Shantila's Inside Logic #16 The Law of Non-contradiction Since the rule RA, like the rule CP, reduces our assumptions by one, RA also can be used to prove logical truths even when CP is not used. For example, consider the proof of the following logical truth: 93 } ~(P&~P)
This might look like a trick, but it is not a trick. It is just a very simple proof. Recall that the rule RA says that if you make an assumption and can prove a contradiction, you can negate the assumption, where the premises of the new line are all those of the contradiction except the number of line negated. We have used RA correctly here. The unusual feature is that we simply assumed a contradiction on line 1, so given our rule RA, the derivation is very simple: the line where we make the assumption (line 1) is exactly the same as the line where we get the contradiction! The logical truth ~(P&~P) that we prove here is known traditionally as the "Law of Non-contradiction". Notice that the proof of this law is simple, given our rule RA, because accepting RA as a rule means pretty much the same thing as accepting ~(P&~P) as a logical truth! Intuitively it makes sense that we should be able to prove this sentence as a logical truth, since it says simply that no sentence P is such that both P and ~P are true. There is a passage in the writings of the Greek philosopher Aristotle, where in discussing this "law" of logic, he says that you can't really have a reasonable discussion with a person who does not accept it -- all you can do is try to beat some sense into him. I think he probably was joking but I'm not sure. * Practice 16.1 Each of the following expresses a logical truth. Symbolize each of them and prove that it is a logical truth. (Note: Each of these is simply reviewing what we mean by "logical truth".) (a) Joe will not both win the lottery and not win it. (J: Joe will win the lottery) (b) It is not true that both Sally lives in Cleveland and does not live in Cleveland. (C: Sally lives in Cleveland) (c) If it is not the case that Joe will not run for President, then Joe will run for President. (P: Joe will run for President) (d) If Joe is an alien from outer space, then he wins the lottery or he is an alien from outer space. (A: Joe is an alien from outer space. L: Joe wins the lottery.) (e) If Joe wins the lottery, then either he wins the lottery or he will marry Jessica. 16.2 Each of the following sequents is valid. Use our Rules to construct derivations for each of them. What we are doing here, of course, is showing that each of these sentences is a logical truth. 91 } (P&Q) > ~(P>~Q) 92 } ~(P&Q) > (P>~Q) 93 } ~(P&~Q) > (P>Q) 94 } P > ~(~P&Q) 95 } ~(PvQ) > ~P |
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