Fitch's paradox and OWA
in [Logic]
|
From: Jesse F. Hughes on 1 Jan 2010 12:36 Jan Hidders <hidders(a)gmail.com> writes: > On 1 jan, 05:26, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> Marshall says... >> >> >Are your comfortable with how step 8 is >> >obtained from step 7 via Rule C as described >> >on this page? >> >http://plato.stanford.edu/entries/fitch-paradox/ >> >> Yes, that's exactly what they are doing. They >> didn't use the |- symbol in step 7, but it is >> clear that (7) is the conclusion of a proof. > > Wow. You are right. They correctly conclude in (7) that |- ~K(p & ~Kp). > > Hmm. I need to think this over. I'm beginning to believe now that the > inference in the paradox is in fact correct. But of course it's correct! Fitch's paradox is perfectly non-controversial, as a matter of purely formal reasoning. It's well-known and well studied by logicians. It would be truly remarkable if you found an error in a famous formal proof of under a dozen lines. -- "The papers are currently at journals. [When published,] make no mistake, there will be no place on this planet where you can hide. Remember, I'm not talking about something vague here. I'm talking about publication in journals." James S. Harris. Wow. Journals.
From: Jan Hidders on 1 Jan 2010 13:00 On 1 jan, 18:36, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Jan Hidders <hidd...(a)gmail.com> writes: > > On 1 jan, 05:26, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> Marshall says... > > >> >Are your comfortable with how step 8 is > >> >obtained from step 7 via Rule C as described > >> >on this page? > >> >http://plato.stanford.edu/entries/fitch-paradox/ > > >> Yes, that's exactly what they are doing. They > >> didn't use the |- symbol in step 7, but it is > >> clear that (7) is the conclusion of a proof. > > > Wow. You are right. They correctly conclude in (7) that |- ~K(p & ~Kp). > > > Hmm. I need to think this over. I'm beginning to believe now that the > > inference in the paradox is in fact correct. > > But of course it's correct! > > Fitch's paradox is perfectly non-controversial, as a matter of purely > formal reasoning. It's well-known and well studied by logicians. It > would be truly remarkable if you found an error in a famous formal > proof of under a dozen lines. What!? Argument from authority? How very unlogical of you! :-) And yes, it was very arrogant of me to think I would have actually found an error there, but I would have been even more ashamed of myself if I had failed to critically examine it. -- Jan Hidders
From: Jesse F. Hughes on 1 Jan 2010 13:15 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > That's not a change of the *semantics*. That's a change of the > *syntax*. My claim is that in the possible worlds semantics, > every predicate (and operator) that can vary from world to world > implicitly is a function of the world. That complexity can usually > be avoided because implicitly we assume that everything is talking > the same world. But when you nest <> and K, it is no longer possible > to make that assumption. Not without restrictions on what can be > said. My point is that the knowability principle doesn't make > any sense without explicit mention of possible worlds. > > It might make sense if we restrict the principle to propositions > p that don't involve the knowability operator. But if we restrict > it that way, we can't carry out Fitch's proof. I haven't worked through the semantic details (at least not recently), but the proof clearly "works" and the intuition behind the proof seems plausible enough. Suppose that p is true, but I don't know it. Then p & ~Kp is true. But surely, I could not know p & ~Kp. That is, I couldn't know "p is true, but I don't know that p is true." After all, if I know that conjunction, then I know that p is true, so how could I know that I don't know that p is true? The argument seems perfectly clear to me, both formally and informally. -- Jesse F. Hughes "To all Leaders of the World, buy or rent the movie 'The Day After'[...] I assure you will have a new perspective on WMDs." -- practical advice from online petitions
From: Jesse F. Hughes on 1 Jan 2010 13:55 Jan Hidders <hidders(a)gmail.com> writes: > On 1 jan, 18:36, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Jan Hidders <hidd...(a)gmail.com> writes: >> > Hmm. I need to think this over. I'm beginning to believe now that the >> > inference in the paradox is in fact correct. >> >> But of course it's correct! >> >> Fitch's paradox is perfectly non-controversial, as a matter of purely >> formal reasoning. It's well-known and well studied by logicians. It >> would be truly remarkable if you found an error in a famous formal >> proof of under a dozen lines. > > What!? Argument from authority? How very unlogical of you! :-) > > And yes, it was very arrogant of me to think I would have actually > found an error there, but I would have been even more ashamed of > myself if I had failed to critically examine it. Well, yes, of course you should critically examine the argument. It just seems to me that, if I were in your shoes and found a step I didn't understand, I would presume an error on my part. But regardless of the presumption, the next step is the same: investigate the proof further to determine where the error *actually* lies. Which is, of course, just what you did. -- Jesse F. Hughes "To [mathematicians] amateur mathematicians are worse than scum, and scarier than nuclear bombs." -- James S. Harris on mathematicians' phobias
From: Jan Hidders on 1 Jan 2010 15:41
On 1 jan, 16:28, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Jan Hidders says... > > > > > > > > >On 31 dec 2009, 18:47, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> 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. > > >True, but you have now fundamentally changed the semantics of the K > >operator in the sense that the model theory now looks very different. > >You have essentially turned K from a unary operator K(p) to a binary > >operator K(w,p). > > That's not a change of the *semantics*. That's a change of the > *syntax*. My claim is that in the possible worlds semantics, > every predicate (and operator) that can vary from world to world > implicitly is a function of the world. That complexity can usually > be avoided because implicitly we assume that everything is talking > the same world. But when you nest <> and K, it is no longer possible > to make that assumption. Not without restrictions on what can be > said. My point is that the knowability principle doesn't make > any sense without explicit mention of possible worlds. Explicit in the formulas? So you reallly do want to change the syntax? If not, I'm a bit puzzled as to how you want to change the semantics. It would help if you could provide a model theory to explain how you want to change the semantics. Right now the model theory I gave already does allow the operator K to be different in possible worlds. So how would your semantics differ from that? -- Jan Hidders |