Kripke semantics

From: Marchal <marchal.domain.name.hidden>
Date: Fri Mar 30 09:19:54 2001

Hi George,

Do you remember Leibniz semantics?

Here is Leibniz semantics for modal propositions.
In any world, []p will be considered true if p is true
in all worlds of W. And in any world, <>p will be considered true,
if there is (at least one) world in W in which p is true.

the problem is that it makes valid at once all the formula

[]p -> p
p -> <>p
[]p -> [][]p
p -> []<>p
<>p -> []<>p
[](p->q) -> ([]p -> []q)

Do you remember our attempt to modelise belief by a modal box?

We would like not have "[]p -> p", beacuse we know that a belief can
be wrong.
But then S5 is certainly not the good system because "[]p -> p" is
among the theorems, and Leibniz semantics is not a good one because
it makes "[]p -> p" valid, true everywhere (in all worlds of all
frames!).


We would like weaker theories and more general semantics,k capable
of making the above formula ([]p -> p, []p -> [][]p, p -> []<>p
<>p -> []<>p, <>p -> []<>p) independent.


Let W be a frame (a set of worlds)

With Leibniz, []A is true in a world w, if A is true in all worlds.

Kripke relativises Leibniz:

[]A is true in a world w, if A is true in all world accessible from w.

And (as you can guess):

<>A is true in a world w, if there is a world, accessible from w,
where A is true.

So a Kripke frame will be a set of worlds + an accessibility relation,
which is just a binary relation defined on the frame.

(W, R) will denoted Kripke frame. W is the set of worlds, and R is
the relation of accessibility. I will write xRy for the world y is
accessible from x.

A model (W, R, V), by definition, is a frame + a valuation (a fonction
which attributes truth value to each atomic proposition in each world).

Try to convince you that []p->p is automatically valid in all
reflexive model. R is reflexive if for all x in W we have xRx, that is
all world are self-accessible.

Try to find a model in which []p->p is falsified.

Convince yourself that Leibniz semantics is a particular case
of Kripke semantics where the relation of accessibility
makes all the world accessible from each other. That is R is an
equivalence relation (a reflexive (xRx), transitive (xRy and yRz entails
xRz), and symetric relation (xRy entails yRx).

You can try to guess the relation R making valid []p -> [][]p,
p -> []<>p, ...).

Take your time, and tell me if everything is ok up to now.

It is not a luxe, for a relativist everythinger, to have a look
on Kripke insight.

Bruno
Received on Fri Mar 30 2001 - 09:19:54 PST