Fitch's paradox and OWA
in [Logic]
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From: Nam Nguyen on 31 Dec 2009 22:24 Marshall wrote: > On Dec 31, 5:31 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Dec 31, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Marshall wrote: >>>>> On Dec 31, 1:08 pm, Barb Knox <Barb...(a)LivingHistory.co.uk> wrote: >>>>>> Marshall <marshall.spi...(a)gmail.com> wrote: >>>>>>> On Dec 30, 8:16 pm, Barb Knox <s...(a)sig.below> wrote: >>>>>>>> Marshall <marshall.spi...(a)gmail.com> wrote: >>>>>>>> By the nature of the construction of predicate logic, every arithmetic >>>>>>>> formula must be either true or false in the standard model of the >>>>>>>> natural numbers. >>>>>>>> But, we have no satisfactory way to fully characterise that standard >>>>>>>> model! We all think we know what the natural numbers are, but Goedel >>>>>>>> showed that there is no first-order way to define them, and I don't know >>>>>>>> of *any* purely formal (i.e., syntactic) way to do do. >>>>>>> I was more under the impression that Goedel showed there >>>>>>> was no complete finite theory of them, rather than no >>>>>>> way to define them. Are you saying those are equivalent? >>>>>> Yes, in this context. Since we are finite beings we need to use finite >>>>>> systems. >>>>> I have no disagreement with the point about finiteness, but I >>>>> don't see how that point leads to saying that a theory is >>>>> the same thing as a definition. That is rather tantamount to >>>>> saying that theories are all there are, and that's just not >>>>> true. There are things such as computational models, >>>>> for examples. It seems entirely appropriate to me to >>>>> use a computational model as the definition of something, >>>>> which is why I gave a computational model of the naturals >>>>> as a definition. >>>> You seemed to have confused between the FOL definition of models of formal >>>> systems in general and constructing a _specific_ model _candidate_. In defining >>>> the naturals, say, from computational model ... or whatever, you're just >>>> defining what the naturals be. It's still your onerous to prove/demonstrate >>>> this definition of the naturals would meet the definition of a model for, >>>> say Q, PA, .... So far, have you or any human beings successfully demonstrated >>>> so, without being circular? Of course not. >>> Showing that the axioms of PA are true in my definition is >>> straightforward, using only structural induction, >> It might be straightforward to you and you might call it "Cheney induction" >> instead of "structural induction" but it's irrelevant and the question is >> the same: how could you demonstrate that your definition would meet the FOL >> standard definition of model of a formal system? Did you already make that >> presentation in the thread and I simply missed it? > > What sort of thing would you accept as an answer? The simple thing that everyone including you would expect and accept: conforming with the standard definition of a model of a formal system. For instance given the language L(e) and the formal system T = {Ax[x=e]}; let's U be the singleton of the empty set U = {{}} and the set M of ordered pairs be defined/constructed as: M = {('A',U), ('e',(e,e))} One doesn't call -or not call- the constructed M a model of T until one verifies it does or doesn't conform with the FOL definition of model, right? An in this case it turns out M meets the definition and therefore Nam or Marshal could call it a model of M, but not before the verification. Naturally. Can you verify that your definition of the naturals meet the definition of formal system model, with say Q is the underlying system at hand, as I did verify M w.r.t to T above? [It's just a pure simple technical question!] > What difficulties do you foresee? Ok. this is a much better and more technical question one could entertain. In a nutshell, one of difficulties that formula such as (1) or (1') presents is that there's no way you could define any model of Q such that a certain expected set of 2-tuples (i.e. _relation_) can be verified to exist. And if you can't, you can't tell whether or not you have would conform to the overall definition of a model of the underlying formal system (say Q in this case). In details, if (1) is true then there would exist an infinite sequence of primes p1, p2, p3, ...., each of which is the maximum prime less than the corresponding counter example of GC. Which means there's a relation "depicted" as: p1 < p2 < p3 < .... pn < ... or, using the definition, there's this relation R: R = {(p1,p2), (p2,p3), ....} The problem is then there's not yet a formal or intuitive way that we could determine R to be empty or not - and there's always the possibility you can never be able to ascertain one way or another. But R is part of what you could define as a model of Q (and the naturals would be such a model). And if you couldn't ascertain the existence of part of the model (naturals or non-standard), how could you know what you have is in fact a model of the formal system? If you understand this difficulty then to say there's a formula we can't assign a truth value in this "model" is equivalent in meta level to saying there's no way to verify this is in fact a model of the system. In a nutshell. > > If you are convinced it is impossible and that nothing will > satisfy you, I'd rather not waste my time. On the other > hand if you have a specific idea as to what a correct > answer would look like, I might be able to satisfy you. I did come up with the requirement that the R above being empty or not should be known: that's the requirement of FOL model definition. If you don't know that, you can't never know if certain formulas would be true or false simply because what you believe as a model fails to be verified as a model.
