On 21 Jul 2006 16:59:48 -0700, "Jiri Lebl" <firstname.lastname@example.org> wrote:
>Lester Zick wrote: >> Tautological combinations such as "A, not A" exhaust all possibilities >> for truth unless I'm very much mistaken. > >Good thing you admit it. Yes you are very much mistaken. For example >take A be the statement: "Is P(N) = aleph_1 true in ZFC".
Not much for transfinite arithmetic so it's a little difficult to evaluate this example. If you're talking compound predicates in A you have a problem. If I say "red, not red" alternatives are exhaustive. But if I say "red idea, not red idea" and the predicate "red" does not apply to the predicate "idea" the situation is problematic at best.
> Simple >enough statement. However, neither "A" nor "not A" is true. You must >be VERY careful what kind of statement you make A. Further the fact >that either A or not A is true is an AXIOM (given that A is a well >defined statement that is either true or false in some system)! It is >an ASSUMPTION! Further some people reject this assumption, for it >leads to non constructive proofs.
Well this point is well taken: the tautological exhaustion of truth is an assumption. However the point I would make in this connection is whether there can be any alternative. In other words can there be an alternative to the tautological exhaustion of truth that is actually "not a tautology"? If so it would seem we nonetheless have the resulting tautology "tautology, not tautology" in order to deny the tautology which as far as I can tell denies the possibility of any alternatives to the tautology whether or not the exhaustion of truth in tautologies is an assumption. In other words whether or not the exhaustion of truth in tautologies is an assumption, there can be no mechanical alternative to tautologies.