False Statement implies Every Statement

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If something is false, then it implies anything.

Formulation 1

\(\ds \neg p\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds p \implies q\) \(\) \(\ds \)

Formulation 2

$\vdash \neg p \implies \paren {p \implies q}$

This apparent paradox can be reconciled by considering the figure of speech in natural language:

If Dilbert passes his Practical Management exam I'll eat my hat.

That is, if statement $p$ is so absurdly improbable as to be a falsehood for all practical purposes, then it can imply an even more absurdly improbable conclusion $q$.

Also see