Fitch's paradox and OWA
in [Logic]
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From: Daryl McCullough on 2 Jan 2010 16:57 In article <46afa0f9-e70d-4d21-bf67-7f49cd16bf32(a)21g2000yqj.googlegroups.com>, vldm10 says... > >On Jan 2, 4:14=A0pm, 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. In the higher-order type theories that I know of, the liar is not expressible (which is good, since it would lead to a contradiction). -- Daryl McCullough Ithaca, NY
From: Jan Hidders on 3 Jan 2010 04:17 On 2 jan, 22:52, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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 purpose of (3) was only to illustrate the translation of formulas in the original logic to your logic. You are right that by itself it does not show the paradox. But if this translation exists then all formulas used in the proof of the paradox will have their equivalents in your logic. If your logic is complete it will also have the equivalents of all the used axioms and principles, and so the proof will still proceed but will just be phrased in a different syntax. For example, on the Stanford page the formulas (4) and (5) both have their equivalents in your logic. You should also have the principle (A) in your logic, but of course translated to your syntax, so in your logic we should be able to derive the equivalent of (5) from the equivalent of (4). Et cetera. -- Jan Hidders
From: Daryl McCullough on 3 Jan 2010 08:09 Jan Hidders says... >The purpose of (3) was only to illustrate the translation of formulas >in the original logic to your logic. You are right that by itself it >does not show the paradox. But if this translation exists then all >formulas used in the proof of the paradox will have their equivalents >in your logic. Yes, but my point is that in the more expressive logic, the knowability principle can be expressed as forall p:P, exists w:W, k(w,p) The original knowability principle, when translated into this new logic, would look something like this: forall f:W -> P, forall w:W, f(w) -> exists w':W, f(w') & k(w',f(w')) The "propositions" of modal logic are actually functions on worlds. >If your logic is complete it will also have the >equivalents of all the used axioms and principles, Yes, but I'm *rejecting* the knowability principle in favor of a more sensible (non-contradictory) principle. I'm suggesting a *different* principle, one that *doesn't* lead to a contradiction. -- Daryl McCullough Ithaca, NY
From: vldm10 on 3 Jan 2010 10:00 On Jan 2, 10:57 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > In article <46afa0f9-e70d-4d21-bf67-7f49cd16b...(a)21g2000yqj.googlegroups.com>, > vldm10 says... > > > > >On Jan 2, 4:14=A0pm, 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. > I guess that some objects can be treated as types if they have some characteristics in common. We expect the propositions which will have same properties of concern to logic, i.e. propositions are types. This is not the case with Liar paradox. The liar paradox contains a sort of self-reference and the predicate - is true and it is applied to name its own sentences. This paradox is important, for example in proving the first incompleteness theorem, Gödel used a slightly modified version of the liar's paradox (see at http://en.wikipedia.org/wiki/Liar_paradox ) > In the higher-order type theories that I know of, the liar is not > expressible (which is good, since it would lead to a contradiction). There are self-references without predicates - is true. Let me give you two examples which are related to other kind of the self- reference: Example1. Here we have two sentence: Tom is a mathematician. Tom is a mathematician. They have the same truth value in any model. Example 2. I have two sheets of paper. One is marked with P1 and another with P2. I will put the following sentence into every of the paper: The sentence which is on paper P1 have red letters. So on each of the mentioned paper there is the same sentences and nothing else. However semantically these sentences are not the same. (These examples are inspired from the following two books: John Burdian on Self-Reference by Hughes, G.E; Classical Mathematical Logic by R.L. Epstein) It is interesting to find models for the proposition that contains self-referencing. Regarding abstract objects it is also interesting the following question: Can the propositions come to an existence and cease to exist? > > -- > Daryl McCullough > Ithaca, NY Vladimir Odeljin
From: Jan Hidders on 3 Jan 2010 11:42
On 3 jan, 14:09, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Jan Hidders says... > > >The purpose of (3) was only to illustrate the translation of formulas > >in the original logic to your logic. You are right that by itself it > >does not show the paradox. But if this translation exists then all > >formulas used in the proof of the paradox will have their equivalents > >in your logic. > > Yes, but my point is that in the more expressive logic, the > knowability principle can be expressed as > > forall p:P, exists w:W, k(w,p) Er, I think you forgot the part where it requires that p is true. But if you fix that, then this is indeed equivalent with the one used in the Stanford page. This one will still lead to the conclusion that all truths are known. > The original knowability principle, when translated into this > new logic, would look something like this: > > forall f:W -> P, forall w:W, f(w) -> exists w':W, f(w') & k(w',f(w')) > > The "propositions" of modal logic are actually functions on worlds. Really? This is actually a stronger principle that implies the previous one since as a particular case I can take for f the function that maps each world to the same predicate p in P. Also, I don't understand what you mean by "propositions are actually functions on world" except that the same proposition can have a different semantics in different worlds, and that was already taken into account in the old semantics. -- Jan Hidders |