In a Consistent System, Can a True Sentence Imply a False One?
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From: Jesse F. Hughes on 29 Jul 2010 14:44 Chris Menzel <cmenzel(a)remove-this.tamu.edu> writes: > On Thu, 29 Jul 2010 08:05:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> said: >> Alan Smaill wrote: >>> 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. >> >> So an inconsistent system is also a consistent one then? > > Obviously not. Since the question has such an obvious answer, I'm > guessing that your intention here is to imply that Alan's answer leads > to the absurd conclusion that a system can be simultaneously consistent > and inconsistent. It doesn't, of course, but why do you think it does? > Probably because he's confusing "disprovable" (which I assume means "refutable") with "not provable". In fact, here's what he says in reply: Because in a T if A is provable but ~A is "disprovable" then T should be consistent. "Provable" means having a proof and "disprovable" means otherwise, in the context of discussing (in)consistency of a system. -- "Customers have come to SCO asking what they can do to respect and help protect the rights of the SCO intellectual property in Linux. SCO has created the Intellectual Property License for Linux in response to these customers needs." -- SCO responds to needs.
From: Nam Nguyen on 29 Jul 2010 22:19 Jesse F. Hughes wrote: > Chris Menzel <cmenzel(a)remove-this.tamu.edu> writes: > >> On Thu, 29 Jul 2010 08:05:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> said: >>> Alan Smaill wrote: >>>> 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. >>> So an inconsistent system is also a consistent one then? >> Obviously not. Since the question has such an obvious answer, I'm >> guessing that your intention here is to imply that Alan's answer leads >> to the absurd conclusion that a system can be simultaneously consistent >> and inconsistent. It doesn't, of course, but why do you think it does? >> > > Probably because he's confusing "disprovable" (which I assume means > "refutable") with "not provable". Why was I the one who got confused when a) even you yourself only assumed (not asserted) "disprovable" would mean "refutable" and b) somebody else introduced (CM) or defined (AS) the term here which face-value seems to be ridiculous because the context is in an inconsistent system where *all formulas* are _provable_ by sheer technical definition, or consequences thereof? > > In fact, here's what he says in reply: > > Because in a T if A is provable but ~A is "disprovable" then T > should be consistent. "Provable" means having a proof and > "disprovable" means otherwise, in the context of discussing > (in)consistency of a system. So, CM still doesn't answer my question above: >>>>>> Actually, what does it mean for a formula to be "disprovable" in an >>>>>> _inconsistent_ formal system? and AS' answer wouldn't make much sense in this context of an inconsistent formal system: all formulas would be _both_ provable and disprovable! -- --------------------------------------------------- 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 30 Jul 2010 06:24 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Chris Menzel wrote: >> On Thu, 29 Jul 2010 08:05:52 -0600, Nam Nguyen <namducnguyen(a)shaw.ca> said: >>> Alan Smaill wrote: >>>> 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. >>> So an inconsistent system is also a consistent one then? >> >> Obviously not. Since the question has such an obvious answer, I'm >> guessing that your intention here is to imply that Alan's answer leads >> to the absurd conclusion that a system can be simultaneously consistent >> and inconsistent. It doesn't, of course, but why do you think it does? > > Because in a T if A is provable but ~A is "disprovable" then T should > be consistent. "Provable" means having a proof and "disprovable" means > otherwise, in the context of discussing (in)consistency of a system. > And Alan was making the definition for an inconsistent theory where all > formulas are supposed to be _provable_ (not disprovable). Right. In an inconsistent theory, all formulas are provable, and all formulas as disprovable also, in the sense I used above. > Of course one could rename something to anything, but that would be an odd > renaming. The term "unprovable" already exists; "disprovable" is normally used as above -- it does not mean the same thing as "unprovable". -- Alan Smaill
From: Aatu Koskensilta on 30 Jul 2010 07:18 Alan Smaill <smaill(a)SPAMinf.ed.ac.uk> writes: > The term "unprovable" already exists; "disprovable" is normally used > as above -- it does not mean the same thing as "unprovable". This is indeed standard usage in mathematical logic. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Marshall on 30 Jul 2010 08:07
On Jul 29, 7:19 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > ... AS' answer wouldn't make much sense in this context of an inconsistent > formal system: all formulas would be _both_ provable and disprovable! Both provable and disprovable! Why, that's hard to imagine. This suggests that such an "inconsistent" theory would be: 1. lacking in harmony between the different parts or elements; self-contradictory: an inconsistent story. 2. lacking agreement, as one thing with another or two or more things in relation to each other; at variance: a summary that is inconsistent with the previously stated facts. 3. not consistent in principles, conduct, etc.: He's so inconsistent we never know if he'll be kind or cruel. etc. Imagine that! A "self-contradictory" theory! We should come up with a name for that, if indeed anyone can demonstrate the possibility of such a thing. Marshall |