But then think about what happens in quantum mechanics, classical logic is wrong when applied to it, especially the law of excluded middle goes out of the window.
32.
The Fregean view is thus committed to the kind of truth value gaps ( and failures of the Law of Excluded Middle ) that the Russellian analysis is designed to avoid.
33.
One can see that the law of excluded middle cannot hold from the following basic theorem ( again, understood in the context of a theory of smooth infinitesimal analysis ):
34.
Comparing the use of the law of excluded middle ( rejected by constructivists ) to wine, Keisler likened Halmos'choice with " choosing a teetotaller to sample wine ".
35.
From a proof-theoretic perspective, Heyting s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination are not valid logical rules.
36.
Your other option is to try the infinitesimals of smooth infinitesimal analysis which don't require augmenting the reals, but instead ask you to drop the law of excluded middle.
37.
The law of non-contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the law of excluded middle, an expression of its jointly exhaustive aspect.
38.
The rule of double negation elimination ( for any proposition P, \ lnot \ lnot P \ to P ) can be seen as a result of the law of excluded middle.
39.
In the context of Aristotle's traditional logic, this is a remarkably precise statement of the law of excluded middle, " P " ( " ?" P ".
40.
However, there are proofs in classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, there are propositions that cannot be proven that way.
