'When Are Relations Neither True Nor False?
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From: Newberry on 31 Mar 2010 00:21 On Mar 30, 3:30 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >On Mar 29, 10:17=A0pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > >> 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 > > >No. > > What do you mean, no? You are proposing to equate truth and provability. > Some sentences of the form "the Diophantine equation D(x1, ..., xn) = 0 > has no solutions" are neither provable nor refutable. In which theory? It follows from > your equating of truth and provability that they are neither true nor > false. > > The reasoning that there are statements that are true, but unprovable > goes like this: > > 1. Let D(x1,...,xn) be a polynomial in the variables x1, ..., xn. > > 2. If m1, ..., mn are n integers such that D(m1,...,mn) = 0, > then the statement "There exists x1, ..., xn such that D(x1,...,xn) = 0" > is provable. We can easily prove this by plugging in m1, ..., mn and > checking to see if the result is 0. > > 3. If the statement "There exists x1, ..., xn such that D(x1,...,xn) = 0" > is not provable, then there are no integers m1, ..., mn such that > D(m1, ..., mn) = 0. This follows immediately from 2. > > 4. Note that if Phi is the formula > "There exists x1, ..., xn such that D(x1,...,xn) = 0", > then 3. has the form: "If Phi is not provable, then ~Phi". > In other words, if Phi is not provable, then the negation of Phi > holds. > > 5. Therefore, if Phi is neither provable nor refutable, then > the negation of Phi holds. So if Phi is neither provable nor > refutable, then Phi is false. ("Phi is false" means the same > thing as "The negation of Phi holds"). > > 6. Therefore, if Phi is neither provable nor refutable, then > there is a statement, Phi, that is false, but not provably false. > There is another statement, ~Phi that is true, but not provable. > > 7. Therefore, if there is a statement Phi (of the appropriate > form) that is neither provable nor refutable, then provability > and truth are not the same. > > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text -
From: Newberry on 31 Mar 2010 00:26 On Mar 30, 6:07 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > 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. > > You have to *show* that this can be done in your system. And to a reasonable degree I have shown it. And I do NOT mean by pointing out that Tarski does not apply. > > And, indeed, the word "equate" is still misleading, since it suggests > that define true to mean "provable". That can certainly be done. I > can say that, hereafter, when I say that a statement of PA is true, I > mean that there is a proof of P in PA. Of course, such semantic play > is unsatisfactory. > > -- > "After years of arguing I realize that your intellects are too limited > to fully grasp my work. [...] Still, no matter how child-like your > minds are, [...] since you have language, [...] there's a chance that > I'll be able to find something that your minds can handle." --JSH- Hide quoted text - > > - Show quoted text -
From: Newberry on 31 Mar 2010 00:34 On Mar 30, 6:04 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > 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. > > You seem to have misrepresented Aatu's claims. Moreover, you're just > wrong. I've argued repeatedly that some sentences of the form > > ~(Ex)(P & Q) > > occur in ordinary mathematical reasoning (and hence are useful), even > when (Ex)P is false. An example occurred in sci.math recently. > > Simon C. Roberts gave a purported proof of FLT[1], by arguing: > > ~(En)(Ea,b,c)(a^n + b^n = c^n & a, b, c are pairwise coprime). > > Let's denote a^n + b^n = c^n by P(a,b,c,n) and a, b, c are pairwise > coprime by Q(a,b,c), so that Simon's argument attempts to show that > > c > > Of course, I am *not* claiming that he proved what he claims. That's > beside my point. A poster named bill replied that (1) is not Fermat's > last theorem[2], which has the form > > ~(En)(Ea,b,c) P(a,b,c,n). (2) > > Arturo responded[3] by proving > > (En)(Ea,b,c) P(a,b,c,n) -> (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) ). (3) > > Hence, a proof of (1) yields a proof of (2) by modus tollens. How about this? (En)(Ea,b,c) P(a,b,c,n). Assumption (En)(Ea,b,c) P(a,b,c,n) -> (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) ). (3) ~(En)(Ea,b,c) P(a,b,c,n). Modus Tollens ~(En)(Ea,b,c) P(a,b,c,n). RAA > > According to you, however, if (2) is true (and I assume we all know > that (2) was proved by Wiles), then (1) is meaningless. Yet, no one > here balked at the claim that (1) could be used to prove (2) (once (3) > was proved). No one here had any trouble understanding what (1) > means. Everyone in the thread accepted this form of mathematical > argument as beyond suspicion -- although the claim that Simon actually > proved (1) is regarded as doubtful. > > So, you're just plain wrong. These statements that you call > meaningless occur in ordinary mathematical reasoning all the time. > > Footnotes: > [1] Message id > <1917288606.455209.1269716329839.JavaMail.r...