Torkel Franzen on truth
in [Logic]
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From: Newberry on 22 Dec 2007 18:38 On Nov 17, 5:01 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Nov 17, 10:26 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> >What I meant is that we can say > >> >(Ex)(Px, #(F)) --> F > >> >leaving the T out. > >> >Yes, it is inconsistent. > > >> No, it's not inconsistent, if you are careful about it. > > >Please explain. You were the one who claimed it was inconsistent in an > >earlier post. > > We talked about a bunch of different things. I don't know > what P is supposed to mean here. Is it the provability predicate? > If so, the provability predicate for what theory? What is F? > Is it just any contradiction? I'll assume it is. > > If you have a theory T1, then you can define a provability > predicate for T1, call it P1(x,y) (meaning "x is a code > for a proof in T1 of a formula whose code is y"). If T1 > is a sound theory (anything it says about arithmetic is > true in the usual interpretation), then the statement > > Ex (P1(x,#F)) -> F > > is a perfectly true sentence. So you can add that as > an axiom to T1 to get a new theory T2. There is no > problem with consistency. T2 can prove G1, the Godel > sentence for T1. But T2 cannot prove G2, the Godel > sentence for T2. You can define a provability predicate > for T2, call it P2, and you can formulate a perfectly > good statement of arithmetic: > > Ex (P2(x,#F)) -> F > > You can add this as an axiom to T2 to get a new theory T3. > And so on. > > This process gives you a way to go from axiomatizable > true theories to more complete axiomatizable true theories. > > Now, if you want to go for the whole ball of wax and > come up with a theory T_ultimate with the following > property: > > T_ultimate proves > Ex (P_ultimate(x,#F)) -> F > > There is no such theory T_ultimate except for an > inconsistent theory. Do you agree though that adding Ex (P_ultimate(x,#F)) -> F would force the theory in the standard model?
From: Nam D. Nguyen on 22 Dec 2007 18:58 Nam D. Nguyen wrote: > > In a technical sense, "2=1+1" is true in all models of some consistent > theories where it is a theorem. I might sound strange, but let me re-phrase it: In a technical sense, "2=1+1", is *supposed to be* true in all models of some consistent theories where it is a theorem. (I know there exists a meta theory called "Completeness").
From: Peter_Smith on 22 Dec 2007 19:33 On 22 Dec, 17:56, george <gree...(a)cs.unc.edu> wrote: > Here is how stupid you sound: > You, the distinguished Prof.Peter Smith, are saying the following: > 2+5=7 doesn't mean anything. Am I? That's odd. I always thought that "2 + 5 = 7" means that two plus five is seven ... but obviously I must have got that wrong.
From: herbzet on 23 Dec 2007 08:19 george wrote: > > "G. Frege" wrote: > > > Hence isn't it the /interpretation/ that assigns > > > truth to an axiom? [...] > On Dec 21, 3:30 pm, herbzet <herb...(a)gmail.com> wrote: > > > > Well, they're true in all their models, by definition. > > But axioms are NOT UNIQUE in that regard! > That is NOT a special property Of Axioms! > EVERY last wff is true in all of ITS models! All true, of course. > MODELS have to be models OF something! > It is, as I was trying to beat through MoeBlee's head a minute ago, > "structures" and "interpretations" that get to be unattached. MoeBlee's assertion that it is a theorem that every structure is a model of some theory is a new thought to me. Even if it is so, it does seem to me to be ill-advised to use "structure" and "model" interchangably (although I'm as slack as anyone else on using "structure" and "interpretation" more or less interchangeably). > If a structure is a model of a wff then the the wff is true in the > structure. Of course. > MoeBlee's point was that people were in a very generalized habit of > using > "model" withOUT requiring it to be a model of anything in PARTICULAR. > That is an observable fact about the brute weight of usage; neither I > nor > anyone else GETS to disagree with it, so MoeBlee probably thought it > would > therefore be safe, or at least defensible, to assert it. He was > entirely wrong > about that. People's general habit of doing this IS BAD. It is > sloppy. > IT NEEDS reform. People NEED to clean up their (usage)ACT. > Instead, MoeBlee chose to DEFEND the fact that people generically tend > to talk this way AS ACCEPTABLE because it is allowed by definitions > that > he can quote from Enderton. That is using a good work to confuse > people > and make discourse in general LESS accurate