Simple yet Profound Metatheorem
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From: Charlie-Boo on 20 Dec 2005 15:09 H. J. Sander Bruggink wrote: > Charlie-Boo wrote: > > One example will do. What is the propositional calculus wff that you > > prove? > 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!) I said you could prove using case analysis any propositional calculus wff that can be proven using the various rules of inference. P => P is ~P v P P ~P ~P v P true false true false true true See, you can prove P => P using case analysis, as I said. Now why don't YOU admit 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? C-B > groente > -- Sander
From: Daryl McCullough on 20 Dec 2005 15:28 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 -- Daryl McCullough Ithaca, NY
From: G. Frege on 20 Dec 2005 18:10 On 20 Dec 2005 12:28:29 -0800, stevendaryl3016(a)yahoo.com (Daryl McCullough) wrote: > > There are formulas that are provable [not really --F.] using truth > tables but are not provable intuitionistically. [Simple] examples > are > > Excluded Middle: A v ~A Double Negation: ~~A -> A > Peirce's Law: ((P -> Q) -> P) -> P > F. -- "I do tend to feel Hughes & Cresswell is a more authoritative source than you." (David C. Ullrich)
From: Charlie-Boo on 21 Dec 2005 01:27 Torkel Franzen wrote: > "Charlie-Boo" <chvol(a)aol.com> writes: > > Please give a proof of a propositional calculus proposition that cannot > > be done using case analysis (examination of the truth tables.) > There are no truth tables for intuitionistic propositional logic. That has no relevance. Please give a proof of a propositional calculus proposition that cannot be done using case analysis (examination of the truth tables) or admit there are none. C-B
From: Torkel Franzen on 21 Dec 2005 01:38
"Charlie-Boo" <chvol(a)aol.com> writes: > That has no relevance. A penetrating observation! Now you only need to take one step further. |