[FOM] possibility and probability

[A. Mani]

On Thursday 03 October 2013 03:40 AM, Lotfi A. Zadeh wrote:
> How would you differentiate, mathematically, between the propositions,
> "It is possible that p," and"It is probable that p," where p isa
> proposition which may be drawn from a natural language? This deceptively
> simple question touches upon the fundamental structure of the concepts
> of possibility and probability.


In that form, the problem is too open ended.

When the concept of possibility has a general rough (includes all fuzzy
contexts), then the problem can be interesting.

A few results on the connection are known in set-theoretical
perspective. Entropy has been related to possibility in particular -
though it is dense. With some work these may be stretched to results in
the desired form over existing modal logics/ other logics. But they
would not look nice :(

The situation changes drastically if we proceed over rough Y-systems
(RYS) introduced a few years ago by me. There I use an axiomatic
approach to granularity and discard the "precision based granules"
paradigm.

This is the form of generalities across some of my results in a
forthcoming paper:

Some set of granular axioms + minimal assumptions on the RYS
ensure the existence of probability measures for the
proposition  p.

"It is possible that p" can always be realized in the RYS through the
approximation operators and mereological assumptions.


As the answer depends on ontological assumptions in the context, the
road appears to be very(0.1) long :)


Regards


A. Mani


A. Mani
CU, ASL, AMS, CLC, CMS
http://www.logicamani.in