Between Hilbert and Gentzen: four-valued consequence systems and structural reasoning

Structural reasoning is simply reasoning that is governed exclusively by structural rules. In this context a proof system can be said to be structural if all of its inference rules are structural. A logic is considered to be structuralizable if it can be equipped with a sound and complete structural proof system. This paper provides a general formulation of the problem of structuralizability of a given logic, giving specific consideration to a family of logics that are based on the Dunn–Belnap four-valued semantics. It is shown how sound and complete structural proof systems can be constructed for a spectrum of logics within different logical frameworks.