We introduce lexicographic logic, an extension of propositional logic that can represent a variety of preferences, most notably lexicographic ones. The proposed logic supports a simple new connective whose semantics can be defined in terms of finite lists of truth values. We demonstrate that, despite the well-known theoretical limitations that pose barriers to the quantitative representation of lexicographic preferences, there exists a subset of the rational numbers over which the proposed new connective can be naturally defined. Lexicographic logic can be used to define in a simple way some well-known preferential operators, like "A and if possible B", and "A or failing that B". We argue that the new logic is an effective formalism for ranking query results according to the satisfaction level of user preferences.