'When Are Relations Neither True Nor False?
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From: Daryl McCullough on 29 Mar 2010 06:30 Newberry says... > >On Mar 28, 4:37=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >> >> >> >> >> >> >On Mar 27, 4:56=3DA0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >> >wrote: >> >> Newberry says... >> >> >> >L: ~T(L) >> >> >> >If v(L) =3D3D ~(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 Yes, and I think it's silly. He wants to associate truth with sentence tokens (occurrences of sentences, rather than sentences themselves), so that two identical sentences may differ in truth values. There is a sense in which that is necessary, when referring expressions are used (for example, if I say "That is a cat", obviously some tokens are true, when I'm pointing to a cat, and some tokens are false, when I'm pointing to a dog). However, if you have two sentence tokens, and they have the same subject, and the same predicate, it's silly to call one true and one false (or more generally, not true). It's a silly resolution that doesn't resolve anything. If "true" is a predicate applying to sentence *tokens*, then we can invent a second predicate, "truthy" applying to sentences: A sentence X is truthy if at least one of its tokens is true. Then you can form a new Liar paradox: This sentence is not truthy. -- Daryl McCullough Ithaca, NY
From: Newberry on 30 Mar 2010 00:09 On Mar 28, 9:01 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(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." It does automatically folow but we can nevertheless do that. In asystem without gaps we cannot. > > 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/
From: Newberry on 30 Mar 2010 00:21 On Mar 29, 3:30 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >On Mar 28, 4:37=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> Newberry says... > > >> >On Mar 27, 4:56=3DA0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >> >wrote: > >> >> Newberry says... > > >> >> >L: ~T(L) > > >> >> >If v(L) =3D3D ~(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 > > Yes, and I think it's silly. He wants to associate truth > with sentence tokens (occurrences of sentences, rather > than sentences themselves), so that two identical sentences > may differ in truth values. There is a sense in which that > is necessary, when referring expressions are used (for example, > if I say "That is a cat", obviously some tokens are true, > when I'm pointing to a cat, and some tokens are false, when > I'm pointing to a dog). > > However, if you have two sentence tokens, and they have > the same subject, and the same predicate, it's silly to > call one true and one false (or more generally, not true). > It's a silly resolution that doesn't resolve anything. If > "true" is a predicate applying to sentence *tokens*, then > we can invent a second predicate, "truthy" applying to > sentences: > > A sentence X is truthy if at least one of its tokens is true. > > Then you can form a new Liar paradox: > > This sentence is not truthy. > There are two issues here. a) The two tokens have the same subject and the same predicate. b) The resolution can be semingly defeated by forcing all tokens into one type. Not sure why you think they are related. Maybe they are but it is not immediately obvious to me. Let's take a) first. Gaifman's evaluation procedure is such that if two tokens have the same subjects and predicates one can nevertheless be true and the other neither true nor false. Now b): This sentence is not truthy. "This sentence is not truthy" is not truthy. These two sentences have the same subjects and predicates. The former is self-referential the latter is not. The former is ~(T v F), the latter is T.
From: Newberry on 30 Mar 2010 00:41 On Mar 28, 9:04 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(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. "The proof by Goedel and Tarski that a language cannot contain its own semantics applied only to languages without truth gaps." Outline of a Theory of Truth Saul Kripke The Journal of Philosophy, Vol. 72, No. 19, Seventy-Second Annual Meeting American Philosophical Association, Eastern Division. (Nov. 6, 1975), p. 714. > > > 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. Indeed. But if we leave out all the vacuous sentences we can still do all the useful arithmetic as we know it. Although all the people on this board believe that such sentences are true nobody argued that they were useful. Aatu even said that they did not belong in ordinary mathematical reasoning. Furthermore there is a reason to think that they are neither true nor false. I cannot think of any good reason for claiming that 1 + 1 = 2 is not true. > > -- > "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: Aatu Koskensilta on 30 Mar 2010 01:17
Newberry <newberryxy(a)gmail.com> writes: > But if we leave out all the vacuous sentences we can still do all the > useful arithmetic as we know it. Although all the people on this board > believe that such sentences are true nobody argued that they were > useful. Aatu even said that they did not belong in ordinary > mathematical reasoning. You're imagining things. > Furthermore there is a reason to think that they are neither true nor > false. I cannot think of any good reason for claiming that 1 + 1 = 2 > is not true. Well, perhaps you might be moved to answer the following query, which I have now presented on several occasions: How are we to apply your ideas about vacuity, meaningfulness, truth, proof, what not, in context of the following mathematical observation: for any consistent theory T extending Robinson arithmetic, either directly or through an interpretation, in which statements of the form "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" can be expressed, there are infinitely many Diophantine equations D(x1, ..., xn) = 0 that have no solutions but for which "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" is not provable in T. On an ordinary understanding, a statement of the form "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" is true just in case D(x1, ...., xn) = 0 has no solutions. According to your account some such statements are neither true nor false -- owing, so I gather, to your eager zeal for equating formal provability with truth, contra G�del and his lackwit lackeys -- regardless of whether the corresponding Diophantine equations are soluble or not. What are we to make of this? In what way, and just how, do your musing connect to our mathematical experience or reasoning? You may of course introduce whatever technical terminology you wish, defining truth and falsity whichever way you see fit, to suit your whim and fancy, but unless answers to questions like I have presented are forthcoming your fiddling will be of no apparent interest in the wider scheme of things in mathematics, in the philosophy of mathematics, in our everyday arithmetical reasoning, thinking, reflection. It's presumptuous of me, but I nonetheless surmise it is at least in part a hope of yours, in these your endless mumblings Usenetical on matters logical, the liar, vacuity, what not, that others come to recognise the wonderful clarity of your theory of meaning, your take on logic, your this and that. On this surmise, I can only suggest you make some attempt to demonstrate the value of your insight to the skeptic masses by showing how some conundrum can be exorcised, some problematic enigma eradicated, by way of transparent and compelling reasoning involving your novel insights, some traditional baffler dissected with particular cunning made possible by the notions you champion, and so on. You could do well to emulate Dummett or Kreisel, neither of whom I'm an unreserved fan, but who never fail to stimulate and inspire, even if it is only to cry out in philosophical exasperation. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |