Torkel Franzen on truth
in [Logic]
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From: george on 22 Dec 2007 12:56 > On 21 Dec, 00:02, george <gree...(a)cs.unc.edu> wrote: > > What assigns meanings to the wffs of the system is DESIGNATING > > SOME OF THEM AS AXIOMS. On Dec 21, 4:03 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > Really? Well that sounds like magic to me. Oh, it is. You should be more than impressed. More seriously, though, obviously there is more than the designation going on. These are FIRST-ORDER formal languages we are talking about and the rules of first-order grammar and the rules OF INFERENCE IN FIRST-ORDER LOGIC are also necessary to cause axiomatizations to have meanings. But my point is, ALL OF THOSE ARE CHARACTERIZABLE SYNTACTICALLY, so, again, SEMANTIC INTERPRETATION is NOT necessary. Something LIKE "interpretation" is necessary for the LOGICAL part of the vocabulary, perhaps, but again, that is performable syntactically. > If I tell you that "mae glo yn du" is true in Welsh [heck, hope I've > remembered that right], you don't thereby get to know what it means. THIS is NOT like THAT! > Telling you the same about some > other Welsh sentences won't help > either. If, however, some of the words were logical connectives and you DID manage to figure out those, and if you had a specification of ALL the provable things in the language, then help might eventually occur, or at least insight. For you to be trying this with a finite number of assertions about the concrete world, when you know that OUR subject matter deals with an infinite number of axioms about abstractions, is, well, I hope it is just a way of designing an introductory lecture on the topic. It is not serious. The natural-language sentences you are whining about will fail here because THOSE SENTENCES ARE ABOUT THE REAL WORLD. But if they are about abstractions or math then it is ENTIRELY different. If pattern is all they are TRYING to communicate IN THE FIRST PLACE, then this can and does work, simply because there is nothing beyond the patterning FOR the symbols TO mean, in the abstract context. > Telling you that in my fave axiomatized system "Fa" is an axiom (or is > an axiom and true), you don't thereby get to know what it means. It's an axiom SYSTEM. This axiom won't occur by itself. > Telling you the same about some other sentences of the system won't > help either. I repeat, set theory, PA, and every other 1st-order theory you know IS A FACTUAL COUNTEREXAMPLE. WhatEVER "epsilon" means, it means in virtue of the axioms of ZFC AND NOTHING ELSE (except maybe of the definition of FOL). Of course, that theory is interpretable into many different structures, but what will you THEN allege? That before the language is interpreted, it means NOTHING? Here is how stupid you sound: You, the distinguished Prof.Peter Smith, are saying the following: 2+5=7 doesn't mean anything. It doesn't say anything about anything. Now, if it were interpreted, if I said 2 cars + 5 cars = 7 cars, or starting with 2 cups of sugar and adding 5 cups of sugar yields 7 cups of sugar, THEN THAT would be saying something. But both of those would be INTERPRETING 2+5=7. UNTIL you do something like that, 2+5=7 can't mean anything. The actual truth of the matter is that it is precisely BECAUSE 2+5=7 DOES mean something by itself (and something we can prove from axioms, at that) that WE CAN KNOW that 2 eggs plus 5 eggs will be 7 eggs, AND ALL THOSE OTHER consequences. The uninterpreted version is actually MORE important. Of course, on your side of the fence, this state of affairs is so intolerable that you will make up abstract entities for 2, 5, and 7 to actually "be" so that you CAN interpret them.
From: george on 22 Dec 2007 12:59 On Dec 20, 8:36 pm, G. Frege <nomail(a)invalid> wrote: > >> "Typically, a /logic/ consists of a formal or informal language together > >> with a deductive system and/or a model-theoretic semantics. The language > >> is, or corresponds to, a part of a natural language like English or > >> Greek. The deductive system is to capture, codify, or simply record > >> which inferences are correct for the given language, and the semantics > >> is to capture, codify, or record the meanings, or truth-conditions, or > >> possible truth conditions, for at least part of the language." > Huh? Hey, george, this was written in 2000 (!), by Stewart Shapiro. In which case I completely agree with it. He is talking about what is typical. This is how it is usually done, yes. This is expository. It is not relevant to the specific formal enterprise that was under debate. And it was cited BY OTHER PEOPLE as MATCHING something that WAS promulgated TECHNICALLY in 1951. 50 years later, everybody's perspective is different. In any case, none of THIS is the POINT! The POINT is that natural language IS NOT RELEVANT PERIOD to this whole enterprise!
