modal logic and probability

From: Wei Dai <weidai.domain.name.hidden>
Date: Thu, 5 Sep 2002 19:05:07 -0700

On Wed, Sep 04, 2002 at 10:48:38AM -0700, Tim May wrote:
> And, putting in a plug for modal/topos logic, the essence of nearly
> every interpretation, whether MWI or Copenhagen or even Newtonian, is
> that observers at time t are faced with unknowable and branching
> futures.

How useful is modal logic in dealing with these unknowable and branching
futures? Modal logic is the logic of possibility and necessity,
but to make decisions you need to reason about probabilities rather than
modalities. How does modal logic "fit in" with or relate to probability
theory? As far as I know, the logic used in probability theory is
classical logic. For example, P(not not A) = P(A) is assumed, and there
are no axioms in probability theory that deal with modal qualifiers. Are
there versions of probability theory based on modal logic or other logics?

Also, can you clarify something else for me. The elements in the poset
that is used to interprete modal logic are the possible worlds, whereas
the elements of the poset in your "causal time" post are space-time
events. If we re-interpret modal logic using the poset of events instead
of the poset of possible worlds, <>A would mean A is true sometime
somewhere in the future lightcone of here-now, and []A would mean A is
true all-the-time everywhere in the future lightcone of here-now. Are you
advocating such usage of modal logic?

I had argued earlier that in the case of possible worlds, we might as well
just use classical logic and talk about possible worlds and the
accessibility relationship directly. That seems to apply even more to the
poset of space-time events. I really fail to see what benefits modal logic
brings in this case.
Received on Thu Sep 05 2002 - 19:06:59 PDT