Torkel Franzen on truth
in [Logic]
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From: MoeBlee on 3 Jan 2008 18:44 On Jan 3, 2:55 pm, george <gree...(a)cs.unc.edu> wrote: > On Jan 2, 5:54 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > 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), > > > > Put up or shut up. > > > The thread you renamed to 'tedious sledding [...]' > > Lying as usual. The argument in that first encounter was about > the existence of "logical axioms", which I deprecated in favor > of inference rules. I was making the point that the usual > inference rules for the quantifier in 1st-order logic have > semantic content. You were claiming that axioms don't have > semantic content, I made no statement that I accept as summarized as "axioms have no semantic content". And your arguments then were not just about axioms and a preference for inference rules but also as to additional concerns regarding syntax and semantics, while, by the way, you were strawmaning me in prosecution of your arguments against things I wasn't even claiming, AS YOU SO USUALLY DO. > but my position then does not contradict > my position now: Like I said, your garbage is best left unsorted. > EVEN THEN, I was insisting that the things > with semantic content were inference rules AND NOT axioms. Bizarre. MoeBlee
From: herbzet on 4 Jan 2008 01:46 MoeBlee wrote: > herbzet wrote: > > MoeBlee wrote: > > > 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. > > > > > What is difficult about it? > > > > > (1) By definition, M is model of a set of sentences G iff M is a > > > structure for the language of G and every member of G is true in M. So > > > every model is a structure, by definition. > > > > Right. > > > > > (By the way, in another post I think I might have allowed G to be a > > > set of formulas. But I think G should be a set of sentences.) > > > > OK. > > > > > (2) Every structure M is a model of the set of valid sentences in the > > > language that M is a structure for. > > > > Yup. > > > > > So every structure is a model. > > > > Of any validity, yes. Can we say further that every structure is a > > model of a theory (other than the null theory containing only validities)? > > I'm not familiar with the expression 'null theory' to refer to the set > of validities, but I'll go along with it. I probably just made it up, although it seems right. It's the theory with no non-logical axioms (ie only validities as theorems). It seems to me that this is a degenerate case of the word "theory": usually I'd expect a "theory" (as opposed to a "logic") to _assume_ a logic and to have some non-logical axioms/theorems. > As to the question, I'd have to think about it. Bingo. That's what's "a new thought to me": that an arbitrary structure is a model of some non-null theory. I don't happen to know whether that's true. > > Shall we agree as to what sets of sentences constitute "a theory" first? > > For simplicity, let's confine to classical first order. > > Authors such as Enderton take a theory to be any set of sentences > closed under entailment (which, thanks to the completeness theorem, is > a set of sentences closed under provability). Authors such as Chang & > Keisler take a theory to be any set of sentences. And some authors > take a theory to be a pair <S E> where S is a set of sentences and E > is the set of sentences entailed by S (which, thanks to the > completeness theorem, is the set of theorems of S). > > I adopt Enderton's definition. OK. > > I usually think of "a theory" as having a recursive (or at least r.e.) > > set of axioms. > > That is a recursively axiomatized theory. There are theories that are > not recursively axiomatized. Yes, they seem a little imaginary to me, but I'm willing to go along with it. > > And as being consistent (i.e. having a model)! > > Those are consistent theories. There are theories that are not > consistent. It would seem that there is but one such theory over any signature: the theory that consists of all sentences in the language. This seems like another degenerate case of "theory". I'm willing to accept the informal locution "inconsistent theory"; I'm willing to accept an "inconsistent theory" as existing under the Enderton definition of "theory". But the extremal cases of "theories" that contain only validities, or that contain every sentence, do seem a little silly to me. I guess it's the price you pay for definitional elegance. > > > > 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). > > > > > In such contexts as I mentioned, what is the harm of using 'model' and > > > 'structure' interchangably, especially the context I mentioned?: > > > > If I recall correctly, this started when you said to Nam: > > > > > > Okay, in a technical sense, '2=1+1' is true relative to models because > > > > it's true in some models but not in others. > > > > To which George objected: > > > > > No, true in some interpretations but not in others. > > > > which might seem like a rather pedantic distinction (or, as you seem > > to think, a false distinction), but I think that in this area with > > Nam you have to be very precise, and the distinction is merited. > > Anyone may state definitions and explicate a discussion on the basis > of those definitions. Meanwhile, what I said is precisely correct > given ordinary defintions: > > '2=1+1' is true in some models and not true in other models. > > That is precisely correct. > > What is cloudy is the terminology 'true relative to models', which is > Nam's terminology. I don't use that terminology. All I said (or meant > to convey) in the passage you mentioned is that I could see a sense in > which that terminology could be understood. > > So, again, to be clear: > > My terminology is 'true in a model' and it is precisely correct that > '2=1+1' is true in some models and not true in other models. Nam's > terminology is 'true relative to a model', and though I do not in any > way claim to arbitrate what HE means by that, my point was just to say > that IF he means 'true in a model', then yes, of course, '2=1+1' is > true in some models and not true in other models. Then, there were the > rest of my remarks. Yuh, I guess the argument turns on whether you want to affirm a) 2 = 1 + 1 is false in no model (of PA) or b) 2 = 1 + 1 is false in some models (of the language of PA). It seems to be a simple ambiguity to be resolved. I personally would prefer (a) and would assume that you are using "model" in the sense of (a) in the absence of an explicit definition otherwise. I concede that it is possible that Nam uses the word "model" in some sense other than (a). I acccept that the literature allows the usage of "model" in the sense of both (a) and (b). But, as I said before, this seems ill- advised, in that it allows this ambiguity to arise. > > > (1) M is a model for a language > > > > > (2) M is model of a set of sentences > > > > > (3) M is structure for a language. > > > > > (4) M is structure of a set of sentences. > > > > > Especially, when the precise definitions are given. > > > > > (2) and (3) are more common than (1) and (4), though (1) is found in, > > > for example, Chang & Keisler, though, (4) is admittedly rather odd > > > sounding and therefore I don't use it, but, as long as my definitions > > > have been clearly stipulated, it wouldn't be harmful if I did use (4) > > > even though I don't prefer it. > > > > Sure, as long as you clearly stipulate your definitions, no problem. > > I think the default usages are, as you point out, (2) and (3). > > More common, not necessarily default. Well, time will tell, perhaps. > > > > > 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. [posted by george] > > > > > > 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. > > > > > Except we don't have an example of anyone disagreeing with George > > > simply because he is otherwise a royal jerk. > > > > On the contrary, I think that people in this forum tend to be argumentative > > when they are contradicted. They often will not take a brusque correction > > with equanimity. They will put uncharitable and even unreasonable > > constructions on what has been said to them. They will move heaven > > and earth to show that they were not, in fact, wrong. Do you want > > documentation? I'd prefer not to name names. Also, that would be > > a very tedious chore. > > I'm not interested in such tedium. But to be convinced I would have to > know what examples you have in mind specifically regarding George, > since I don't know of an instance of someone disagreeing with George > merely for his being a jerk. Such instances would, of course, require an inference as to someone's motives, since no one's going to assert that they are disagreeing just to be contrary. That will remain, irredeemably, a matter of opinion. I'd rather just state my opinions and leave it at that. > > I'd like to take this opportunity to say that George has never been > > a jerk to me. Of course I attribute this to his acute perception of > > my sterling character, but it's more probably that I'm not edjicated > > enough to merit abuse. I hope one day to be smart enough to rate > > an "oh, SHUT UP" from George. In general I find his explanations > > of things to be quite patient and lucid. > > I find his explanations usually to be bizarre. Not usually, but sometimes. Usually there is a core of something insightful even in the seemingly bizarre assertions. > > Of course, he is occasionally wrong. So what? > > I think he's more than occasionally wrong, and worse, he's a jerk > while being wrong. Actually, I find it even more annoying when someone is being a jerk while being right! -- hz
From: MoeBlee on 4 Jan 2008 12:08 > > On Jan 2, 6:25 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > and were not claimed to be definitions. And it occurred to me, even if the formulas were offered as definitions (though they were not, at least not by the posters who corrected the original poster), such a formula as: 0 = {x | ~x=x} is found as a definition in certain widely referenced textbooks (as well as a theorem in many others). MoeBlee
