A conventional expansion of first-order Belnap-Dunn logic.

Abstract

This paper concerns an expansion of first-order Belnap-Dunn logic whose connectives and quantifiers all have a counterpart in classical logic. The language and logical consequence relation of this logic are defined, a proof system for this logic is presented, and the soundness and completeness of this proof system is established. The minor differences between the presented proof system for the defined logic and a sound and complete proof system for the version of classical logic with the same language illustrates the close relationship between the logical consequence relations of these logics. A clear characterization of the classical nature of the connectives and quantifiers of the defined logic is given by means of classical laws of logical equivalence. Moreover, a simple embedding of this logic in classical logic is presented and the potential of the logic for dealing with inconsistencies and incompletenesses in inductive machine learning is discussed.