Hm. I don't know if the first one was OK but anyway let's look at the second one.
A Kripke multiverse (W, R) is said transitive if R is transitive. That is alpha R beta, and beta R gamma entails alpha R gamma, for all alpha beta and gamma in W. Show that (W, R) respects []A -> [][]A if and only R is transitive, I think "[]A -> [][]A" means (for a world alpha in (W,R)) that if A is true in all worlds accessible from alpha, then it's true in all worlds reachable from alpha that A is true in all worlds reachable from alpha. That's a bit - I don't know - recursive? I can feel a bit of boggling starting in my mind. Let's try to keep things (very, very) simple. Consider a world alpha in which p is true. I assume I can use p since I'm used to typing []p by now! And suppose we have beta and gamma as above. So []p implies that p is true in beta because alpha R beta... OK so far... Hang on, does transitive imply reflexive? This is hard to think about, having 3 things! For ALL a,b,c, in (W,R) we have (aRb & bRc) -> aRc. Specifically if a,b,c are the same (aRa & aRa) -> aRa, so we (kind of redundantly) get reflexivity too. I think. By the way, I suspect that the 3-fold nature of the transitivity rule somehow connects with the 3 []s in the thing I'm trying to prove! But I have no idea why or how that works, if it does. Maybe I should stop for a coffee break and let this percolate around my brain for a bit. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
