The End of an Era - That of The Natural Numbers
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From: Daryl McCullough on 13 Jun 2010 07:39 Nam Nguyen says... > >Daryl McCullough wrote: >> >> No, it doesn't, but I don't actually care what Shoenfield or Tarski >> said. What I care about is having a non-stupid definition of "truth >> in a model" that applies to models with empty domain. > >But you shouldn't have worried about that: because that "non-stupid >definition" could only render falsehood. No, a nonstupid definition of "truth in a model" makes some statements true, and the rest false. >Why would you get so concerned about "truths" of the emptiness, anyway? I've explained it before. I'm interested in *submodels*. Suppose you start with a structure S for a language L and you restrict attention to a substructure S' for a sublanguage L' in the following way: The domain U' of S' consists of all elements of S that satisfy some unary predicate D(x). The denotations of the relation and function symbols for S' are the same as in S, except the domains are restricted to elements of U'. The constant symbols have the same denotations they had in S'. Then with one way of defining "truth in a model", truth in the submodel has a simple relationship to truth in the original model. You just *relativize* the formulas. Let Phi' be the formula Phi relativized to predicate D. Then Phi' is related to Phi through (Ax Phi)' = Ax D(x) -> Phi' (Ex Phi)' = Ex D(x) & Phi' (Phi & Psi)' = Phi' & Psi' (~Phi)' = ~Phi' etc. Then we have the nice relationship between truth in the model and truth in the submodel: Phi is true in submodel S' <-> Phi is in the sublanguage L', and Phi' is true in the model S This notion of truth in a submodel can be used for *any* predicate D(x) whatsoever, *even* one for which no elements satisfy D(x). There is no reason to make a special case for empty domains. The other nice thing about the nonstupid definition of truth in a model is that it gives a nice way to state that a domain is empty: ~Ex x=x That is a formula that is only true in the empty model. So it becomes possible to state "Nothing exists", and that's actually true in the model with empty domain. >> You want to propose that we use some particular definition of truth >> in the model with empty domain, say *why* you want to use that definition. >> Is there any point? Arguing that Tarski did it is no good, unless you >> can reproduce Tarski's reasoning for doing it that way. > >I _already_ did mention one reason for the "why": the relativity nature of >reasoning in FOL. So, the point of your definition is that it allows you to win arguments in a way that a nonstupid definition wouldn't? What I meant by a reason why is the reason that someone who is interested in model theory would prefer your definition over mine. There are technical advantages to my definition (the nice relationship between truth in a model, and truth in a submodel, and the fact that it is possible to state a formula with the interpretation "the domain is empty"). There seem to be only emotional reasons for adopting your definition. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 13 Jun 2010 11:31 Daryl McCullough wrote: > Nam Nguyen says... >> Daryl McCullough wrote: >>> No, it doesn't, but I don't actually care what Shoenfield or Tarski >>> said. What I care about is having a non-stupid definition of "truth >>> in a model" that applies to models with empty domain. >> But you shouldn't have worried about that: because that "non-stupid >> definition" could only render falsehood. > > No, a nonstupid definition of "truth in a model" makes > some statements true, and the rest false. The caveat here is when I said "could only render falsehood" I meant that only in the cases you had refereed as "models with empty domain". Iow, you had complained (above) >>> What I care about is having a non-stupid definition of "truth >>> in a model" that applies to models with empty domain. What I was saying is that the "non-stupid definition" (which I take it to be the normal definition of truth through set-membership) would only render falsehood in the case of empty domain (U = {}). > >> Why would you get so concerned about "truths" of the emptiness, anyway? > > I've explained it before. I'm interested in *submodels*. Suppose you > start with a structure S for a language L and you restrict attention to > a substructure S' for a sublanguage L' in the following way: > > The domain U' of S' consists of all elements of S that satisfy some > unary predicate D(x). We can just top it right here. If the U of S is empty, so is the U' of S': what is the point of going further? > The denotations of the relation and function > symbols for S' are the same as in S, except the domains are restricted > to elements of U'. The constant symbols have the same denotations they > had in S'. But why bother with "except the domains are restricted to elements of U'", when both U and U' are empty? Your argument here about "sub-models" has lost me. I've never said there's no