Fitch's paradox and OWA
in [Logic]
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From: Jan Hidders on 2 Jan 2010 03:30 On 2 jan, 00:14, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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. Doesn't that imply that you want to reformulate it in a different 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? > > 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. But is that not what the given model theory already does? It uses set theory rather then FOL, but since you want to talk about possible worlds and statements about statements, that seems more appropriate to me anyway. The given model theory still seems to contain the paradox, so you will want to change it. Can you show how? -- Jan Hidders
From: Daryl McCullough on 2 Jan 2010 10:14 Jan Hidders says... > >On 2 jan, 00:14, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> 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. > >Doesn't that imply that you want to reformulate it in a different >syntax? I wouldn't say that I *want* to; I'm just saying that if I wanted to assert the knowability principle, then I would formulate it in something other than modal logic. >> 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. > >But is that not what the given model theory already does? It uses set >theory rather then FOL, but since you want to talk about possible >worlds and statements about statements, that seems more appropriate to >me anyway. I don't think the model theory is rich enough. If you are going to allow nested instances of the knowability operator, then there is the issue of *who* knows what. The fact that proposition p is not known in world w1 is itself a proposition, and that proposition can be known, but *not* in w1. Another world, w2 could know that p is not known in w1. But you can't express that without world indices on the knowability operator. Now, it could be that we are not interested in what *another* world knows about this world. So we restrict our attention to one-world claims (all knowability operators refer to the same world). That's fine, and in that case, the knowability principle is just false in any nontrivial model of modal logic. >The given model theory still seems to contain the paradox, >so you will want to change it. Can you show how? Now that I think about it, it seems that it would be a mess to formalize. The problem is that if knowability is a two-place predicate (as opposed to an operator), then that means that formulas have to serve double-duty: both as formulas and as terms (that can be arguments to the knowability predicate). In higher-order type theory, I think we can do it this way: Introduce new types W = the type of possible worlds A = the type of atomic propositions P = the type of all propositions (the propositions are closed under the operations of and, or, implies, negation, universal and existential quantification) t : W x A --> P t(w,a) says "a is true in world w" k : W x P --> P k(w,p) says "p is known in world w" Then the knowability principle could be formalized as: forall p:P, (p -> exists w:W, k(w,p)) (any true proposition is known to be true in some world). I think it would be a lot of work to nail down all the details here, but my point is that the knowability principle can be formulated in a way that isn't susceptible to Fitch's proof. -- Daryl McCullough Ithaca, NY
From: vldm10 on 2 Jan 2010 11:52 On Jan 2, 4:14 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > W = the type of possible worlds > A = the type of atomic propositions > P = the type of all propositions I am not sure that propositions are types??? Let me give you the following example: This sentence is false. -- Vladimir Odrljin
From: Jan Hidders on 2 Jan 2010 16:13 On 2 jan, 16:14, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Jan Hidders says... > > > > >On 2 jan, 00:14, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> 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. > > >Doesn't that imply that you want to reformulate it in a different > >syntax? > > I wouldn't say that I *want* to; I'm just saying that if I wanted > to assert the knowability principle, then I would formulate it in > something other than modal logic. > > >> 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. > > >But is that not what the given model theory already does? It uses set > >theory rather then FOL, but since you want to talk about possible > >worlds and statements about statements, that seems more appropriate to > >me anyway. > > I don't think the model theory is rich enough. If you are going > to allow nested instances of the knowability operator, then there > is the issue of *who* knows what. The fact that proposition p is > not known in world w1 is itself a proposition, and that proposition > can be known, but *not* in w1. Another world, w2 could know that > p is not known in w1. But you can't express that without > world indices on the knowability operator. > > Now, it could be that we are not interested in what *another* > world knows about this world. So we restrict our attention to > one-world claims (all knowability operators refer to the same > world). That's fine, and in that case, the knowability principle > is just false in any nontrivial model of modal logic. > > >The given model theory still seems to contain the paradox, > >so you will want to change it. Can you show how? > > Now that I think about it, it seems that it would be a mess > to formalize. The problem is that if knowability is a two-place > predicate (as opposed to an operator), then that means that > formulas have to serve double-duty: both as formulas and as > terms (that can be arguments to the knowability predicate). > > In higher-order type theory, I think we can do it this way: > Introduce new types > > W = the type of possible worlds > A = the type of atomic propositions > P = the type of all propositions > > (the propositions are closed under the operations of > and, or, implies, negation, universal and existential > quantification) > > t : W x A --> P > t(w,a) says "a is true in world w" > > k : W x P --> P > k(w,p) says "p is known in world w" > > Then the knowability principle could be formalized as: > > forall p:P, (p -> exists w:W, k(w,p)) > > (any true proposition is known to be true in some world). > > I think it would be a lot of work to nail down all the > details here, but my point is that the knowability > principle can be formulated in a way that isn't susceptible > to Fitch's proof. Ok. I think I get what you want to do. But I'm afraid I don't think that will work. The reason is that in your logic you can still express the same things that could be expressed in the old logic. Take for example the following proposition in the old model theory: (1) K(p & ~K(p)) You can still express this in your logic. You can do this by using a predicate CW(w) that expresses that w is (equivalent to) the current world. You can express this as follows: (2) CW(w) =def= For all p, ( t(w,p) <-> p ) With that you can write (1) in your logic as: (3) Forall w : W, ( CW(w) -> k(w, (p & ~k(w, p))) ) This can be done for all for all formulas in the old logic and so it seems to me that you will still have the same paradox but written down differently. -- Jan Hidders
From: Daryl McCullough on 2 Jan 2010 16:52
Jan Hidders says... >But I'm afraid I don't think that will work. The reason is that in >your logic you can still express the same things that could be >expressed in the old logic. Take for example the following proposition >in the old model theory: > >(1) K(p & ~K(p)) > >You can still express this in your logic. Yes, but with the correct axiomatization of knowability predicate, the corresponding proposition will not be true. >You can do this by using a predicate CW(w) that expresses >that w is (equivalent to) the current world. You can express >this as follows: > >(2) CW(w) =def= For all p, ( t(w,p) <-> p ) > >With that you can write (1) in your logic as: > >(3) Forall w : W, ( CW(w) -> k(w, (p & ~k(w, p))) ) > >This can be done for all for all formulas in the old logic and so it >seems to me that you will still have the same paradox but written down >differently. I don't see how it is a paradox. Your proposition (3) will (with the appropriate axiomatization of the knowability predicate) be provably false. The only reason in the original proof of Fitch's paradox to believe (1) (the claim K(p & ~K(p))) is because it follows from the knowability principle and the principle of non-omniscience. In the logic that I sketched, I don't believe it follows from those. 1. Knowability principle: forall p:P, p -> exists w:W (k(w,p)) 2. Non-omniscience principle: forall w:W, exists p:P, p & ~k(w,p) Your statement (3) above does not follow from my 1. and 2. At least, I don't see how. -- Daryl McCullough Ithaca, NY |