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When you are proving something using proof by contradiction, are you actually constructing a vacuously true statement?

Let's say that you want to prove $P$ via proof by contradiction. First, you assume that $\lnot P$ is true. Then via direct proof, you show that $\lnot P \Longrightarrow \bot$. Since the only way for the statement $\lnot P \Longrightarrow \bot$ to be true is vacuously, we conclude that $P$ is true.

Is this reasoning false?

ryang
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Jon
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3 Answers3

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Hmm, your reasoning is almost valid (not "true/false").

I'd say that a proof by contradiction of the statement $P$ derives, rather than constructs, the vacuously true implication $\lnot P \Longrightarrow \bot.\tag*{}$ However, I'd qualify that its vacuousness (i.e., the falseness of its supposition) is not logically inevitable, and is deduced rather than by default/construction/assumption. Specifically, the proof derives an implication that turns out to be vacuous, and certainly does not rely on $P$ being true.

ryang
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  • "$\lnot P \implies \bot$" can be true without $\lnot P$ being false": how so? $\lnot P \implies \bot$ is precisely the definition of "$\lnot P$ is false". – Jean Abou Samra Oct 14 '24 at 11:56
  • @JeanAbouSamra I've just revamped the answer; thanks for pointing that out. – ryang Oct 14 '24 at 13:15
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The only way for $\neg P \implies C$ to be true is for $\neg P$ to be false. Therefore, by the Law of Excluded Middle, $P$ is true.

Colin Tan
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No. The vacuousness of a statement is about how it is not necessary for proving any (subjectively?) concrete statement. It is not about how it was proved, as there are any number of ways of proving any provable statement.

Assume $T$ is a provable statement. Then $\lnot T \vdash \bot$ should be constructible from the previous proof of $T$. Whether or not $T$ is vacuous.

DanielV
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  • The OP isn't asking whether a statement proven by contradiction is necessarily vacuous. On a separate note, a tautology is self-evidently true but not necessarily vacuous. – ryang Oct 15 '24 at 05:51
  • @ryang Reading his question literally, it looks to me like he is asking whether proof by contradiction necessarily construct a vacuous statement. And a tautology is a statement that is true under all variable assignments (in context). It has nothing to do with being "self evident" whatever that means. It doesn't have anything to do with the question that I can see. – DanielV Oct 15 '24 at 06:17
  • My second point was referring to your sentence "The vacuousness of a statement is about how it is not necessary for proving any concrete statement", and merely pointing out that not every self-evident statement (i.e., one that's unnecessary to prove) is vacuous (though the converse certainly holds). – ryang Oct 15 '24 at 07:26