Torkel Franzen on truth
in [Logic]
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From: Peter_Smith on 19 Dec 2007 18:26 On 19 Dec, 18:53, george <gree...(a)cs.unc.edu> wrote: > Prof.Smith and I have been talking, for example, about the > intended model vs. the formal language. He is the one who > said that he didn't think formal languages should even be referred > to as a language. That is considerably less defensible than anything > *I* have ever said. What I said was that uninterpreted syntax is just that, uninterpreted. So not (yet) a vehicle for communicating anything. And so, in the ordinary sense of the term, not a language. Hardly an indefensible view. If some logicians, and according to George, many/most computer scientists do talk of uninterpreted syntax as a language (without qualification) then fine as long as the jargon is made clear: but it *is* in that usage specialist jargon, and not -- to my mind -- entirely happy jargon as it is potentially misleading in various ways. I hesitate to add "as evidenced here".
From: Peter_Smith on 19 Dec 2007 18:45 On 19 Dec, 19:21, G. Frege <nomail(a)invalid> wrote: > Well..., Alonzo Church [you know that guy?] also mentioned such a view, > in one of his papers. He writes: > > "We distinguish between a /logistic system/ and a /formalized language/ > on the basis that the former is an abstractly formulated calculus for > which no interpretation is fixed, and thus has a syntax and no > semantics; but the latter is a logistic system together with an > assignment of meanings to its expressions. [...] > > In order to obtain a formalized language it is necessary to add to > these /syntactical rules/ of the logistic system, /semantical rules/ > assigning meanings (in some sense) to the well-formed expressions of the > system." > > (Alonzo Church, The Need for Abstract Entities, 1951) Exactly the distinction that well-brought-up logicians still make (if not necessarily using those words) :-)))
From: MoeBlee on 19 Dec 2007 19:43 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. Okay, in a technical sense, '2=1+1' is true relative to models because it's true in some models but not in others. 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'? MoeBlee
From: G. Frege on 19 Dec 2007 19:51 On Wed, 19 Dec 2007 15:26:03 -0800 (PST), Peter_Smith <ps218(a)cam.ac.uk> wrote: >> >> Prof. Smith and I have been talking, for example, about the >> intended model vs. the formal language. He is the one who >> said that he didn't think formal languages should even be referred >> to as a language. That is considerably less defensible than anything >> *I* have ever said. >> > What I said was that uninterpreted syntax is just that, uninterpreted. > So not (yet) a vehicle for communicating anything. And so, in the > ordinary sense of the term, not a language. Hardly an indefensible > view. > > If some logicians, and according to George, many/most computer > scientists do talk of uninterpreted syntax as a language (without > qualification) > Which indeed is the case. See for example http://plato.stanford.edu/entries/logic-classical/ "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." > > then fine as long as the jargon is made clear: but it > *is* in that usage specialist jargon, and not -- to my mind -- > entirely happy jargon as it is potentially misleading in various ways. > I hesitate to add "as evidenced here". > Well, how about the following approach? Those /formal languages/ (considered in formal logic) are (usually) at least "capable" of "bearing meaning". F. -- E-mail: info<at>simple-line<dot>de
From: tchow on 19 Dec 2007 20:05
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? What else can (3) be relative to? Or to ask the question another way, explain to us a sense in which (3) can be false. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences |