Quantitative fuzzy semantics

The point of departure in this paper is the definition of a language, L, as a fuzzy relation from a set of terms, T= {x}, to a universe of discourse, U = {y}. As a fuzzy relation, L is characterized by its membership function c(=: T x CJ --f [O,l], which associates with each ordered pair (x,y) its grade of membership, &,y), in L. Given a particular x in T, the membership function &x,y) defines a fuzzy set, M(x), in U whose membership function is given by p ,&y) = p&y). The fuzzy set M(x) is defined to be the meaning of the term X, with x playing the role of a name for M(x). I f a term x in T is a concatenation of other terms in 7’, that is, x = x1 * . * x., x, E T, i=l ,...,n, then the meaning of x can be expressed in terms of the meanings of x,,...,xn through the use of a lambda-expression or by solving a system of equations in the membership functions of the x, which are deduced from the syntax tree of x. The use of this approach is illustrated by examples.