Boolean Valued Models and Incomplete Specifications

Logic programming has favored universal Horn theories for several reasons. An advantage of consistent universal Horn theories, compared with arbitrary consistent theories, is the fact that they possess a “typical” model-the least Herbrand model. More precisely, in order to verify that a sentence of the form 3 X, . . . 3 x, A, where A is a conjunction of atomic formulas (i.e. positive literals), is a consequence of a universal Horn theory .7, it suffices to prove that this sentence holds in the least Herbrand model of 7. Moreover, this distinguished model of 7 can be characterized algebraically as the initial object in the category of all models of 7 which has homomorphisms as morphisms. It is the aim of this paper to show that every consistent universal theory has a Boolean valued model with similar properties. This model will be described, and its principal properties will be discussed. Boolean valued models provide a natural tool for modeling incompletely specified objects, such as those arising from resolution proofs of existential formulas from non-Horn theories. Green’s procedure [6] for extracting such specifications from proofs is refined to yield a complete specification of an object in a Boolean valued model.