Simple yet Profound Metatheorem
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From: Daryl McCullough on 29 Dec 2005 17:24 Charlie-Boo says... >Change "you are not interested in" to "there's nothing gained in the >final analysis", then the answer is: It shows that Frege is wasting his >time since nobody has ever come up with a wff that he produces that >case analysis does not. Propositional intuitionistic logic is a *refinement* of case analysis. It's not supposed to prove more wffs, it is supposed to prove *fewer* wffs, and it succeeds. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 29 Dec 2005 17:27 Charlie-Boo says... > >Daryl McCullough wrote: >> Intuitionistic analysis is not a "variation in syntax", >> it has a different *semantics* than classical logic, and >> case analysis is not a valid way to decide validity in >> intuitionistic analysis. > >I am talking about the net result - the output - the wffs >(formulas, sentences) that are ultimately proven. Yes, and case analysis is *invalid* for intuitionistic logic. It proves intuitionistically *incorrect* formulas. -- Daryl McCullough Ithaca, NY
From: David C. Ullrich on 30 Dec 2005 05:55 On 29 Dec 2005 12:08:43 -0800, "Charlie-Boo" <chvol(a)aol.com> wrote: >David C. Ullrich wrote: >> On 20 Dec 2005 05:35:47 -0800, "Charlie-Boo" wrote: > >> > 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. > >It's just a cop-out to talk about variables with no meaning rather than >actual assertions about something. You're a funny guy. You post a "theorem" that's obviously false. Of course you include a "proof" of the obviously false theorem, which of course says something about your understanding of what a proof is, and something about why people just laugh when you claim to have proved other things without including the proof. But I digress. You post a "proof" of an _obviously_ false theorem. Where "obviously false" means that it takes no effort whatever to find a counterexample, counterexamples are simply obvious. And in reply to an utterly simple counterexample all you can say is that it's not a very good counterexample because it's too trivial. Huh. Hint: An utterly trivial counterexample here is better because it better illustrates how _obviously_ false the theorem was. Another hint: Credibility-wise it doesn't seem like a good idea for you to point out that a "Godel sentence" gives a counterexample. Because all this started with your amazing profundities about Godel this and Godel that - if I were you I'd hope that nobody _noticed_ that what I was saying was obviously inconsistent with Godel's theorems. >By that logic, you can say that a >system is incomplete because it has a propositional variable with no >meaning and thus no proof of it or its negation. Yes, if the question were whether propositional logic was complete the answer would be obviously not, because an atomic formula is neither a tautology nor an anti-tautology. >That's not what we're talking about. Giggle. Translation: Let's not talk about the really really obvious counterexamples to my assertions. Instead let's restrict discussion to counterexamples that depend on non-trivial results in logic - that way we'll still sound smart, even though what we said was obviously wrong. Tee-hee. >C-B > >> ************************ >> >> David C. Ullrich ************************ David C. Ullrich
From: H. J. Sander Bruggink on 30 Dec 2005 07:25 Charlie-Boo wrote: > H. J. Sander Bruggink wrote: > >>Charlie-Boo wrote: >> >>>H. J. Sander Bruggink wrote: > > >>>>Please show, by a "case analysis", that P->P is >>>>intuitionistically valid. >>> >>>I didn't say anything about "intuitionistically valid". (Got it? >>> Good!) >> >>No of course not. You didn't even know what it was. >>(I suppose you looked it up in wikipedia, by now, right?) > > You are attacking the messenger.I will not be dragged into that. What are you the messenger of, then? > Just answer the mathematical question. There was no mathematical question in the part I replied to, so what are you talking about? > (Actually, I usually use > mathworld.) Ok, mathworld is fine. >>>I said you could prove using case analysis any propositional >>>calculus wff that can be proven using the various rules of inference. >> >>Yes, you said that, and because you said it in a subthread >>about *intuitionistic* propositional logic, and even explicitly >>denied that you were talking about *classical* propositional >>logic, what you said was WRONG. > > > I denied that? Yes you did: [quote author="Charlie-Boo"] Torkel Franzen wrote: > "Charlie-Boo" <chvol(a)aol.com> writes: > >> What do you disagree with in the above? > > You mistakenly take the argument to be about classical > propositional logic. So "~(A<->~A) is a propositional calculus wff" is wrong? (I never used the word "classical".) C-B [/quote] [snip] >>Because it isn't true. It's only true for *classical* >>propositional logic, and you explicitly denied that you were >>talking about that. > > Quote, please. (Then what was I talking about?) See above. [snip] groente -- Sander
From: H. J. Sander Bruggink on 30 Dec 2005 07:33
Charlie-Boo wrote: > Thanks. These endless variations in syntax not only gain you nothing, > they do even less than simple case analysis. Aah, so you think that logics that prove more theorems are always better? In that case, I suggest you look into logics known as "inconsistent logics", because you can prove anything you want in those logics. > > And still the idiots try to defend it. LOL And still those idiots try to defend classical logic. LOL. (You can't even prove P & ~P in that!) :-) groente -- Sander |