Fitch's paradox and OWA
in [Logic]
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From: Daniel Pitts on 1 Jan 2010 17:11 Jesse F. Hughes wrote: > 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. It actually is possible for "p & ~Kp" may be true, in which case "~K(p & ~Kp)" would also be true. There are a whole islands of truths that can't be known to be true, but are indeed true. This is basically G�del's theorem. Fitch's proof (at least by your description) is using the proof as its own premise. p & ~Kp can be true without knowing it, therefore you still don't know p is true. -- Daniel Pitts' Tech Blog: <http://virtualinfinity.net/wordpress/>
From: Daryl McCullough on 1 Jan 2010 18:00 Jesse F. Hughes says... >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. I agree. My point is not about the proof, it's about the "knowability principle" that if something is true, then it is possible that it is knowable. That's not a reasonable thing to assume unless we either restrict what sort of propositions we are talking about, or be more explicit about *who* knows what. I don't have any problem with the proof of Fitch's paradox. It's a valid proof, but I take it as evidence for rejecting the knowability principle. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 1 Jan 2010 18:14 Jan Hidders says... >Explicit in the formulas? So you really do want to change the syntax? I'm not advocating a change in the syntax, I'm just saying that the syntax of modal logic is inadequate to capture the intuition behind the knowability principle. >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? I would just use first-order logic semantics, and allow explicit quantification over possible worlds. The point about modal logic is that it is a simpler fragment of full first-order logic, but I think that it is not expressive enough to talk about complex issues of necessity and knowability. Fitch's paradox shows its limitations. -- Daryl McCullough Ithaca, NY
From: Jan Hidders on 1 Jan 2010 18:43 On 1 jan, 19:15, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > stevendaryl3...(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. Yes, it does. Thanks. Very nicely formulated. It did strike me that you formulated K as "I know that P". For some reason it made me realize that it was formulated on the Stanford page as "Somebody at some time knows that P". Under the latter interpretation it seems now indeed a bit strange to me to require that all facts, and specifically those of the form ~Kp, are possibly known. I can imagine there are facts p for which we can never establish definitively that they will not be known to somebody at some time, except after waiting until we run out of time or persons, and by then there will be nobody left to know this. So ~Kp might very well be both true and unknowable. -- Jan Hidders
From: Jesse F. Hughes on 1 Jan 2010 20:16
stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > Jesse F. Hughes says... > >>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. > > I agree. My point is not about the proof, it's about the > "knowability principle" that if something is true, then > it is possible that it is knowable. That's not a reasonable > thing to assume unless we either restrict what sort of propositions > we are talking about, or be more explicit about *who* knows what. > > I don't have any problem with the proof of Fitch's paradox. It's > a valid proof, but I take it as evidence for rejecting the knowability > principle. Er, I thought that was the point, too. Obviously, the counterexample suggests that a restriction to the knowability principle. The principle seems perfectly sensible, until you realize that it can be applied to sentences like "p & ~Kp". -- "All intelligent men are cowards. The Chinese are the world's worst fighters because they are an intelligent race[...] An average Chinese child knows what the European gray-haired statesmen do not know, that by fighting one gets killed or maimed." -- Lin Yutang |