Torkel Franzen on truth
in [Logic]
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From: MoeBlee on 2 Jan 2008 18:55 On Dec 22 2007, 9:41 am, george <gree...(a)cs.unc.edu> wrote: > MoeBlee's point was that people were in a very generalized habit of > using > "model" withOUT requiring it to be a model of anything in PARTICULAR. No that was NOT my point. I didn't say anything about a generalized habit or any habit. And I did not say that M can be a model without being a model of some set of sentences. As to the word 'particular', I simply observed that we have the 2-place predicate: M is a model of G and that we can also have the 1-place predicate M is a model by defining M is a model iff there is a set of formulas G such that M is a model of G. And doing so harms nothing. And doing so helps to see why such as Chang & Keisler say such things as M is a model for the language L where other authors say M is a structure for the language L. Chang & Keisler's use is clear since 'model' and 'structure' can be be used interchangably in such contexts since M is a model iff M is a structure. > That is an observable fact about the brute weight of usage; neither I > nor > anyone else GETS to disagree with it, so MoeBlee probably thought it > would > therefore be safe, or at least defensible, to assert it. He was > entirely wrong > about that. People's general habit of doing this IS BAD. It is > sloppy. > IT NEEDS reform. People NEED to clean up their (usage)ACT. > Instead, MoeBlee chose to DEFEND the fact that people generically tend > to talk this way AS ACCEPTABLE because it is allowed by definitions > that > he can quote from Enderton. That is using a good work to confuse > people > and make discourse in general LESS accurate and it is not > intellectually > acceptable behavior. There is nothing misleading or damaging to mathematics by my usage, especially as I defined it explictily and as it is used even in such an authoritative works as Chang & Keisler (I mean the kind of thing as 'M is model for the language L' ). MoeBlee
From: MoeBlee on 2 Jan 2008 19:12 On Dec 23 2007, 5:19 am, herbzet <herb...(a)gmail.com> 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. (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.) (2) Every structure M is a model of the set of valid sentences in the language that M is a structure for. So every structure is a model. > 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?: (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. > > 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. MoeBlee
From: MoeBlee on 2 Jan 2008 19:18 On Dec 23 2007, 9:45 am, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > In any rate, sometimes in the past, I mentioned to MoeBlee and others > that if one *loosely* considers the set m above as a "structure" then > one could see a model of a theory is nothing but a mere _subjective_ mapping > M between m and a *chosen*theory. Yeah, well, I DON'T loosely take 'structure' defined as you do. I use the technical definition given by a certain text (or, if other people use a different formulation from another text, then I'm happy to use the other formulation while I note the way in which it is just a different technical way to achieve a formalization of the same notion). > Moeblee seemed > to be bogged-down with a lot of terminology-technicalities to see > the "bigger picture" of this model subjectivity. Your 'bigger picture' is a mural of confusion. And my being careful as to technical definitions does not disallow me from seeing a coherent bigger picture. Moeblee
From: herbzet on 3 Jan 2008 02:14 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)? Shall we agree as to what sets of sentences constitute "a theory" first? I usually think of "a theory" as having a recursive (or at least r.e.) set of axioms. And as being consistent (i.e. having a model)! > > 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. > (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). > > > 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'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. Of course, he is occasionally wrong. So what? -- hz
From: MoeBlee on 3 Jan 2008 12:35
On Jan 2, 11:14 pm, herbzet <herb...(a)gmail.com> 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. As to the question, I'd have to think about it. > 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. > 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. > And as being consistent (i.e. having a model)! Those are consistent theories. There are theories that are not consistent. > > > 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. > > (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. > > > > 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. > 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. > 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. MoeBlee |