In order to build expert systems and other human-behaviour-like systems one needs reasoning schemes that are richer than the classical two-valued logic. This paper starts with a linguistic justification that intermediate degrees of truth are needed. Up to an isomorphism the unit interval [0,1] with its natural order is taken as the domain of possible values for the truth of a proposition. The main part of this paper concerns the generalization of the classical implication operator. The implication \(A\to B\) quantifies the degree by which B is at least as true as A. Starting from some intuitively acceptable axioms, introduced by Trillas and Valverde, it is proved that the implication must be a continuous archimedean nilpotent S-conorm satisfying a supplementary condition. The last paragraph treats the mathematical structure of conjunction and disjunction operators as well as some additional theorems about the implication.