Re: Fitch's paradox and OWA
Date: Thu, 31 Dec 2009 20:24:48 -0700
Message-ID: <7Wd%m.554$XU.447_at_newsfe03.iad>
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.
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.
Received on Fri Jan 01 2010 - 04:24:48 CET
>
> 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.