case in which the standard definition of model truths a la Tarski and set-membship would yield x=x being true. My position all along is that such definition would yield x=x being false in the degenerated case of U = {}. What you seem to have assumed here with "sub-models" is U (of S) is non-empty, which is not the case when x=x would be false! > > Then with one way of defining "truth in a model", truth in the submodel > has a simple relationship to truth in the original model. You just > *relativize* the formulas. Let Phi' be the formula Phi relativized to > predicate D. Then Phi' is related to Phi through > > (Ax Phi)' = Ax D(x) -> Phi' > (Ex Phi)' = Ex D(x) & Phi' > (Phi & Psi)' = Phi' & Psi' > (~Phi)' = ~Phi' > etc. > > Then we have the nice relationship between truth in the model > and truth in the submodel: > > Phi is true in submodel S' > <-> Phi is in the sublanguage L', and Phi' is true in the model S > > This notion of truth in a submodel can be used for *any* predicate > D(x) whatsoever, *even* one for which no elements satisfy D(x). There > is no reason to make a special case for empty domains. > > The other nice thing about the nonstupid definition of truth in a model > is that it gives a nice way to state that a domain is empty: > > ~Ex x=x Whatever you have there is _not_ an application of the _standard_ model truth definition being applied to the degenerated case of U = {}. So I have no interest in it. (I'm only defending my position in the degenerated case using the _standard_ definition). > > That is a formula that is only true in the empty model. So it becomes > possible to state "Nothing exists", and that's actually true in the > model with empty domain. > >>> You want to propose that we use some particular definition of truth >>> in the model with empty domain, say *why* you want to use that definition. >>> Is there any point? Arguing that Tarski did it is no good, unless you >>> can reproduce Tarski's reasoning for doing it that way. >> I _already_ did mention one reason for the "why": the relativity nature of >> reasoning in FOL. > > So, the point of your definition is that it allows you to win arguments > in a way that a nonstupid definition wouldn't? No. As long as you keep making the distinction between "my" definition and the _standard_ definition then no matter what you say or ask, you'd still not get it: there might be different wordings but there's only ONE definition involved. (Of course I've never said you yourself couldn't map a formula to anything you'd like to: it's all interpretation which is subjective here). > > What I meant by a reason why is the reason that someone who is interested > in model theory would prefer your definition over mine. I do believe you and I use the same _standard_ definition. It's just your argument was just wrong in the case of U = {}. > There are technical > advantages to my definition (the nice relationship between truth in a model, > and truth in a submodel, and the fact that it is possible to state a > formula with the interpretation "the domain is empty"). If by "nonstupid definition" you meant the standard definition then whatever you meant to say here (which I'm still not sure what it is) doesn't change the fact that x=x is false in the case U = {}. Remember no contingent truth implies no logical truth? > There seem to > be only emotional reasons for adopting your definition. Look Daryl: "nonstupid definition" is _your own word_ so you must realize the one who has emotional reasons is _you_ !
From: Marshall on 13 Jun 2010 11:58 On Jun 13, 8:31 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Daryl McCullough wrote: > > Nam Nguyen says... > >> Daryl McCullough wrote: > >>> No, it doesn't, but I don't actually care what Shoenfield or Tarski > >>> said. What I care about is having a non-stupid definition of "truth > >>> in a model" that applies to models with empty domain. > >> But you shouldn't have worried about that: because that "non-stupid > >> definition" could only render falsehood. > > > No, a nonstupid definition of "truth in a model" makes > > some statements true, and the rest false. > > The caveat here is when I said "could only render falsehood" I meant > that only in the cases you had refereed as "models with empty domain". > Iow, you had complained (above) > > >>> What I care about is having a non-stupid definition of "truth > >>> in a model" that applies to models with empty domain. > > What I was saying is that the "non-stupid definition" (which I take > it to be the normal definition of truth through set-membership) would > only render falsehood in the case of empty domain (U = {}). > > > > >> Why would you get so concerned about "truths" of the emptiness, anyway? > > > I've explained it before. I'm interested in *submodels*. Suppose you > > start with a structure S for a language L and you restrict attention to > > a substructure S' for a sublanguage L' in the following way: > > > The domain U' of S' consists of all elements of S that satisfy some > > unary predicate D(x). > > We