'When Are Relations Neither True Nor False?
in [Math]
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From: Aatu Koskensilta on 24 Feb 2010 12:17 Newberry <newberryxy(a)gmail.com> writes: > Anyway, whethere GIT suggests that two valued logic is impossible or > not is not essential to my main point. Well, I don't really have any interest in your main point. I was just wondering why the incompleteness theorem suggests to you that "two valued logic is impossible". Apparently it's impossible to say. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Herman Jurjus on 24 Feb 2010 13:52 James Burns wrote: > Newberry wrote: > >> You are assuming classical logic. I am not. > > If you are not assuming classical logic, why are you > writing about Goedel's incompleteness theorem? That > is classical. > > Do you know of a version of GIT that uses a > non-classical logic? If you do, I would be grateful > for what information you have about it, references, > etc. That sounds interesting. Do you mean something like this? T. McCarthy, Self-Reference and Incompleteness in a Non-Monotonic setting, J. Phil. Logic 23 (1994) 423-449. -- Cheers, Herman Jurjus
From: Daryl McCullough on 24 Feb 2010 15:31 Newberry says... > >On Feb 24, 7:14=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >Goedel's theorem states that there is a sentence G such that neither >> >it nor ~G are provable (in a rather large class of formal systems.) >> >This to me suggests non-bivalence. >> >> *Why* does it suggest non-bivalence? If you have a theory that is >> incapable of proving 0 ~=3D 1 (and also incapable of proving its >> negation), does that suggest non-bivalence? Not to me; it only >> suggests that the theories axioms are not strong enough to prove >> some interesting true statements. At what point would you ever >> be justified in saying, for any formula Phi, "It's not that our >> axioms are inadequate to prove Phi or ~Phi, it's that Phi is >> neither true nor false"? > >a) Goedel shows that the axioms CANNOT be made strong enough. I would say that there is a quantifier ordering ambiguity at work here: Godel's theorem says that for every theory T of the right type, there is a formula Phi such that neither Phi nor ~Phi is provable in T. It doesn't say: There is a formula Phi such that for any theory T of the right type, neither Phi nor ~Phi is provable. In other words, the fact that neither Phi nor ~Phi is provable in a *particular* theory T does not suggest that Phi is "neither true nor false", because there will always be some other theory T' in which it *is* provable. No formula is *absolutely* unprovable (that is, unprovable by any sound theory). -- Daryl McCullough Ithaca, NY
From: Frederick Williams on 24 Feb 2010 17:13 Daryl McCullough wrote: > No formula is *absolutely* unprovable (that is, unprovable by > any sound theory). Is that what you meant?
From: Frederick Williams on 24 Feb 2010 17:17
Newberry wrote: > Goedel's theorem states that there is a sentence G such that neither > it nor ~G are provable (in a rather large class of formal systems.) > This to me suggests non-bivalence. If it was a case of neither G nor ~G are true it might. |