On 5 March 2014 20:59, Bruno Marchal <[email protected]> wrote: > > You have to show two things: > > 1) R is transitive -> (W,R) respects []A -> [][]A > > and > > 2) (W,R) respects []A -> [][]A -> R is transitive > > Let us look at "1)". To show that "R is transitive -> (W,R) respects > []A -> [][]A", you might try to derive a contradiction from > R is transitive, and (W,R) does not respect []A -> [][]A. > > What does it mean that (W,R) does not respect a formula? It can only mean > that in some (W,R,V) there is world alpha where that formula is false. > To say that "[]A->[][]A" is false in alpha means only that []A is true in > that world and that [][]A is false in that world. >
OK. I'm not sure where V came from, but anyway... So as you say a contradiction is t -> f (because f -> x is always true, as it t -> t) So []A is true in a world alpha. Hence if alpha is transitive, and if []A is true in all worlds reachable from alpha, let's call one beta, then []A is also true in all worlds reachable from beta. We don't know if alpha is reachable from beta, but we do know that if []A is true in beta then it's true in all worlds reachable from beta. > > I let you or Brent continue, or anyone else. I don't want to spoil the > pleasure of finding the contradiction. Then we can discuss the "2)". > Surely the pleasure of NOT finding a contradiction? Oh dear I don't think my brain can take this! Maybe a diagram would help. Anyway I have to go now :) > > It is almost more easy to find this by yourself than reading the solution, > and then searching the solution is part of the needed training to be sure > you put the right sense on the matter. > > Keep in mind the semantic definitions. We assume some illuminated (W,R,V) > > Atomic proposition (like the initial p, q, r, ...) is true in a world > alpha , iff V(p) = 1 for that word alpha. > Classical propositional tautologies are true in all worlds. > []A is true at world alpha iff A is true in all worlds accessible from > alpha. > > (W,R,V) satisfies a formula if that formula is true in all worlds in W > (with its R and V, of course). > (W,R) respects a formula if that formula is satisfied for all V. So the > formula is true in all worlds of W, whatever the valuation V is. > > Courage! > > Bruno > > > > > > > -- > 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. > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- > 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. > -- 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.
