'When Are Relations Neither True Nor False?
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From: Newberry on 2 Apr 2010 00:32 On Apr 1, 6:05 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Mar 31, 5:57 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> 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. > > An assumption so silent that no one but you noticed it? > > > > > > >> 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. > > A similar syntactic transformation proves > > ~ (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) ) (1) > > This syntactic transformation does not apparently involve any > assumptions at all. It seems to me that you're coming close to > advocating an unsound theory: one which proves (1), despite the fact > that you've said that (1) is not true. > > In any case, I really think you ought to think these things through. > Up until now, you seem to have assumed that statements like (1) are > useless if they're vacuously true. Now you've seen several proofs > involving statements like (1) -- proofs which entail that these > statements *are* vacuously true! > > It seems to me that quick responses to this observation are unlikely > to work. Why not take the time and figure out what you think counts > as a proof and see where that takes you? Glad we finally agree on something. I do not know if assuming existence i.e. a presupposition of another formula, then drawing a conclusion from the formula that contradicts the initial existential assumption is a valid argument. But I am not convinced that it is not invalid either. Of course these things need to be worked out. I actually started to work on a derivation system for predicate calculus, basically trying to mimic the traditional syllogism as Strawson has shown that the logic of prsuppositions and the traditional syllogism are compatible or identical. I was thinking of appending it to my t-relevant logic exposition paper. But it is not anywhere near publishable stage. There are still a few issues to be worked out. But we will get there in due course. It may not even be that difficult. I do not know how it will turn out. I forgot who proved that the square root of 2 was irrational and what his proof looked like. Maybe your version is something concocted by the modern mathematicians who take classical logic for granted. Maybe it will turn invalid, maybe valid with some modifications or added assumptions. Mind you the Greeks did not have the concept that the vacuous sentences were true. The traditional syllogism presupposes that the subject class is non- empty. > > -- > Jesse F. Hughes > ... one of the main causes of the fall of the Roman Empire was that, > lacking zero, they had no way to indicate successful termination of > their C programs. -- Robert Firth- Hide quoted text - > > - Show quoted text -
From: Newberry on 2 Apr 2010 00:33 On Apr 1, 5:59 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > Let > > > ~(Ex)(Ey)(Pxy & Qy) (1) > > > be Goedel's sentence. [Pxy: x is a proof of y, Qy is such that only > > one y=m satisfies it & m = #(1)]. You are absolutely convinced that PA > > is consistent i.e. (1) is true. That is, the search for x and y will > > never terminate. How do you know that the search for x and y will > > never terminate? > > How do I know that Peano arithmetic is consistent? I know it the way I > know any mathematical theorem I have personally proved. You proved PA consistent? > Perhaps you'd > now be willing to say whether you agree that for any formal theory T > extending Robinson arithmetic, either directly or through an > interpretation, there are infinitely many true statements (of the form > "the Diophantine equation D(x1, ..., xn) = 0 has no solutions") which > are unprovable in T if T is consistent? > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Newberry on 2 Apr 2010 00:35 On Apr 1, 3:09 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >Let me ask you something. Let > > >~(Ex)(Ey)(Pxy & Qy) (1) > > >be Goedel's sentence. [Pxy: x is a proof of y, Qy is such that only > >one y=m satisfies it & m = #(1)]. You are absolutely convinced that PA > >is consistent i.e. (1) is true. That is, the search for x and y will > >never terminate. How do you know that the search for x and y will > >never terminate? > > Suppose you find two natural numbers m and n such that, > P(n,m) & Q(m). Then you can easily prove (Ex) (Ey) (Pxy & Qy). > You can also "decode" n to get a proof of ~(Ex)(Ey)(Pxy & Qy). > So you would, in that case, have a proof of a contradiction. > > If your axioms are consistent, the the above case cannot happen. > To say that "the above case cannot happen" is to say that there > are no natural numbers m and n such that P(n,m) & Q(m). Which > is formalized as ~(Ex)(Ey)(Pxy & Qy). > > So the assumption that your system is consistent directly leads > to conclusion (1). What I was getting at is how we know that the system is consistent. And if we do know it then we also know that certain seraches will never terminate. Can we apply this knowledge to Diophantine equations?
From: Newberry on 2 Apr 2010 00:37 On Mar 31, 5:12 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >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. > > In general, any *procedure* used to evaluate the truth of sentences > in a self-referential language is incomplete, in the sense that there > are true sentences that are not evaluated as true by the procedure. > > This is easy to see: Let P be some procedure to evaluate the truth > of sentences. Then consider the sentence > > "When procedure P is applied to this sentence, the result is not true" And what procedure would it be? It cannot be Gaifman's procedure because the sentence above does not have the form "The sentence written in/on ... is true" > > Contrary to your claims about sentences that fail to express a possible > state of the world, the above sentence makes a perfectly definite claim: > That a certain procedure applied to a certain sentence does not produce > a certain result. If the procedure is actually applied to that sentence, > and run to completion, then we can just check to see what the result is. > > What this shows is that for any sound procedure that purports to evaluate > the truth of sentences, there is a true (in the sense of corresponding to > the facts) sentence that the procedure fails to evaluate as true. So *every* > procedure for evaluating truth of sentences is incomplete. A procedure > can be complete for a certain *class* of sentences, for example, those > expressible in a limited language, but there will always be an extended > language that the procedure fails on. > > -- > Daryl McCullough > Ithaca, NY
From: Daryl McCullough on 2 Apr 2010 06:19
Newberry says... > >On Mar 31, 5:12=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> In general, any *procedure* used to evaluate the truth of sentences >> in a self-referential language is incomplete, in the sense that there >> are true sentences that are not evaluated as true by the procedure. >> >> This is easy to see: Let P be some procedure to evaluate the truth >> of sentences. Then consider the sentence >> >> "When procedure P is applied to this sentence, the result is not true" > >And what procedure would it be? Well, for example, the search for a proof for the statement. Or Gaifman's procedure. >It cannot be Gaifman's procedure because the sentence above does >not have the form "The sentence written in/on ... is true" Then, as I said, it's a true sentence that Gaifman's procedure does not return true for. -- Daryl McCullough Ithaca, NY |