If we show that a claim is equivalent to a tautology (which is stronger than showing the claim implies a tautology), how come that isn't a valid method of proof?
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Who says it isn't valid? In what context? For what purpose? In some (actually most) mathematical contexts, for some (actually most) mathematical purposes, demonstrating that a formula is an instance of a tautology is a perfectly cromulent method of proving it. Among the exceptions is if the context is a course in formal logic and the purpose is to gain or show familiarity with a particular formal proof system and how it works. In that (fairly narrow) circumstance, appealing to tautology obviously misses the point, at least until you have formally proved in general that every tautology has a proof in the proof system at hand. |
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If $T$ is a tautology, $(P\Rightarrow Q)\Leftrightarrow T$ is enough to prove $P\Rightarrow Q$, but it's overkill. All you need is $(P\Rightarrow Q)\Leftarrow T$. $(P\Rightarrow Q)\Rightarrow T$ is always true because $T$ is a tautology - it holds whether $P\Rightarrow Q$ is true or not, so it is a tautology in and of itself. On the other hand, $(P\Rightarrow Q)\Leftarrow T$ is only true when $P\Rightarrow Q$ is true, and in fact is equivalent to $P\Rightarrow Q$. |
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