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muxol
yuletide Usergroup: Members Joined: Dec 07, 2003 Total Topics: 1 Total Posts: 1932 |
 #21
I still don't get your proof. You are getting 3 from 1 by using something equivalent to MMp -> Mp (or its rule form), while the step to 3 (from apparently 2) is completely mysterious. Natural deduction modal systems outside those for S5 and S4 are not entirely straightforward unless they are labelled. But since yours appears unlabelled I can only imagine that, probably, you are attempting to prove a contradiction from the negation of MMp -> Mp in S5! (It is a theorem, usually axiom in a Hilbert setting, of S5.) That is rather silly. |
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the.yangist
Graduate  Usergroup: Members Joined: Sep 30, 2007 Location: Tempe, AZ Total Topics: 9 Total Posts: 188 |
#22
I would think that "it is possible that p" implies "It is possible that not p". Then you'd have to accept that the LNC is possibly not the case, which is nonsense. 1.) [](~(p & ~p) :ASS (but actually, a fully derivable proof) 2.) ~(p & ~p) :[]E, 1 3.) <>~(p & ~p) :<>I, 2 4.) <>~~(p & ~p) :Wacky Inference <>p |- <>~p, 3 5.) ~<>~~(p & ~p) :~<>~p ≡ []p, 1 6.) <>~~(p & ~p) & ~<>~~(p & ~p) :Conj., 4,5: CONTRADICTION Part 4., using your allowance, is where this proof goes completely awry. muxol wrote: No, it is not adequate.. That original proof that <><>p |- <>p actually followed from GL, so I don't know where the claims to inadequacy arise. Is this just growing into a critique about using a richer modal logic, or is there something actually wrong with the proof? "If it were not for the laughter, the Way would not be what it is." -- Laozi |
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muxol
yuletide Usergroup: Members Joined: Dec 07, 2003 Total Topics: 1 Total Posts: 1932 |
#23
the.yangist wrote: That original proof that <><>p |- <>p actually followed from GL, so I don't know where the claims to inadequacy arise. Is this just growing into a critique about using a richer modal logic, or is there something actually wrong with the proof? The people here are insanely thick. A proof of <><>p -> <>p (rather than the sequent you are citing) in GL is not surprising; it is well known. The original poster is asking for some plausible reasoning for rejecting it under some interpretation of '<>' as 'possibly'. Showing that it is provable in GL is hardly an answer, and even less interesting. *That's* why your response is horrifically inadequately. Try again. |
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7
Assistant Professor Usergroup: Members Joined: Mar 23, 2008 Total Topics: 12 Total Posts: 340 |
#24
muxol wrote: I still don't get your proof. You are getting 3 from 1 by using something equivalent to MMp -> Mp (or its rule form), while the step to 3 (from apparently 2) is completely mysterious. Natural deduction modal systems outside those for S5 and S4 are not entirely straightforward unless they are labelled. But since yours appears unlabelled I can only imagine that, probably, you are attempting to prove a contradiction from the negation of MMp -> Mp in S5! (It is a theorem, usually axiom in a Hilbert setting, of S5.) That is rather silly. It's a tableau. All of the rules applied are obvious. It does not work in K, as I said, because the last line cannot be derived from the second. ~p can only be added at world #2. BTW, it IS labeled. The numbers are prefixes, not line numbers. |
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muxol
yuletide Usergroup: Members Joined: Dec 07, 2003 Total Topics: 1 Total Posts: 1932 |
#25
7 wrote: It's a tableau. All of the rules applied are obvious. It does not work in K, as I said, because the last line cannot be derived from the second. ~p can only be added at world #2. BTW, it IS labeled. The numbers are prefixes, not line numbers. So it appears. I looked hastily and carelessly at it--apologies. But at least you see that the formula fails in a number of normal systems, one of which includes K. |
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the.yangist
Graduate Usergroup: Members Joined: Sep 30, 2007 Location: Tempe, AZ Total Topics: 9 Total Posts: 188 |
#26
muxol wrote: What are the reasons for supposing that "It is possible that it is possible that p" doesn't imply "it is possible that p"? (Sorry, my computer won't let me post the diamond symbol.) Some logics don't allow it. I hope this is as enlightening for you as it was for me! muxol wrote: The people here are insanely thick. A proof of <><>p -> <>p (rather than the sequent you are citing) in GL is not surprising; it is well known. The original poster is asking for some plausible reasoning for rejecting it under some interpretation of '<>' as 'possibly'. Showing that it is provable in GL is hardly an answer, and even less interesting. *That's* why your response is horrifically inadequately. Try again. I'm dreadfully sorr(il)y. I don't mean to be so horrifically inadequate(ly), and I'm sorr(il)y I wasted your precious(ly) time with my meager(ly) proof that possibility of possibility of a proposition implies more simply its possibility in GL. And you're right. A modal logic with fewer axioms and rules not being able to derive a proof of <><>p |- <>p is enlightening(ly) and insightful(ly), and not just a dumb(ly), obvious(ly) answer that provides no insight into the interpretation of modal(ly) operators, nor issues a somehow less horribly inadequate(ly) response to the question at hand, which to you is apparently more philosophical(ly) than I had originally assumed. I'll be sure(ly) to try my best in the future thanks in part to your admirable(/y) genius. "If it were not for the laughter, the Way would not be what it is." -- Laozi |
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muxol
yuletide Usergroup: Members Joined: Dec 07, 2003 Total Topics: 1 Total Posts: 1932 |
#27
I didn't write what you are alleging that I did in the first quote. So what are you going on about? I might forgive your inadequacy one day. |
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aufbau87
Student  Usergroup: Members Joined: Jun 08, 2008 Location: Southern California Total Topics: 3 Total Posts: 87 |
#28
makerowner wrote: What are the reasons for supposing that "It is possible that it is possible that p" doesn't imply "it is possible that p"? (Sorry, my computer won't let me post the diamond symbol.) Whatever the reason, it is fallacious. Here's a proof (in a quasi-truth-tree format): 1. <><>p (w) / :. <>p 2. ~<>p (w) assume 3. []~p (w) from 2, modal negation 4. <>p (k) from 1, modal instantiation 5. p (l) from 4, modal instantiation 6. ~p (l) from 3, modal instantiation X In other words, if we start with the premise that it is possible that it is possible that p and deny it is possible that p, then we contradict ourselves. '<><>p' entails '<>p'. Notes: The (w) refers to world w, (k) to world k, and (l) to world l. When we instantiate from a diamond operator (for possibility) we have to start a new world distinct from all previous worlds referred to. When instantiation occurs from a box operator (for necessity) we can use any world to refer to since it implies that its operand is true in all worlds, not just at least one. |
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