# How come proof by tautology is not acceptable?

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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Showing that the claim implies a tautology doesn't tell you anything, since a tautology should be true anyhow. It is sufficient to show that your claim follows from a tautology, so if you show that your claim is equivalent to a tautology, you have shown that it is true. – Brett Frankel Nov 4 '12 at 3:21
What text or class did this come from? – Doug Spoonwood Nov 4 '12 at 3:29

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 $P\Rightarrow Q$.