Re: Fitch's paradox and OWA

In article
<e3cb76b6-8d77-4a92-bd71-7cd6e163d061_at_k17g2000yqh.googlegroups.com>,  Marshall <marshall.spight_at_gmail.com> wrote:

> On Dec 30, 6:22 pm, Nam Nguyen <namducngu..._at_shaw.ca> wrote:
> >
> > One of the shortcomings of modern mathematical logic is that it assumes
> > every single formula written in the language of arithmetic "must be"
> > arithmetically either true or false.

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. (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.)

So, we can say we have a fully-pinned-down notion of arithmetical truth, but only in terms of a background (the Standard Model) which we can't fully pin down.

> 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 sci.philoisophy.tech), 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.

> Do you have any reason to believe that there exist statements
> of arithmetic that *don't* fall in to one of those two categories?
> Note that not being able to know which one it is is not the same
> thing as it actually being something other than true or false.
>
> (I'm guessing you actually disagree with that last sentence,
> though.)
>
>
> Marshall

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