Toward Formalizing Usefulness in Propositional Language

In this paper, we attempt to capture the notion of usefulness in propositional language. We believe that classical implication captures a certain kind of usefulness, and name it strict usefulness. We say that a formula P is strictly useful to a formula Q under a formula set Γ if and only if P implies Q under Γ in classical propositional logic. We also believe that classical implication is too strict to capture the whole notion of usefulness. Therefore, we extend it in two ways. The first one is partial usefulness, which means that if P is true, then Q will be partially true under the background of Γ . The second one is probabilistic usefulness, which means that the probability of Q is true will increase by given P is true under Γ . This paper provides semantic definitions of them respectively in propositional language, and discusses the fundamental properties of them.