Torkel Franzen on truth
in [Logic]
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From: Nam D. Nguyen on 17 Dec 2007 20:13 MoeBlee wrote: > On Dec 17, 3:27 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: >> tc...(a)lsa.umich.edu wrote: >>> In article <xLw9j.7491$hQ3.4060(a)pd7urf3no>, >>> Nam D. Nguyen <namducngu...(a)shaw.ca> wrote: > >> From what I gather, we don't call that "beliefs". We call it _interpretation > > That we have a method for interpreting formal languages does not > contradict that one may believe certain things about mathematics. > >> which model basically is, and in which truths are true or false. > > "in which truths are true or false". Are you sure you've eaten > breakfast this morning? I usually have a glass of milk every morning. Some *interpret* that as having breakfast; others would have opposite interpretation. Actually my interpretation on that might vary from years to year. What's about yours interpretation on this? > >> The problem >> of this model-truth is over the same "structure" there could be opposite >> interpretation. > > No, that's completely wrong. Given a structure, there is only one > interpretation associated with that structure. Given the structure "Nam's having a glass of milk every morning", how many interpretations would one have for the sentence "Nam has breakfast every morning"? > >> Religious truth on the other hand is supposed to *believed* >> as true whether or not there is a model to reflect the truth. > > I've never seen such a description of religious belief. I'm sure there are descriptions that are very much *similar*! > >> That's why >> belief doesn't have much of relevance in reasoning. > > Belief may or may not have relevance in reasoning, but the confusions > you just posted don't lead to any conclusion on the matter. Whose "confusions" are you talking about? I don't seem to have any here. > >>> Then how do you become convinced that *anything* is true? >> As I've explained above. > > No you didn't. That's one opinion of course. > >>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? >> On the basis of model that "sqrt(2) is irrational" is true, of course. > > Maybe you mean, on the basis that there is a model in which "sqrt(2) > is irrational" is true. "Maybe"? My answer to Tim Chow's question is a straightforward short-one-liner answer and you seemed to not understand? > And there is a model in which it is false also. So far I don't see what your point here is! > What about operations on finite strings? What about them? > Don't you believe, for > example, irrespective of any model, that the string "0011" is the same > as the string "0022 [with 1 substituted for 2]"? In what context are you talking about "sameness", "substitute", etc... Sorry your question is too vague in semantic and consequently is subject to different interpretations. > >>> On the basis of the proof? >> No, not on the basis of proof: what is true or false is based strictly on model. >> Syntactical provability is actually in a different (and independent) paradigm, >> not withstanding Completeness. >> >>> But the proof starts with some axioms. >> Of course. >> >>> On what basis do you become convinced of the correctness of the axioms? >> What exactly does "correctness of the axioms" mean? >> >>> Or are you *not* convinced of the axioms? >> The only senses for which we could talk about axioms are: >> >> (a) They be independent from each other. >> (b) They don't contradict each other. > > No, there are lots of other properties of axioms. One, for example, is > that of a certain model being a model of the axioms. Of course there are other properties: axioms' being finite formulas, etc... All of these (and what you've mentioned) are utterly trivial not worth being mentioned, it seems. So why are you mentioning here? Is it because it has something to do with the purported "correctness of the axioms" that is being discussed between me and the other poster? (Besides, to be to be a property of axioms it has to apply to all axioms in all circumstances, e.g. being finite formulas. Your "certain model being a model of the axioms" is not applicable to all axioms!) > >> So, again, what does it mean to be "convinced of the axioms"? >> >>> But if you're not convinced of the axioms, then what good is a proof of >>> "sqrt(2) is irrational" from those axioms? >> Proofs of course are good as a mechanism of assisting us in preventing >> our reasoning from being inconsistent. Of course. > > Except if the axioms are inconsistent. Actually, (first order) proof > doesn't ensure consistency but rather entailment. Agree. Except that I only said "as a mechanism of assisting": I never said anything about proofs guaranteeing/ensuring consistency. Of course not, in general. But in some particular circumstances, proof would help consistent reasoning. > > MoeBlee >
From: Nam D. Nguyen on 17 Dec 2007 20:47 MoeBlee wrote: > On Dec 17, 5:13 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > >> I usually have a glass of milk every morning. Some *interpret* that >> as having breakfast; others would have opposite interpretation. >> Actually my interpretation on that might vary from years to year. >> What's about yours interpretation on this? > > That your postings make apparent that for you a class of milk is not > sufficient nutrition to maintain a healthy working brain. > Of course talking about mathematical reasoning with you seems fruitless, based on many of your posts in the past: they're seem to be more about "attacking", "character assassination", or what have we, and less about the *genuine discussing* about underlying points of contention. So hope you don't mind my stopping dialog with you at this point in time. > > MoeBlee
