"Harlan Messinger" <email@example.com> wrote in message news:<a4tua8$34a32$1@ID-114100.news.dfncis.de>... > "Virgil" <firstname.lastname@example.org> wrote in message > news://vmhjr2-9E34CB.email@example.com... > > In article <a4revv$2cs7s$1@ID-114100.news.dfncis.de>, > > "Harlan Messinger" <firstname.lastname@example.org> wrote: > > > > > > It is quite possible to do large amounts of mathematics in systems > > > > whose axioms do not include or allow a law of the excluded middle. > > > > > > > > And there is a school of thought that rejects anything that cannot > > > > be done in such systems. > > > > > > In other words, are these people who refuse to see the difference that > most > > > people implicitly understand to exist between "short" and "not tall"? > > > > More accurately, they refuse to accept that everyone is either tall > > or not tall without some sort of constructive proof. > > Why does it not suffice for them to say that that's how "not" is *defined*?
Because that *isn't* how "not" is defined in intuitionistic logic. The notions of truth, falsity, negation, ... all have intuitionistic definitions that are somewhat different from the classical definitions you may be accustomed to.
Under intuitionism, saying that A is true means something like this: There is a constructive procedure for determining whether A holds or not, and it gives a positive result. Saying that A is false means that there is such a constructive procedure and it gives a negative result.
Now you can see that from this point of view, A could be neither true nor false. In particular, if there is no constructive procedure for determining whether A holds or not, then an intuitionist would say that neither A nor not-A is true.