On 6 March 2014 21:44, Bruno Marchal <[email protected]> wrote: > > On 05 Mar 2014, at 23:06, LizR wrote: > > 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... > > > W = the set of worlds > R = the binary relation (of accessibility) > > (W, R) = the multiverse, or the "frame" > > (W, R, V) is the same as the multiverse, except that now, in each worlds > of W, the sentence letters p, q, r, ... got a value 1, or 0. And so, all > formula can be said to be true or false in each world, by the use of > classical logic and of the semantic of Kripke (the fact that []A is > determined in alpha by the value of A in its accessible worlds). >
So V is an illumination? > > > So as you say a contradiction is t -> f (because f -> x is always true, as > it t -> t) > > > Like a tautology is true in all worlds, a contradiction is a proposition > false in all worlds, like f, or (A & ~A), or "0 = 1" (in arithmetic). f -> > x is a tautology, yes, and x -> t also. > > > > So []A is true in a world alpha. > > > I guess you assume []A -> [][]A is false in alpha, which belong to a > transitive multiverse, and you want to show that we will arrive at a > contradiction. > > > > Hence if alpha is transitive, > > > I understand what you mean. But of course it is R which 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. > > > It looks like now you suppose []A -> [][]A is true in alpha. So I am no > more sure of what you try to prove. > > > > 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? > > > No, the pleasure of finding a contradiction from > []A -> [][]A is false in alpha > and > R is transitive > > I was suggesting you to prove P -> Q, by showing that P & ~Q implies a > contradiction. it is the easiest way, although there are (infinitely many) > other ways to proceed. > > > > > > > Oh dear I don't think my brain can take this! > > Maybe a diagram would help. Anyway I have to go now :) > > > > Diagram would help a lot. I teach this basically every years since a long > time, and I only draw diagrams on the black board, not one symbols, except > for the sentence letters true or false in the worlds. > > Take it easy, we have all the time. My feeling is that you are impatient > with yourself. Just calm down. > It seems like I have no time! (But you sound like Charles :-) > You will eventually NOT understand why you ever did find this difficult. > But this takes times and work, that's normal. > I hope you're right. -- 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/d/optout.