and it is not > intellectually > acceptable behavior. I agree that it's a bad habit that needs to be reformed, and I'm willing to do my bit to be the annoying guy who keeps pointing out the misuse. Assuming MoeBlee is correct in asserting that Enderton's definitions allow calling every structure a model, I would guess that that sort of usage would vary by context -- one would have to take some care not use one term for the other indiscriminately. > > Last time George and I spoke on this point, he was of the opinion, > > if I understood him correctly, that interpreting the axioms in > > structures which falsify (one or more of) them doesn't count -- > > axioms are always taken as true regardless of how they are > > interpreted; interpretation is otiose. > > Close, yeah. > It amazes me that some people can just tune into my > wavelength (if they feel like it) while others must insist > that I'm just evil. Like anyone else, when I'm in an argument I'm inclined to reject EVERYTHING my opponent says, no matter how innocuously and obviously true some of it may be. This is so obviously a form of ad hominem (If this jerk says X, then X must be false) that it's particularly embarassing, as a psuedo-logician, to fall prey to it. It's rhetorically bad, too, to be caught denying what's plainly true. > > I'll admit it seems like a rather pointless exercise to interpret > > axioms in structures in which they are false, it just happens > > to be involved in the Tarskian conception of logical consequence: > > in those structures too the axioms imply their theorems /and > > nothing else/. > > The verb "imply" is the key word in that sentence. Yes, and that's _my_ little hobby-horse. > Material implication is convenient in some ways and misleading in > others; please let us NOT have another thread about "vacuously true". That's not my intent, nor am I asserting that axioms (or any other sentence) "vacuously imply" their consequences when interpreted in structures in which they (the axioms or sentences) are false (with the unfortunate exception of contradictions -- which "vacuously imply" every sentence, regardless of the structure in which they are interpreted, according to the standard (i.e. Tarskian) definition of logical consequence). -- hz
From: Nam D. Nguyen on 23 Dec 2007 12:45
herbzet wrote: > > MoeBlee's assertion that it is a theorem that every structure is a > model of some theory is a new thought to me. Even if it is so, > it does seem to me to be ill-advised to use "structure" and "model" > interchangably (although I'm as slack as anyone else on using > "structure" and "interpretation" more or less interchangeably). While at the issue of structure/interpretation/model let me point a couple of related points, which is basically that, in any one of the 3 just mentioned ,a degree of subjectivity (relativity) must necessarily exist. For instance, given the language L(a,b,<) we'd have 2 theories say T1 = {a < b} and T2 = {~(a < b)}. Now given a particular ZF ordered pair say m = (a',b'). The point is in and of itself, the *set* m is neither an model nor an interpretation (nor a "structure", depending on one's definition of the word). The missing link here is that the mental (hence subjective/relative) mapping M between m and one of the T1 or T2 must necessarily exist. "Necessarily" here is because in strict Hilbert's syntactical formalism, there are just [syntactical] axioms, rules of inference, and (syntactical) provability. So literally we couldn't prove any undecidability by by mentioning the word "true" or "false", because there is none of such word in this Hilbert's strictly syntactical paradigm. In a nutshell, Godel introduced an addendum to this strict paradigm by introducing the _subjectivity_ of "arithmetic truths". (We could discuss this if that's desired). In any rate, sometimes in the past, I mentioned to MoeBlee and others that if one *loosely* considers the set m above as a "structure" then one could see a model of a theory is nothing but a mere _subjective_ mapping M between m and a *chosen*theory. So when *one* talks about a particular model, one should be prepared for the fact that others just might have another *different* mapping M' between m and whatever the other theory be. So in a slack usage, *with some appropriate explanation*, there's nothing wrong to say when one considers a model for a theory, others might not consider it as a model of the same theory (subjectively). Moeblee seemed to be bogged-down with a lot of terminology-technicalities to see the "bigger picture" of this model subjectivity. > > -- > hz |