Have you ever encoutered a proof like this one :
If $Q$ is true then $P$ is true.
If $Q$ is false then $P$ is true.
Therefore $P$ is true.
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Here's an example: take $Q$ to be $\sqrt{2}^\sqrt{2}\in\Bbb Q$, and $P$ to be $\exists a,\,b\in\Bbb R\setminus\Bbb Q(a^b\in\Bbb Q)$. If $Q$ is false (it is), take $a=\sqrt{2}^\sqrt{2},\,b=\sqrt{2}$.
J.G.
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See also https://en.wikipedia.org/wiki/Gelfond-Schneider_constant#Properties – lhf Oct 22 '19 at 17:23
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Ohh I remember that one. If $\sqrt{2}^\sqrt{2} \in \mathbb{Q} $ then we're done otherwise $ (\sqrt{2}^\sqrt{2})^\sqrt{2}= \sqrt{2}^2 = 2 \in \mathbb{Q} $. Quite smart. – LIR Oct 22 '19 at 17:26
