A Logical Study of Partial Entailment

We introduce a novel logical notion–partial entailment–to propositional logic. In contrast with classical entailment, that a formula P partially entails another formula Q with respect to a background formula set Γ intuitively means that under the circumstance of Γ, if P is true then some “part” of Q will also be true. We distinguish three different kinds of partial entailments and formalize them by using an extended notion of prime implicant. We study their semantical properties, which show that, surprisingly, partial entailments fail for many simple inference rules. Then, we study the related computational properties, which indicate that partial entailments are relatively difficult to be computed. Finally, we consider a potential application of partial entailments in reasoning about rational agents.