Simple yet Profound Metatheorem
in [Logic]
|
Prev: Obections to Cantor's Theory (Wikipedia article)
Next: Why Has None of Computer Science been Formalized?
From: Daryl McCullough on 9 Jan 2006 18:48 Charlie-Boo says... > >> >Daryl McCullough wrote & Charlie-Boo says... >> Your point was that you shouldn't use case analysis if you >> are interested in finding out which propositional formulas >> are intuitionistically valid? > >No, I never said to not use case analysis. Yes, but that's the correct conclusion. Case analysis is not valid for intuitionistic logic. -- Daryl McCullough Ithaca, NY
From: Charlie-Boo on 9 Jan 2006 23:13 > >> >Daryl McCullough wrote & Charlie-Boo says... > > >> Your point was that you shouldn't use case analysis if you > >> are interested in finding out which propositional formulas > >> are intuitionistically valid? > > > >No, I never said to not use case analysis. > > Yes, but that's the correct conclusion. Case analysis is > not valid for intuitionistic logic. It is also not valid for creating inconsistent systems or redundantly expressing the same deduction in numerous equivalent ways. Yes, there is a lot that case analysis does not do. (That is a variation on the old trick of bringing in a red herring - referring to something without establishing it is needed.) Oh, BTW How do you know that case analysis is not applicable? I mean, just use it in the right way and maybe you'll find out how useful it really is! Can't there be any way to take advantage of the finite universe of possibilities that case analysis provides (as opposed to that infinite set of theorems)? C-B > Daryl McCullough > Ithaca, NY
From: Torkel Franzen on 10 Jan 2006 01:50 "Charlie-Boo" <chvol(a)aol.com> writes: > Oh, BTW How do you know that case analysis is not applicable? From G?del's proof that intuitionistic propositional logic cannot be characterized in terms of truth tables using a finite number of truth values.
From: Daryl McCullough on 10 Jan 2006 06:24 Charlie-Boo says... > >> >> >Daryl McCullough wrote & Charlie-Boo says... >> >No, I never said to not use case analysis. >> >> Yes, but that's the correct conclusion. Case analysis is >> not valid for intuitionistic logic. > >It is also not valid for creating inconsistent systems or redundantly >expressing the same deduction in numerous equivalent ways. Yes, there >is a lot that case analysis does not do. Well, then why are you advocating case logic as an alternative to intuitionistic logic? That doesn't make any sense. >(That is a variation on the old trick of bringing in a red herring - >referring to something without establishing it is needed.) I don't know what your point is well enough to know what's a red herring. >Oh, BTW How do you know that case analysis is not applicable? Because case analysis allows you to prove statemsnts that are intuitionistically invalid. >I mean, just use it in the right way and maybe you'll find out >how useful it really is! Can't there be any way to take advantage >of the finite universe of possibilities that case analysis provides >(as opposed to that infinite set of theorems)? Intuitionistic logic has an infinite number of different "truth values", so it cannot be decided using case analysis over a finite number of cases. -- Daryl McCullough Ithaca, NY
From: David C. Ullrich on 10 Jan 2006 08:05
On 9 Jan 2006 14:58:32 -0800, "Charlie-Boo" <chvol(a)aol.com> wrote: >David C. Ullrich wrote: > >> In formal Logic we don't use one symbol to denote >> two diffferent things in the same expression. The >> fact that I knew that is exactly why I had no idea >> that you meant what you say you meant by that >> expression. > >I did not change the semantics, only the syntax in a one-to-one manner >from that used in Modal Logic. Thus your criticism applies equally >well to the standard use of Modal Logic. Huh? The bit you cite here was in reply to the following comment of yours: >>Don't you see the parallel? If |- (|-Q == Q) is extremely bad >>notation because the first |- means |- while the second |- refers to >>some encoding of provability, then [] ( []Q == Q ) is extremely bad >>notation because the first [] means |- while the second [] refers to >>some encoding of provability. No, in standard modal logic people do not use the symbol [] to mean two different things. >> > A better >> >counterexample for you (than meaningless propositional variables) would >> >be P is any Godel sentence and Q is FALSE. >> >> A counterexample that depends on an _actual_ deep theorem is >> "better" than a simple and totally elementary counterexample? >> Fascinating. > >Your "proof" says, "suppose P (is an) atomic formula in the >predicate calculus. Then P is not provable." This is not >well-formed. (It is meaningless.) You're very confused about a lot of things. Have you noticed that? >If P is an Atomic Formula then it consists of a Predicate Letter >applied to terms. But then P == |-P has no meaning since P is not >interpreted. P has no proof - nor does it have any truth value. >When we talk about true, provable, unprovable etc. we are talking about >interpreted wffs, not uninterpreted Predicate Letters. Atomic Formulas >have no proof nor truth value, and so talking about whether P is == |-P >or not is meaningless. (Mendelson 1964 pg 46-53) > >As I then pointed out, after trying combinations of Godel sentences, I >realized that P is a Godel sentence and Q is FALSE refutes this earlier >version of my theorem, not a trivial misuse of concepts of Mathematical >Logic. > >C-B > >> ************************ >> >> David C. Ullrich ************************ David C. Ullrich |