Meaning, Presuppositions, Truth-relevance, Goedel's Theorem and
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From: Jesse F. Hughes on 12 Aug 2010 12:56 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > 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. Well, almost. He says the conjunction of a *necessarily* (not contingently) true sentence with a false sentence is neither true nor false. -- Jesse F. Hughes. Me: It's very sad when one's husband or wife dies. Quincy (Age 4 1/2): Yeah. You might want to tell them something and you just can't. [Long pause] Like "Take out the trash."
From: Daryl McCullough on 12 Aug 2010 13:15 Jesse F. Hughes says... >stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > >> 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. > >Well, almost. > >He says the conjunction of a *necessarily* (not contingently) true >sentence with a false sentence is neither true nor false. Well, in the case of arithmetic, presumably everything that is true is necessarily true, right? Everything that is false is necessarily false. -- Daryl McCullough Ithaca, NY
From: Newberry on 12 Aug 2010 23:30 On Aug 12, 10:15 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Jesse F. Hughes says... > > >stevendaryl3...(a)yahoo.com (Daryl McCullough) writes: > > >> 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. > > >Well, almost. > > >He says the conjunction of a *necessarily* (not contingently) true > >sentence with a false sentence is neither true nor false. > > Well, in the case of arithmetic, presumably everything that is true > is necessarily true, right? Everything that is false is necessarily > false. Right, that is why >~(Qm & (Ex)Pxm) is ~(T v F) if (Ex)Pxm is false. I recommend reading section 2.2. Truth-relevance is a very simple concept. > > -- > Daryl McCullough > Ithaca, NY
From: Newberry on 12 Aug 2010 23:37 On Aug 12, 8:35 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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? Because ~(Ex)Pxm. The sentence above is equivalent to (x)(Pxm -> ~Qm) It is clearly "vacuously true" (in classcal logic), is it not? And as such it is equivalen to (x)((x # x) -> ~Qm) > > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text -
From: Daryl McCullough on 13 Aug 2010 07:03
Newberry says... > >On Aug 12, 8:35=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> 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? > >Because ~(Ex)Pxm. The sentence above is equivalent to > >(x)(Pxm -> ~Qm) > >It is clearly "vacuously true" (in classcal logic), is it not? It seems to me that all true formulas of arithmetic are vacuously true. It's not an interesting concept. -- Daryl McCullough Ithaca, NY |