Metalurgia

In Mathematical, the principle of the third excluded affirms that the proposal “ $\ phi$ or (not $\ phi$ )” is true, for all proposal $\ phi$ . That means that for any proposal $\ phi$ , one must accept either $\ phi$ , or its negation.

The principle of the excluded third was introduced by Aristote like consequence of the principle of non-contradiction , whereas these two principles are different. The principle of noncontradiction stipulates that for any proposal $\ phi$ one cannot have $\ phi$ and not $\ phi$ at the same time.

In Logical traditional, the principle of the third excluded results from elimination from the double negation (not (not (R)) = R):

not contradiction: not (R and (not (R)) <=> not (R) or not (not (R)) <=> not (R) or R (third excluded)

It is not however the case in all the logical formalisms, and in particular in Logique intuitionalist, which preserves the principle of non-contradiction but does not use the “principle” of the excluded third. In logic intuitionalist, we cannot say that “R or (not R)” is true a priori for any proposal R, it should be shown for each proposal R (and, in certain cases, that will be impossible without introducing a new axiom). For mathematician intuitionalist, principle of third excluded is at best useless (for the demonstrations which one could make without using it), in the worst case sterilizing (it slices proposals indécidables, without that resulting from a conscious and deliberated decision, knowing that each time one can create several distinct and potentially fertile formalisms).

An example of reasoning calling upon third-excluded is the following: we want to show the implication

\ forall has, B \ in \ mathbb R, ab=0 \ Rightarrow a=0 \ {\ rm or} \ b=0

For that, we can consider the proposal R: “a=0” and to use the principle of the third excluded for R. There are then two cases to examine (a third case being excluded):

is R is true: then the proposal “R or b=0” is checked
is R is false i.e a≠0 then while simplifying by with the relation ab=0, we obtain b=0, and “R or b=0” is checked.

In the preceding implication, a mathematician intuitionalist will refuse to conclude that a=0 or b=0, because it cannot prove that “a=0 or not (a=0)” is true (equality on realities being indécidable).

However, logic intuitionalist is not basically more weak that traditional logic: for any proposal R provable in traditional logic, there exists a proposal R' (which can be identical to R) such as R and R' are equivalent within the meaning of traditional logic, and R' is provable in logic intuitionalist. In our example, this R' proposal would be “not (ab=0 and not (a=0) and not (b=0)) ”.

The Raisonnement by the absurdity rests on the principle of the excluded third. Indeed, it functions on the following mechanism: I want to prove R. For that, I suppose “not (R)” and I fall on a Contradiction: it is thus that “not (R)” is false, and according to the principle of the excluded third, that “R” is true. In logic intuitionalist, this last stage is impossible: of “not (R)” is false, one can just conclude “not (not (R)) ” is true, but it is not equivalent to “R” is true, as in traditional logic. There is just the Implication $R \ Rightarrow {\ rm not} ({\ rm not} (R))$ , but not its Reciprocal.

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