Re: Fitch's paradox and OWA

From: Marshall <>
Date: Thu, 31 Dec 2009 10:39:02 -0800 (PST)
Message-ID: <>

On Dec 30, 8:16 pm, Barb Knox <s..._at_sig.below> wrote:
>  Marshall <> 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?

> (The usual ways
> to define them are not fully syntactic, but rely on "the full semantics"
> of 2nd-order logic, or "a standard model" of set theory, both of which
> are more complicated than just relying on "the Standard Model" of
> arithmetic in the first place.)

Here's a possible definition:

nat := 0 | succ nat

x + 0 = x
x + succ y = succ x+y

x * 0 = 0
x * succ y = x + (x * y)

Is there some way this definition is not fully syntactic? It uses no quantifying over predicates, so it can't be using second order logic.

It certainly seems to me that the above is fully syntactic, and is a complete definition of basic arithmetic. Are there statements that are true of this definition that can't be captured by any finite theory? Sure there are, but that has nothing to do with whether it's a proper syntactic definition. To say it's not a syntactic definition, you have to point out something about it that's not syntactic, or not correct as a model of the naturals.

> > If it's actually the case (that every statement of basic arithmetic
> > is either true or false) then it's not a shortcoming to say so.
> > On the contrary, that would be a virtue.
> Speaking philosophically (since I'm posting from,
> entities which in some sense exist but are thoroughly inaccessible seem
> to be of little value.  This applies to the truth values of any
> statements which can never be known to be true or false.

While I have sympathy for that position, I don't think it's tenable in the long run. Or anyway, it's not tenable to go from "of little value" to suggesting that we should, say, not attend to the real numbers because of the existence of uncomputable numbers, or suggest that statements that are undecidable one way or the other are somehow neither true nor false. What they are is undecidable.

Marshall Received on Thu Dec 31 2009 - 19:39:02 CET

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