other person is Bill Gates, who does not know how to program a computer and is not a computer scientist. Now the following sentences are true: All computer scientists are rich. Anyone who knows how to program a computer is a computer scientist. Bill Gates is rich. However, it is not true that Bill Gates knows how to program a computer. In other words, we have shown that there is a possible world in which all the premises are true and the conclusion is false. If this was not obvious to you, take a close look at the ﬁrst premise, which has the form ‘All As are Bs.’ This means that if someone is an A, then they are a B. It does not mean that if someone is a B, they are also an A. So all computer scientists are rich does not mean that all rich people are computer scientists. 2. Write formal proofs for the following, as you would using the program Fitch . You can use the following rules: =Intro , =Elim , Reit , Ana Con . Make sure you cite sentences to justify your use of the rule in each case. 1

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(a) (Ex 2.24) 1 Larger(b,c) 2 Smaller(b,d) 3 SameSize(d,e) 4 Larger(e,c) We need to use Ana Con here a bunch of times, since there are no premises using the identity relation. Make sure you understand why each inference using Ana Con holds, and that you can explain in words why it holds. Make especially sure you can see why I have cited the lines I have for each use of Ana Con . Here is a possible proof: 1 Larger(b,c) 2 Smaller(b,d) 3 SameSize(d,e) 4 Larger(d,b) Ana Con: 2 5 Larger(d,c) Ana Con: 4,1 6 Larger(e,c) Ana Con: 5,3 (b) (Ex 2.27) 1 SameRow(b,c) 2 SameRow(a,d) 3 SameRow(d,f) 4 FrontOf(a,b) 5 FrontOf(f,c) Here is a possible proof. Again, make sure you understand why I have cited the lines I have for each use of Ana Con . 2
1 SameRow(b,c) 2 SameRow(a,d) 3 SameRow(d,f) 4 FrontOf(a,b) 5 SameRow(a,f) Ana Con: 2,3 6 FrontOf(f,b) Ana Con: 4,5 7 FrontOf(f,c) Ana Con: 6,1 2 Something slightly harder, if there’s time. 1.

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