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Transitivity of possibility

makerowner
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Posted 06/16/08 - 01:56 PM: Subject: Transitivity of possibility	#1
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.)
For philosophy, Socrates, if pursued in moderation and at the proper age, is an elegant accomplishment, but too much philosophy is the ruin of human life. Even if a man has good parts, still, if he carries philosophy into later life, he is necessarily ignorant of all those things which a gentleman and a person of honour ought to know.

unenlightened
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Posted 06/16/08 - 03:09 PM:	#2
I would think that "it is possible that p" implies "It is possible that not p". So "it is possible that it is possible that p" implies "it is possible that it is not possible that p" In which case it is impossible that it could imply that it is possible that p, because it might not be. possibly?
...most of our actions are the result of the past, or according to a future ideal. That's not action, that is just conformity. J Krishnamurti "Philosophy, to the Philistine, is an evolutionary process, watched over by some sort of brisk dynamic Providence, and culminating in the supreme insight of modern thought." John Cowper Powys

makerowner
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Posted 06/16/08 - 03:30 PM:	#3
unenlightened wrote: I would think that "it is possible that p" implies "It is possible that not p". So "it is possible that it is possible that p" implies "it is possible that it is not possible that p" In which case it is impossible that it could imply that it is possible that p, because it might not be. possibly? Ah, that makes sense. Thanks.
For philosophy, Socrates, if pursued in moderation and at the proper age, is an elegant accomplishment, but too much philosophy is the ruin of human life. Even if a man has good parts, still, if he carries philosophy into later life, he is necessarily ignorant of all those things which a gentleman and a person of honour ought to know.

the.yangist
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Posted 06/17/08 - 10:45 PM:	#4
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.) Meh? There is such a derivation. Want to see it?
"If it were not for the laughter, the Way would not be what it is." -- Laozi

unenlightened
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Posted 06/18/08 - 04:34 AM:	#5
the.yangist wrote: Meh? There is such a derivation. Want to see it? Yes please.
...most of our actions are the result of the past, or according to a future ideal. That's not action, that is just conformity. J Krishnamurti "Philosophy, to the Philistine, is an evolutionary process, watched over by some sort of brisk dynamic Providence, and culminating in the supreme insight of modern thought." John Cowper Powys

muxol
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Posted 06/18/08 - 07:06 AM:	#6
unenlightened wrote: I would think that "it is possible that p" implies "It is possible that not p". That it is possible that p v ~p does not imply that it is possible that ~(p v ~p). Tautologies are possible but their negations are not. If you take an epistemic/doxastic reading of the possibility operator <>, then it seems plausible that the 4 axioms you cite (which defines a transitive accessibility relation) fails. For in this case it expresses the highly contentious KK principle that if S knows that p, then S knows that he knows that p. If you read <> metaphysically, then it may fail according to some arguments put forth by origin essentialists. Here is the basic idea. It is possible that table x could've been composed of matter y slightly different from what it in fact was composed of (at world w). In that other world (where it is composed of x) it is possible that it could've been composed yet again of slightly different matter. Rerun this argument sufficiently many times. It is clear that at some point the original table at w could not have been composed of nearly entirely different matter than it was (according to origin essentialism). But if accessibility is transitive then surely it is possible. Hence accessibility is not transitive.

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Posted 06/19/08 - 10:51 AM:	#7
Interesting! I'm not sure I exactly understand, but there is something odd in your interpretation of 'possible'. Let p = "I win the lottery today.' Let q = "I buy a lottery ticket today." If q, it is possible that p. It is possible that q. Therefore, it is possible that it is possible that p. But if not q, (which is also possible), then it is not .possible that p. This is the kind of thing I was thinking of. There is an imbalance between 'It is possible that p.' and its negation. I think to express it as 'p or not p' cannot be right, becaue the negation says unequivocally 'not p'. I think we need to say that 'It is possible that p.' asserts something (as a possiblity), and is not a mere tautology. It is always the case that I either will or will not win the lottery on any day, but it is only possible that I will win on the days I buy a ticket.
...most of our actions are the result of the past, or according to a future ideal. That's not action, that is just conformity. J Krishnamurti "Philosophy, to the Philistine, is an evolutionary process, watched over by some sort of brisk dynamic Providence, and culminating in the supreme insight of modern thought." John Cowper Powys

the.yangist
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Posted 06/19/08 - 11:46 PM:	#8
unenlightened wrote: the.yangist wrote: Meh? There is such a derivation. Want to see it? Yes please. I always hate all of those letter rules, so I'll just use the now conventional "introduction" and "elimination" terminology It is possible that it is possible that p: <><>P. It is possible that p: <>P 1.) <><>P, :ASS 2.) ~[]~~[]~p :<>p â‰¡ ~[]~p, 1 3.) \| []~p :ASS 4.) \| \| ~[]~p :ASS 5.) \| \| ~[]~p & []~p :Conj.,3,4 6.) \| ~~[]~p :~I, 4-5 7.) \| []~~[]~p :[]I, 3-6 8.) \| []~~[]~p & ~[]~~[]~p :Conj., 3,4 9.) ~[]~p :~I, 3-8 C.) <>p :<>p â‰¡ ~[]~p, 9 I hope that is adequate. Edited by the.yangist on 06/19/08 - 11:53 PM
"If it were not for the laughter, the Way would not be what it is." -- Laozi

unenlightened
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Posted 06/20/08 - 07:19 AM:	#9
Thanks for that. Now I just have to work out what it means, and why I don't like it. My formal logic has not improved with age, and it was a sickly child in the first place.
...most of our actions are the result of the past, or according to a future ideal. That's not action, that is just conformity. J Krishnamurti "Philosophy, to the Philistine, is an evolutionary process, watched over by some sort of brisk dynamic Providence, and culminating in the supreme insight of modern thought." John Cowper Powys

muxol
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Posted 06/20/08 - 02:01 PM:	#10
the.yangist wrote: Yes please. I always hate all of those letter rules, so I'll just use the now conventional "introduction" and "elimination" terminology It is possible that it is possible that p: <><>P. It is possible that p: <>P 1.) <><>P, :ASS 2.) ~[]~~[]~p :<>p â‰¡ ~[]~p, 1 3.) \| []~p :ASS 4.) \| \| ~[]~p :ASS 5.) \| \| ~[]~p & []~p :Conj.,3,4 6.) \| ~~[]~p :~I, 4-5 7.) \| []~~[]~p :[]I, 3-6 8.) \| []~~[]~p & ~[]~~[]~p :Conj., 3,4 9.) ~[]~p :~I, 3-8 C.) <>p :<>p â‰¡ ~[]~p, 9 I hope that is adequate.[/quote] No, it is not adequate.

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