Meaning, Presuppositions, Truth-relevance, Goedel's Theorem and the Liar Paradox
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From: Jesse F. Hughes on 12 Aug 2010 12:30 Newberry <newberryxy(a)gmail.com> writes: > On Aug 12, 6:41 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> "Jesse F. Hughes" <je...(a)phiwumbda.org> writes: >> >> > Newberry <newberr...(a)gmail.com> writes: >> >> >> Goedel's sentence is not true because it is vacuous, and we do not >> >> regard vacuous sentences as true. >> >> > It is funny, then, that the overwhelming majority of respondents here >> > *do* regard vacuous sentences like Goedel's theorem as true. >> >> What's vacuous about Gödel's theorem or the Gödel sentence of a theory? > > Nothing vacuous about Gödel's theorem. At least I would not put it > that way. Yes, I meant Goedel's sentence, not theorem. Sorry. -- "By initially making it virtually impossible to maintain a heterogenous environment of Word 95 and Word 97 systems, Microsoft offered its customers that most eloquent of arguments for upgrading: the delicate sound of a revolver being cocked somewhere just out of sight." --Dan Martinez
From: Jesse F. Hughes on 12 Aug 2010 12:54 Newberry <newberryxy(a)gmail.com> writes: >> > Wow! We are making a lot of progress. (Actually it is ~(T v F) >> > regardless if Qm is true.) >> >> No, we're not. > > It is disappointing. You need to read chapter 2.2. Oh, I recall (now) your odd beliefs about vacuity. I just think they're unmotivated. Take a simple deductive rule: v-elimination, that is: |- P v Q |- ~Q --------------- |- P. It seems to me that this rule is very hard to understand, given your ideas. As soon as I prove that P is true, P v Q is neither true nor false ... despite the fact that I have (by assumption) proved P v Q prior to deducing P. I suppose I have to retroactively judge that my purported proof of P v Q was not a proof after all, since P v Q is not true. -- Jesse F. Hughes "[I]f gravel cannot make itself into an animal in a year, how could it do it in a million years? The animal would be dead before it got alive." --The Creation Evolution Encyclopedia
From: Jesse F. Hughes on 13 Aug 2010 09:00 Newberry <newberryxy(a)gmail.com> writes: > On Aug 12, 9:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> >> > Wow! We are making a lot of progress. (Actually it is ~(T v F) >> >> > regardless if Qm is true.) >> >> >> No, we're not. >> >> > It is disappointing. You need to read chapter 2.2. >> >> Oh, I recall (now) your odd beliefs about vacuity. I just think >> they're unmotivated. >> >> Take a simple deductive rule: v-elimination, that is: >> >> |- P v Q |- ~Q >> --------------- >> |- P. >> >> It seems to me that this rule is very hard to understand, given your >> ideas. As soon as I prove that P is true, P v Q is neither true nor >> false ... despite the fact that I have (by assumption) proved P v Q >> prior to deducing P. I suppose I have to retroactively judge that my >> purported proof of P v Q was not a proof after all, since P v Q is not >> true. > > Truth-relevant logic is not classicl logic, and this rule is probably > not compatible with it. I think you need to reconsider this judgment. The same reasoning applies to, for instance, Modus Ponens. If P -> Q is provable and P is provable, then Q is provable and hence Q is necessarily true. But if Q is necessarily true, then P -> Q is vacuously true (right?) and hence neither true nor false. Similarly for Modus Tollens. So, it seems as if MP and MT are not rules of your truth-relevant logic. Does that seem right to you? -- Jesse F. Hughes "[M]oving towards development meetings for new release class viewer 5.0 and since [I]'m the only developer, easy to schedule." --James S. Harris tweets on code development
From: Jesse F. Hughes on 13 Aug 2010 11:12
Newberry <newberryxy(a)gmail.com> writes: > On Aug 13, 6:00 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> > On Aug 12, 9:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> Newberry <newberr...(a)gmail.com> writes: >> >> >> > Wow! We are making a lot of progress. (Actually it is ~(T v F) >> >> >> > regardless if Qm is true.) >> >> >> >> No, we're not. >> >> >> > It is disappointing. You need to read chapter 2.2. >> >> >> Oh, I recall (now) your odd beliefs about vacuity. I just think >> >> they're unmotivated. >> >> >> Take a simple deductive rule: v-elimination, that is: >> >> >> |- P v Q |- ~Q >> >> --------------- >> >> |- P. >> >> >> It seems to me that this rule is very hard to understand, given your >> >> ideas. As soon as I prove that P is true, P v Q is neither true nor >> >> false ... despite the fact that I have (by assumption) proved P v Q >> >> prior to deducing P. I suppose I have to retroactively judge that my >> >> purported proof of P v Q was not a proof after all, since P v Q is not >> >> true. >> >> > Truth-relevant logic is not classicl logic, and this rule is probably >> > not compatible with it. >> >> I think you need to reconsider this judgment. The same reasoning >> applies to, for instance, Modus Ponens. >> >> If P -> Q is provable and P is provable, then Q is provable and hence >> Q is necessarily true. But if Q is necessarily true, then P -> Q is >> vacuously true (right?) and hence neither true nor false. >> >> Similarly for Modus Tollens. >> >> So, it seems as if MP and MT are not rules of your truth-relevant >> logic. Does that seem right to you? > > Theorem: The rule of modus ponens is compatible with TR: > |- A, |- A -> B > ----------------- > |- B > Proof: > A -> B is equivalent to. ~A v B. A and ~A v B are t-relevant by > hypothesis. ~A is false. > Assume B is not t-r. Then there exist a variable q such that q = U and > B = T (for all > possible valuations of B.) Then ~A v B = T contrary to the assumption. > This is the case even if q is relevant in A. Then for q = U, A can > take either U or T. But in either case ~A v B = T. Hence q is not > relevant in ~A v B. QED. Seems to me that your above argument uses v-elimination, which you just suggested is not compatible with T-relevant logics. In any case, I thought that if Q is necessarily true, then P -> Q is vacuous. Is this incorrect? -- "Every major result I have has been preceded by lots of stupid mistakes. And I mean REALLY stupid mistakes. So making stupid mistakes tells me that I'm in pursuit of something." -- James S. Harris |