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This is something I was wondering about for quite a while.

Is it possible to construct a statement that can only be proven by using 'proof by contradicition' or contraposition? Or to put it differently: If we do not use contradictions or contrapositions, would we be able to build the same mathematical framework that we would get without using them?

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    To whoever may know, is the concept of 'direct proof' even definable? – Git Gud Jul 17 '14 at 22:21
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    Sure, you can define "direct proof" as any proof that doesn't use certain steps of deduction. It depends on your rules of logic which those are. Essentially, it's any proof that refuses to use excluded middle. @GitGud – Thomas Andrews Jul 17 '14 at 22:25
  • The pages on constructive/nonconstructive proofs and constructivism at Wikipedia seems relevant here. It sounds like you're asking whether all theorems can proved constructively. – Semiclassical Jul 17 '14 at 22:25
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    The OP defined it well. 'direct proof = 'not a proof by contradiction'. – Elimination Jul 17 '14 at 22:25
  • @ThomasAndrews Not quite, contraposition doesn't use the excluded middle. – Git Gud Jul 17 '14 at 22:28
  • There are a couple other threads on this topic. I'll leave you with the task of finding them. – Asaf Karagila Jul 17 '14 at 22:29
  • @Elimination We are not reading the same question. – Git Gud Jul 17 '14 at 22:29
  • @Git Gud: to prove the contrapositive of a conditional is equivalent to the original conditional, you need double negation elimination. I take it Thomas would rule this out in dismissing "essentially" anything that refuses to use excluded middle. – symplectomorphic Jul 18 '14 at 11:46
  • @symplectomorphic I agree. But you don't need equivalence, one direction suffices. The theorem $(A\to B)\to (\neg B\to \neg A)$ can be proved without double negation. So if the goal is to prove $\neg B\to \neg A$, you can instead prove $A\to B$ without using the double negation. – Git Gud Jul 18 '14 at 12:25
  • @symplectomorphic Let me rephrase what I said. You don't need equivalence, you need only one direction (obviously). One of them doesn't use double negation, the other one uses. If we're lucky we can manage without it. – Git Gud Jul 18 '14 at 12:34
  • @Git Gud: that's true -- if we're lucky. – symplectomorphic Jul 18 '14 at 13:21

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