Simple yet Profound Metatheorem
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From: H. J. Sander Bruggink on 21 Dec 2005 06:50 H. J. Sander Bruggink wrote: > Charlie-Boo wrote: >> New Question: Just curious - is there even a wff such that you can >> prove it to be intuitionistically valid but I can't prove the wff >> using case analysis? > > No, there isn't. Sorry, I read the question as: "is there even a wff such that you can prove it to be intuitionistically valid but I can't prove the wff *to be classically valid* using case analysis? Now, I see the question really is the same as the previous question, so the answer should be: "Yes, *any* intuitionistically valid wff." groente -- Sander
From: Daryl McCullough on 21 Dec 2005 06:49 Charlie-Boo says... >No I don't. My original point remains undisputed. Neither you nor >anyone else has come up with a propositional calculus wff that can be >proven in some way that can't be proven using case analysis. There *aren't* any such wffs. Nobody is disputing that. They are disputing its *significance*. Here's an analogy: Suppose a gardener is looking for some way to kill the pesky Japanese beetles that are eating his flowers. You come along and say: If we drop Napalm on your garden, that will kill the beetles. That's true, but it's not what the gardener had in mind---he wants something that will kill the beetles but *not* destroy his garden. Similarly, it is not good enough to be able to prove every valid sentence, you have to make sure you don't prove any *invalid* sentences. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 21 Dec 2005 06:55 Charlie-Boo says... > >Torkel Franzen wrote: >> There are no truth tables for intuitionistic propositional logic. > >That has no relevance. If you are interested in intuitionistic propositional logic, then it is certainly relevant. If you are *not* interested in intuitionistic propositional logic, then what is the point of your claim that every wff that is provable in intuitionistic logic is also provable using truth tables? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 21 Dec 2005 06:58 G. Frege says... >stevendaryl3016(a)yahoo.com (Daryl McCullough) wrote: >> There are formulas that are provable [not really --F.] using truth >> tables but are not provable intuitionistically. I don't understand your "not really" here. -- Daryl McCullough Ithaca, NY
From: David C. Ullrich on 21 Dec 2005 08:30
On 20 Dec 2005 05:35:47 -0800, "Charlie-Boo" <chvol(a)aol.com> wrote: >David C. Ullrich wrote: >> On 19 Dec 2005 07:31:22 -0800, "Charlie-Boo" wrote: > >> I don't know whether you've noticed this, but others have: > >Name 2. > >> You complain that I'm just quibbling about syntax, >> and at the same time you ignore most of my comments >> about the substance of your Simple yet Profound Metatheorem... > >Glad you agree it's profound. > >> Oh for heaven's sake. How was I supposed to know that that two >> []'s meant entirely different things? > >Sorry, I thought you knew all about Logic. You really enjoy making a fool of yourself, eh? 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. >> >> > Thm. ((|- P) = (|- Q)) => |- (P=Q) >> > >> >> This is obviously false. >> >> If neither P nor Q is provable (so that >> >> ((|- P) = (|- Q)) is true) it certainly >> >> does not follow that P==Q _is_ provable. >> > >> >Let's suppose that P is "1>2" and Q is "2>3". Then P==Q is >> >(1>2)==(2>3) which IS provable. > >> Huh? > >I said, let's suppose that P is "1>2" and Q is "2>3". Then P==Q is >(1>2)==(2>3) which IS provable. > >> I didn't say that it is never the case that P==Q is >> provable. I said that this does not follow from the assumption >> that neither P nor Q is provable. > >I was just illustrating my thinking with a specific example. You mean illustrating why it is you got this wrong? Got it. Hint: The rest of us don't look at one example and then claim that we've proved a theorem because it seems to be right in that one example. > 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. >> David C. Ullrich ************************ David C. Ullrich |