Fitch's paradox and OWA
in [Logic]
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From: Marshall on 30 Dec 2009 22:30 On Dec 30, 6:22 pm, Nam Nguyen <namducngu...(a)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. 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
From: Daryl McCullough on 30 Dec 2009 22:51 Nam Nguyen says... > >Daryl McCullough wrote: >> By the way, I haven't thought about it a huge amount, but I >> don't have any problems with the paradox, because I don't >> accept the premise: Every true proposition is potentially knowable. > >> It seems to me that sufficiently complex true propositions may never >> be known. > >But how can we know it's true in the first place, when its being true >can't be known? I didn't say that we can *know* it is true. That's my point---something can be true without anyone knowing that it is true. It might be true, for example, that there is an even number of grains of sand in the world, but we may never find that out. Is e^pi rational? We may never find out. -- Daryl McCullough Ithaca, NY
From: Marshall on 30 Dec 2009 23:14 On Dec 30, 7:51 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Nam Nguyen says... > >Daryl McCullough wrote: > >> By the way, I haven't thought about it a huge amount, but I > >> don't have any problems with the paradox, because I don't > >> accept the premise: Every true proposition is potentially knowable. > > >> It seems to me that sufficiently complex true propositions may never > >> be known. > > >But how can we know it's true in the first place, when its being true > >can't be known? > > I didn't say that we can *know* it is true. That's my point---something > can be true without anyone knowing that it is true. It might be true, > for example, that there is an even number of grains of sand in the world, but we > may never find that out. Is e^pi rational? We may never find out. I believe Nam is roughly of the opinion that if we can't know which one of {true, false} a sentence is, then we have no basis for saying it must be one or the other. I seem to recall being less than completely clear on that point myself sometime in the past, in re the halting problem, and getting a sound sci.logic thrashing by some guy as a result. His name was Darren McColor, or anyway it was something like that. Boy was I embarrassed! Marshall
From: Barb Knox on 30 Dec 2009 23:16 In article <e3cb76b6-8d77-4a92-bd71-7cd6e163d061(a)k17g2000yqh.googlegroups.com>, Marshall <marshall.spight(a)gmail.com> wrote: > On Dec 30, 6:22�pm, Nam Nguyen <namducngu...(a)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 -- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum videtur. | BBB aa a r bbb | -----------------------------
From: Nam Nguyen on 30 Dec 2009 23:37
Marshall wrote: > On Dec 30, 6:22 pm, Nam Nguyen <namducngu...(a)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. |