Re: Question about proof by contradiction

On Mon, 10 Mar 2008 15:58:33 +0100, G. Frege <nomail@invalid> wrote:


- in general, how can you tell whether any given proof technique is
valid?

By using techniques of so called "mathematical logic" we can identify
valid rules. (RAA is one of them.)

Hint:

(A -> (B & ~B)) -> ~A

is a theorem (and also a tautology) in classical logic.

So _if_ we can derive

A -> (B & ~B)

we may conclude (by an application of MPP)

~A.

With other words, if we derive a contradiction of the form

B & ~B

starting with the ASSUMPTION

A

we may conclude

~A

(while "discharging" the assumption "~A" at the same time).


All this is "combined" in the rule of derivation /reductio ad absurdum/
(RAA). This rule has the following form:

[A] Assumption
:
B & ~B Contradiction!
------ RAA
~A

Explanation: We start with the assumption A. After reaching a
contradiction of the form B & ~B, we may derive/conclude ~A.


F.

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