From: Nam Nguyen on 31 Dec 2009 22:35 Nam Nguyen wrote: > Marshall wrote: >> On Dec 31, 5:31 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> Marshall wrote: >>>> On Dec 31, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>> Marshall wrote: >>>>>> On Dec 31, 1:08 pm, Barb Knox <Barb...(a)LivingHistory.co.uk> wrote: >>>>>>> Marshall <marshall.spi...(a)gmail.com> wrote: >>>>>>>> On Dec 30, 8:16 pm, Barb Knox <s...(a)sig.below> wrote: >>>>>>>>> Marshall <marshall.spi...(a)gmail.com> wrote: >>>>>>>>> By the nature of the construction of predicate logic, every >>>>>>>>> arithmetic >>>>>>>>> formula must be either true or false in the standard model of the >>>>>>>>> natural numbers. >>>>>>>>> But, we have no satisfactory way to fully characterise that >>>>>>>>> standard >>>>>>>>> model! We all think we know what the natural numbers are, but >>>>>>>>> Goedel >>>>>>>>> showed that there is no first-order way to define them, and I >>>>>>>>> don't know >>>>>>>>> of *any* purely formal (i.e., syntactic) way to do do. >>>>>>>> I was more under the impression that Goedel showed there >>>>>>>> was no complete finite theory of them, rather than no >>>>>>>> way to define them. Are you saying those are equivalent? >>>>>>> Yes, in this context. Since we are finite beings we need to use >>>>>>> finite >>>>>>> systems. >>>>>> I have no disagreement with the point about finiteness, but I >>>>>> don't see how that point leads to saying that a theory is >>>>>> the same thing as a definition. That is rather tantamount to >>>>>> saying that theories are all there are, and that's just not >>>>>> true. There are things such as computational models, >>>>>> for examples. It seems entirely appropriate to me to >>>>>> use a computational model as the definition of something, >>>>>> which is why I gave a computational model of the naturals >>>>>> as a definition. >>>>> You seemed to have confused between the FOL definition of models of >>>>> formal >>>>> systems in general and constructing a _specific_ model _candidate_. >>>>> In defining >>>>> the naturals, say, from computational model ... or whatever, you're >>>>> just >>>>> defining what the naturals be. It's still your onerous to >>>>> prove/demonstrate >>>>> this definition of the naturals would meet the definition of a >>>>> model for, >>>>> say Q, PA, .... So far, have you or any human beings successfully >>>>> demonstrated >>>>> so, without being circular? Of course not. >>>> Showing that the axioms of PA are true in my definition is >>>> straightforward, using only structural induction, >>> It might be straightforward to you and you might call it "Cheney >>> induction" >>> instead of "structural induction" but it's irrelevant and the >>> question is >>> the same: how could you demonstrate that your definition would meet >>> the FOL >>> standard definition of model of a formal system? Did you already make >>> that >>> presentation in the thread and I simply missed it? >> >> What sort of thing would you accept as an answer? > > The simple thing that everyone including you would expect and accept: > conforming with the standard definition of a model of a formal system. > For instance given the language L(e) and the formal system T = {Ax[x=e]}; > let's U be the singleton of the empty set U = {{}} and the set M of ordered > pairs be defined/constructed as: > > M = {('A',U), ('e',(e,e))} > > One doesn't call -or not call- the constructed M a model of T until one > verifies > it does or doesn't conform with the FOL definition of model, right? An > in this case > it turns out M meets the definition and therefore Nam or Marshal could > call it > a model of M, but not before the verification. Naturally. > > Can you verify that your definition of the naturals meet the definition of > formal system model, with say Q is the underlying system at hand, as I did > verify M w.r.t to T above? [It's just a pure simple technical question!] > >> What difficulties do you foresee? > > Ok. this is a much better and more technical question one could entertain. > > In a nutshell, one of difficulties that formula such as (1) or (1') > presents is > that there's no way you could define any model of Q such that a certain > expected > set of 2-tuples (i.e. _relation_) can be verified to exist. And if you > can't, > you can't tell whether or not you have would conform to the overall > definition of a > model of the underlying formal system (say Q in this case). > > In details, if (1) is true then there would exist an infinite sequence > of primes > p1, p2, p3, ...., each of which is the maximum prime less than the > corresponding > counter example of GC. Which means there's a relation "depicted" as: > > p1 < p2 < p3 < .... pn < ... > > or, using the definition, there's this relation R: > > R = {(p1,p2), (p2,p3), ....