From: george on 20 Dec 2007 19:02 > > Well..., Alonzo Church [you know that guy?] also mentioned such a view, He mentioned it in 1951. This was well before people were in the habit of talking about regular languages, context-free languages, or anything else in a hierarchy of PURELY formal languages. NObody, NOT EVEN Alonzo Church, NOWadays, gets to try to put THAT genie back in the bottle. There simply now IS a restricted sense of "formal language" in which certain sets of strings are formal languages. > > In order to obtain a formalized language Well, certainly, if you start out with Peter Smith's sense of "language" and then try to make THAT "more" formal, then, yes, > > 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) If on the other hand you just start with a purely formal language to begin with, the question of a "formalized language" is simply moot. The language didn't NEED to be formalIZED because it was already totally formal. What it needed to be, according all three of Fritsche, Smith, and Church, here, was semanticized or interpreted. On Dec 19, 6:45 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > Exactly the distinction that well-brought-up logicians still make (if > not necessarily using those words) :-))) This is not a distinction and you are lapsing into lying and ignorance. What assigns meanings to the wffs of the system is DESIGNATING SOME OF THEM AS AXIOMS. This is an act that does have semantic content simply because axioms have to be true, but that is not the point. The point is that the axioms are characterizABLE syntactically AS OPPOSED to semantically and that they confer NOT semantic truth BUT RATHER syntactic provability. As a result of which they do NOT need to be interpreted in order to communicate.
From: george on 20 Dec 2007 19:03 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. In this case, we are talking about PA and models are models of PA. And 2=1+1, being a THEOREM of PA, is true in ALL models (of PA). So do NOT go around giving Nam any "Okay"s. Nam is an idiot.
From: george on 20 Dec 2007 19:15
> >> 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. Prof Smith how dishonestly defends: > > What I said was that uninterpreted syntax is just that, uninterpreted. > > So not (yet) a vehicle for communicating anything. Yes, you did indeed say that, but that is not germane to the point I was just making, which is that you ALSO said, > you are thinking of a formalized language as a purely syntactic system. > Why? I know that logic books occasionally talk like this, but this has > always seemed to me simply to be a misuse of the word "language". THAT is the point under debate. I offered what should've been a clear defense, which is simply that on the computing side, people do legitimately care about languages as purely syntactic collections and do legtimately care about how hard they are to define/parse. It is a legitimate arena of inquiry. It is normal to talk and think about purely formal languages AND TO CALL THEM LANGUAGES. It is not "simply a misuse of" ANYthing. The fact that you choose to call it such makes you simply abusive. > > And so, in the ordinary sense of the term, > > not a language. YOU DIDN'T SAY "In the ordinary sense of the term". YOU DID SAY "simply a misuse". Simply choosing a more restricted technical meaning for a term in a context of discourse IS NOT simply misusing the term. > > Hardly an indefensible view But hardly recognizable as the view were promoting. > > If some logicians, and according to George, many/most computer > > scientists do talk of uninterpreted syntax as a language (without > > qualification) Hardly. We call them formal languages. But just as surely as a yellow banana is a banana, a formal language is a language. > Which indeed is the case. FF is being stupid here. He is claiming to agree with you while citing an article that agrees with YOUR point, but DISagrees with what people who were studying complexity-classes of formal languages would do. > 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. Nowadays we have clear definitions of all this. > > 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, good hesitation. My point is simply that Prof.Smith's contention that formal languages are too degenerate to be really called languages is fundamentally ignorant. He is not a linguist. His claim that they need to be interpreted in order to communicate is factually false and every decidable categorical first- order theory constitutes a living counter-example to it. The fact that is currently debating against that says a lot more about his character than it does about his intellect. > Well, how about the following approach? Those /formal languages/ > (considered in formal logic) are (usually) at least "capable" of > "bearing meaning". That is not under contention. If you interpret them then of course they have meaning AFTER that. My point was simply that you don't have to do anything SEMANTIC in order to CAUSE them to "realize" their "capability" of bearing meaning BECAUSE FIRST-ORDER CONSEQUENCE IS SYNTACTIC, because there is a completeness theorem. |