On 3 April 2014 04:37, Bruno Marchal <[email protected]> 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 [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.
