Meaning, Presuppositions, Truth-relevance, G�del's Theorem and the Liar Paradox
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From: Daryl McCullough on 9 Aug 2010 08:24 In article <6b4407fe-f953-48eb-a22c-f9ab8de8519c(a)g6g2000pro.googlegroups.com>, Newberry says... > >On Aug 8, 1:54=A0pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> > This sentence: >> >> > =A0 =A0 ~(Ex)(Ey)(Pxy & Qy). =A0 =A0 =A0 (3.3.1) >> >> > Pxy means that x is the proof of y, where x and y are G=F6del numbers o= >f >> > wffs or sequences of wffs. Q has been constructed such that only one y >> > =3D m satisfies it, and m is the G=F6del number of (3.3.1). >> >> Proof in what theory? > >How does it matter? If it doesn't matter, then why would you object to my assuming that we are talking about PA? I've run rings around you, logically. -- Daryl McCullough Ithaca, Ny
From: Daryl McCullough on 9 Aug 2010 09:33 Newberry says... >The question was what is Goedel sentence. The formula I exhibited is >Goedel's formula in many kinds of logic including PA. If it is sufficiently similar to the Godel formula for PA, then it is nonsensical to say that it is neither true nor false. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 9 Aug 2010 10:59 In article <3f8cee92-6dbb-4f44-beb7-0878d02b9b8d(a)p11g2000prf.googlegroups.com>, Newberry says... > >On Aug 9, 6:33=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> Newberry says... >> >> >The question was what is Goedel sentence. The formula I exhibited is >> >Goedel's formula in many kinds of logic including PA. >> >> If it is sufficiently similar to the Godel formula for PA, then it >> is nonsensical to say that it is neither true nor false. > >Would you care to define "sufficiently similar" and show how your >conclusion follows? The main ideas behind Godel's proof is 1. Invent a coding for formulas so that every formula is associated with a natural number (or an element of whatever the domain of the theory is about) 2. Define a formula Pr(x) such that Pr(x) holds of a natural number x if and only if x is the code of a provable formula of whatever theory we are talking about. 3. Construct a sentence G such that G <-> ~Pr(#G) is a theorem, where #G means the code for G. 1-3 is what I consider the essential features of what it means for G to be a "Godel sentence". There a few details that can be tweaked---for instance, 3 presupposes that there are constant terms (e.g. numerals) for each element of the domain. That's not essential; instead, we can have a formula Q(x) such that G <-> Ax (Q(x) -> ~Pr(x)) and such that Q(x) holds if and only if x is the code for G. Anyway, in terms of 1-3, it is nonsensical to say that G is neither true nor false. G is a specific formula. If that formula is provable, then Pr(#G) holds (by definition, Pr(#G) holds if G is provable). But G is the negation of that formula. So G is the negation of a true sentence, and so is a false sentence. So if you say that G is not false, then it follows that G is not provable, and from that it follows that ~Pr(#G) is true, and from that, it follows that G is true. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 10 Aug 2010 06:55 Newberry says... >But it does not. Just because > >~(Ex)(Px#G) > >is true it does not follow that > >~(Ex)(Ey)(Pxy & Qy) (G) > >is true. In normal (nonstupid) semantics, it does follow. You can certainly make up whatever semantics you like for first order logic, but what is the point here? If there is only one number, #G, for which Qy holds, then Ey (Pxy & Qy) means the same thing as Px#G. Why in the *world* would you want that not to be the case? You seem to be going out of your way to block the usefulness of first-order logic for reasoning. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 10 Aug 2010 11:20
Newberry says... > >On Aug 10, 3:55=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> If there is only one number, #G, for which Qy holds, > >At least you are beginning to get the point. > >> then >> Ey (Pxy & Qy) means the same thing as Px#G. Why in the *world* >> would you want that not to be the case? > >Two reasons >a) ~(Ex)(Ey)(Pxy & Qy) is a hierarchy of vacuous sentences. In what sense is it vacuous? If there actually *is* a proof of the Godel sentence, then that statement would be provably *false*. So that statement *could* be false, so it certainly is not vacuously true. Maybe you mean that *if* it is true, then it is vacuously true. That's stretching the meaning of "vacuously true" beyond the breaking point. >Some of us think those are really not true. (1) It's not vacuous, and (2) vacuously true sentences are *true*. >b) We get a semantically consistent system. >> You seem to be going out of your way to block the usefulness >> of first-order logic for reasoning. > >How so? Let me go through an example. Suppose I don't know whether Goldbach's conjecture is true, or not. But I can prove the following two statements: (1) "If x is a counterexample to GC, then x is a multiple of 3" (2) "If x is a counterexample to GC, then x/2 is a prime number" In ordinary logic, (1) and (2) imply the conclusion: "There are no counterexamples to GC". Assuming GC is true, then (1) and (2) are vacuous. But those two vacuous statements imply the nonvacuous statement, Goldbach's conjecture. If you want to say that vacuous statements are not true, then presumably you are blocked from proving (1) or (2), because neither of those "lemmas" are true, in your semantics. Worse, since we don't know whether Goldbach's conjecture is true, or not, we don't know whether (1) and (2) are vacuous or not. So your semantics gets in the way of doing ordinary mathematical proofs. -- Daryl McCullough Ithaca, NY |