can just top it right here. If the U of S is empty, so is the U' of > S': what is the point of going further? > > > The denotations of the relation and function > > symbols for S' are the same as in S, except the domains are restricted > > to elements of U'. The constant symbols have the same denotations they > > had in S'. > > But why bother with "except the domains are restricted to elements of U'", > when both U and U' are empty? > > Your argument here about "sub-models" has lost me. I've never said > there's no case in which the standard definition of model truths a la > Tarski and set-membship would yield x=x being true. My position all along > is that such definition would yield x=x being false in the degenerated > case of U = {}. What you seem to have assumed here with "sub-models" > is U (of S) is non-empty, which is not the case when x=x would be false! > > > > > > > Then with one way of defining "truth in a model", truth in the submodel > > has a simple relationship to truth in the original model. You just > > *relativize* the formulas. Let Phi' be the formula Phi relativized to > > predicate D. Then Phi' is related to Phi through > > > (Ax Phi)' = Ax D(x) -> Phi' > > (Ex Phi)' = Ex D(x) & Phi' > > (Phi & Psi)' = Phi' & Psi' > > (~Phi)' = ~Phi' > > etc. > > > Then we have the nice relationship between truth in the model > > and truth in the submodel: > > > Phi is true in submodel S' > > <-> Phi is in the sublanguage L', and Phi' is true in the model S > > > This notion of truth in a submodel can be used for *any* predicate > > D(x) whatsoever, *even* one for which no elements satisfy D(x). There > > is no reason to make a special case for empty domains. > > > The other nice thing about the nonstupid definition of truth in a model > > is that it gives a nice way to state that a domain is empty: > > > ~Ex x=x > > Whatever you have there is _not_ an application of the _standard_ model > truth definition being applied to the degenerated case of U = {}. So I > have no interest in it. (I'm only defending my position in the degenerated > case using the _standard_ definition). > > > > > That is a formula that is only true in the empty model. So it becomes > > possible to state "Nothing exists", and that's actually true in the > > model with empty domain. > > >>> You want to propose that we use some particular definition of truth > >>> in the model with empty domain, say *why* you want to use that definition. > >>> Is there any point? Arguing that Tarski did it is no good, unless you > >>> can reproduce Tarski's reasoning for doing it that way. > >> I _already_ did mention one reason for the "why": the relativity nature of > >> reasoning in FOL. > > > So, the point of your definition is that it allows you to win arguments > > in a way that a nonstupid definition wouldn't? > > No. As long as you keep making the distinction between "my" definition > and the _standard_ definition then no matter what you say or ask, you'd > still not get it: there might be different wordings but there's only > ONE definition involved. (Of course I've never said you yourself couldn't > map a formula to anything you'd like to: it's all interpretation > which is subjective here). > > > > > What I meant by a reason why is the reason that someone who is interested > > in model theory would prefer your definition over mine. > > I do believe you and I use the same _standard_ definition. It's just your > argument was just wrong in the case of U = {}. > > > There are technical > > advantages to my definition (the nice relationship between truth in a model, > > and truth in a submodel, and the fact that it is possible to state a > > formula with the interpretation "the domain is empty"). > > If by "nonstupid definition" you meant the standard definition then > whatever you meant to say here (which I'm still not sure what it is) > doesn't change the fact that x=x is false in the case U = {}. Remember > no contingent truth implies no logical truth? > > > There seem to > > be only emotional reasons for adopting your definition. > > Look Daryl: "nonstupid definition" is _your own word_ so you must realize > the one who has emotional reasons is _you_ ! WHOOSH! Marshall
From: Nam Nguyen on 13 Jun 2010 12:32 Marshall wrote: > On Jun 13, 8:31 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Daryl McCullough wrote: >>> Nam Nguyen says... >>>> Daryl McCullough wrote: >>>>> No, it doesn't, but I don't actually care what Shoenfield or Tarski >>>>> said. What I care about is having a non-stupid definition of "truth >>>>> in a model" that applies to models with empty domain. >>>> But you shouldn't have worried about that: because that "non-stupid >>>> definition" could only render falsehood. >>> No, a nonstupid definition of "truth in a model" makes >>> some statements true, and the rest false. >> The caveat here is when I said "could only render falsehood" I meant >> that only in the cases you had refereed as "models with empty domain". >> Iow, you had complained (above) >> >> >>> What I care about is having a non-stupid definition of "truth >> >>> in a model" that applies to models with empty domain. >> >> What I was saying is that the "non-stupid definition" (which I take >> it to be the normal definition of truth through set-membership) would >> only render falsehood in the case of empty domain (U = {}). >> >> >> >>>> Why would you get so concerned about "truths" of the emptiness, anyway? >>> I've explained it before. I'm interested in *submodels*. Suppose you >>> start with a structure S for a language L and you restrict attention to >>> a substructure S' for a sublanguage L' in the following way: >>> The domain U' of S' consists of all elements of S that satisfy some >>> unary predicate D(x). >> We can just top it right here. If the U of S is empty, so is the U' of >> S': what is the point of going further? >> >>> The denotations of the relation and function >>> symbols for S' are the same as in S, except the domains are restricted >>> to elements of U'. The constant symbols have the same denotations they >>> had in S'. >> But why bother with "except the domains are restricted to elements of U'", >> when both U and U' are empty? >> >> Your argument here about "sub-models" has lost me. I've never said >> there's no case in which the standard definition of model truths a la >> Tarski and set-membship would yield x=x being true. My position all along >> is that such definition would yield x=x being false in the degenerated >> case of U = {}. What you seem to have assumed here with "sub-models" >> is U (of S) is non-empty, which is not the case when x=x would be false! >> >> >> >> >> >>> Then with one way of defining "truth in a model", truth in the submodel >>> has a simple relationship to truth in the original model. You just >>> *relativize* the formulas. Let Phi' be the formula Phi relativized to >>> predicate D. Then Phi' is related to Phi through >>> (Ax Phi)' = Ax D(x) -> Phi' >>> (Ex Phi)' = Ex D(x) & Phi' >>> (Phi & Psi)' = Phi' & Psi' >>> (~Phi)' = ~Phi' >>> etc. >>> Then we have the nice relationship between truth in the model >>> and truth in the submodel: >>> Phi is true in submodel S' >>> <-> Phi is in the sublanguage L', and Phi' is true in the model S >>> This notion of truth in a submodel can be used for *any* predicate >>> D(x) whatsoever, *even* one for which no elements satisfy D(x). There >>> is no reason to make a special case for empty domains. >>> The other nice thing about the nonstupid definition of truth in a model >>> is that it gives a nice way to state that a domain is empty: >>> ~Ex x=x >> Whatever you have there is _not_ an application of the _standard_ model >> truth definition being applied to the degenerated case of U = {}. So I >> have no interest in it. (I'm only defending my position in the degenerated >> case using the _standard_ definition). >> >> >> >>> That is a formula that is only true in the empty model. So it becomes >>> possible to state "Nothing exists", and that's actually true in the >>> model with empty domain. >>>>> You want to propose that we use some particular definition of truth >>>>> in the model with empty domain, say *why* you want to use that definition. >>>>> Is there any point? Arguing that Tarski did it is no good, unless you >>>>> can reproduce Tarski's reasoning for doing it that way. >>>> I _already_ did mention one reason for the "why": the relativity nature of >>>> reasoning in FOL. >>> So, the point of your definition is that it allows you to win arguments >>> in a way that a nonstupid definition wouldn't? >> No. As long as you keep making the distinction between "my" definition >> and the _standard_ definition then no matter what you say or ask, you'd >> still not get it: there might be different wordings but there's only >> ONE definition involved. (Of course I've never said you yourself couldn't >> map a formula to anything you'd like to: it's all interpretation >> which is subjective here). >> >> >> >>> What I meant by a reason why is the reason that someone who is interested >>> in model theory would prefer your definition over mine. >> I do believe you and I use the same _standard_ definition. It's just your >> argument was just wrong in the case of U = {}. >> >>> There are technical >>> advantages to my definition (the nice relationship between truth in a model, >>> and truth in a submodel, and the fact that it is possible to state a >>> formula with the interpretation "the domain is empty"). >> If by "nonstupid definition" you meant the standard definition then >> whatever you meant to say here (which I'm still not sure what it is) >> doesn't change the fact that x=x is false in the case U = {}. Remember >> no contingent truth implies no logical truth? >> >>> There seem to >>> be only emotional reasons for adopting your definition. >> Look Daryl: "nonstupid definition" is _your own word_ so you must realize >> the one who has emotional reasons is _you_ ! > > WHOOSH! In all that _Marshall still doesn't have any valid argument_ for his statement that x=x is true in _all_ contexts of FOL reasoning.
