Simple yet Profound Metatheorem
in [Logic]
|
Prev: Obections to Cantor's Theory (Wikipedia article)
Next: Why Has None of Computer Science been Formalized?
From: Charlie-Boo on 21 Dec 2005 01:48 Daryl McCullough wrote: > Charlie-Boo says... > >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, it's the other way around. There are formulas that are provable > using truth tables (case analysis) but are not provable > intuitionistically. The examples are > > Excluded Middle: A or ~A > Pierce's Law: ((P -> Q) -> P) -> P Thanks. These endless variations in syntax not only gain you nothing, they do even less than simple case analysis. And still the idiots try to defend it. LOL > Daryl McCullough > Ithaca, NY
From: Charlie-Boo on 21 Dec 2005 01:59 Torkel Franzen wrote: > "Charlie-Boo" <chvol(a)aol.com> writes: > > That has no relevance. > A penetrating observation! Now you only need to take one step > further. 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. If one understands case analysis, they know that you can't, as case analysis will reveal any conclusions there may be. C-B
From: Torkel Franzen on 21 Dec 2005 02:06 "Charlie-Boo" <chvol(a)aol.com> writes: > No I don't. My original point remains undisputed. It is indisputable.
From: H. J. Sander Bruggink on 21 Dec 2005 06:32 Charlie-Boo wrote: > H. J. Sander Bruggink wrote: >>Here's an intuitionistic proof: >> >>1. | P >> |---- >>2. | P (rep) >>3. P -> P (->I) >> >>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?) > 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. > P => P is ~P v P This is not a valid equivalence in IL. > > P ~P ~P v P > > true false true > false true true > > See, you can prove P => P using case analysis, as I said. No, you didn't prove anything, because proof tables are not valid for IL. (In fact, ~P v P isn't valid in IL). (But at least three people mentioned this already, right, so you could have expected this answer.) > > Now why don't YOU admit that? Because it isn't true. It's only true for *classical* propositional logic, and you explicitly denied that you were talking about that. > 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. groente -- Sander
From: Daryl McCullough on 21 Dec 2005 06:32
Charlie-Boo says... > >Daryl McCullough wrote: >> No, it's the other way around. There are formulas that are provable >> using truth tables (case analysis) but are not provable >> intuitionistically. The examples are >> >> Excluded Middle: A or ~A >> Pierce's Law: ((P -> Q) -> P) -> P > >Thanks. These endless variations in syntax not only gain you nothing, >they do even less than simple case analysis. What "variations in syntax" are you talking about? 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. The inuitionistic meaning of P -> Q is *not* the same as the intuitionistic meaning of ~P or Q so converting the first into the second and performing a case analysis doesn't make any sense, intuitionistically. Case analysis is valid intuitionistically provided that for every atomic proposition P we can prove P or ~P. -- Daryl McCullough Ithaca, NY |