Re: Fitch's paradox and OWA
From: Nam Nguyen <namducnguyen_at_shaw.ca>
Date: Thu, 31 Dec 2009 20:35:54 -0700
Message-ID: <w4e%m.3262$_H7.2656_at_newsfe24.iad>
>
> 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!]
>
>
> 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.
>
>
> I did come up with the requirement that the R above being empty or not
> should be known:
Date: Thu, 31 Dec 2009 20:35:54 -0700
Message-ID: <w4e%m.3262$_H7.2656_at_newsfe24.iad>
Nam Nguyen wrote:
> Marshall wrote:
>> On Dec 31, 5:31 pm, Nam Nguyen <namducngu..._at_shaw.ca> wrote: >>> Marshall wrote: >>>> On Dec 31, 4:03 pm, Nam Nguyen <namducngu..._at_shaw.ca> wrote: >>>>> Marshall wrote: >>>>>> On Dec 31, 1:08 pm, Barb Knox <Barb..._at_LivingHistory.co.uk> wrote: >>>>>>> Marshall <marshall.spi..._at_gmail.com> wrote: >>>>>>>> On Dec 30, 8:16 pm, Barb Knox <s..._at_sig.below> wrote: >>>>>>>>> Marshall <marshall.spi..._at_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.
Received on Fri Jan 01 2010 - 04:35:54 CET