In a Consistent System, Can a True Sentence Imply a False One?
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From: Nam Nguyen on 28 Jul 2010 22:07 Chris Menzel wrote: > On Tue, 27 Jul 2010 19:36:54 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> said: >> Chris Menzel wrote: >>> On Sat, 24 Jul 2010 00:27:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> >>> said: >>>> ... >>>> Surely you must know, for example, it's impossible to disprove a >>>> formula by rules of inference, because rules of inference can only >>>> lead to proof, not to a disproof. That's how! >>> You don't think a proof of ~A is simultaneously a disproof of A? >>> >> Of course not because in general there cases where there are proofs >> for _both_ A and ~A. > > Then you simply have an inconsistent system in which everything is both > provable and disprovable. So what? Actually, what does it mean for a formula to be "disprovable" in an consistent formal system? -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Nam Nguyen on 28 Jul 2010 22:10 Nam Nguyen wrote: > Chris Menzel wrote: >> On Tue, 27 Jul 2010 19:36:54 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> >> said: >>> Chris Menzel wrote: >>>> On Sat, 24 Jul 2010 00:27:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> >>>> said: >>>>> ... >>>>> Surely you must know, for example, it's impossible to disprove a >>>>> formula by rules of inference, because rules of inference can only >>>>> lead to proof, not to a disproof. That's how! >>>> You don't think a proof of ~A is simultaneously a disproof of A? >>>> >>> Of course not because in general there cases where there are proofs >>> for _both_ A and ~A. >> >> Then you simply have an inconsistent system in which everything is both >> provable and disprovable. So what? > > Actually, what does it mean for a formula to be "disprovable" in an > consistent > formal system? Sorry for a typo: I meant to ask: > Actually, what does it mean for a formula to be "disprovable" in an > _inconsistent_ formal system? -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Nam Nguyen on 29 Jul 2010 00:55 Nam Nguyen wrote: > Chris Menzel wrote: >> On Tue, 27 Jul 2010 19:36:54 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> >> said: >>> Chris Menzel wrote: >>>> On Sat, 24 Jul 2010 00:27:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> >>>> said: >>>>> ... >>>>> Surely you must know, for example, it's impossible to disprove a >>>>> formula by rules of inference, because rules of inference can only >>>>> lead to proof, not to a disproof. That's how! >>>> You don't think a proof of ~A is simultaneously a disproof of A? >>>> >>> Of course not because in general there cases where there are proofs >>> for _both_ A and ~A. >> >> Then you simply have an inconsistent system in which everything is both >> provable and disprovable. So what? > > So: > > a) However much we _believe_ a formal system be consistent, especially > one that is "sufficiently complex" (whatever that might mean) there's > always a chance our belief might turn out to be simply incorrect! > > Iow, we shouldn't trust intuition and what's "clearly evident" too much! > > b) There's a chance that it's impossible (even in principle) to determine > if a formula is provable in any extension of a formal system and the > underlying extension is still consistent! and c) The concept of the natural numbers is a relative concept: at least relative to whether or not a certain formulas, e.g. pGC, are true or false. -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Alan Smaill on 29 Jul 2010 06:00 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Sorry for a typo: I meant to ask: > >> Actually, what does it mean for a formula to be "disprovable" in an >> _inconsistent_ formal system? That there is a proof of its negation. -- Alan Smaill
From: Aatu Koskensilta on 29 Jul 2010 06:24
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > b) There's a chance that it's impossible (even in principle) to determine > if a formula is provable in any extension of a formal system and the > underlying extension is still consistent! "There's a chance"? We know that in general the problem of determining whether a formula is provable in a theory is undecidable. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |