assuming the conclusion

Question

A natural deduction proof goes from premmisses to conclusion, and under normal circumstances you will not assume the conclusion.

Sometimes you may assume the negation of the conclusion and do some reductio ad absurdum and that ilk, but never assume the conclusion unnegated.

But recently I saw a wrong proof where a student did assume the conclusion.

So (internally) I was ranting "never, never , never in a proof assume the conclusion, it is a crime against logic, prooftheory . Where is my broad red marker, and all that."

But then I became reflective: "is this really always the case? Are there really no valid proofs where assuming the conclusion is needed or where assuming the conclusion speeds up or shortens the proof?"

Any proofs why assuming the conclusion cannot help, or examples where it is useful welcome.

Answer 1

You can't assume the conclusion. If you could, you could prove any statement P as follows:

1. Assume P.
2. Then P.
3. Done.

It's the mathematical equivalent of saying that something is true "because it is".

That said, in a formal system, you might have to do a subproof premised upon a statement that happens to be the conclusion of the overall proof; for example, to prove $P$ from $P \vee ((P \rightarrow Q) \rightarrow P)$, you might have to use "assume $P$" and "assume $((P \rightarrow Q) \rightarrow P)$" cases and prove $P$ in both cases. However, the "assume $P$" case is trivial. In a proof with words, we'd say something like "we only need to consider the second case", or perhaps devote one sentence to "If P, then we're done." This is quite different from unconditionally assuming the desired result.

Answer 2

If you assume the negation of the conclusion, $\neg P$, and arrive at a contradiction, $\bot$, you have proven the statement $$\neg P \implies \bot.$$ This statement has some weight. It is not trivial. from it, you can conclude that $P$ is true: using the fact that $A\implies B$ is equivalent to $\neg B\implies \neg A$, this means you have proven that $\top\implies P$, and since $\top$ is true, this means you have proven $P$.

On the other hand, if you assume $P$ and prove $P$, the only statement you have proven is $$P\implies P,$$ which is a tautology. No statement follows from this statement and you can definatelly not prove $P$ with it.

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Another way of thinking why assuming $P$ to prove $P$ is wrong: Take $P="1=2\text{ in }\mathbb N"$. If I assume $P$ is true, then $P$ is true, and I have just proven that $1$ equals $2$.

If this is a valid method of proof, then I can prove anything with it, even untrue statements. That tells you there is something wrong with the method...