Mathematical Logic and the Strengthened Liar's Paradox
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From: tchow on 4 Apr 2010 18:31 In article <wR7un.39091$ao7.4351(a)newsfe21.iad>, Nam Nguyen <namducnguyen(a)shaw.ca> wrote: <tchow(a)lsa.umich.edu wrote: <> In article <23946bf8-1be3-4aa1-9947-7e36b3cc9ddb(a)r27g2000yqn.googlegroups.com>, <> cplxphil <cplxphil(a)gmail.com> wrote: <>> S = "~Provable('S') and ~Provable('For all sentences T, (T<-- <>>> Provable(T))-->S')" <>> Why doesn't that express soundness? <> <> I'm not even sure what you're trying to do here. You can talk about <> provability until you're blue in the face but it's not going to express <> the concept of truth. < <I think that's a bit too strong a sentiment. We certainly can define <a formula being true in a formal system if it's provable there and <the system is syntactically consistent. Whether or not we know <the system be syntactically consistent is a different matter, but <we can express the concept of truth using provability. Iow, FOL "truth" <is a dispensable concept. Phil, you may have seen enough of Nam Nguyen's posts to have figured this out for yourself by now, but you should just ignore what he says here. As usual, he is letting his philosophical views about truth confuse what is actually just a technical mathematical point. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: cplxphil on 4 Apr 2010 18:46 On Apr 4, 5:30 pm, tc...(a)lsa.umich.edu wrote: > In article <23946bf8-1be3-4aa1-9947-7e36b3cc9...(a)r27g2000yqn.googlegroups..com>, > > cplxphil <cplxp...(a)gmail.com> wrote: > >S = "~Provable('S') and ~Provable('For all sentences T, (T<-- > >>Provable(T))-->S')" > > >Why doesn't that express soundness? > > I'm not even sure what you're trying to do here. You can talk about > provability until you're blue in the face but it's not going to express > the concept of truth. > -- > Tim Chow tchow-at-alum-dot-mit-dot-edu > The range of our projectiles---even ... the artillery---however great, will > never exceed four of those miles of which as many thousand separate us from > the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences I meant to write, 'For all sentences T, (Provable(T) --> T) --> S'. I was trying to express the idea of soundness by saying that for all sentences T, if T is provable, then T is true. After reading the link you posted, though, I understood that you can't express the concept of truth. I was just a little confused on what is meant by truth and what is meant by soundness. -Phil
From: Nam Nguyen on 4 Apr 2010 18:53 tchow(a)lsa.umich.edu wrote: > In article <wR7un.39091$ao7.4351(a)newsfe21.iad>, > Nam Nguyen <namducnguyen(a)shaw.ca> wrote: > <tchow(a)lsa.umich.edu wrote: > <> In article <23946bf8-1be3-4aa1-9947-7e36b3cc9ddb(a)r27g2000yqn.googlegroups.com>, > <> cplxphil <cplxphil(a)gmail.com> wrote: > <>> S = "~Provable('S') and ~Provable('For all sentences T, (T<-- > <>>> Provable(T))-->S')" > <>> Why doesn't that express soundness? > <> > <> I'm not even sure what you're trying to do here. You can talk about > <> provability until you're blue in the face but it's not going to express > <> the concept of truth. > < > <I think that's a bit too strong a sentiment. We certainly can define > <a formula being true in a formal system if it's provable there and > <the system is syntactically consistent. Whether or not we know > <the system be syntactically consistent is a different matter, but > <we can express the concept of truth using provability. Iow, FOL "truth" > <is a dispensable concept. > > Phil, you may have seen enough of Nam Nguyen's posts to have figured this > out for yourself by now, but you should just ignore what he says here. > As usual, he is letting his philosophical views about truth confuse what > is actually just a technical mathematical point. What could *you* find technical wrong with defining a formula being true in a formal system as I've just mentioned? Or is it just your habit of dismissing anything you don't like to hear as "philosophical"?
From: tchow on 4 Apr 2010 19:11 In article <tL8un.92861$gF5.19960(a)newsfe13.iad>, Nam Nguyen <namducnguyen(a)shaw.ca> wrote: >What could *you* find technical wrong with defining a formula being >true in a formal system as I've just mentioned? Or is it just your habit >of dismissing anything you don't like to hear as "philosophical"? You can of course define words however you want to. But this won't help Phil understand the mathematical content of Tarski's theorem. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: Nam Nguyen on 4 Apr 2010 19:50
tchow(a)lsa.umich.edu wrote: > In article <tL8un.92861$gF5.19960(a)newsfe13.iad>, > Nam Nguyen <namducnguyen(a)shaw.ca> wrote: >> What could *you* find technical wrong with defining a formula being >> true in a formal system as I've just mentioned? Or is it just your habit >> of dismissing anything you don't like to hear as "philosophical"? > > You can of course define words however you want to. But this won't help > Phil understand the mathematical content of Tarski's theorem. You might be right here and it might not have helped. But it also might have helped and, for what it's worth, I wouldn't pretend to know whether or not in Phil's mind he actually got confused between what I said and the content of Tarski's theorem: both seem to be about the undefinability of "truth" after all, despite certain intuition. I just found your "just ignore" below is either to "strong" or borderline "attacking". >> but you should just ignore what he [Nam] says here. >> As usual, he is letting his philosophical views about truth confuse what >> is actually just a technical mathematical point. |