[FOM] Possibility and probability/ Wenmackers

[Lotfi A. Zadeh]

Dear Sylvia:

I read your comment with great interest. What you wrote is very much to 
the point. It is obvious that you have given a great deal of thought to 
the issues under discussion.

Let me restate the question which I posed. How would you differentiate 
between the propositions: (a) It is possible that p, and (b) It is 
probable that p? p is a proposition which may be drawn from a natural 
language. For convenience, let us refer to (a) as a 
possibility-qualified proposition, written as possibly p, and to (b) as 
a probability-qualified proposition, written as probably p. It is my 
belief that to differentiate between (a) and (b), it is necessary to 
precisiate the meaning of p, that is, express the meaning of p in a 
mathematically well-defined form which lends itself to serving as an 
object of computation.

The real challenge is to addressthe posed question when p is a 
proposition drawn from a natural language. Typically, a proposition 
drawn from a natural language is a fuzzy proposition. A fuzzy 
proposition is a proposition which contains fuzzy predicates, e.g., 
tall, fast, rich; and/or fuzzy quantifiers, e.g., many, most, many more; 
and/or fuzzy probabilities, e.g., likely, unlikely, very probable; 
and/or fuzzy possibilities, e.g., quite possible, almost impossible, 
more or less possible; and/or fuzzy truth values, e.g., quite true, 
almost true, not very true. Simple example. Robert is much taller than 
most of his friends.

     In your comment, you attempt to deal with fuzzy propositions within 
the conceptual structure of systems of reasoning which are based on 
bivalent logic, including probability theory and modal logic. Underlying 
my question is my belief that this cannot be done. Traditional 
bivalent-logic-based theories provide no means for precisiating the 
meaning of fuzzy propositions. What is needed for this purpose is fuzzy 
logic--the logic of classes with unsharp (fuzzy) boundaries. Very 
briefly, in fuzzy logic a class with unsharp boundaries, A, is 
precisiated through association with a membership function--a function 
which assigns to each object in a space, U, its grade of membership in 
A. A precisiated class with unsharp boundaries is a fuzzy set. Grade of 
membership in a fuzzy set is allowed to be a fuzzy set. In fuzzy logic, 
everything is, or is allowed to be, a matter of degree. Fuzzy logic and 
multi-valued logic have different agendas.

     What are needed for precisiation of meaning are three logic-based 
formalisms: possibility theory (Zadeh 1978 
<http://www.cs.berkeley.edu/%7Ezadeh/papers/Fuzzy%20sets%20as%20a%20basis%20for%20a%20theory%20of%20possibility-1978.pdf>; 
Dubois and Prade, Possibility Theory: An Approach to Computerized 
Processing of Uncertainty 1988), generalized probability theory (Zadeh 
2002 
<http://www.cs.berkeley.edu/%7Ezadeh/papers/Toward%20a%20perception-based%20theory%20of%20probabilistic%20reasoning%20with%20imprecise%20probabilities-2002.pdf>, 
2006 
<http://www.cs.berkeley.edu/%7Ezadeh/papers/GTU--Principal%20Concepts%20and%20Ideas-2006.pdf>) 
and a theory of meaning (Zadeh 2013 
<http://www.cs.berkeley.edu/%7Ezadeh/papers/Toward%20a%20Restriction-Centered%20Theory%20of%20Truth%20and%20Meaning%20FINAL.pdf>). 
In possibility theory, possibility is a matter of degree and possibly is 
a fuzzy set. This fuzzy set is interpreted as a restriction on the 
possibility measure of p. In generalized probability theory, probably is 
a fuzzy set, and so are fuzzy events. This fuzzy set is a restriction on 
the probability measure of p. This, in essence, is what differentiates 
(a) from (b). What should be noted is that in fuzzy logic, possibly and 
probably are not viewed as operators. Details may be found in the cited 
references.

     Regards,

     Lotfi

-- 
Lotfi A. Zadeh
Professor Emeritus
Director, Berkeley Initiative in Soft Computing (BISC)

Address:
729 Soda Hall #1776
Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720-1776
zadeh at eecs.berkeley.edu  
Tel.(office): (510) 642-4959
Tel.(home): (510) 526-2569
Fax (home): (510) 526-2433
URL:http://www.cs.berkeley.edu/~zadeh/

BISC Homepage URLs
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