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Are there any limitations to a proof of contradiction?

For eg: Proving $p\sqrt q$ rational, given that $p$ is rational and $\sqrt q$ is irrational.

We go by the usual technique of contradiction and assume it to be rational.

But it becomes rational if it is squared. So we can't go on proving it irrational if we square it before.

Why this problem? Where is the flaw and where did I go wrong?

khaxan
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  • Formally, a proof by contadiction works like this: 1) a bunch of statements, which we may for simplicity summarize in a single statement $P$, have already been proved; 2) you call $T$ your thesis; 3) you know that $T\lor\neg T$ is an axiom; 4) $P\land T$ proves $T$; 5) you show that $P\land \neg T$ proves $\neg P$; 6) since $P\land \neg P$ proves every statement by the principle of explosion, we have that $P\land \neg P$ proves $T$, and therefore that $P\land \neg T$ proves $T$; 7) since $P\land T$ and $P\land\neg T$ prove $T$, we have that $P\land (T\lor\neg T)$ proves $T$. – Sassatelli Giulio Feb 02 '23 at 17:08
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    Proving a statement by contradiction generalizes the statement in the same degree any other proof would, that is, to the extent that it shows that the statement is true. It is hard to conceive two correct proofs being one more "generalizing" than the other. The only difference is that instead of showing that something is true directly, you show that it is impossble for it to be false. But the "extent to which it generalizes the statement" is the same. – lafinur Feb 02 '23 at 17:11
  • proof's by contradiction tend to be nonconstructive. You prove a result without constructing an example of the thing you are proving. So, in that sense it can be less insightful than a direct proof. – Michael Carey Feb 02 '23 at 20:31

1 Answers1

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Not sure I understand your question right. But generally proof by contradiction is 'more powerful' than direct proofs. i.e. there are statements which can be proven by contradiction but not in a direct manner. However direct proofs are considered to be more useful, since they contain more information e.g. in the form of computational content which can be extracted to turn the proof into an algorithm.

StiftungWarentest
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    But what if there are results that can be proved directly but not by contradiction? Perhaps denying the conclusion is less useful than running with the hypotheses outright? – Randall Feb 02 '23 at 17:27
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    That can never happen, every direct proof can be turned into a proof by contradiction by simply assuming the opposite first, then carrying out the direct proof which then contradicts the assumption. – StiftungWarentest Feb 02 '23 at 18:04
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    That's a decent argument. – Randall Feb 02 '23 at 18:09
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    LOL I second both of @Randall's comments. – ryang Feb 02 '23 at 18:50