From: Marshall on 13 Jun 2010 13:08
On Jun 13, 9:32 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Jun 13, 8:31 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Daryl McCullough wrote: > >>> Nam Nguyen says... > >>>> Daryl McCullough wrote: > >>>>> No, it doesn't, but I don't actually care what Shoenfield or Tarski > >>>>> said. What I care about is having a non-stupid definition of "truth > >>>>> in a model" that applies to models with empty domain. > >>>> But you shouldn't have worried about that: because that "non-stupid > >>>> definition" could only render falsehood. > >>> No, a nonstupid definition of "truth in a model" makes > >>> some statements true, and the rest false. > >> The caveat here is when I said "could only render falsehood" I meant > >> that only in the cases you had refereed as "models with empty domain". > >> Iow, you had complained (above) > > >> >>> What I care about is having a non-stupid definition of "truth > >> >>> in a model" that applies to models with empty domain. > > >> What I was saying is that the "non-stupid definition" (which I take > >> it to be the normal definition of truth through set-membership) would > >> only render falsehood in the case of empty domain (U = {}). > > >>>> Why would you get so concerned about "truths" of the emptiness, anyway? > >>> I've explained it before. I'm interested in *submodels*. Suppose you > >>> start with a structure S for a language L and you restrict attention to > >>> a substructure S' for a sublanguage L' in the following way: > >>> The domain U' of S' consists of all elements of S that satisfy some > >>> unary predicate D(x). > >> We can just top it right here. If the U of S is empty, so is the U' of > >> S': what is the point of going further? > > >>> The denotations of the relation and function > >>> symbols for S' are the same as in S, except the domains are restricted > >>> to elements of U'. The constant symbols have the same denotations they > >>> had in S'. > >> But why bother with "except the domains are restricted to elements of U'", > >> when both U and U' are empty? > > >> Your argument here about "sub-models" has lost me. I've never said > >> there's no case in which the standard definition of model truths a la > >> Tarski and set-membship would yield x=x being true. My position all along > >> is that such definition would yield x=x being false in the degenerated > >> case of U = {}. What you seem to have assumed here with "sub-models" > >> is U (of S) is non-empty, which is not the case when x=x would be false! > > >>> Then with one way of defining "truth in a model", truth in the submodel > >>> has a simple relationship to truth in the original model. You just > >>> *relativize* the formulas. Let Phi' be the formula Phi relativized to > >>> predicate D. Then Phi' is related to Phi through > >>> (Ax Phi)' = Ax D(x) -> Phi' > >>> (Ex Phi)' = Ex D(x) & Phi' > >>> (Phi & Psi)' = Phi' & Psi' > >>> (~Phi)' = ~Phi' > >>> etc. > >>> Then we have the nice relationship between truth in the model > >>> and truth in the submodel: > >>> Phi is true in submodel S' > >>> <-> Phi is in the sublanguage L', and Phi' is true in the model S > >>> This notion of truth in a submodel can be used for *any* predicate > >>> D(x) whatsoever, *even* one for which no elements satisfy D(x). There > >>> is no reason to make a special case for empty domains. > >>> The other nice thing about the nonstupid definition of truth in a model > >>> is that it gives a nice way to state that a domain is empty: > >>> ~Ex x=x > >> Whatever you have there is _not_ an application of the _standard_ model > >> truth definition being applied to the degenerated case of U = {}. So I > >> have no interest in it. (I'm only defending my position in the degenerated > >> case using the _standard_ definition). > > >>> That is a formula that is only true in the empty model. So it becomes > >>> possible to state "Nothing exists", and that's actually true in the > >>> model with empty domain. > >>>>> You want to propose that we use some particular definition of truth > >>>>> in the model with empty domain, say *why* you want to use that definition. > >>>>> Is there any point? Arguing that Tarski did it is no good, unless you > >>>>> can reproduce Tarski's reasoning for doing it that way. > >>>> I _already_ did mention one reason for the "why": the relativity nature of > >>>> reasoning in FOL. > >>> So, the point of your definition is that it allows you to win arguments > >>> in a way that a nonstupid definition wouldn't? > >> No. As long as you keep making the distinction between "my" definition > >> and the _standard_ definition then no matter what you say or ask, you'd > >> still not get it: there might be different wordings but there's only > >> ONE definition involved. (Of course I've never said you yourself couldn't > >> map a formula to anything you'd like to: it's all interpretation > >> which is subjective here). > > >>> What I meant by a reason why is the reason that someone who is interested > >>> in model theory would prefer your definition over mine. > >> I do believe you and I use the same _standard_ definition. It's just your > >> argument was just wrong in the case of U = {}. > > >>> There are technical > >>> advantages to my definition (the nice relationship between truth in a model, > >>> and truth in a submodel, and the fact that it is possible to state a > >>> formula with the interpretation "the domain is empty"). > >> If by "nonstupid definition" you meant the standard definition then > >> whatever you meant to say here (which I'm still not sure what it is) > >> doesn't change the fact that x=x is false in the case U = {}. Remember > >> no contingent truth implies no logical truth? > > >>> There seem to > >>> be only emotional reasons for adopting your definition. > >> Look Daryl: "nonstupid definition" is _your own word_ so you must realize > >> the one who has emotional reasons is _you_ ! > > > WHOOSH! > > In all that _Marshall still doesn't have any valid argument_ for his statement > that x=x is true in _all_ contexts of FOL reasoning. It's true in all contexts in which there isn't anything that is not equal to itself. Can you find a context where x is not equal to x? Please show me an x, any x, that is not equal to itself. Go on, Potato Chip, show me one. Oh, and because I'm sure you don't know what WHOOSH means, it means that it's hilarious just how totally you failed to understand what Daryl was saying. You didn't follow him in the big points, nor in the little ones. Marshall |