Re: Fitch's paradox and OWA

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.

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