From: george on 22 Dec 2007 15:41 On Dec 22, 12:46 pm, G. Frege <nomail(a)invalid> wrote: > > The Axioms are the things that you just presume true FROM THE > > BEGINNING, ... > > Oh, wait a second. Aren't you mixing up TRUTH with PROVABILITY here? :-o It's not MIXING if they never got SEPARATED TO BEGIN with! It is if you DO have structures/interpretations/semantics that the concepts are SEPARATE and you have to draw a distinction. If there simply ARE No models or interpretations at all then the ONLY way ANYthing ever gets to be true is BY being proved. (P v ~P) is tautological and therefore axiomatic and therefore true, but withOUT models, what can you say about the truth of P if it is not a theorem, EVEN though you know (P v ~P) ? NOTHING, THAT'S what. My point is that we are obviously in two different worlds here. One is the usual world where semantics is relevant, and the other is the post-completeness-theorem world where one has THE OPTION of simply ignoring first-order semantics altogether, SINCE THE COMPLETENESS THEOREM EXPLAINS how to re- do that SYNTACTICALLY. It MATTERS WHETHER "it's true" VERSUS "it's provable" IN THE WORLD THAT HAS semantics. Since I am inviting people into the world THAT DOESN'T have semantics, that distinction is NOT stressable versus ME!
From: Nam D. Nguyen on 22 Dec 2007 16:40 tchow(a)lsa.umich.edu wrote: > In article <Myhaj.20387$Tx.15878(a)pd7urf3no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: > <tchow(a)lsa.umich.edu wrote: > <> (3) In every model of PA, there are no nonzero integers m and n such that > <> m^2 = 2 n^2. > [...] > <Apparently you've missed my last post where I made the correction: > < > < > Let me re-phrase it: what might be "integers" or "standard" to one, > < > might not be to the others. > < > <Any rate, so (3) is true, *relative* to PA's models: the *truth* of (3) > <is a relative value, and the relativity is still there! > > What do you mean that (3) is true relative to PA's models? I might just have been too quick here. Yes my statement "(3) is true, *relative* to PA's models" sounds a bit obscure in meaning. So let me explain it. But first, iirc, the context of our debate is you (and others) believe certain mathematical truths, and just like religion beliefs, are *absolutely* true that we could not believe or see otherwise. On the other hand I believe that in so far as any mathematical statement has to be stated/worded (whether it's a FOL formula or a meta statement), there's *nothing absolute* about it's truth: there is always hierarchy of framework contexts that you could traverse upward one level and go down in the other direction to see the opposite truth *for the very same stated statement*! (But religion statement is not supposed to be as such!). In other words, there is *no intrinsic (built-in) semantic for a sentence* (FOL formula, meta statement, or any sentence for that matter). Its semantic and hence truth is always relative to some context! > What else can (3) be relative to? Given what I've just stated above, then there are more than one way the truth of (3), as a meta statement, could be altered. For an example, if we simply change some of the reasoning framework contexts, such as rules of inference, the truth of (3) in principle could be changed. > Or to ask the question another way, explain to us a sense in which (3) can be false. For another example, if we change by what we mean by "PA" then (3) is not necessarily true. So, sure, one could accuse me of talking what would typically considered as "frivolous" here. On the other hand, talking about mathematical "absolute" beliefs/truths as if they were religious ones then I think everything is a fair game!
From: Nam D. Nguyen on 22 Dec 2007 17:03
MoeBlee wrote: > On Dec 19, 3:11 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: >> tc...(a)lsa.umich.edu wrote: >>> In article <q8gaj.19679$Tx.4697(a)pd7urf3no>, >>> Nam D. Nguyen <namducngu...(a)shaw.ca> wrote: >>>> tc...(a)lsa.umich.edu wrote: >>>>> (1) There are no nonzero integers m and n such that m^2 = 2 n^2. >>>>> (2) In the standard model of the integers, there are no nonzero integers >>>>> m and n such that m^2 = 2 n^2. >>> [...] >>>>> Does that mean that (2) is absolutely, unconditionally true? >>>> No. What is a standard model of the integers might not be the "standard" >>>> model to others! >>> All right, then, let's try this one: >>> (3) In every model of PA, there are no nonzero integers m and n such that >>> m^2 = 2 n^2. >>> Most people would agree with (3), since the proof of the irrationality of >>> sqrt(2) can be formalized in PA, and therefore the statement holds in all >>> models of PA, standard or nonstandard. >>> Do you believe (3)? Is (3) absolutely true? Whether or not your "standard" >>> model is the same as mine makes no difference, since the assertion holds in >>> every model, so I don't see where relativism enters. >> Apparently you've missed my last post where I made the correction: >> >> > Let me re-phrase it: what might be "integers" or "standard" to one, >> > might not be to the others. >> >> Any rate, so (3) is true, *relative* to PA's models: the *truth* of (3) >> is a relative value, and the relativity is still there! > > (1) No, you completely dodged the point. No I'm not. See my latest response to TC. > > Okay, in a technical sense, '2=1+1' is true relative to models because > it's true in some models but not in others. In a technical sense, "2=1+1" is true in all models of some consistent theories where it is a theorem. > > But the challenge put to you was to say in what way (3) is relative to > models, since it's not a matter of being true in some models and not > in others, but rather of being true PERIOD, since it is a statement > about ALL models. To reinforce that it is a statement about all > models, please recognize that it is of the form, Given ANY model, if > it is a model of PA, then [...]. > > (2) What is the relativity in '0011' being the same string as '0022 > with 1 substituted for 2'? Quite a few ways of relativity! For example, what did *you* mean by "substituted"? Were you referring to some kind of _mapping_? > > MoeBlee > |