sole1 - ANALYSIS EXERCISE 1SOLUTIONS 1(a Let B The main...

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ANALYSIS EXERCISE 1–SOLUTIONS 1. (a) Let B The main course will be beef. F The main course will be fish. P The vegetable will be peas. C The vegetable will be corn. The logical form of the argument is B F P C ¬ ( F C ) ¬ ( B P ) The premises are B F , P C , ¬ ( F C ); the conclusion is ¬ ( B P ). The argument is invalid. If B P is true, then the premises are true, but the conclusion is false. It might help to consider the possibilities ( B P , etc.). (b) Let S Sales will go up. H The boss will be happy. E Expenses will go up. The logical form of the argument is ( S H ) ( E ∧ ¬ H ) ¬ ( S E ) The premise is ( S H ) ( E ∧ ¬ H ); the conclusion is ¬ ( S E ). The argument is invalid. Suppose S E H . Then the premise is valid and the conclusion false. 2. (a) P Q P + Q T T F T F T F T T F F F (b) ( P ∧ ¬ Q ) ( ¬ P Q ). 3. ¬ ( ¬ P Q ), for example. 4. (b), (c), (d). 5. (a) A B C ¬ C B ∨ ¬ C A ( B ∨ ¬ C ) T T T F T T T T F T T T T F T F F F T F F T T T F T T F T F F T F T T F F F T F F F F F F T T F 1
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2 ANALYSIS EXERCISE 1–SOLUTIONS (b) A B ¬ A ¬ B A B ¬ A ∨ ¬ B ( A B ) ( ¬ A ∨ ¬ B ) T T F F T F F T F F T T T T F T T F T T T F F T T F T F 6. (a) [( A B ) ( B C )] ( A C ) . Let us prove that this is a law by constructing its truth table. (Denote by D the above propositional form.) A B C A B B C ( A B ) ( B C ) A C D T T T T T T T T T T F T F F F T T F T F T F T T T F F F T F F T F T T T T T T T F T F T F F T T F F T T T T T T F F F T T T T T (b) [ ¬ C ( A
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