Functional dependencies, supervenience, and consequence relations

An analogy between functional dependencies and implicational formulas of sentential logic has been discussed in the literature. We feel that a somewhat different connexion between dependency theory and sentential logic is suggested by the similarity between 'Armstrong's axioms' for functional dependencies and Tarski's defining conditions for consequence relations, and we pursue aspects of this other analogy here for their theoretical interest. The analogy suggests, for example, a different semantic interpretation of consequence relations: instead of thinking of B as a consequence of a set of formulas {A₁,..., A_n} when B is true on every assignment of truth-values on which each A_i is true, we can think of this relation as obtaining when every pair of truth-value assignments which give the same truth-values to A₁, the same truth-values to A₂,..., and the same truth-values to A_n, also make the same assignment in respect of B. We describe the former as the consequence relation 'inference-determined' by the class of truth-value assignments (valuations) under consideration, and the latter as the consequence relation 'supervenience-determined' by that class of assignments. Some comparisons will be made between these two notions.