Fitch's paradox and OWA
in [Logic]
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From: Daryl McCullough on 31 Dec 2009 00:02 Marshall says... >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. But typically, for some statements such as "The Greek philosopher Plato was left-handed" I don't know whether the statement is true or not, and I also don't know whether anyone else knows whether it is true or not, and I don't know whether it is *possible*, at this late date, to find out whether it is true or not. But surely, it's either true or false, right? -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 31 Dec 2009 00:39 Daryl McCullough wrote: > Marshall says... > >> 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. > > But typically, for some statements such as "The Greek philosopher > Plato was left-handed" I don't know whether the statement is true > or not, and I also don't know whether anyone else knows whether it > is true or not, and I don't know whether it is *possible*, at this > late date, to find out whether it is true or not. > But surely, it's either true or false, right? No. Not surely. Since by our assumption here is nobody would know about his handed-ness, his nervous system to both arms might not have functioned at all to begin with and hence whether or not he was left-handed is moot and is not-truth assignable. As well, there are people are strong equally on both arms and therefore handed-ness is not applicable to them.
From: Nam Nguyen on 31 Dec 2009 01:26 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. Don't want to beat a dead horse so to speak but not knowing a truth because its proof (knowledge) is _finitely_ larger than what one can possibly know is *not* the same as not knowing a truth value because the statement is not *genuinely* truth-assigned-able. The "sand in the world" being an even number example above is of the 1st kind: not the 2nd kind.
From: Nam Nguyen on 31 Dec 2009 03:18 Bob Badour wrote: > Nam Nguyen wrote: > >> Daryl McCullough wrote: >> >>> Marshall says... >>> >>>> 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. >>> >>> But typically, for some statements such as "The Greek philosopher >>> Plato was left-handed" I don't know whether the statement is true >>> or not, and I also don't know whether anyone else knows whether it >>> is true or not, and I don't know whether it is *possible*, at this >>> late date, to find out whether it is true or not. >> >>> But surely, it's either true or false, right? >> >> No. Not surely. Since by our assumption here is nobody would know about >> his handed-ness, his nervous system to both arms might not have >> functioned >> at all to begin with and hence whether or not he was left-handed is moot >> and is not-truth assignable. As well, there are people are strong equally >> on both arms and therefore handed-ness is not applicable to them. > > The term is ambidextrous and ambidextrous is not left-handed so the > predicate would be false if that were the case. The _analogy_ was under the assumption that we'd logically live under a binary world where the negation of "left-handed" is "right-handed". I don't think we were arguing about precise meanings of biological/physiological matters. My point still stands: if it's _impossible_ (as opposed to just being difficult) to assign truth values to a formula then the formula is neither true nor false, which means that collectively the naturals isn't a _complete_ model of Q or its extensions. > > It doesn't get tricky until handedness is equally strong in both arms > but not for the same things like a person who writes left-handed but > shoots right-handed etc. >
From: Jan Hidders on 31 Dec 2009 05:15
On 31 dec, 01:07, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Jan Hidders says... > > >If we reformulate the meaning of (C) in the model theory we get: > > >(mC) If (W,w) |- f then (W,w) |- []f. > > >Given the semantics of []f this is equivalent with: > > >(mC') If (W,w) |- f then (W,w') |- f for all w' in W. > > I don't think that that is correct. Rule (C) says that > if p is a *theorem* (that is, p is provable) then it is > necessarily true (and so is true in all worlds). My apologies. Everywhere where I wrote (W,w) |- f I actually meant (W,w) ||- f. So what I wanted to say with the above is the following. You are of course right that what (C) really says is: (C) if |- f then |- []f And, assuming that for all f it holds that |- f iff ||- f, this is in fact confirmed by the model theory. However, in the inference process of the paradox as described on the Stanford page the rule is used as if it says f |- []f or |- f->[]f, and that would have the much stronger model-theoretic meaning that I described. Their reasoning can be simplified to this: (1) p & ~Kp (assumption, for arbitrary variable p) (2) <>Kp (from (1) using KP) (3) []~Kp (from (1) using (C)) (4) ~<>Kp (from (3) using (D) (5) ~(p & ~Kp) (from (1) and contradicting (2) and (4)) (6) Forall p (~(p & ~Kp)) (forall introduction) (7) Forall p (p -> Kp) (propositional reasoning) The error in the reasoning is caused by the omission of |- before each formula. If you add that, it is clear that at step (5) it is concluded erroneously that |- ~(p & ~Kp) but it should have said that "it is not true that |- (p & ~Kp)", which is of course not the same thing. -- Jan Hidders |