Tuesday, January 3, 2012

Fundamental Modes of Reasoning.

I would argue that all argument, reason and logic is fundamentally reducible to proof by contradiction and proof by exhaustion. As a toy heuristic argument for the former's fundamentality I will here prove, via contradiction, that everything that can be proved can be proved via contradiction.

1. Suppose some proposition S has a proof P, but is such that it cannot
be proven by contradiction.
2. Suppose ~S
3. Reiterate ~S
4. P
5. Therefore S
6. Contradiction
7. Thus ~~S
8. S
9. But then S is proven by lines 2-8 through contradiction, which contradicts line 1
10. Therefore there does not exist such a proposition S.

Therefore if anything can be proven it can be proven by contradiction.

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