Perucho my friend, I thought you, someone grounded in applied theory, would appreciate the freedom of not having just true and false -- aren't the mechanical engineering applications what drove the invention of fuzzy logic? [Fuzzy logic is an honest math in which true and false are 1 and 0 but statements can take on any value in the continum [0,1]. It is really useful in practice.]
A counter-example to a statement P, in logic, is a proof that determines P is false, or more precisely, that there exist an x for which P(x) is false, that is, that it is not true that for all x, P(x) is true. This is not using PEM because it is a proof that P is false. (true and false still exist without PEM, but they may not be the only values.)
However, a proof by contradiction requires PEM. So you can have one without the other.
It is all axiomatic, so a belief is involved somewhere. By default we assume everyone believes the same axioms, until they specify otherwise. It is not different that assuming + is a commutative. Some people will use + for non-commutative, they are out of the ordinary, and so they, not the rest us, should specify how and why they want to use + in a different manner, but they are within their mathematical rights to do so and it doesn't result in throwing out anything.
Actually, to the contrary, it adds to the subject to see alternate proofs to things that don't use contradiction, or better yet, to see problems that simply require such a proof. It tells us what fundamental assumptions must be made (or can be omitted). It makes the subject more general and robust to future changes.
[Just doing my part to make the discussion go on even longer! Sorry.] |
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