Meaning, Presuppositions, Truth-relevance, Godel's Theorem and
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From: Daryl McCullough on 8 Aug 2010 14:03 Newberry says... > >On Aug 8, 9:43=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> So to say that the Godel sentence is neither true nor false means >> that it is neither true nor false that PA is consistent. That is >> nonsensical. > >The paper is NOT about PA. That doesn't matter. For any theory with a notion of a Godel sentence, the same argument applies. -- Daryl McCullough Ithaca, NY
From: Newberry on 8 Aug 2010 15:55 On Aug 8, 11:03 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Aug 8, 9:43=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> So to say that the Godel sentence is neither true nor false means > >> that it is neither true nor false that PA is consistent. That is > >> nonsensical. > > >The paper is NOT about PA. > > That doesn't matter. For any theory with a notion of a Godel sentence, > the same argument applies. Proof?
From: Daryl McCullough on 9 Aug 2010 08:22 Newberry says... > >On Aug 8, 11:03=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >> >> >On Aug 8, 9:43=3DA0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrot= >e: >> >> So to say that the Godel sentence is neither true nor false means >> >> that it is neither true nor false that PA is consistent. That is >> >> nonsensical. >> >> >The paper is NOT about PA. >> >> That doesn't matter. For any theory with a notion of a Godel sentence, >> the same argument applies. > >Proof? If a theory is inconsistent, then the corresponding Godel sentence is false (because it asserts the unprovability of a certain formula, and every sentence is provable in an inconsistent theory). Therefore, to claim that the Godel sentence is not false is to claim that the corresponding theory is consistent. The Godel sentence is provably equivalent to the claim that the theory is consistent. So the claim that the Godel sentence is not false implies that the theory is consistent, which implies that the Godel sentence is true. Claiming that the Godel sentence is neither true nor false is contradictory. -- Daryl McCullough Ithaca, NY
From: Newberry on 9 Aug 2010 09:18 On Aug 9, 5:22 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >On Aug 8, 11:03=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> Newberry says... > > >> >On Aug 8, 9:43=3DA0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrot= > >e: > >> >> So to say that the Godel sentence is neither true nor false means > >> >> that it is neither true nor false that PA is consistent. That is > >> >> nonsensical. > > >> >The paper is NOT about PA. > > >> That doesn't matter. For any theory with a notion of a Godel sentence, > >> the same argument applies. > > >Proof? > > If a theory is inconsistent, then the corresponding Godel sentence > is false (because it asserts the unprovability of a certain formula, > and every sentence is provable in an inconsistent theory). This is not the case for every kind of logic. > Therefore, > to claim that the Godel sentence is not false is to claim that the > corresponding theory is consistent. The Godel sentence is provably > equivalent to the claim that the theory is consistent. > > So the claim that the Godel sentence is not false implies that > the theory is consistent, which implies that the Godel sentence is > true. > > Claiming that the Godel sentence is neither true nor false is > contradictory. > > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text -
From: Daryl McCullough on 9 Aug 2010 09:30
Newberry says... >On Aug 9, 5:22=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> If a theory is inconsistent, then the corresponding Godel sentence >> is false (because it asserts the unprovability of a certain formula, >> and every sentence is provable in an inconsistent theory). > >This is not the case for every kind of logic. In any case, if a sentence asserts the unprovability of something, and that something is unprovable, then the sentence asserts a truth. To say otherwise is to divorce truth from meaning. -- Daryl McCullough Ithaca, NY |