'When Are Relations Neither True Nor False?
in [Math]
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From: Newberry on 27 Mar 2010 00:53 On Mar 26, 4:24 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Mar 25, 3:38=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> Newberry says... > > >> >If you take the position that there are truth value gaps then the Liar > >> >papradox is solvable in English. > > >> What does it mean to be "solvable" and why do you want it to be solvable? > > >It means that there is a plausible explanation why there is no > >inconsistency. I do not like inconsistencies. > > I don't see how truth gaps help in the Liar paradox. > > Suppose you have a truth predicate T(x) and you have a sentence L > (with Godel number #L) of the form > > forall x, A(x) -> ~T(x) > > and you have a theorem > > forall x, A(x) <-> x=#L > > Then L cannot be true, and cannot be false. So L falls into a "truth gap".. > But then what about the sentence > > "L is not true" > > which is formalized by > > ~T(#L) > > Is that true, or does that have a truth gap, as well? We just > agreed that L was not true, so if T(x) is a truth predicate, > that should be formalized by ~T(#L). From that, surely it follows > that > > "forall x, if x=#L, then ~T(x)" > > Since x=#L <-> A(x), then surely it follows that > > "forall x, A(x) -> ~T(x)" > > So L follows from the claim that L falls in the truth gap. Truth > gaps *don't* help with the Liar paradox. If you do not mind I will leave the Goedel numbers out as they are not applicable to the natural language. It wil also streamaline the discussion. Then let L: ~T(L) If v(L) = ~(T v F) then there is no contradiction. L is not true. The argument usually goes "but that is what L says." But L does not say anything.
From: Daryl McCullough on 27 Mar 2010 07:56 Newberry says... >L: ~T(L) > >If v(L) = ~(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. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 27 Mar 2010 07:59 Newberry says... > >On Mar 26, 3:49=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> The Liar sentence is not *expressible* in any standard mathematical >> theory (PA or ZFC). So you don't have to do anything to keep the Liar >> from spoiling the consistency of those languages. > >Why you think you have to tell me that I do not know. If you lool a >few lines above you will see that I was talking about the Liar paradox >in the natural language. So your theory of truth gaps is only for natural language? So you agree that formal languages such as arithmetic don't require any truth gaps? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 27 Mar 2010 08:02 Nam Nguyen says... > >Alan Smaill wrote: > >>> I'm sure your belief in the "absolute" truth of G(PA) is subjective, which >>> you'd need to overcome - someday. Each of us (including Godel) coming to >>> mathematics and reasoning has our own subjective "baggage". >> >> Why on earth do you think I have some belief in the " "absolute" truth " >> of G(PA) ? I don't even know what that *means* . > >OK. Then on the meta level, do you think it's correct to say that >G(PA) can be arithmetically false? It is a *relative* truth. It's true in the standard interpretation of the language of PA. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 27 Mar 2010 09:26
Newberry <newberryxy(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 is incapable of proving that 2 + 3 = 5. I see now that your theory is superior and will alter my brain accordingly. (Honestly, I have no idea what you're talking about. You seem to see a disadvantage in classical arithmetic that I simply don't see. Why not explain your point?) -- Jesse F. Hughes "I post for many reasons [...] and there's no reason to think that I'll stop." -- James S. Harris |