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I watched a video lecture that talked a bit about when proof by contradiction is and isn't a useful approach. This was useful, but it was from a heuristic approach which caused me to wonder is it the case that "P is provable" implies "there is a proof by contradiction of P"? I suspect this may vary by logic system. If that is the case I would still be interested in know what major systems do/don't have this property.

Hunter
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    Technically , you can turn every proof into a proof by contradiction. But if there is a direct proof , there is no need to do that. – Peter Aug 06 '23 at 10:56
  • Thanks. I think I have an intuition of how that would work and it does satisfy my curiosity about if it would limit what can be proven if it was somehow required to use proof by contradiction for every proof. – Hunter Aug 06 '23 at 11:00
  • Example of how it works: Prove $1 < 3$ by contradiction: Assume $3 \le 1$. Now by the theorem $n < n+1$, we have $1 < 2$ and $2 < 3$. By the transitivity of "$<$", $1 < 3$, contradicting our assumption that $3 \le 1$. Thus $1 < 3$. – Paul Sinclair Aug 07 '23 at 19:01

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