'When Are Relations Neither True Nor False?
in [Math]
|
From: Newberry on 28 Mar 2010 23:36 On Mar 28, 5:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(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." Did I say it follows? I meant that it is possible. In classical logic withuot gaps it is impossible. Why did you not interpret what I said this way? > > -- > Jesse F. Hughes > > "C is for Cookie. That's good enough for me." > Cookie Monster- Hide quoted text - > > - Show quoted text -
From: Newberry on 28 Mar 2010 23:39 On Mar 28, 5:50 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(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. Do you agree that Tarki's theorem does not apply to systems with gaps? DO you agree that if we say that the vacuous sentences are neither true nor false that we will have gaps? > > -- > 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- Hide quoted text - > > - Show quoted text -
From: Newberry on 28 Mar 2010 23:41 On Mar 28, 4:37 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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. Did you read this? http://www.columbia.edu/~hg17/gaifman6.pdf > > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text -
From: Jesse F. Hughes on 29 Mar 2010 00:04 Newberry <newberryxy(a)gmail.com> writes: > On Mar 28, 5:50 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> 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. > > Do you agree that Tarki's theorem does not apply to systems with > gaps? You keep saying so. Although I haven't looked up the reference, I assume that you're not mistaken. > DO you agree that if we say that the vacuous sentences are neither > true nor false that we will have gaps? Sure. I also think that if we simply say "1 + 1 = 2" is neither true nor false (while every other formula is interpreted in the standard way), then we have a theory with gaps. It does not follow, of course, that truth and provability are the same in this theory, nor that my new and improved notion of truth is sensible. -- "There's lots of things in this old world to take a poor boy down. If you leave them be, you can save yourself some pain. You don't have to live in fear, but you best have some respect, For rattlesnakes, painted ladies and cocaine." -- Bob Childers
From: Jesse F. Hughes on 29 Mar 2010 00:01
Newberry <newberryxy(a)gmail.com> writes: > On Mar 28, 5:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> >> > 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." > > Did I say it follows? I meant that it is possible. In classical logic > withuot gaps it is impossible. Why did you not interpret what I said > this way? "We can then simply equate truth with provability." Hmm... No idea why I thought that meant it was a simple task, indeed, that one could just equate truth with provability. I guess I'm just a bit slow. -- Jesse F. Hughes "[Lancelot] sighed, defeated. 'It is as practical to hurry an acorn toward treeness as to urge a damsel when her mind is set.'" -- John Steinbeck, /The Acts of King Arthur and His Noble Knights/ |