Meaning, Presuppositions, Truth-relevance, Godel's Theorem and
in [Math]
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From: Newberry on 9 Aug 2010 10:20 On Aug 9, 6:30 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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. But the sentence does NOT assert the provability of something.
From: Daryl McCullough on 9 Aug 2010 11:04 Newberry says... > >On Aug 9, 6:30=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> Newberry says... >> >> >On Aug 9, 5:22=3DA0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrot= >e: >> >> 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. > >But the sentence does NOT assert the provability of something. I said "unprovability". The Godel sentence asserts the unprovability of something. -- Daryl McCullough Ithaca, NY
From: Newberry on 9 Aug 2010 23:45 On Aug 9, 8:04 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >On Aug 9, 6:30=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> Newberry says... > > >> >On Aug 9, 5:22=3DA0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrot= > >e: > >> >> 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. > > >But the sentence does NOT assert the provability of something. > > I said "unprovability". The Godel sentence asserts the unprovability > of something. I meant to say that the sentence does not assert the unprovability of something. ~(Ex)Px#G does asserts the unprovability of G, but ~(Ex)(Ey)(Pxy & Qy) does not.
From: Daryl McCullough on 10 Aug 2010 06:38 Newberry says... >What I am saying is that >a) you can do arithmetic in the logic of presuppositions >b) you get a more well behaved system There *could* be a point of doing arithmetic with presuppositions, especially if you wanted to allow possibly non-denoting terms (e.g., if you wanted to formalize statements such as "the smallest counterexample to Goldbach's Conjecture is a multiple of 3"---that presupposes the existence of a counterexample). But the Godel statement is not that type of claim. As for the second point, what kind of misbehavior are you talking about? Has classical logic been knocking over garbage cans in your neighborhood? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 10 Aug 2010 06:48
Newberry says... > >On Aug 9, 8:04=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> >> 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. >> >> >But the sentence does NOT assert the provability of something. >> >> I said "unprovability". The Godel sentence asserts the unprovability >> of something. > >I meant to say that the sentence does not assert the unprovability of >something. >~(Ex)Px#G does asserts the >unprovability of G, but >~(Ex)(Ey)(Pxy & Qy) does not. Well, it is logically equivalent to Ay (Qy -> ~Ex Pxy) "Any y such that Qy holds is unprovable". You want to mess with the semantics of first-order logic so that these two formulas are not equivalent. WHY? In any case, the Godel sentence is the latter, which definitely says something about provability. If you want to have a logic of presupposition, then make your presuppositions explicit. Don't try to reverse-engineer them from the first-order logic. That's a truly weird thing to want to do. Again, you seem to be in the business of *muddying* things that are already clear. There are no hidden presuppositions in the Godel sentence (other than the existence of the naturals, I suppose). -- Daryl McCullough Ithaca, NY |