Brent, Liz, others,
I sum up the main things, and give a lot of exercises, or meditation
subject.
Liz we can do them one at a time, even one halve. Ask questions if the
question asked seems unclear.
***
A Kripke frame, or multiverse, is a couple (W, R) with W a non empty
set of worlds, and R a binary relation of accessibility.
An illuminated, or valued, multiverse (W,R, V), is a Kripke multiverse
together with an assignment V of a truth value (0, or 1) to each
propositional letter for each world. We say that p is true in that
world, when V(p) = 1, for that world. If you want V is a collection of
functions V_alpha in {0, 1}, one for each world alpha.
***
Some class of multiverses will play some role.
A Kripke multiverse (W, R) is said reflexive if R is reflexive. alpha
R alpha, for all alpha in W.
A Kripke multiverse (W, R) is said transitive if R is transitive. That
is
alpha R beta, and beta R gamma entails alpha R gamma, for all alpha
beta and gamma in W.
A Kripke multiverse (W, R) is said symmetric if R is symmetric. alpha
R beta entails beta R alpha, for all alpha in W.
A Kripke multiverse (W, R) is said ideal if there are no cul-de-sac
worlds. For all alpha, there is beta such that alpha R beta.
A Kripke multiverse (W, R) is said realist if all non cul-de-sac
worlds can access to a cul-de-sac world.
***
Finally: (The key thing)
I say that a Kripke multiverse (W,R) respects a modal formula if that
formula is true in all worlds in W, and this for any valuation V.
***
Show that
(W, R) respects []A -> A if and only if R is reflexive,
(W, R) respects []A -> [][]A if and only R is transitive,
(W, R) respects A -> []<>A if and only R is symmetrical,
(W,R) respects []A -> <>A if and only if R is ideal,
(W, R) respects <>A -> ~[]<>A if and only if R is realist.
You can try to find small counter-examples, and guess the pattern of
what happens when you fail.
Of course proving that (W, R) respects []A -> A if and only if R is
reflexive, consists in proving both
(W, R) respects []A -> A if R is reflexive,
and
(W, R) respects []A -> A only if R is reflexive, that is
R is reflexive if (W, R) respects []A -> A
That's a lot of exercises. 10 exercises.
We can do them one at a time. Who propose a proof for
(W, R) respects []A -> A if R is reflexive, That is:
R reflexive -> (W, R) respects []A -> A
?
Bruno
Oh! I forget this one:
Show that all the Kripke multiverses (W, R), whatever R is, respect []
(A -> B) -> ([]A -> []B).
http://iridia.ulb.ac.be/~marchal/
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