As instructed I will have a look at Brent's proofs and see if I follow them, and agree...
On 2 April 2014 15:45, meekerdb <[email protected]> wrote: > On 4/1/2014 7:40 AM, Bruno Marchal wrote: > >> BTW, are you OK in the math thread? Are you OK, like Liz apparently, that >> the Kripke frame (W,R) respects A -> []<>A iff R is symmetrical? >> >> Should I give the proof of the fact that the Kripke frame (W,R) respects >> []A -> [][]A iff R is a transitive? >> >> Bruno >> > > Here's the ones I've done so far. One more to go. Hold off on that proof > (or put a warning in the subject line so I can avoid reading it). > > Brent > > > ******************* > > Show that > > > > (W, R) respects []A -> A if and only if R is reflexive, > > R is reflexive implies (alpha R alpha) for all alpha. []A in alpha > implies A is true in all beta where (alpha R beta), which includes the case > beta=alpha. So R is reflexive implies (W,R) respects []A->A. > I like more words, but I think I follow that and it comes out right. > > Assume R is not reflexive. Then there exists at least one world beta such > that (alpha R beta) and ~(beta R alpha). Consider a valuation such that > p=f in alpha and p=t in all beta. Then []p is true in alpha but p is false > so []A->A is false in alpha for some A. R not reflexive implies []A->A is > not respected for all alpha and all valuations. > Yes that seems right, too. Brent obviously has a far more logical mind than I do, but I guess I already knew that. > > > (W, R) respects []A -> [][]A if and only R is transitive, > > R is transitive means that for all beta such that (alpha R beta) and all > gamma such that (beta R gamma), (alpha R gamma). So every []A implies A=t > in all beta and also A=t in all gamma. But A=t in all gamma means []A is > true in beta, which in turn means [][]A is true in alpha. So R is > transitive implies (W,R) respects []A->[][]A. > > Suppose R is not transitive, so for all beta (alpha R beta) and there are > some gamma such that [(beta R gamma) and ~(alpha R gamma)]. Let A=t in > beta, A=f in gamma. Then []A is true in alpha but []A isn't true in beta, > so [][]A isn't true in alpha. So (W, R) respects []A -> [][]A implies R is > transitive. > > Yes, again, I eventually managed to follow that. You make it seem so easy. > > > (W, R) respects A -> []<>A if and only R is symmetrical, > > R symmetrical means that if (alpha R beta) then (beta R alpha). Suppose A > is true in alpha; then <>A is true in beta (by symmetry of R) and this > holds for all alpha and beta so []<>A in alpha. > > Suppose R is not symmetrical, so there is a pair of worlds (alpha R beta) > and ~(beta R alpha). So consider V such that A=t in alpha and A=f in all > worlds gamma such that (beta R gamma) then ~<>A in beta. So it would be > false that []<>A in alpha. > > Again I an overawed. > > > (W,R) respects []A -> <>A if and only if R is ideal, > > R is ideal, means that for every alpha there is a beta such that (alpha R > beta). Suppose []A is true in alpha, then A must be true in every world > beta (alpha R beta) and there is a least on such beta, so <>A is true in > alpha. > > Suppose R is not ideal, then there is a cul-de-sac alpha. For alpha []A > is vacously true for all A, but <>A is false so []A-><>A is false. > > Yes. > > > (W, R) respects <>A -> ~[]<>A if and only if R is realist. > > R is realist means that for every world alpha there is a world beta such > that (alpha R beta) and beta is cul-de-sac. Suppose A is true in beta, > then <>A is true in alpha but <>A=f in beta so []<>A cannot be true in > alpha. Hence <>A->~[]<>A in alpha where alpha is any non cul-de-sac world. > Then consider a cul-de-sac world like beta; <>A is always false in beta so > <>A->X is true in beta for any X, including ~[]<>A. > I think my brain is starting to melt down, I can't work out if that proves "if and only if" ? By the way why "realist" ? -- 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.
