Validity, dialetheism and self-reference

Abstract

It has been argued recently (Beall in Spandrels of truth, Oxford University Press, Oxford, 2009; Beall and Murzi J Philos 110:143–165, 2013) that dialetheist theories are unable to express the concept of naive validity. In this paper, we will show that (Formula presented.) can be non-trivially expanded with a naive validity predicate. The resulting theory, (Formula presented.) reaches this goal by adopting a weak self-referential procedure. We show that (Formula presented.) is sound and complete with respect to the three-sided sequent calculus (Formula presented.). Moreover, (Formula presented.) can be safely expanded with a transparent truth predicate. We will also present an alternative theory (Formula presented.), which includes a non-deterministic validity predicate.