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From: Daryl McCullough on 12 Aug 2010 08:18 Newberry says... >If (Ex)Pxm is necessarily false then according to the principles of >truth-relevant logic > >~(Qm & (Ex)Pxm) > >is ~(T v F). No, it's ~(T & F). >The reason is that it is analogous to > >~(Q & (P & ~P)) > >Please see section 2.2 of my paper. So you are saying that ~(Qm & (Ex) Pxm) IS vacuously true? Look, if m is the Godel number of G, and G is not provable, then (Ex) Pxm is false, right? Then (Qm & (Ex)Pxm) is also false, right? But its negation is neither true nor false? That's weird. -- Daryl McCullough Ithaca, NY
From: Newberry on 12 Aug 2010 09:12 On Aug 12, 5:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >If (Ex)Pxm is necessarily false then according to the principles of > >truth-relevant logic > > >~(Qm & (Ex)Pxm) > > >is ~(T v F). > > No, it's ~(T & F). > > >The reason is that it is analogous to > > >~(Q & (P & ~P)) > > >Please see section 2.2 of my paper. > > So you are saying that ~(Qm & (Ex) Pxm) IS vacuously true? I am saying it is ~(T v F). > > Look, if m is the Godel number of G, and G is not provable, > then (Ex) Pxm is false, right? Right. > Then (Qm & (Ex)Pxm) is also > false, right? Wrong. It is ~(T v F). > But its negation is neither true nor false? Right. > That's weird. > > -- > Daryl McCullough > Ithaca, NY
From: Daryl McCullough on 12 Aug 2010 11:35 Newberry says... > >On Aug 12, 6:41=A0am, 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 G=F6del's theorem or the G=F6del sentence of a theor= >y? > >Nothing vacuous about G=F6del's theorem. At least I would not put it >that way. > >Let > > ~(Ex)(Ey)(Pxy & Qy) (G) > >be G=F6del's sentence, where Pxy means x is the proof of y, and only one >y = m satisfies Q, m being the G=F6del 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. That's just bizarre. With the interpretation that Qm holds only if m is the Godel number of the Godel sentence G, then (Pxm & Qm) says "x is a code for a proof of the formula whose code is m and m is the code for G" which is just an indirect way of saying "x is code for a proof of G". So ~(Ex) (Pxm & Qm) is an indirect way of saying "There is no proof of G". Calling it vacuous is just bizarre. Why in the world would you want to do that? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 12 Aug 2010 11:50 Newberry says... > >On Aug 12, 5:18=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >If (Ex)Pxm is necessarily false then according to the principles of >> >truth-relevant logic >> >> >~(Qm & (Ex)Pxm) >> >> >is ~(T v F). >> >> No, it's ~(T & F). >> >> >The reason is that it is analogous to >> >> >~(Q & (P & ~P)) >> >> >Please see section 2.2 of my paper. >> >> So you are saying that ~(Qm & (Ex) Pxm) IS vacuously true? > >I am saying it is ~(T v F). Well, it's not. It's of the form ~(T & F). -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 12 Aug 2010 11:54
Newberry says... > >On Aug 12, 5:18=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >If (Ex)Pxm is necessarily false then according to the principles of >> >truth-relevant logic >> >> >~(Qm & (Ex)Pxm) >> >> >is ~(T v F). >> >> No, it's ~(T & F). >> >> >The reason is that it is analogous to >> >> >~(Q & (P & ~P)) >> >> >Please see section 2.2 of my paper. >> >> So you are saying that ~(Qm & (Ex) Pxm) IS vacuously true? > >I am saying it is ~(T v F). >> >> Look, if m is the Godel number of G, and G is not provable, >> then (Ex) Pxm is false, right? > >Right. > >> Then (Qm & (Ex)Pxm) is also >> false, right? > >Wrong. It is ~(T v F). Like Jesse, I misunderstood your notation. So you are saying that the conjunction of a true sentence with a false sentence is neither true nor false? That is completely bizarre. If I say: "Newberry posted a message to sci.logic, and then later robbed a bank" I've said something that is neither true nor false? You can't possibly be serious! -- Daryl McCullough Ithaca, NY |