'When Are Relations Neither True Nor False?
in [Math]
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From: Aatu Koskensilta on 31 Mar 2010 10:45 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > In article <87eij05ztb.fsf(a)dialatheia.truth.invalid>, Aatu Koskensilta > says... > >>Newberry <newberryxy(a)gmail.com> writes: >> >>> You never answered my question what you ment by "Goedel." >> >>You have asked me what I mean by "Goedel"? > > When people write "Goedel", they usually mean "G�del". This is true, but I don't think I've ever used the spelling "Goedel". -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frederick Williams on 31 Mar 2010 15:45 Aatu Koskensilta wrote: > This is true, but I don't think I've ever used the spelling "Goedel". Thank heavens for the use-mention distinction! -- I can't go on, I'll go on.
From: Newberry on 31 Mar 2010 23:57 On Mar 31, 4:23 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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. Indeed. A sentence of the form "The sentence written in/on ... is true" says that the sentence written at ... is true. Naturally then a sentence of the form "The sentence written in/on ... is true" is true If that if the sentence written at ... is true, otherwise the former is false. > 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". I do not know. Yourr definition seems circular. > 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
From: Newberry on 1 Apr 2010 00:11 On Mar 31, 5:57 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Mar 30, 6:04 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > >> 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 > > How about it? That is certainly valid reasoning classically. I have > no idea whether it's valid in your proposed system, since you haven't > said. > > But that is certainly not the argument that was offered in the other > thread. The statement > > (En)(Ea,b,c) P(a,b,c,n) -> (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) ) > > was proved independently of your assumption, and no one balked. Thus, > my question remains: was the argument I gave invalid? It is valid but it makes a silent assumption. > > A second question comes to mind: what happened to your imagination > test? You've said that (because FLT is true) you cannot picture > (E a,b,c,n)( P(a,b,c,n) & Q(a,b,c) ). Yet, once you assume (contrary > to fact) that FLT is false, you *can* picture it? No. > > You can't picture a green round triangle, right? What if I say: > assume a round triangle exists. Can you picture a green round > triangle *then*?[1] No. > That is, does the act of making an assumption > change your capacities for imagining stuff? No. > If not, then your > argument above doesn't work. Why not. Assume there are such things as meaningless sentences. These sentences are well formed. Well formed sentences can go through syntactic transformations. > If so, well, then your powers of > imagination are different than mine. > > Footnotes: > [1] Green round triangles puzzle me. As far as I can tell, you find > the statement "No green triangles are round" meaningful, but the > statement "No round triangles are green" meaningless. > > -- > "[I]n mathematics there are two types of integers: primes and > composites. [...] It's like how in the world there are mostly two > kinds of people: male and female [...] and lots of reasons for > interest in the differences." -- JSH on math/biology- Hide quoted text - > > - Show quoted text -
From: Daryl McCullough on 1 Apr 2010 00:28
Newberry says... > >On Mar 31, 4:23=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> >Gaifman: >> >> >QUOTE >> >Line 1: The sentence on line 1 is not true. >> >Line 2: The sentence on line 1 is not true. .... >> 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 procedur= >e, >> 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". > >I do not know. Your definition seems circular. It's not *my* definition. It's Gaifman's. He describes a procedure for evaluating sentences, which either returns "true", "false", or nothing at all. The self-referential sentence "Gaifman's evaluation procedure when applied to this sentence does not return 'true'" certainly defines a "possible state of affairs". -- Daryl McCullough Ithaca, NY |