On 3 April 2014 04:37, Bruno Marchal <marc...@ulb.ac.be> wrote: > > 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)]. > > > I cannot parse that sentence, I guess some word are missing. R is not > transitive means that there exist alpha, beta and gamma, such that > alpha R beta, and beta R gamma, and ~(alpha R gamma). I will guess that > this is what you meant. >
That's what I took it to mean. (I didn't realise that wasn't what it said!) OK Liz? Others? Feel free to ask definitions or explanations. > Yes, at least at the point where I think very hard about each one, they all seem to make sense. > > The next one is important, as it plays a role in the 'derivation of > physics'. > > > > (W, R) respects A -> []<>A if and only R is symmetrical, > > R symmetrical means that if (alpha R beta) then (beta R alpha). > > > Yes, for all alpha and beta in W. > > 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. > > And so A -> []<>A is true in alpha. (Here we are using the deduction rule > in the CPL context, which is valid. Later we will see it is not valid in > the modal context). > > > 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. > > Liz told me this already! OK. > Phew. > > (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. > > > OK. > > > 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, all cul-de-sac world are counterexample of []A -> <>A. In the Kripke > semantics, they are counterexamples of <>#, with # put for any proposition. > > > (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. > > > For every *transitory* world alpha. OK. The cul-de-sac world are still > world! > > > > 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. > > > OK. Nice. > > So you proved that R is realist implies that (W, R) respects <>A -> ~[]<>A. > > But you have still not prove that if R is *not* realist, (W,R) does not > respect <>A -> ~[]<>A (unlike all other cases). OK? > > You proved: "(W, R) realist" implies "respects <>A -> ~[]<>A", but not yet > the converse, that "respects <>A -> ~[]<>A" implies " (W, R) realist". > > I let you search, and might justify this (with pre-warning to avoid > spoiling!). > > And what about the euclidian multiverse? May be you did them? > > R is euclidian, or euclidean, if (aRb and aRc) implies bRc, for all a, b > and c in W. (I use "a" for the greek *alpha*!) > > Proposition: (W,R) respects <>A -> []<>A iff R is euclidian. > > Hmm. I'll think about that later. -- 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/d/optout.
