Fitch's paradox and OWA
in [Logic]
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From: Jan Hidders on 30 Dec 2009 06:37 On 26 dec, 22:39, Marshall <marshall.spi...(a)gmail.com> wrote: > On Dec 26, 4:36 am, Jan Hidders <hidd...(a)gmail.com> wrote: > > > > > Fitch's paradox is no more. :-) > > From plato.stanford.edu: > > "As for the knowability proof itself, there continues to be no > consensus on whether and where it goes wrong." > > So ... you gonna publish that? Give those modal logic guys > a kick in the pants? :-) Only after checking carefully whether I'm right. :-) Unfortunately it turns out that I was wrong. But in the process I did find what I think is really the problem with the paradox. You can "check" the reasoning by writing down a model theory for the logic and see if all the inferences are actually valid there. In this case the model theory is a bit more complicated than usual because we are reasoning about possible worlds. Basically a model will now look like a pair (W,w) where W is the set of all possible worlds and w is the world we are in right now. In normal propositional logic a single world is described by the set of true atomic propositions. In the setting of the paradox it also contains propositions of the form Kf where f is some formula, which say that f is known in that world. So a world could for example be {a, b, Ka} where a and b are true, but only a is actually known. A complete model could for example be ({w1, w2, w3}, w1) where w1 = {a,b,Ka}, w2 = {a,Ka} and w3 = {b,Kb}. For normal logic operators the model theory is straightforward: - (W,w) |- f1 & f2 iff (W,w) |- f1 and (W,w) |- f2 - (W,w) |- ~f1 iff not (W,w) |- f1 etc. For the basic proposition and the K-facts: - (W,w) |- a iff a in w - (W,w) |- Kf iff Kf in w For the modal operators: - (W,w) |- []f iff (W,w') |- f for all w' in W - (W,w) |- <>f iff (W,w') |- f for some w' in W It can be verified that in this model theory all the inference steps that are used in the paradox are always valid. For the assumptions (A), (B), (C) and (D) on the Stanford page it holds that (D) follows from the model theory. So we can question the validity of (A), (B) and (C). As it turns out (I will not explain that here) you don't need (A) and (B) to get the result of the paradox, so I'm going to focus on (C). If we reformulate the meaning of (C) in the model theory we get: (model-C) If (W,w) |- f then (W,w) |- []f. Given the semantics of []f this is equivalent with: (model-C') If (W,w) |- f then (W,w') |- f for all w' in W. Note that in particular this will hold for f's that are basic propositions or negations of basic propositions. Since the basic propositions hold for (W,w') iff they are elements of w', and their negation only holds if they are not elements of w', it follows that w and w' must always contain exactly the same elements, and therefore in fact the same world. In other words, (C) says effectively that there is always only one possible world. Knowing this, it is not surprising we get such unintuitive results, because it means that everything that is possible, i.e., true in one of the possible worlds, is actually necessary, i.e., true in all possible worlds. This also explains how, from the assumption that every true fact is possibly known, we can come to the conclusion that every true fact is necessarily known. Conclusion: axiom (C) is not a modest modal assumption at all, and in fact quite absurd. -- Jan Hidders
From: Jan Hidders on 30 Dec 2009 06:54 On 26 dec, 22:39, Marshall <marshall.spi...(a)gmail.com> wrote: > On Dec 26, 4:36 am, Jan Hidders <hidd...(a)gmail.com> wrote: > > > > > Fitch's paradox is no more. :-) > > From plato.stanford.edu: > > "As for the knowability proof itself, there continues to be no > consensus on whether and where it goes wrong." > > So ... you gonna publish that? Give those modal logic guys > a kick in the pants? :-) > > Marshall Only after checking carefully whether I'm right. :-) Unfortunately it turns out that I was wrong. But in the process I did find what I think is really the problem with the paradox. You can "check" the reasoning by writing down a model theory for the logic and see if all the inferences are actually valid there. In this case the model theory is a bit more complicated than usual because we are reasoning about possible worlds. Basically a model will now look like a pair (W,w) where W is the set of all possible worlds and w is the world we are in right now which must of course be an element of W. In normal propositional logic a single world is described by the set of true atomic propositions. In the setting of the paradox it also contains propositions of the form Kf where f is some formula, which say that f is known in that world. I will call those atomic propositions and Kf propositions collectively basic propositions. So a world could for example be {a, b, Ka} where a and b are true, but only a is actually known. A complete model could for example be ({w1, w2, w3}, w1) where w1 = {a,b,Ka}, w2 = {a,Ka} and w3 = {b,Kb}. For normal logic operators the model theory is straightforward: - (W,w) |- f1 & f2 iff (W,w) |- f1 and (W,w) |- f2 - (W,w) |- ~f1 iff not (W,w) |- f1 etc. For the basic proposition and the K-facts: - (W,w) |- a iff a in w - (W,w) |- Kf iff Kf in w For the modal operators: - (W,w) |- []f iff (W,w') |- f for all w' in W - (W,w) |- <>f iff (W,w') |- f for some w' in W It can be verified that in this model theory all the inference steps used in the paradox are always valid. For the assumptions (A), (B), (C) and (D) on the Stanford page <http://plato.stanford.edu/entries/ fitch-paradox/> it holds that (D) is always valid in the above model theory. So we can question the validity of (A), (B) and (C). As it turns out (I will not explain that here) you don't need (A) and (B) to get the result of the paradox, so I'm going to focus on (C). If we reformulate the meaning of (C) in the model theory we get: (mC) If (W,w) |- f then (W,w) |- []f. Given the semantics of []f this is equivalent with: (mC') If (W,w) |- f then (W,w') |- f for all w' in W. Note that in particular this will hold for f's that are basic propositions or negations of basic propositions. Since the basic propositions hold for (W,w') iff they are elements of w', and their negation only holds if they are not elements of w', it follows that w and w' must always contain exactly the same elements, and therefore in fact be the same world. In other words, (C) says effectively that there is always only one possible world. Knowing this, it is not surprising we get such unintuitive results, because it means that everything that is possible, i.e., true in one of the possible worlds, is actually necessary, i.e., true in all possible worlds. This also explains how, from the assumption that every truth is possibly known, we can come to the conclusion that every truth is necessarily known. Conclusion: axiom (C) is not a modest modal assumption at all, and in fact quite absurd. -- Jan Hidders
From: Daryl McCullough on 30 Dec 2009 19:07 Jan Hidders says... >If we reformulate the meaning of (C) in the model theory we get: > >(mC) If (W,w) |- f then (W,w) |- []f. > >Given the semantics of []f this is equivalent with: > >(mC') If (W,w) |- f then (W,w') |- f for all w' in W. I don't think that that is correct. Rule (C) says that if p is a *theorem* (that is, p is provable) then it is necessarily true (and so is true in all worlds). In Kripke semantics, we distinguish between what is true in one world and what is provable. So you should be writing (W,w) ||- f to mean f is true in world w (where W is the set of all possible worlds) instead of (W,w) |- f I'm not sure what the latter would mean. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 30 Dec 2009 19:22 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. Certainly there are candidate mathematical truths, such as Goldbach's conjecture, that we have no idea how to ever prove, so it seems plausible (to me) that we may never come to know that they are true. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 30 Dec 2009 21:22
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? > Certainly there are candidate mathematical truths, such > as Goldbach's conjecture, that we have no idea how to ever prove, > so it seems plausible (to me) that we may never come to know that > they are true. Let me add more to what you've said. One of the shortcomings of modern mathematical logic is that it assumes every single formula written in the language of arithmetic "must be" arithmetically either true or false. There is a class of formulas (written in the language) whose arithmetic truths or falsehoods can't be established as a matter of principle. [The existence of this class could be demonstrated]. GC and the formula "There are infinite counter examples of GC" are candidates of being in such class. |