Meaning, Presuppositions, Truth-relevance, Godel's Theorem and
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From: Newberry on 10 Aug 2010 09:14 On Aug 10, 3:38 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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? It is semantically incomplete. > -- > Daryl McCullough > Ithaca, NY
From: Newberry on 10 Aug 2010 09:22 On Aug 10, 3:48 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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. To get a semantically complete system. > > 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. I am not following this. > > 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). Let's get this straight. I never said there were. I said we could do arithmetic in the logic of presuppositions. Then given a suitable valuation (Ex)Px#G (1) is the presupposition of ~(Ex)(Ey)(Pxy & Qy) (G) So if the negation (1) is true then (G) is neither true nor false. > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text -
From: Daryl McCullough on 10 Aug 2010 11:26 Newberry says... > >On Aug 10, 3:48=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> 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. > >To get a semantically complete system. What is the point of that? >I said we could do arithmetic in the logic of presuppositions. Then >given a suitable valuation >(Ex)Px#G (1) >is the presupposition of >~(Ex)(Ey)(Pxy & Qy) (G) Why in the WORLD would that be the case? >So if the negation (1) is true then (G) is neither true nor false. That sounds like a reducto ad aburdum for your system. You are basically saying that if sentence (G) is true, (That is, there is no pair (x,y) such that Pxy and Qy), then it is neither true nor false. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 10 Aug 2010 11:48 Newberry says... > >On Aug 10, 3:38=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> 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? > >It is semantically incomplete. It is a fact that there are statements that we don't have any procedure for answering. Given an arbitrary computer program computing a function on naturals, we cannot tell, in general, whether it will halt on input 0. That's a fact. The halting problem is not solvable. The semantic incompleteness of first-order logic is *unavoidable* given the unsolvability of the halting problem. If I axiomatize computer programs and naturals, then there will be statements of the form "Program P does not halt on input 0" that are not provable and not disprovable. -- Daryl McCullough Ithaca, NY
From: Newberry on 10 Aug 2010 23:38
On Aug 10, 8:48 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >On Aug 10, 3:38=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> 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? > > >It is semantically incomplete. > > It is a fact that there are statements that we don't have any > procedure for answering. Given an arbitrary computer program > computing a function on naturals, we cannot tell, in general, > whether it will halt on input 0. That's a fact. The halting > problem is not solvable. > > The semantic incompleteness of first-order logic is *unavoidable* > given the unsolvability of the halting problem. If I axiomatize > computer programs and naturals, then there will be statements of > the form "Program P does not halt on input 0" that are not provable > and not disprovable. Yet you are convinced that some of these unprovable sentences are true. I still do not understand how you arrive at the conclusion that something is true without deriving that it is true. |