We have designed a programming logic based on an integration of functional programming languages with classical set theory. The logic merges a classical view of equality with a constructive one by using equivalence classes, while at the same time allowing computation with representatives of equivalence classes. Given a programming language and its operational semantics, a logic is obtained by extending the language with the operators of set theory and classical logic, and extending the operational semantics with "evaluation" rules for these new operators. This operational approach permits us to give a generic design. We give a general formalism for specifying evaluation semantics, and parameterize our design with respect to languages specifiable in this formalism. This allows us to prove, once and for all, important properties of the semantics such as the coherence of the treatment of equality.