Predicate Logics on Display

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem's modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap's display logic 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.

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