In a Consistent System, Can a True Sentence Imply a False One?
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From: Nam Nguyen on 30 Jul 2010 10:16 Aatu Koskensilta wrote: > 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. Which standard textbook/source did you have in mind that would show an example of a formula being both provable and disprovable in a 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 30 Jul 2010 10:45 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> 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. Please don't say "Right" when you're not fully aware of what has been said. What I said above includes "(not disprovable)" and what you said below includes "disprovable also"; so we're actually NOT in agreement here to warrant the word "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; Right. To be more precise, "unprovable" in technical definition is negating "provable". > "disprovable" is normally used as above -- > it does not mean the same thing as "unprovable". It actually is, in the context where it's supposed to be used: the context of a consistent theory. In such case, the set of disprovable formulas and the set of unprovable ones are _identical_ which is disjoint from the set of provable formulas. In the case of an inconsistent system, by sheer definition, the set of provable formulas is equal to the set of _all_ formulas. One is free in this case to rename _this one_ set anything one would like to, say "Bush-disprovable" "Obama-unprovable", "Clinton-provable-and-unprovable", .... But why would one need to come up with such terms, technically speaking? -- --------------------------------------------------- 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 30 Jul 2010 11:00 Marshall wrote: > 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. Don't tell me but tell Alan that: because that's what his definition would render in the case of an inconsistent theory! If whatever you said below makes sense then the audience wouldn't be Nam: I'm not the one who introduced (CM), defined (AS), or defended the word-usage "disprovable" in the context of an inconsistent theory! Fwiw, technically speaking, "provable", "unprovable" aren't context- sensitive definitions. But "disprovable", "refutable" are! (And all that is a trivial notion). > 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 -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: Marshall on 30 Jul 2010 12:42 On Jul 30, 8:00 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > 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. > > Don't tell me but tell Alan that: because that's what his definition > would render in the case of an inconsistent theory! > > If whatever you said below makes sense then the audience wouldn't > be Nam Oh please! Don't make it THAT easy. It should be a *little* challenge. Marshall
From: Nam Nguyen on 30 Jul 2010 12:56
Marshall wrote: > On Jul 30, 8:00 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> 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. >> Don't tell me but tell Alan that: because that's what his definition >> would render in the case of an inconsistent theory! >> >> If whatever you said below makes sense then the audience wouldn't >> be Nam > > Oh please! Don't make it THAT easy. It should be a *little* challenge. Like whatever you've said here really has any technical merits. As usual. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt ----------------------------------------------------------- |