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SolutionsI - ENGRI 126 SOLUTIONS TO HOMEWORK I Spring 2007...

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ENGRI 126 SOLUTIONS TO HOMEWORK I Spring 2007 1. The idea is that a situation is paradoxical if it leads inevitably to a logical contradic- tion of the form “ A implies not A .” If there’s some set of circumstances that leads to no such contradiction, then it’s not a paradox. (a) Certainly, I could not possibly be telling the truth, since in that case the indicated statement would be false, meaning that I am not telling the truth after all. I.e., I am telling the truth implies I am not telling the truth. However, I could be lying. In other words, it is possible that I do tell the truth sometimes, but not when I say this sentence to you. In other words, if I am lying when I speak the indicated sentence, there’s no logical contradiction. (b) Could Lisa be telling the truth? If so, Bart must be lying, which means that Lisa is not telling the truth after all. So Lisa could not be telling the truth. Could she be lying? If so, we conclude that Bart must tell the truth sometimes. Might he be telling the truth now? If so, Lisa could not be lying as we assumed, so he must actually be lying now. If he is lying now, then Lisa is lying now — which is exactly what we assumed to start this chain of reasoning. Conclusion: there’s no logical contradiction if Lisa is lying and Bart is also lying. Hence the situation is not a paradox. (c) Could Lisa be telling the truth? If so, what Bart says in response must be a lie, which means that Lisa is not telling the truth after all. So Lisa could not be telling the truth. Could Lisa be lying? If so, we conclude that what Bart says in response must be true. This implies Lisa is not lying as we assumed. Hence, we have a paradox. Regardless of whether Lisa is lying or telling the truth (and these two possibilities exhaust the set of possible circumstances), we get a logical contradiction. (Note: I did not intend for you to worry about the time-delay feature of this example. Compare part (e).) (d) If (A) is true, then (B) must be false, meaning that (A) must be false — a contradiction. If (A) is false, then (B) is true, meaning that (A) is true — a contradiction. Thus we have a paradox. This is exactly like (c) when you think about it. (e) I don’t have a good answer for this one. I was just hoping to have you think about what’s going on in situations with linked declarative sentences that are uttered at different times. The field of temporal logic, which is relevant to theoretical computer science among other things, addresses situations of this kind. One reasonable way to approach the question is to assert that the sentence I utter before I go to bed, despite its declarative nature, does not have a truth value when I utter it because it refers to an event in the future. Only after that event takes place can we hope to (attempt to) assign a truth value to that first sentence.
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