[FOM] possibility and probability

[Sylvia Wenmackers]

Dear Professor Zadeh,


   In recent years I have been working on the foundations of probability
theory, including aspects of this very question.

*POSSIBLE*
   Possibilty is a categorical (yes-no) property; so is 'possible'.
(Constructions such as 'degree of possibility', 'equally possible', ... are
usually taken to be category mistakes. For instance, Laplace's formulation
of the principle of insufficient reason has been criticized for this
reason.)
   A mathematical model for possibility can be found in the event space of
classical probability theory, which can be represented equivalently as a
set algebra or as a sentential algebra. Using the set algebra terminology,
the sample space is a set of all elementary possibilities and the event
space is a collection of subsets of the sample space (usually a
sigma-algebra). The empty set corresponds to impossibility, all the other
elements of the event space are possible.
   Of course, also modal logic models the modality of possibility.
   Since you specified in your question that the propositions of interest
are drawn from a natural language, these models may be too idealized to
your taste. Although I suppose adaptations can be developed, this
idealization need not be a crucial defect if you only want to study the
contrast of 'possible' versus 'probable'.

*PROBABLE*
   Probability is a qualified (graded) property, but 'probable' can be used
in two distinct ways: (1) to express *how* probable something is (graded),
or (2) simply to say *that* something is probable (yes-no).
   (1) A mathematical model for probability (the graded concept) is a
probability function on the event space.* It will assign value zero to the
impossible event (empty set).
   I assume that possibility is conceptually prior to probability. Hence,
it would desirable if a probability measure would at least reflect the
fundamental distinction between impossible and possible events (as well as
many other gradations among possible events). However, it is well-known
that standard probability theory assigns the same value *(i.e.*, zero) to
the impossible event (empty set) as it assigns to many possible events (*
e.g.*, singleton events in a continuum-size sample space). In contrast,
some philosophers of probability have argued for a norm of rationality
called 'Regularity'; one formulation of it: one ought to assign probability
zero only to the impossible event.
   Together with Vieri Benci and Leon Horsten, I have recently published a
paper on Non-Archimedean probability (NAP) theory [1], which has regularity
as an axiom. Of course, this requires infinitesimals in the range of our
probability functions. (We are currently finishing a follow-up paper that
details the philosophical aspects of NAP, which I can send you upon
request.)
   I have tried to connect this work to the literature on branching time
models [2], where I connect non-zero probability (in a NAP model) to the
modal notion of possibility.
*: To be complete, there are also theories of probability that only model
it as a relative property (partial order of events), without assigning a
numerical probability value to the events; this is sometimes called
"qualitative probability" in contrast to the more common "quantative
probability" theory.
   (2) A popular mathematical model for probability in the second sense,
uses standard probability theory and then imposes a threshold (that can be
agent- and context-dependent) on the probability. In this case, I think the
idealization of using a sharp threshold does not work so well. In relation
to the Lottery Paradox, I have proposed an alternative model that embraces
some of the vagueness of the term 'probable' [3]. It uses relative analysis
(a form of non-standard analysis) to model probabilities that are
'sufficiently close' to 1.
   Models for defeasible reasoning also come to mind (Adams conditionals,
default logic, ... see, *e.g.*,
http://plato.stanford.edu/entries/reasoning-defeasible/).

   I always assumed that a nice model for the vague word 'probable' could
be achieved using fuzzy sets as well, but maybe you can enlighten us about
this option?


*References*
[1] V. Benci, L. Horsten, S. Wenmackers "Non-Archimedean probability
theory" Milan Journal of Mathematics 81, 121–151 (2013)
http://dx.doi.org/10.1007/s00032-012-0191-x (or
http://arxiv.org/abs/1106.1524).
[2] S. Wenmackers "Real and hyperreal possibility: infinitesimal
probabilities in branching time structures" (draft)
http://www.sylviawenmackers.be/documents/.Bristol/SWenmackersPaper.pdf.
[3] S. Wenmackers "Ultralarge lotteries: analyzing the Lottery Paradox
using non-standard analysis" Journal of Applied Logic (in press)
http://dx.doi.org/10.1016/j.jal.2013.03.005.


Best wishes,
   Sylvia

--------
dr. dr. Sylvia Wenmackers
http://www.sylviawenmackers.be/

University of Groningen
Faculty of Philosophy
Oude Boteringestraat 52
9712 GL Groningen
The Netherlands
E-mail: s.wenmackers at rug.nl


> How would you differentiate, mathematically, between the propositions,
"It is possible that p," and "It is probable that p," where p is a
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.
>
> Lotfi Zadeh
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