From: MoeBlee on 4 Jan 2008 12:44 On Jan 3, 10:46 pm, herbzet <herb...(a)gmail.com> wrote: > MoeBlee wrote: > > I'm not familiar with the expression 'null theory' to refer to the set > > of validities, but I'll go along with it. > > I probably just made it up, although it seems right. It's the theory > with no non-logical axioms (ie only validities as theorems). It seems > to me that this is a degenerate case of the word "theory": usually > I'd expect a "theory" (as opposed to a "logic") to _assume_ a logic > and to have some non-logical axioms/theorems. You may consider it a degenerate or trivial case, but it is still a theory by such ordinary defiinitions of 'theory' that I mentioned. > > As to the question, I'd have to think about it. > > Bingo. That's what's "a new thought to me": that an arbitrary structure > is a model of some non-null theory. I don't happen to know whether > that's true. I'd have to clear some technicalities, but I suspect it is true. But in any case, every structure is a model of the set of valid sentences (in the language that the structure is a structure for). And that set of valid sentences is a theory, even if only trivially so. Therefore, every structure is a model of some theory, even if only of the theory that is the set of valid sentences in the language that the structure is a structure for. > > > Shall we agree as to what sets of sentences constitute "a theory" first? > > > For simplicity, let's confine to classical first order. > > > Authors such as Enderton take a theory to be any set of sentences > > closed under entailment (which, thanks to the completeness theorem, is > > a set of sentences closed under provability). Authors such as Chang & > > Keisler take a theory to be any set of sentences. And some authors > > take a theory to be a pair <S E> where S is a set of sentences and E > > is the set of sentences entailed by S (which, thanks to the > > completeness theorem, is the set of theorems of S). > > > I adopt Enderton's definition. > > OK. > > > > I usually think of "a theory" as having a recursive (or at least r.e.) > > > set of axioms. > > > That is a recursively axiomatized theory. There are theories that are > > not recursively axiomatized. > > Yes, they seem a little imaginary to me, but I'm willing to go along > with it. They are famous and important. Most famous in that way is perhaps the theory that is the set of sentences true in the standard model for the language of PA. It is a famous theorem that that theory is not recursively axiomatized (or 'not recursively axiomatizable' if you prefer). That there are theories that are not recursively axiomatized (or 'not recursively axiomatizable' if you prefer) is a famous and important subject in mathematical logic. > > > And as being consistent (i.e. having a model)! > > > Those are consistent theories. There are theories that are not > > consistent. > > It would seem that there is but one such theory over any signature: > the theory that consists of all sentences in the language. Right. > This > seems like another degenerate case of "theory". If you prefer to think of it that way. > I'm willing to > accept the informal locution "inconsistent theory"; I'm willing > to accept an "inconsistent theory" as existing under the Enderton > definition of "theory". > > But the extremal cases of "theories" that contain only validities, > or that contain every sentence, do seem a little silly to me. I > guess it's the price you pay for definitional elegance. Whether it seems silly, it still is indeed the case, from the definitions. > > My terminology is 'true in a model' and it is precisely correct that > > '2=1+1' is true in some models and not true in other models. Nam's > > terminology is 'true relative to a model', and though I do not in any > > way claim to arbitrate what HE means by that, my point was just to say > > that IF he means 'true in a model', then yes, of course, '2=1+1' is > > true in some models and not true in other models. Then, there were the > > rest of my remarks. > > Yuh, I guess the argument turns on whether you want to affirm > > a) 2 = 1 + 1 is false in no model (of PA) > > or > > b) 2 = 1 + 1 is false in some models (of the language of PA). I affirm both. But the particular remark I made didn't mention PA. > It seems to be a simple ambiguity to be resolved. I personally would > prefer (a) and would assume that you are using "model" in the sense > of (a) in the absence of an explicit definition otherwise. I concede > that it is possible that Nam uses the word "model" in some sense > other than (a). There is no conflict among a), b), and my use of the word 'model'. As to Nam's nottions, that's a whole other matter. > I acccept that the literature allows the usage of "model" in the > sense of both (a) and (b). But, as I said before, this seems ill- > advised, in that it allows this ambiguity to arise. There is not an ambiguity in what I said. '2 = 1 + 1' is true in some models and false in other models Is easily formalized as: EBC(B is a model & C is a model & '2 = 1 + 1' is true in B & '2 = 1 + 1' is false in C) with B is a model <-> EG(G is a set of sentences & B is a model of G) And even without such