Predicate Logics on Display

TL;DR: The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic and considers predicate logics obtained by adopting van Behthem's modal perspective on first-order logic.

Abstract: This chapter provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Benthem’s modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap’s display logic, DL by introduction rules for the existential and the universal quantifier. These rules for ∀x and ∃x are analogous to the display introduction rules for the modal operators □ and ◊ and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal ‘modal’ predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules. The chapter has two main aims, namely 1. presenting a uniform proof-theoretic schema for both substructural subsystems of classical first-order logic, FOL and various subsystems of FOL obtained by relaxing Tarski’s truth definition for the existential and universal quantifiers, and 2. introducing these quantifiers into the framework of DL.