Meaning, Presuppositions, Truth-relevance, Goedel's Theorem and the Liar Paradox
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From: Jesse F. Hughes on 11 Aug 2010 09:59 Newberry <newberryxy(a)gmail.com> writes: > A lot of people are absolutely convinced that e.g. PA is consistent. > But anyway the point is that IF > ~(Ex)Pxm > then > ~(Ex)[Pxm & ((x = x &) Qm)] > is vacuous. I do not know what you are trying to argue here. By > "vacuous" I mean that the subject class is empty. That's a deep and exciting result, of course, since the formula ~(Ex)[Pxm & ((x = x) & Qm)] plays such an important and widespread role in the literature. But, back to the earlier formulas: ~(Qm & (Ex)Pxm) and ~(Ex)(Pxm & Qm) Are these two formulas equivalent, in your view? If so, the second formula is not vacuous, right? -- "Another factor one has got to look at is the amount of liquidity in the system. In other words, is there enough liquidity to enable markets to be able to correct? And I am told there is enough liquidity in the system to enable markets to correct." -- Guess who.
From: Newberry on 12 Aug 2010 00:12 On Aug 11, 6:59 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > A lot of people are absolutely convinced that e.g. PA is consistent. > > But anyway the point is that IF > > ~(Ex)Pxm > > then > > ~(Ex)[Pxm & ((x = x &) Qm)] > > is vacuous. I do not know what you are trying to argue here. By > > "vacuous" I mean that the subject class is empty. > > That's a deep and exciting result, of course, since the formula > > ~(Ex)[Pxm & ((x = x) & Qm)] > > plays such an important and widespread role in the literature. > > But, back to the earlier formulas: > > ~(Qm & (Ex)Pxm) and ~(Ex)(Pxm & Qm) > > Are these two formulas equivalent, in your view? If so, the second > formula is not vacuous, right? If (Ex)Pxm is necessarily false then according to the principles of truth-relevant logic ~(Qm & (Ex)Pxm) is ~(T v F). The reason is that it is analogous to ~(Q & (P & ~P)) Please see section 2.2 of my paper.
From: Jesse F. Hughes on 12 Aug 2010 09:20 Newberry <newberryxy(a)gmail.com> writes: > On Aug 11, 10:22 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> > On Aug 11, 6:59 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> Newberry <newberr...(a)gmail.com> writes: >> >> > A lot of people are absolutely convinced that e.g. PA is consistent. >> >> > But anyway the point is that IF >> >> > ~(Ex)Pxm >> >> > then >> >> > ~(Ex)[Pxm & ((x = x &) Qm)] >> >> > is vacuous. I do not know what you are trying to argue here. By >> >> > "vacuous" I mean that the subject class is empty. >> >> >> That's a deep and exciting result, of course, since the formula >> >> >> ~(Ex)[Pxm & ((x = x) & Qm)] >> >> >> plays such an important and widespread role in the literature. >> >> >> But, back to the earlier formulas: >> >> >> ~(Qm & (Ex)Pxm) and ~(Ex)(Pxm & Qm) >> >> >> Are these two formulas equivalent, in your view? If so, the second >> >> formula is not vacuous, right? >> >> > If (Ex)Pxm is necessarily false then according to the principles of >> > truth-relevant logic >> >> > ~(Qm & (Ex)Pxm) >> >> > is ~(T v F). The reason is that it is analogous to >> >> > ~(Q & (P & ~P)) >> >> > Please see section 2.2 of my paper. >> >> Surely, if (Ex)Pxm is necessarily false, then we all agree that >> ~(Qm & (Ex)Pxm) is ~(T v F) (given that Qm is true). > > Wow! We are making a lot of progress. (Actually it is ~(T v F) > regardless if Qm is true.) No, we're not. I realize now that you meant the formula is neither true nor false. I thought you meant that the formula evaluates to ~(T v F) (but, of course, it really evaluates to ~(T & F). -- "Now I'm informing all of you that the people arguing against me are EVIL, yes they are real, live EVIL people as mathematics is that important, so it's important enough for Evil itself to send minions like them." -- James Harris on Evil's interest in Algebraic Number Theory
From: Jesse F. Hughes on 12 Aug 2010 09:36 Newberry <newberryxy(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. Something wrong with pretty much everyone but you, huh? -- "I've noticed [...] I routinely have been putting up flawed equations with my surrogate factoring work. My take on it is that I have some deep fear that the work is too dangerous and am sabotaging myself." -- James S. Harris
From: Aatu Koskensilta on 12 Aug 2010 09:41
"Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > Newberry <newberryxy(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 G�del's theorem or the G�del sentence of a theory? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |