'When Are Relations Neither True Nor False?
in [Math]
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From: Daryl McCullough on 26 Mar 2010 06:40 Newberry says... >Are you saying in a roundabout manner that in classical logic (P & ~P) >-> Q is a tautology? Well we know that. But what does it have to do >with anything? I'm saying that none of the "paradoxes of material implication" are paradoxical. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 26 Mar 2010 06:43 Newberry says... > >On Mar 25, 3:37=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >In my logic the Liar paradox can be expressed as follows. >> >> > =A0 ~(Ex)(Ey)(Pxy & Qy) =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0= > =A0 =A0(L) >> >> >where Pxy means that x is a proof of y, Q is satisfied by only one y =3D >> >m, and m is Goedel number of (L). >> >> That's not the Liar sentence. > >The sentence seemingly says about itself that it is not provable. That's a Godel sentence, not the Liar sentence. >SInce Tarski's theorem does not apply we can equate provability with >truth. Why does that follow? You can't possibly do that. What is provable depends on what axioms you have. Different axioms leads to different notions of provable. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 26 Mar 2010 06:45 Newberry says... > >On Mar 25, 3:49=A0am, 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. That doesn't follow. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 26 Mar 2010 06:49 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 mean that there is a plausible explanation why there is no >inconsistency. I do not like inconsistencies. 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. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 26 Mar 2010 06:54
Nam Nguyen says... >Daryl McCullough wrote: > >> >> Ultimately, the people on this newsgroup who object to standard >> mathematics are really objecting to the idea that there can be >> such a thing as a counter-intuitive result. > >That's a grossly erroneous over-generalization. It's many of those who >defend standard mathematics who erroneously object to the counter-intuitive >_but real nature_ of mathematics: relativity of truth and provability. Huh? Standard mathematics perfectly well takes into account the "relativity of truth". Truth is relative to an interpretation. So your objection makes no sense. -- Daryl McCullough Ithaca, NY |