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An example of a non-strictly non-constructive proof: given a constructive proof of p, transform to a new proof of p via the lemma p & q where q is a fixed sentence that is classically but non constructively valid. By strictly non-constructive proof I mean that you prove p without constructing it.

donnoh
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  • Does this answer your question? Can every proof by contradiction also be shown without contradiction? – Natalie Clarius Jul 06 '20 at 23:31
  • No. I know there are proposition that can be proven in classical logic but non constructively, what I am asking is: if it’s known that p can be constructed, can I prove p without constructing it for every p? It’s almost the opposite question but without trivial answers – donnoh Jul 06 '20 at 23:38
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    Not very clear... From a "classical" point of view a constructive proof is valid. Thus, every constructive proof is also classical. Having said that, we can trivially transform $\Gamma \vdash \varphi$ into $\Gamma, \lnot \varphi \vdash \varphi$ and thus we have $\Gamma \vdash \lnot \lnot \varphi \vdash \varphi$ that is not constructively valid, because uses DN. – Mauro ALLEGRANZA Jul 07 '20 at 08:17
  • but $ \Gamma, \lnot \varphi \vdash \varphi $ is still a constructive proof if you use $\Gamma \vdash \varphi$ to prove it and the latter is constructive, right? In this case $\Gamma \vdash \lnot \lnot \varphi \vdash \varphi$ is not strictly non-constructive in the sense that I defined it before – donnoh Jul 07 '20 at 10:49

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