Stochastic and fuzzy logics

It is shown that it is possible to regard stochastic and fuzzy logics as being derived from two different constraints on a probability logic: statistical independence (stochastic) and logical implication (fuzzy). To contrast the merits of the two logics, some published data on a fuzzylogic controller is reanalysed using stochastic logic and it is shown that no significant difference results in the control policy. Fuzzy logic as probability logic The literature on fuzzy logic (Lee, 1972; Zadeh 1973) treats it very much as a new concept, distinct from that of probability logic (Rescher, 1969), even though both ascribe to events numbers in the interval [0, 1]. This suggests that a new theoretical framework is required in which to analyse the results of practical applications of fuzzy logic. This letter is to demonstrate that fuzzy logic may be treated in terms of probability theory. This is possible because probability logic is itself not truth functional (Rescher, 1969)—the truth value of a logical expression is not uniquely determined by those of its components, and additional assumptions are necessary to determine it. It will be shown that, if a relationship of logical implication is assumed between variables, the rules of fuzzy logic apply. Conversely, if these rules do apply, there is necessarily logical implication between the variables (if they are probabilistic). This is in contrast to the more common assumption of statistical independence of variables, giving what is here termed a stochastic logic. Fuzzy logic is an extension of Boolean logic based on Zadeh’s (1965) fuzzy set theory in which the usual binary truth values (0 and 1) are extended to include any degree of membership in the closed interval of reals [0, 1]. The normal logic operations are defined in terms of arithmetic functions on these degrees of membership. That is, taking a capital letter as a logic variable, and the corresponding lower case letter as its degree of membership, C = A AND B => c = min(a, b) (1) C = A OR B => c = max(a, b) (2) C= NOT B => c =1-b (3) These definitions coincide with the normal logic functions for the two extreme values (TRUE = 1, FALSE = 0). Using this notation, but regarding, for example, a as being not only a degree of membership but also the actual probability of occurrence of event A, one may derive the probabilistic equivalents of equations 1-3. It is assumed that the events themselves are binary in nature and either occur or do not occur. Equation 3 still applies (as usual ¬A means the nonoccurrence of A); for C = NOT B, c = p(C) = p(¬B) = 1-p(B) = 1-b (4)