'When Are Relations Neither True Nor False?
in [Math]
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From: Newberry on 28 Mar 2010 01:16 On Mar 27, 11:10 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > Daryl McCullough wrote: > >> Nam Nguyen says... > >>> Daryl McCullough wrote: > > >>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation > >>>> of the language of PA. > >>> So you've agreed "G(PA) can be arithmetically false"? > > >> It is false in nonstandard models of PA. > > > Why don't we make it more precise. When we say F, a formula written > > in the language of arithmetic, is true or false _by default_ we > > mean it's being arithmetically true or false: i.e. true or false > > in the natural numbers. So we're *not* talking about F is being > > true/false in any general kind of models here. > > You haven't (and can't) give me an effective way to use > this definition to decide truth or falsity in the natural numbers. But I can. In a system with gaps Tarski's theorem does not apply. We can then simply equate truth with provability. > > The point is Alan said he wouldn't know what an absolute truth > > of G(PA) would mean > > and I've implicitly defined it for him, and > > here is the explicit version: > > > There's _no other_ context in which the meta statement > > "G(PA) is arithmetically true in the natural number" would > > be false. > > I'm not obliged to accept any particular definition of yours. > > FWIW you can look at Bourbaki's account of Goedel's incompleteness > theorem to note that they studiously avoid saying that the goedel > sentence is true (arithaally, absolutely, or in any other way). > > > The question I was hoping you'd answer one way or another is > > whether or not there's a context in which the meta statement: > > > "G(PA) is arithmetically true in the natural number" > > > would be _false_ ? > > > If your answer is "yes", then the [arithmetically-in-the-natural- > > number] truth of G(PA) is a relative notion. Otherwise it's an > > absolute notion. > > Why should I believe the only answers are no or yes? > Your realist assumptions are showing. > > > Which answer would you have? And perhaps why? > > you need a bit more Zen. > > -- > Alan Smaill- Hide quoted text - > > - Show quoted text -
From: Daryl McCullough on 28 Mar 2010 07:37 Newberry says... > >On Mar 27, 4:56=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >L: ~T(L) >> >> >If v(L) =3D ~(T v F) then there is no contradiction. L is not true. >> >> But *if* T is a truth predicate, then "L is not true" is formalized >> by the statement ~T(L). >> >> >The argument usually goes "but that is what L says." But L does not >> >say anything. >> >> It says "L is not true". >> >> So your proposed resolution is complete nonsense. > >It contains the string "L is not true", but it does not "say" that L >is not true That's completely silly. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 28 Mar 2010 08:50 Newberry <newberryxy(a)gmail.com> writes: > On Mar 27, 6:26 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> > On Mar 26, 3:50 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> Newberry <newberr...(a)gmail.com> writes: >> >> > On Mar 25, 3:49 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >> >> > wrote: >> >> >> Newberry says... >> >> >> >> >Tarski's theorem does not apply to formal systems with gaps. I think >> >> >> >it is preferable. >> >> >> >> If you the way you express Tarski's theorem is like this, then truth >> >> >> gaps don't change anything: >> >> >> >> There is no formula T(x) such that if x is a Godel code of a true >> >> >> sentence, then T(x) is true, and otherwise, ~T(x) is true. >> >> >> >> Anyway, *why* is it preferable to have a formal system for which Tarki's >> >> >> theorem does not apply? Preferable for what purpose? >> >> >> > If truth is expressible then truth can be equivalent to provabilty. >> >> >> So, you'd like to redefine truth (so that vacuously *true* statements >> >> aren't true) and also redefine provability (so trivially provable >> >> statements aren't provable) in such a way that truth is equivalent to >> >> provability. >> >> >> Then what have you accomplished? Hell, I can do that simply by >> >> requiring that nothing is true and nothing is provable. My "fix" is >> >> better than yours, insofar as we can see that it actually "works". >> >> > My theory has some significant advantages over yours. I can go to a >> > grocery store and count how many tomatoes and bananas I have picked. >> > If I have picked 2 small tomatoes and three large tomatoes my theory >> > can prove that I have 5 tomatoes. Also at the checkout counter I can >> > calculate the total price. Can your theory do that? >> >> No. You're right. The classical theory of arithmetic > > I thought that we were talking about your theory where "nothing is > true and nothing is provable." Ah, my mistake! Sorry, I didn't read the context. My theory is pretty good at what it does, though. It can't tell you that 2 + 3 = 5, but that's okay, since with my new definition of truth, 2 + 3 = 5 is not true. Anyway, I won't really defend my theory. My point is: you claim that your approach may yield a theory in which truth and provability are equivalent. Ignoring the fact that this is wishful thinking thus far, so what? You do so only by redefining what truth means, so that vacuously true statements are not true. I don't see any advantage to that. -- If you like high adventure, come with me. If you like the stealth of intrigue, come with me. If you like blood and thunder, come with me. But first listen to a word from our sponsor. -- Adventures by Morse
From: Jesse F. Hughes on 28 Mar 2010 08:54 Newberry <newberryxy(a)gmail.com> writes: > On Mar 27, 11:10 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: >> Nam Nguyen <namducngu...(a)shaw.ca> writes: >> > Daryl McCullough wrote: >> >> Nam Nguyen says... >> >>> Daryl McCullough wrote: >> >> >>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation >> >>>> of the language of PA. >> >>> So you've agreed "G(PA) can be arithmetically false"? >> >> >> It is false in nonstandard models of PA. >> >> > Why don't we make it more precise. When we say F, a formula written >> > in the language of arithmetic, is true or false _by default_ we >> > mean it's being arithmetically true or false: i.e. true or false >> > in the natural numbers. So we're *not* talking about F is being >> > true/false in any general kind of models here. >> >> You haven't (and can't) give me an effective way to use >> this definition to decide truth or falsity in the natural numbers. > > But I can. In a system with gaps Tarski's theorem does not apply. We > can then simply equate truth with provability. Your second sentence does not follow. You have to show that you have a logic in which provability turns out to be equivalent to truth. Tarski's theorem may not preclude this possibility, but it doesn't follow that you can then "simply equate truth with provability." -- Jesse F. Hughes "C is for Cookie. That's good enough for me." Cookie Monster
From: Aatu Koskensilta on 28 Mar 2010 09:34
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Let me rephrase the question: > >>> Outside Nam's paragraph, is G(T), to you, a formula written in L(T) >>> or in the language of arithmetic [i.e. L(PA)]? > > How I explain "encoded(G(T))" to you would depend on your answering > this question. G(T) on its own is just an arbitrary piece of notation; what it means depends on context. In the statement of your theorem, G(T) is said to be undecidable in T so we would naturally take G(T) to be a formula in the language of T. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |