Torkel Franzen on truth
in [Logic]
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From: george on 20 Dec 2007 19:16 > On 2007-12-18, in sci.logic, george wrote: > > The completeness theorem is fundamentally a proof that first-order > > semantics SIMPLY DOESN'T EXIST. On Dec 19, 7:23 am, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > Right. Just as the fundamental theorem of analysis establishes no-one knows > how to ride a bicycle. Abusive as usual. The completeness theorem is at least ABOUT first-order semantics. Where are the bicycles in analysis?
From: MoeBlee on 20 Dec 2007 20:19 On Dec 20, 4:03 pm, george <gree...(a)cs.unc.edu> wrote: > On Dec 19, 7:43 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > Okay, in a technical sense, '2=1+1' is true relative to models because > > it's true in some models but not in others. > > No, true in some interpretations but not in others. > Models, as OPPOSED to interpretations, have to be models > OF something. This terminological point has been discussed before in threads of not too distant past. The phrase 'true in a model' is routine and vastly understood. And though less routine, upon just a bit of explanation, it can be seen that it turns out (it is a theorem) that every structure for a language is a model (of some theory) and, of course, by definition, every model of a theory is a structure (for some language). Thus, theorem: M is a model iff M is a structure. This is derivable from defintions of these terms as given in Enderton's 'A Mathematical Introduction To Logic', though perhaps he does not make a point of mentioning it. Moreover, the use of the word 'interpretation' seems to to depend on the author (as do the particular definitions of 'model' and 'structure'). I haven't used the word 'interpretation' as technically defined. But some authors take the interpretation to be the function from the symbols to the object, or as a system of functions, or other arrangements. Meanwhile, I simply use 'structure' and 'model' with the precise definitions as in Enderton's book. And I distinguish between: a structure FOR a language and a model OF a theory which, given that every model is a structure, can be written by interchanging the words 'model' and 'structure' in any of the four possible combinations in the above. > In this case, we are talking about PA and > models are models of PA. A model of PA may be also a model of a different theory from PA. Of course, if we stipulate that we're talking only of models of PA, then '2=1=1' is true in those models. But my remark to Nam was not premised with the stipulation that we're talking only of models of PA. > And 2=1+1, being a THEOREM > of PA, is true in ALL models (of PA). Yes, of course. > So do NOT go around > giving Nam any "Okay"s. What I said is correct. The sentence '2=1+1" is true in some models and false in others, and there is nothing misleading in my saying that, especially in the rest of the context of my remark. > Nam is an idiot. He's got some conceptual blocks. You, on the other hand, are impossible in your own very special way. I just let go by uncontested thousands of words of yours, some of them HILARIOUSLY ill-conceived (I just love your 0 = {x | Ay ~yex}, which you posted to CORRECT a bunch of people who had ALREADY CORRECTLY observed various equations with 0 !!!) and also intellectually hypocritical (the line you're arguing about semantics and axioms lately is the EXACT NEGATION of the line you argued, rather by spraying your mouth-foam in my face, when we first exchanged posts), since I learned a while ago that not only is there no point trying to get through to you but doing so is an ESPECIALLY unpleasant endeavor. MoeBlee
From: G. Frege on 20 Dec 2007 20:36 On Thu, 20 Dec 2007 16:15:01 -0800 (PST), george <greeneg(a)cs.unc.edu> wrote: >> >> See for examplehttp://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." >> > This is all old by now and it does not matter what was typical in 1951. > Huh? Hey, george, this was written in 2000 (!), by Stewart Shapiro. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > > Nowadays we have clear definitions of all this. > You may be pretty sure that this guy knows what he's talking about. F. -- E-mail: info<at>simple-line<dot>de
From: MoeBlee on 20 Dec 2007 21:16 On Dec 20, 5:36 pm, G. Frege <nomail(a)invalid> wrote: > On Thu, 20 Dec 2007 16:15:01 -0800 (PST), george <gree...(a)cs.unc.edu> > wrote: > > > > >> See for examplehttp://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." > > > This is all old by now and it does not matter what was typical in 1951. > > Huh? Hey, george, this was written in 2000 (!), by Stewart Shapiro. G. Frege, I basically agree with you and Peter that Peter is well within his intellectual prerogative to give a stipulative definition of 'formal language' so that a formal language has both a formal syntax and formal semantics, which also is in general lines of agreement with the Church quote you gave. But, just to be clear, the above quote doesn't say that a language has both a syntax and semantics. Rather, the above quote says that a LOGIC has a language, a deduction system, and a semantics. MoeBlee
From: G. Frege on 20 Dec 2007 21:44
On Thu, 20 Dec 2007 18:16:17 -0800 (PST), MoeBlee <jazzmobe(a)hotmail.com> wrote: >> >> Huh? Hey, george, this was written in 2000 (!), by Stewart Shapiro. >> > G. Frege, I basically agree with you and Peter that Peter is well > within his intellectual prerogative to give a stipulative definition > of 'formal language' so that a formal language has both a formal > syntax and formal semantics, which also is in general lines of > agreement with the Church quote you gave. But, just to be clear, the > above quote doesn't say that a language has both a syntax and > semantics. Rather, the above quote says that a LOGIC has a language, a > deduction system, and a semantics. > Sure. I now that. I posted it to back up /george's/ position (just to be fair). Here's the quote with some context: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > > If some logicians, and according to George, many/most computer > scientists do talk of uninterpreted syntax as a [formal] 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." ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It's indeed a fact that "we" (today?) _do_ talk about "formal languages" without taking into account any semantical considerations. In brief: "[...] a formal language is a recursively defined set of strings on a fixed alphabet." http://plato.stanford.edu/entries/logic-classical/ george: "There simply now IS a restricted sense of "formal language" in which certain sets of strings are formal languages." --- You see, he's right. F. -- E-mail: info<at>simple-line<dot>de |