Fitch's paradox and OWA
in [Logic]
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From: Jan Hidders on 5 Jan 2010 02:40 On 4 jan, 17:56, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Jan Hidders says... > > > > >On 3 jan, 20:55, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> >But my claim is that you do get a contradiction for the > >> >simple reason that your logic contains the old logic. > > >> It doesn't contain the same *axioms*. In particular, I'm > >> rejecting the "knowability principle" in favor of a variant > >> principle that is (as far as I can see) consistent. > > >Well, I'm not so sure. Your new variant look very similar to how the > >principle is formulated in my model theory. And there I got the > >contradiction. > > Well, as I said, I don't see how the proof of a contradiction > could go through. The variant looks similar to your version, > because I *intended* it to be the closest variant that did > not lead to the contradiction. The main thing that is different > is that in my variant, knowledge is about *non-modal* propositions, > rather than modal propositions. The distinction is this: If I say > "It is raining", that's a modal statement; it's true in some > circumstances and false in others. If I say "It is raining on > July 12, 2006 in New York City", then that statement is non-modal. > If it is ever true, then it is always true. > > So my formulation of the principle of knowability is that if > a *non-modal* proposition is true, then it is known in some > possible world. Now, I can easily come up with statements that > make this principle false, as well, using self-reference: > > "This statement is not known to be true in any possible world" > > But within the syntax that I'm suggesting, such self-reference > isn't obviously possible. > > > > > > >> Let's try to make this more explicit. > >> You have a set W of possible worlds, a set MP of > >> modal propositions, and for each world w, a set S_w of > >> the elements of MP true in world w. The set S_w is constrained > >> by the following rules: > > >> 1. If Kp is in S_w, then p is in S_w (you can only know true > >> statements) > >> 2. And(p,q) is in S_w iff p is in S_w and q is in S_w > >> 3. Or(p,q) is in S_w iff p is in S_w or q is in S_w. > >> 4. Not(p) is in S_w iff p is not in S_w > >> 5. Implies(p,q) is in S_w iff p is not in S_w or q is in S_w > >> 6. <>p is in S_w iff for some w', p is in w' > >> 7. []p is in S_w iff for all w', p is in w' > > >That already looks close enough to a model theory to me. > > Sorry for the confusion. I'm trying to paraphrase *your* > model theory. > > >A model could > >be a pair (W, S) with W the set of possible worlds and S : W -> 2^F > >where F is the set of formulas and satisfies the rules 1-7. I strongly > >conjecture that those models would be isomorphic to the models in my > >formulation of the model theory and lead to the same formulas being > >true. > > That was my intention. > > > > > > >Your mapping to type theory is a bit hard for me to get my head > >around, so I'll assume for the moment that the above is your model > >theory. > > >> Now, to capture this semantics in type theory, we use > >> the following translations: > > >> 1. Introduce a type, W, of all possible worlds. > >> 2. Introduce a type, A, of all atoms (atomic modal propositions). > >> 3. Introduce the predicate t(w,a) saying which atoms are true in > >> which possible worlds. > >> 4. Introduce a predicate k(w,p) saying which propositions > >> are known in which worlds. > >> 5. Define MP, the type of all modal propositions, to be the type of > >> functions from W into P. > > >You didn't define / postulate P yet. > > P was already introduced in another post. It's the type of all > (non-modal) propositions. If you like, you can think of a > proposition as a (closed) formula. > > >But a deeper problem is that I don't see why you let modal propositions > >be different propositions in different worlds. > > I'm trying to model facts that vary from world to world using a > logic in which statements have definite truth values. It's no > different from using set theory to give a semantics to modal logic. > > Let's take an example: Plants are green. If there are two worlds, > w1 and w2, then "Plants are green in world w1" is a *different* > proposition than "Plants are green in world w2". One could be > false, while the other could be true. To say "It is possible > that plants could be purple" is to say: "exists w:W Plants are > purple in world w". > > The statement "Plants are green" without reference to which > world you are talking about is an incomplete proposition. It > becomes a proposition when you supply a world w. So it is a > function from worlds to propositions. > > In terms of your syntax: > > w ||- f > > I would write this as > > f(w) > > Once you've made the world explicit, as is the case with > w ||- f > you no longer have a modal proposition, but just an ordinary > proposition. > > >Why is it not enough that their truth value can be > >different in different worlds? > > You can think of propositions as truth values, if you like. In > a classical logic, there are two propositions, "true" and "false". > I'm specifically using a non-classical notion of proposition, > in which we *don't* identify statements that have the same > boolean truth value because knowledge doesn't work that way. > If I know that "Superman is 6 feet tall" that doesn't mean that > I know that "Clark Kent is 6 feet tall". > > >It also makes it hard for me to see whether this formulation is > >equivalent withe the above one that it is supposed to capture. > >> 6. For each atom a, we associate a corresponding element of MP: > >> p_a == that function f such that f(w) = t(w,a). > >> 7. Define the operator K as follows: > >> Kf == that function g such that g(w) = k(w,p) > > >Kf should be Kp? > > Right. > > >> Look, once again, I'm formalizing the knowability principle > >> as: > > >> forall p:P, p -> exists w:W, k(w,p) > > >In my model theory the semantics of the formula that represented it > >can be formulated as: (with M being the set/class of valid models) > > >Forall (W,w_1) in M, forall w_2 in W, forall f in F, (W,w_2)||-f -> > >exists w_3 in W, (W,w_3)||-Kf > > Yes. I'm claiming that this is *not* a sensible formulation of > the knowability principle in the case in which f itself involves > the knowability operator K. If f is the formula p & ~Kp, then > your principle above gives us: > > (W,w_2) ||- p & ~Kp > -> > exists w_3 > (W,w_3) ||- K(p & ~Kp) > > which is a contradiction. The problem is that the knowability > principle should not (in my opinion) be about modal propositions. > To give the simplest example, suppose p is true in exactly one > world. Further, suppose that p is not *known* to be true in that > world. In that case, it would be ridiculous to say: Since p is > true in one world, then it is known to be true in another world. > p *isn't* true in any world, so it can't be known to be true in > any other world. > > But if we deal with nonmodal propositions (propositions of > the form w ||- p), then we can certainly have the case that > p is true only in world w1, but the *fact* that p is true in > world w1 is known in world w2. > > >If you fix W we can simplify this to: > > >(JH-KP) forall w_2 in W, forall f in F, w_2||-f -> exists w_3 in W, > >w_3||-Kf > > >Doesn't that look similar to you? > > Similar, but just different enough that your formulation leads > to a contradiction, and mine doesn't. My two-place "knowledge" operator > acts on *non-modal* propositions. In your syntax, the entire > expression (w_2 ||- f) is the nonmodal proposition corresponding > to my f(w_2). > > I would write, instead: > > forall w_2 in W, forall f in F, w_2 ||- f -> exists w_3 in W, > w_3 ||- K(w_2 ||- f) First let me say thanks for your patience and taking the time to explain this to me. I'm afraid I can only give a short reply now because life and work are getting busier again. I think I see now better your point about the fact that in different worlds we might use the same description to refer to things that are actually different facts. Your example being "it rains" which refers to something different if the different worlds correspond to different days. But I would argue that this is from the perspective of someone who is outside the model and has some way to identify the different worlds independent of what facts hold in them. When you are inside the model and in a certain world the only way to distinguish them is by looking which facts hold in them. For the rain example it could be that in your vocabulary you can express what day it is, and then you can distinguish the different days, but then you could have formulated the fact that you had in mind as "it rains and it is today 5 January 2010". If the date in your world is not in your vocabulary then you have no way of describing the differences between the "it rains" proposition in different worlds. For me the meaning of a proposition is in its pragmatics. If "it rains" means that I will get wet when I go outside and I need to take my umbrella with me, then I don't care what day it is, so it will in that respect be the same proposition each day. Another example might be "all mushrooms are edible" which might mean something different when I'm in different forests, but if I have no way of knowing in which forest I am, and if the pragmatics are the same for me (I will eat them), then from my perspective these are the same facts. I think I now understand also better how you want to distinguish in your model theory between modal and nonmodal facts, and why you want to restrict the K operator to nonmodal facts. Briefly put, since p & ~Kp is a modal fact, we then simply cannot formulate K(p & ~Kp) and get the contradiction. Although I may not fully agree with the philosophy behind this restriction, I agree now that this strategy, when executed properly, could indeed avoid the contradiction. That's it for now. As I said, it is possible that I will not be able to reply quickly in the future, but I will certainly try to follow the tread. Kind regards, -- Jan Hidders
From: Daryl McCullough on 5 Jan 2010 07:28
Jan Hidders says... >I think I see now better your point about the fact that in different >worlds we might use the same description to refer to things that are >actually different facts. Your example being "it rains" which refers >to something different if the different worlds correspond to different >days. But I would argue that this is from the perspective of someone >who is outside the model and has some way to identify the different >worlds independent of what facts hold in them. Yes, that's true. In ordinary discourse about possibility, we don't explicitly talk about *specific* other worlds. However, I think that for some uses of modal talk, we *do* have an explicit way to characterize the other possible worlds. For example, in a deterministic physical theory (such as Newtonian physics), the various possible worlds are characterized by initial conditions, which are determined by a point in phase space. >When you are inside the model and in a certain world the >only way to distinguish them is by looking which facts hold in >them. For the rain example it could be that in your vocabulary >you can express what day it is, and then you >can distinguish the different days, but then you could have formulated >the fact that you had in mind as "it rains and it is today 5 January >2010". If the date in your world is not in your vocabulary then you >have no way of describing the differences between the "it rains" >proposition in different worlds. Yes, that's true. But on the other hand, we *do* have the language of "possible, necessary" even though there is no way for us to know what is possible and what is necessary (without knowledge of other possible worlds). So if you just stick to what we can know in *this* world, it seems to me that the only notion of "possibility" is logical consistency. That's a very uninteresting notion of possibility. To go beyond logical consistency, we have to have some theory about what the other possible worlds are. >For me the meaning of a proposition is in its pragmatics. Yes, I understand that. I think for the pragmatics of modal language, the "possibilities" are actually found with one physical world (or the history of that world). -- Daryl Mc |