On an ordinary 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 paradefinite logic are defined, a sequent calculus proof system for this logic is presented, and the soundness and completeness of this proof system is established. It is shown that the defined logic distinguishes itself from the many other paradefinite logics that are usually considered equally classical by the classical laws of logical equivalence that hold for it. It is further argued that the defined logic is the most natural paradefinite logic relative to the version of classical logic with the same language. Moreover, a simple embedding of the defined logic in that version of classical logic is presented and the potential of the logic for dealing with inconsistencies and incompletenesses in inductive machine learning is discussed.