Meaning, Presuppositions, Truth-relevance, Godel's Theorem and
in [Logic]
|
Prev: open challenge to Chandler Davis; Iain Davidson reminds one of Flath -- Euclid's IP proof #5.01 Correcting Math
Next: challenge to Chandler Davis to post a valid Euclid Infinitude of Primes proof to sci.math #5.02 Correcting Math
From: Newberry on 11 Aug 2010 00:22 On Aug 10, 8:26 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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? To simplify let us pick y = #G = m. We obtain ~(Ex)(Pxm & Qm) According to Strawson's logic of presupposition Pxm must be non-empty if the above is to be T v F. Hence (Ex)Pxm is a presupposition of the formula above. > >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. I am not saying that. I have already told you that my paper was not about classical logic.
From: Tim Little on 11 Aug 2010 00:42 On 2010-08-11, Newberry <newberryxy(a)gmail.com> wrote: > Yet you are convinced that some of these unprovable sentences are > true. They are only unprovable in the specific formal system being considered. There is no such thing as an unqualifed "unprovable sentence", as for every sentence there is always a system in which it is provable. - Tim
From: Daryl McCullough on 11 Aug 2010 07:59 Newberry says... > >On Aug 10, 8:48=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote >> 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. We know this because we have the following situation: If Program P actually *does* halt on input 0, then *eventually* we will be able to prove this fact. In other words, (P halts on 0) -> (there is a proof of "P halts on 0") The contrapositive of this is: ~(there is a proof of "P halts on 0") -> ~(P halts on 0) In other words (there is no proof of "P halts on 0") -> (P does not halt on 0) You want to disallow this kind of reasoning, which is basically the use of logic. That's why I said that your goal seems to be to block the use of logic. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 11 Aug 2010 08:46 Newberry says... > >On Aug 10, 8:48=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> 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. Suppose I flip a coin and don't show you the result. Then you know that *one* of the following statements is true: 1. The coin landed heads up. 2. The coin landed tails up. You have no way of knowing which one is true, but you know one of them is. The things that are provable are just different from the things that are true. There is no reason to think that they should be the same. It would be nice if they were, but why should you expect it? -- Daryl McCullough Ithaca, NY
From: Newberry on 11 Aug 2010 09:20
On Aug 11, 4:59 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Aug 10, 8:48=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote > >> 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. > > We know this because we have the following situation: > > If Program P actually *does* halt on input 0, then *eventually* > we will be able to prove this fact. > > In other words, > > (P halts on 0) -> (there is a proof of "P halts on 0") > > The contrapositive of this is: > > ~(there is a proof of "P halts on 0") -> ~(P halts on 0) > > In other words > > (there is no proof of "P halts on 0") -> (P does not halt on 0) > > You want to disallow this kind of reasoning, which is basically > the use of logic. That's why I said that your goal seems to be > to block the use of logic. I do not want to disalow anything. If this is how you prove that certain underivable sentences are true that is what I want to alow. > > -- > Daryl McCullough > Ithaca, NY |