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 @m"L:T x U -> [0,1], which associates with each ordered pair (x,y) its grade of membership, @m"L(x,y), in L. Given a particular x in T, the membership function @m"L(x,y) defines a fuzzy set, M(x), in U whose membership function is given by @m"M"("x")(y) = @m"L(x,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). If a term x in T is a concatenation of other terms in T, that is, x = x"1 ... x"n, x"i @e T, i = 1,...,n, then the meaning of x can be expressed in terms of the meanings of x"1,...,x"n through the use of a lambda-expression or by solving a system of equations in the membership functions of the x"i which are deduced from the syntax tree of x. The use of this approach is illustrated by examples.