Re: modal logic and probability

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Fri, 6 Sep 2002 17:19:49 +0200

Wei Dai wrote:

>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.

This is true only for the antic aristotelian alethic modal logic.
This one is approximated by the system S5.

Modern modal logic is used to build a lot of very different systems, by the
use of different modal logics. The more common are T, S4, D, B, etc.

Those system try to approximate common modalities we use everyday:

exemple:

In alethic modal logic [] = it is necessary that
                    and <> = it is possible that

In temporal logic [] means always, and <> means once,

In spatial logic [] means everywhere, and <> means somewhere,

In deontic logic [] means obligatory, and <> means permitted,

In formal provability logic [] means formally provable, and <>
means formally consistent.

etc. Of course the meaning here is the "intended" meaning. The
goal is to capture such intended meaning axiomatically by the
choice or isolation of set of axioms and rules... For exemple if some
proposition is true everywhere we want it to be true somewhere,
so that for spatial logic []p -> <>p should be an axiom, or a
theorem if we choose a stronger axiom.

Note that in all those classical modal logic [] = -<>- and
          ===> <> = -[]- .

But this is false for intuitionist modal logic. (Yes, that exists!)


>How does modal logic "fit in" with or relate to probability
>theory?

The probability ONE interpretation of [], need at least []p -> <>p.


> As far as I know, the logic used in probability theory is
>classical logic.

It depends. Intuitionist mathematicians will use intuitionist
mathematics, etc. (Cf Edward Nelson wrote a book on probability
theory which relies in some way on intuitionist logic.) That can
help for the use of non classical infinitesimal numbers, which
can simplify the proofs.

Classical Modal logic approach of classical probability theory
are proposed as substitution of probability theory.



>For example, P(not not A) = P(A) is assumed,


You point on a real diificulty, because in P(A) A is an event,
and in "-A", A is a proposition.


>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?

I know works by Fatarossi and Barnaba. They use Kripke semantics,
with weighted accessibility relations. The goal is not to build a new
probability theory, but to make more rigorous attempts to clear some
paradox in some use of probability theory.


>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,


I let Tim adding remarks. But in a "time" modal logic, events are the
possible world. Remember that for a logician a "possible world" is just
any element of a set, or of a more sophisticated mathematical structure.
(locale, quantale, lattice, category, ... ..., quantum group, ...)

><>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 mention an interesting paper by Golblatt: " Dioderean Modality in
Minkowski Space-time". It is shown that the modality []p is for "it is
and will always be the case that p". It shows that the dioderean
axiom <>[]p -> []<>p formalises branching time in four dimensional
Minkowki space-time. It seems to work also for two and three dimensions.


>
>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.

Which one. What do you mean by a world?


>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.


Modal logic is just a mathematical tool for making clearer form of
reasoning. It is based on the idea of relativising truth on ... any
mathematical structures, hoping you can show the independence of
a formula from a set of formula.
Kripke geometries are used for showing the NON provability of a formula,
Like polynoms are used in topology for showing you cannot go continuously
form one space to another space. Modal logic in philosophy is like
tensor algebra in physics: it makes possible to shorten big amount
of researchers' work in small formula: like

           Hx = Ex Schroedinger equation

           <>p -> -[]<>p Godel second incompleteness theorem

Of course you need to do an amount of math for getting some meaning
(vector space and linear algebra for Schroedinger equation, and
metamathematics + some modal logic for Godel second incompleteness theorem.

George Levy asks recently "Could somebody incorporate complementarity
in a thought experiment in the style of Bruno's duplication
experiment?"

This is an interesting proposal and I would be glad if someone manage
to present one. Just that it is *because* duplication-like experiment leads
quickly to obscurities and misleading intuitions, *that* modal logic
appears to be a fruitful investment, even if it is not the only one.
Categories and knots are full of promises too. It's also a question of
taste. In Counter-intuitive land, micro or macro, real or virtual,
it would be naive to believe you can go deep without some mathematics.

Bruno
Received on Fri Sep 06 2002 - 08:23:33 PDT