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Logic

When we prove a statement, we show it is true. Since contradictions (statements such as "P and not-P.") are never true, we can't ever prove a contradiction. But that's precisely what we do in a proof by contradiction - we show a contradiction to be true, before declaring it absurd. This must mean we are doing something wrong. It must mean that we can't even assume a false statement to begin with. This makes sense because when we assume a statement, we pretend that it's true, but we can't pretend that a false statement is true. It's a logical impossibility. That would be like saying "1 + 1 = 4" is true. Does this mean the "proof by contradiction" method is flawed? In other words, to prove proposition P, we assume not-P and show this leads to a contradiction. But if P is true to begin with (as we want to prove it), and therefore not-P false, how can we even assume not-P is true? It's false. We can't assume it as true. It's logically impossible. For example, it's logically impossible for the square root of 2 to be rational. How then can we even assume it's rational? That's impossible to begin with. It's just not right. To me it seems as if this shows that the proof by contradiction method, while it may seem nice, useful, and helpful, is actually an invalid form of argument. Am I right? If not, how is my own reasoning incorrect?
Accepted:
June 5, 2011

Comments

Alexander George
June 7, 2011 (changed June 7, 2011) Permalink

When we prove something "by contradiction", we are not proving a contradiction true. We are showing that some assumption, call it X, logically implies a contradiction, that is, logically implies a statement that is logically false. The correct lesson to draw is not that the contradiction is after all true, but rather that our assumption X is incorrect. In other words, we infer not-X. This is what it is to prove not-X by contradiction.

You write that if X is false, then it's impossible to assume that it is true? Why? Is there something incoherent about supposing that Hillary Clinton is President of the U.S. and asking what would follow from that?

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