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I'm comfortable using a direct proof to prove ##P → Q## type statements when I have a ##P## that is either always true (e.g ##x=x##) or can be true (e.g. ##x > 3##).

But what about when ##P## is definitely false, (e.g. ##x \neq x##), or definitely false in relation to an earlier statement (e.g. If we've already been given the fact that we're dealing with a group G, and ##P## is the statement "there does not exist an identity element in G")? Can you use direct proof then? By direct proof I mean proving the statement by assuming ##P## is true, then showing through legal logical inferences that ##Q## must be true as well.

I hope that's made sense. Thanks :)

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# Vacuous "If then" statements: Can you use direct proofs?

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