On 3/2/2014 9:57 AM, Bruno Marchal wrote:
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,
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.
Assume (W,R) respects []A->A, so that []A->A is true in all W. That means that every
world has another R-accessible world and whatever valuation any formula A has in the
world, it has the same valuation in the R-accessible world. Hmm? I don't see how that
implies R is reflexive, unless I can say that any two worlds whose valuation is the same
for every formula are just the same world.
Brent
(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/ <http://iridia.ulb.ac.be/%7Emarchal/>
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