Re: Fitch's paradox and OWA
Date: Wed, 30 Dec 2009 21:37:35 -0700
Message-ID: <qUV_m.28$pL2.14_at_newsfe25.iad>
Marshall 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.
>
> 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.
It _would_ be a virtue, yes, but only, as you said, "_If_ it's actually the case"! But is it?
>
> Do you have any reason to believe that there exist statements
> of arithmetic that *don't* fall in to one of those two categories?
Yes. There are statements written in the lanaguage of arithmetic that no one could possibly assign a truth value to them. For example:
(1) There are infinite counter examples of GC.
Tell me what you'd even suspect as a road-map to assign true or false to (1)?
> 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.
Similarly as in provably-undecidable case (though not identical), there's a 3rd scenario: you can't assign arithmetic truth or falsehood a a certain formula, and in which case the formula is neither true or false! (Of course in such case you could assume it's true or false - but not both - at will.)
>
> (I'm guessing you actually disagree with that last sentence,
> though.)
Of course. But I've also cited reasons. Received on Thu Dec 31 2009 - 05:37:35 CET