Fitch's paradox and OWA
in [Logic]
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From: Daryl McCullough on 31 Dec 2009 10:10 Jan Hidders says... >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. I don't see a rule saying f |- []f. Where did you see that? I don't think that's a sensible modal logic rule. That is essentially saying that there is no difference between f and []f. (Usually, the accessibility relation on worlds is set up so that []f -> f. So if we add f -> []f, then f and []f are logically equivalent.) >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)) No, we don't have |- ~Kp. We only have (W,w) ||- ~Kp. So we can't conclude |- []~Kp. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 31 Dec 2009 10:17 Nam Nguyen says... > >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. That was my point. There can be statements that are true, but which we will never know that they are true. There can also be statements that are true, but which we have no way of ever knowing that they are true. For example, I flip a coin, and before I see whether it lands heads up or tails up, it is run over by train, smashing it into a flat, smooth chip of metal. Now, there is no way of ever knowing whether it was heads-up or tails-up. But it is possible that "It was heads-up before it was smashed" is true. Statements can be true even if there is no way to ever know that they are true. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 31 Dec 2009 10:19 Nam Nguyen says... > >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. Well, it is certainly *possible* that "Plato was left-handed" is a statement that is both true and unknowable (at this late date). -- Daryl McCullough Ithaca, NY
From: Marshall on 31 Dec 2009 12:21 On Dec 30, 11:15 pm, Bob Badour <bbad...(a)pei.sympatico.ca> 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. > > 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. Bob, Nam is a kook; you can safely ignore anything he says. Marshall PS. Ah, the years of history! Too bad no one on sci.logic will get it.
From: Daryl McCullough on 31 Dec 2009 12:47
Okay, I've thought about it a little more, and I have come to the conclusion that Fitch's paradox is invalid. Or perhaps the statement of the knowability principle is wrong. Here's the proof of the contradiction: 1. (Knowability principle) For all p: p -> <> K(p) where <>Phi means "Phi is possibly true" and K(Phi) means "Phi is known". 2. (Non-omniscience principle) For some p: p & ~K(p) 3. Letting p0 be the true but unknown proposition, we have p0 & ~K(p0) 4. From 1&3, we have <>K(p0 & ~K(p0)) At this point, let me switch to possible world semantics: <> Phi means "Phi is true in some world". So let's switch to the world in which K(p0 & ~K(p0)) is true. In that world we have 5. K(p0 & ~K(p0)) From this it follows: 6. K(p0) & K(~K(p0)) But only true things are knowable, so from K(~K(p0)) it follows that ~K(p0). So we have 7. K(p0) & ~K(p0) which is a contradiction. The mistake becomes clearer if we explicitly introduce possible worlds. Let's use w ||- Phi to mean "Phi is true in world w" and K_w(Phi) to mean "Phi is known in world w". Let's introduce w0 to mean "our world". Then the proof becomes the following: 1. (Knowability principle) for all p: (w0 ||- p) -> exists w, K_w(p) In other words, if p is true in our world, then there exists another world in which p is knowable. 2. (Non-omniscience principle) for some p: w0 ||- p & ~K_w0(p) 3. Introducing the constant p0 for this unknown proposition, we have: w0 ||- p0 & ~K_w0(p0) 4. From 1&3, we have exists w, K_w(p0 & ~K_w0(p0)) 5. Letting w' be a name for some world making the existential true, we have: K_w'(p0 & ~K_w0(p0)) From this it follows: 6. K_w'(p0) & K_w'(~K_w0(p0)) Since only true things are knowable, we have: 7. K_w'(p0) & ~K_w0(p0) That's no contradiction at all! The proposition p0 is known in one world, w', but not in another world, w0. It only becomes a contradiction when you erase the world suffixes. -- Daryl McCullough Ithaca, NY |