In article <a4k5a2$tlnc$1@ID-114100.news.dfncis.de>, "Harlan Messinger" <firstname.lastname@example.org> wrote:
> "Virgil" <email@example.com> wrote in message > news://vmhjr2-35D8AB.firstname.lastname@example.org... > > Jon Miller <email@example.com> writes: > > > > > There are even (intelligent) people who (claim that they) are unable > > > to believe in infinite sets at all. To them, Cantor's argument > > > makes no sense. > > > > Why is that? Cantor's argument does not presuppose the existence > > of any infinite set. > > > > > There are people who deny the law of the excluded middle. To them, > proving > > > that not-A is false does not prove that A is true. > > > > There are people who will deny that (P or (not P)) is necessarily > > true, but those same people will insist that (P and (not P)) is > > necessarily false. It is the second form which is relevant to > > Cantor's proof, not the first. > > I don't get it. In symbolic logic, isn't "not" *defined* such that (P or > (not P)) is necessarily true and (P and (not P)) is necessarily false? >
For most people, yes, but there are those that insist that the falseness of P is insufficient to guarantee the truth of not P and vice versa. They deny what is generally known as the law of the excluded middle, and say that there may be something in the middle, "between" P and not P.