Re: Kripke semantics

From: George Levy <GLevy.domain.name.hidden>
Date: Tue, 03 Apr 2001 11:30:49 -0700

Hi Marchal,

My binder is getting fatter but I am catching up. Thank you for all this
wonderful information...I have a couple of questions:

Could you please define belief and knowledge in terms of a machine's states
or potential states...(for example would belief be the actual states of the
machine and knowledge, the potential states the machine could achieve?)

Is there a relationship between belief, knowledge, consistency and
provability?

Marchal wrote:

> 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.
>

In English it means:
If p is true in all worlds accessible from world A then it is true in world
B.

In the physical world, if I see a supernova, then all the worlds within the
light cone of the supernova see it. Worlds outside the supernova's light
cone do not.

Accessibility could also be defined in terms of information flow in a social
or economic environment. In a perfectly competitive market, perfect
information must be available to all buyers and sellers. So for example, the
(lowest) price P for a particular item is known to be X. .Let this be p
defined as P=X. Then []p = p. When information flow is restricted, buyers
and sellers may not have the same information, and therefore []p = p for a
particular market segment but not for another.

> 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).

Yes, obviously, Leibnitz semantics is a special case of Kripke's semantics
for reflexive worlds.

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

I think it means in English, if p is true in all worlds then "p is true in
all worlds" is true in all worlds. This statement I think depends on
accessibility. It is certainly true in the Leibnitz semantics but not in the
Kripke's semantics. So R would have to be the reflexive case.

> 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

Yes, you are definitely right... in a relative way :-)

George
Received on Tue Apr 03 2001 - 11:49:00 PDT