pedantic formulations, it's just common sense in mathematical logic what is meant by: B is a model '1+1=2' is true in B. '1+1=2' is false in B. Nothing more than quite ordinary meanings in mathematical logic are needed to understand that '1+1=2' is true in some models and false in others. I could have said it equivalently as '1+1=2' is a contingent sentence. This is utterly straightforward stuff. > > > On the contrary, I think that people in this forum tend to be argumentative > > > when they are contradicted. They often will not take a brusque correction > > > with equanimity. They will put uncharitable and even unreasonable > > > constructions on what has been said to them. They will move heaven > > > and earth to show that they were not, in fact, wrong. Do you want > > > documentation? I'd prefer not to name names. Also, that would be > > > a very tedious chore. > > > I'm not interested in such tedium. But to be convinced I would have to > > know what examples you have in mind specifically regarding George, > > since I don't know of an instance of someone disagreeing with George > > merely for his being a jerk. > > Such instances would, of course, require an inference as to someone's > motives, since no one's going to assert that they are disagreeing just > to be contrary. That will remain, irredeemably, a matter of opinion. > I'd rather just state my opinions and leave it at that. And I know of no evidence that anyone has disagreed with George on some point or another merely because he is a jerk. > > > I'd like to take this opportunity to say that George has never been > > > a jerk to me. Of course I attribute this to his acute perception of > > > my sterling character, but it's more probably that I'm not edjicated > > > enough to merit abuse. I hope one day to be smart enough to rate > > > an "oh, SHUT UP" from George. In general I find his explanations > > > of things to be quite patient and lucid. > > > I find his explanations usually to be bizarre. > > Not usually, but sometimes. Usually there is a core of something > insightful even in the seemingly bizarre assertions. I find it usual. Often, at least. And occasionally there is some bit of insight mixed in with his bizarre declamations, though hardly enough to justify. Also, of course, many of his statements are not bizarre and are good points. But I find that he's usually, or at least much much too often bizarre, especially with his continual out-of-the- blue decrees as to how things must be formulated, defined, regarded, or conceived to be correct. > > > Of course, he is occasionally wrong. So what? > > > I think he's more than occasionally wrong, and worse, he's a jerk > > while being wrong. > > Actually, I find it even more annoying when someone is being a jerk > while being right! I guess that's a joke or something. Of course, it's subjective what annoys a person. But it does seem disconnected to me to be more annoyed by someone who is at least correct. MoeBlee
From: Alan Smaill on 4 Jan 2008 13:30
Newberry <newberryxy(a)gmail.com> writes: > On Dec 8, 2:04 pm, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> > The cornerstone of TF's argument is that the consistency of ZFC is >> > provable in ZFC + an axiom of infinity, which is no longer manifestly >> > true. We hesitate as we go higher up in the chain of theories, hence - >> > according to TF - we are not better than any machine because we cannot >> > say about an arbitrary system that its Goedel formula is true. Yet he >> > is absolutely sure that PA and ZFC are consistent. >> >> Where did TF claim that he was "absolutely sure that ... ZFC [is] >> consistent"? >> >> I see no evidence that that was his view in the cite you posted >> earlier in the thread. > > He has an entire chapter in his book "Skepticism and Confidence", > where he refutes the skeptics. Having dug out the book in question, I now have the context of the following. > "And given that the axioms of ZFC are so utterly compelling, so > obviously true in the world of sets, we can do no better than to adopt > these axioms as our starting point. Since the axioms are true, they > are also consistent." [p.105] The very next sentence says: "Again, the point at issue is not whether such a view of the axioms of ZFC is justified, but whether it makes good sense to appeal to the incompleteness theorem in criticism of it." The context is exactly a critique of the argument that suggests that *if* someone has certain knowledge of the truth of the axioms of a system *then* they will run into trouble from the incompleteness theorem. So the view expressed are *hypotheical*, not TF's at all. And having set up the context clearly, TF reminds the reader of the context afterwards. > "if the axioms of ZFC are manifestly true, they are obviously > consistent" [p.105] As I said before, this is a conditional statement. Why on earth do you think it expresses a commitment to the antecedent being true? At this rate someone will have to give us "The uses and abuses of TF's writings". (And, while I'm here, he does not refute scepticism in this chapter.) -- Alan Smaill |