} > > The problem is then there's not yet a formal or intuitive way that we > could > determine R to be empty or not - and there's always the possibility you > can never > be able to ascertain one way or another. > > But R is part of what you could define as a model of Q (and the naturals > would be such > a model). And if you couldn't ascertain the existence of part of the > model (naturals > or non-standard), how could you know what you have is in fact a model of > the formal > system? > > If you understand this difficulty then to say there's a formula we can't > assign a truth > value in this "model" is equivalent in meta level to saying there's no > way to verify > this is in fact a model of the system. In a nutshell. > >> >> If you are convinced it is impossible and that nothing will >> satisfy you, I'd rather not waste my time. On the other >> hand if you have a specific idea as to what a correct >> answer would look like, I might be able to satisfy you. > > I did come up with the requirement that the R above being empty or not > should be known: I meant "I didn't come up..." > that's the requirement of FOL model definition. If you don't > know that, > you can't never know if certain formulas would be true or false simply > because > what you believe as a model fails to be verified as a model.
From: Nam Nguyen on 31 Dec 2009 22:40 Gene Wirchenko wrote: > On Thu, 31 Dec 2009 18:10:33 -0700, Nam Nguyen <namducnguyen(a)shaw.ca> > wrote: > >> Marshall wrote: > > [snip] > >>> There is simply no issue here to respond to. Everything you've >>> said here is either false or else it's the same as the conclusion >>> you're trying to establish. >> Great "refute" you seem to have had here! Among "everything" I've said here >> are a) and b). Why do you think they're false? Or you just said so out of the >> habit of saying things with no back-up reasons? >> >> Btw, usually "conclusion" is "the same" thing as what one would be "trying to >> establish". You seemed to be surprise of that. Why? > > I am impressed with the speed that you showed yourself a fool > worthy of killfiling. This is a public forum and hence fwiw I don't have interest or concern about someone is being killfiled by anybody or not. All I'm doing in here is listing to people's rationale to see what is what in mathematical reasoning. Regards, Nam Nguyen > > Sincerely, > > Gene Wirchenko
From: Daryl McCullough on 31 Dec 2009 23:26 Marshall says... >However what I was referring to was specifically >how they get from step 7 to step 8 within that >RAA proof. Your response does not seem to >address that particular issue. That's exactly the step that I was talking about. Steps (4), (5) and (6) and (7) constitute a proof of ~K(p ∧ ~Kp). Therefore, we have |- ~K(p ∧ ~Kp) By C, if you have |- f, then you have |- [] f. Letting f = ~K(p ∧ ~Kp), it follows that |- [] ~K(p ∧ ~Kp) which is step (8). >Are your comfortable with how step 8 is >obtained from step 7 via Rule C as described >on this page? >http://plato.stanford.edu/entries/fitch-paradox/ Yes, that's exactly what they are doing. They didn't use the |- symbol in step 7, but it is clear that (7) is the conclusion of a proof. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 31 Dec 2009 23:30
Barb Knox says... > >In article ><a3f061ed-3838-4be9-b73a-836141dc640f(a)u7g2000yqm.googlegroups.com>, > Marshall <marshall.spight(a)gmail.com> wrote: >> I was more under the impression that Goedel showed there >> was no complete finite theory of them, rather than no >> way to define them. Are you saying those are equivalent? > >Yes, in this context. Since we are finite beings we need to use finite >systems. I don't agree. What Godel's theorem says is that we can't know all the truths about the natural numbers, but it doesn't imply that there is any fuzziness in what we mean by natural numbers. All the nonstandard models of the naturals contain infinite objects. We're not likely to mistake such an object for an actual natural. As you say, we are finite beings, so any natural we can write down is finite. -- Daryl McCullough Ithaca, NY |