Sequent Calculi for First-order $$\textrm{ST}$$

Abstract

Strict-Tolerant Logic (\(\textrm{ST}\)) underpins naïve theories of truth and vagueness (respectively including a fully disquotational truth predicate and an unrestricted tolerance principle) without jettisoning any classically valid laws. The classical sequent calculus without Cut is sometimes advocated as an appropriate proof-theoretic presentation of \(\textrm{ST}\). Unfortunately, there is only a partial correspondence between its derivability relation and the relation of local metainferential \(\textrm{ST}\)-validity – these relations coincide only upon the addition of elimination rules and only within the propositional fragment of the calculus, due to the non-invertibility of the quantifier rules. In this paper, we present two calculi for first-order \(\textrm{ST}\) with an eye to recapturing this correspondence in full. The first calculus is close in spirit to the Epsilon calculus. The other calculus includes rules for the discharge of sequent-assumptions; moreover, it is normalisable and admits interpolation.