Meaning, Presuppositions, Truth-relevance, Goedel's Theorem and the Liar Paradox
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From: Jesse F. Hughes on 12 Aug 2010 01:22 Newberry <newberryxy(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). 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?) -- 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
From: Jesse F. Hughes on 12 Aug 2010 09:05
"Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > Newberry <newberryxy(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). > 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?) -- Jesse F. Hughes "[M]eta-goedelisation as the essence of the globalised dictatorship by denial of sense." -- Ludovico Van makes some sort of point. |