This paper presents a modelling framework for discrete optimization problems that relies on a logic representation in which mixed-integer logic is represented through disjunctions, and integer logic through propositions. It is shown that transformation of the logic formulation into the equation form is not always desirable, and that therefore there is a need to address the solution of mixed-integer programming problems where some of the mixed-integer relationships are expressed in disjunctions while others are expressed as algebraic constraints. A theoretical characterization of disjunctive constraints is proposed which can serve as a criterion for deciding whether a disjunction should be transformed into equation form. A solution algorithm that generalizes the method of Raman and Grossmann (Computers & Chemical Engineering, 17, 909, 1993) for handling mixed-integer disjunctions symbolically is also proposed. Several examples are presented to illustrate the proposed modelling framework and the potential of the solution method.