(a)gallium.mathforum.org>, > in the thread "Another Proof of Fermats Last Theorem". > > [2] Message id > <50f09d88-a96b-464c-aec5-be000f0be...(a)x23g2000prd.googlegroups.com>. > > [3] Message id > <e6768d43-7706-41f4-bff8-8e666d693...(a)j21g2000yqh.googlegroups.com>. > > -- > "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: Newberry on 31 Mar 2010 00:46 On Mar 30, 3:51 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >There are two issues here. > >a) The two tokens have the same subject and the same predicate. > >b) The resolution can be seemingly defeated by forcing all tokens into > >one type. > > >Not sure why you think they are related. > > Any theory of truth that is worth considering, if two sentence > tokens have the same subject and same predicate, then they have > the same truth value. Otherwise, your notion of truth is unconnected > with the meaning of sentences. Why are you stating this so categorically? Look at this excerpt from Gaifman: QUOTE Line 1: The sentence on line 1 is not true. Line 2: The sentence on line 1 is not true. The standard evaluation rule for a sentence of the form The sentence written in/on ... is true is roughly this: (*) Go to ... and evaluate the sentence written there. If that sentence is true, so is The sentence written in ... is true , else the latter is false. END OF QUOTE If you apply this procedure to 1 it will never terminate, so neither T nor F will be assigned to 1. If we apply the same procedure to 2, knowing that 1 is ~(T v F) we obtain that 2 is true. Another way to see this is that 1 is not expressing any possible state of affairs. 2 is. It expresses the state of affairs that 1 does not correspond to an actual state of affairs. > > >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. > > Using Godel coding, you can eliminate direct self-reference and thereby > make the two sentences identical. Then it is a contradiction to say that > one is truthy and the other is not. > > -- > Daryl McCullough > Ithaca, NY
From: Daryl McCullough on 31 Mar 2010 07:23
Newberry says... > >On Mar 30, 3:51=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Any theory of truth that is worth considering, if two sentence >> tokens have the same subject and same predicate, then they have >> the same truth value. Otherwise, your notion of truth is unconnected >> with the meaning of sentences. > >Why are you stating this so categorically? I'm just stating what truth *means. A sentence makes (or attempts to make) a claim about something. To understand a sentence means to understand what is being claimed, and about what. Look at this excerpt from >Gaifman: > >QUOTE >Line 1: The sentence on line 1 is not true. >Line 2: The sentence on line 1 is not true. Yes, it's a silly notion of truth that gives these two sentences different truth values. >The standard evaluation rule for a sentence of the form "The sentence >written in/on ... is true" is roughly this: >(*) Go to ... and evaluate the sentence written there. If that >sentence is true, so is "The sentence written in ... is true", else >the latter is false. >END OF QUOTE To me, the truth of a sentence is determined by what it *says*, not be the result of an evaluation procedure. Now, of course, you could use an evaluation procedure to *define* a property of sentences. That's what proof within a mathematical theory does. It's an evaluation procedure for sentences. Sentences that pass the evaluation are called "theorems". If you are proposing a more sophisticated evaluation procedure, then you're extending the notion of "theorem". But you're not defining truth. >Another way to see this is that 1 is not expressing any possible state >of affairs. Sure it does. You are using the word "true" to mean "evaluates to true after applying Gaifman's evaluation procedure". So the meaning of 1 is: "The sentence on line 1 does not evaluate to true under Gaifman's evaluation procedure" That's a perfectly meaningful state of affairs, and it happens to be the case. -- Daryl McCullough Ithaca, NY |