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 [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
