Topics Logic

Question Why is that if P entails not-Q and Q (a contradiction) do we conclude not-P? I understand that this a reductio ad absurdum and that because of the law of bivalence P either has to be true or false so if it entails a contradiction it is proved not true therefore false. But that last step is what I can't seem to justify...why does it become Not-P if it entails a contradiction? If I had to guess it's because contradictions don't exist in real life so if P were true and it entailed something that could never exist then it must be the case that P is not true (and this is true because of modus tollens: not-Q entails not-P). But we are dealing with symbols in the case of formal logic so how does this apply? Is formal logic an analogy of real life? I hope the question is clear after this rant!

Accepted:
October 13, 2009

Comments

I'll confess that I'm not sure I have your question right. You've given a pretty good explanation of why P can't be true if it entails a contradiction. I'd rephrase the way you put it, however. Instead of saying "contradictions don't exist in real life," I'd say "contradictory statements are never true." But as you in effect note, if a statement entails a contradiction, then the statemetn could only be true if a contradiction were true. That can't happen, so the statement must be false.

So far so good. Your worry has to do with that fact that we are dealing with symbols and formal logic rather than "real life. " But the point of the the symbols is just to let us talk in general. The schema is (roughly) that whenever P entails a contradiction, P is false. That's shorthand for saying that whenever a statement entails a contradiction, the statement can't be true. In other words,

Pick any "real life" statement you like that entails a contradiction. Then the statement is false.

Notice that this version doesn't use any symbols and isn't a matter of formal logic. The formal version provides a way to state the principle more tidily.

And so unless I'm missing something, it's not clear that the difference between formal, symbolic logic and real life is really at issue here. Formal principles bear on real cases - on cases that are instances of the formal principle. It' snot that formal logic is an analogy of "real life." It is (among other things) an abstraction from particular cases. If a principle gives the wrong answer for the particular cases, the principle is wrong. In the case at hand, there's no reason to believe that this problem arises.