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From: golden@FRODO.STANFORD.EDU (Richard Golden)
Newsgroups: comp.ai.digest
Subject: FUZZY LOGIC VS. PROBABILITY THEORY
Message-ID: <8802180658.AA11175@ucbvax.Berkeley.EDU>
Date: 16 Feb 88 04:46:33 GMT
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I am not an expert in Fuzzy Logic or Probability Theory but I have examined
the literature regarding the foundations of Probability Theory and the 
derivation of these foundations from basic principles of deductive logic.
The basic theoretical result is that selecting a "most probable" conclusion
for a given set of data is the ONLY RATIONAL selection one can make in 
an environment characterized by uncertainty.  (Rational selection in this
case meaning consistency with the classic deductive/symbolic logic - boolean
algebra.)  Thus, one could argue that if one constrains the class of 
possible inductive logics to be consistent with the laws of deductive logic
then Probability Theory is the MOST GENERAL type of inductive logic.

The reference from which these arguments are based is given
by Cox (1946). Probability Frequency and reasonable expectation.
American Journal of Statistical Physics, 14, 1-13.
The argument is based upon the following hypotheses:

(i) The belief of the event B given A may be represented by a
    real-valued function F(B,A).

(ii) F(~B,A) may be computed from F(B,A)

(iii) F(C and B,A) may be computed from F(C,B and A) and F(B,A)
      Note this assumption's similarity to Bayes Rule but the 
      multiplicative property is not assumed.
(iv)  Assumptions (i), (ii), and (iii) must be consistent with the
      laws of Boolean Algebra (i.e., deductive/symbolic logic).

From these assumptions one can prove that F(B,A) must be equivalent
to the conditional probability of B given A. That is, F(B,A) must 
lie between a maximum and minimum value (say 1 and 0) and the sum of 
all possible values for B for a particular value of A must equal the
maximum value (1).  Note that we are taking the subjectivist view of
probability theory and we are NOT interpreting the probability of
an event as the limiting value of the relative frequency of an event.

To my knowledge, the axioms of Fuzzy Logic can not be derived from
consistency conditions generated from the deductive logic so I conclude
that Fuzzy Logic is not appropriate for inferencing.  Any comments?!!!

				Richard Golden
				Psychology Department
				Stanford University
				Stanford, CA 94305   GOLDEN@PSYCH.STANFORD.EDU

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