Mathematical Foundations of Fuzzy Logic

Abstract

Our motivation to enter into the foundations of fuzzy logic is based on five aspects. First, fuzzy logic complements parity logic in a unique way and represents currently the most versatile branch of approximate and causal reasoning, in particular with respect to fuzzy cognitive maps in almost every field of psychology. Second, the space B ^l is of fundamental importance to fuzzy logic, because it provides the search space for optimizing fuzzy unit (fit) vectors A = (a _l, a ₂, ..., a _n ) in the unit hypercube I ⁿ. The point at issue is that linguistic variables mean different things to different people. Even experts differ in categorizing the values of information and control variables. This is a problem of meaning, and it is solvable in principle by subjecting fit-vectors of length n to evolutionary genetic optimization, i.e. to submit them to special parity feedback machines which localize satisficing or optimal fit-vectors A in I ⁿ, whose artificial genotypes are l-dimensional bit-vectors in B ^l which encode these fit-vectors.