Meaning, Presuppositions, Truth-relevance, Goedel's Theorem andthe Liar Paradox
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From: Newberry on 12 Aug 2010 09:08 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.) > > But you weren't quite explicit in your answer. Are the two formulas > > ~(Qm & (Ex)Pxm) and ~(Ex)(Pxm & Qm) > > equivalent or not? (Or are there situations in which they are > equivalent and other situations in which they are not?) Since they are both ~(T v F) they are obviously equivalent. > -- > One these mornings gonna wake | Ain't nobody's doggone business how > up crazy, | my baby treats me, > Gonna grab my gun, kill my baby. | Nobody's business but mine. > Nobody's business but mine. | -- Mississippi John Hurt- Hide quoted text - > > - Show quoted text -
From: Newberry on 12 Aug 2010 09:24 On Aug 12, 6:05 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > "Jesse F. Hughes" <je...(a)phiwumbda.org> writes: > > > > > > > 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). > > ^^^^^^^^ > > As Daryl points out, it should be ~(T & F) we all agree to, rather > than ~(T v F). It should be ~(T v F). According to the principles of truth-relevant logic if A is necessarily false the ~(A & B) is ~(T v F). PLease see section 2.2 of my paper. > > > But you weren't quite explicit in your answer. Are the two formulas > > > ~(Qm & (Ex)Pxm) and ~(Ex)(Pxm & Qm) > > > equivalent or not? Yep. > (Or are there situations in which they are > > equivalent and other situations in which they are not?) > > -- > Jesse F. Hughes > > "[M]eta-goedelisation as the essence of the globalised dictatorship by > denial of sense." -- Ludovico Van makes some sort of point.- Hide quoted text - > > - Show quoted text -
From: Newberry on 12 Aug 2010 10:13 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. Let ~(Ex)(Ey)(Pxy & Qy) (G) be Gödel's sentence, where Pxy means x is the proof of y, and only one y = m satisfies Q, m being the Gödel number of G. I will now simplify for the sake of brevity. (More details in Section 3 of my paper.) Let us pick y = m. We obtain ~(Ex)(Pxm & Qm) The above is vacuous ("vacuously true" according to classical logic) since ~(Ex)Pxm. Let's not get off on a tangent and argue that we do not know that ~(Ex)Pxm. My argument does not hinge on that. IF ~Pxm then ~(Ex)(Pxm & Qm) is vacuous. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Newberry on 12 Aug 2010 10:20 On Aug 12, 6:36 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > 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. > Something wrong with pretty much everyone but you, huh? Do you think that the truth of mathematics should determined by the majority vote of the members of sci.logic? What if we put the the question Is the folowing sentence true "All unicorns have two horns"? to the general population? > > -- > "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: Newberry on 12 Aug 2010 10:22
On Aug 12, 6:20 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(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. It is disappointing. You need to read chapter 2.2. > 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- Hide quoted text - > > - Show quoted text - |