VALIDITY, DIALETHEISM AND SELF-REFERENCE (PREPRINT) FEDERICO MATIAS PAILOS Abstract. It has been argued recently (in Beall (2009) and Beall and Murzi (2013)) that dialetheist theories are unable to express the concept of naive validity. In this paper, we will show that LP can be non-trivially expanded with a naive validity predicate. The resulting theory, LP Val reaches this goal by adopting a weak self-referential procedure. We show that LP Val is sound and complete with respect to the three-sided sequent calculus SLP Val . Moreover, LP Val can be safely expanded with a transparent truth predicate. We will also present an alternative theory LP Val * , which includes a non- deterministic validity predicate. KEYWORDS: Validity, Self-reference, Paradoxes, 1. Introduction Dialetheists argue that the acceptance of contradictions is the best way to solve the paradoxes while achieving a semantically closed language. In recent years, Beall (2009) and Beall and Murzi (2013) tried to show that dialetheism is unable to express the concept of naive validity. The inexpressibility result goes as follows. Let Val be a naive validity predicate, characterized by the following rules and meta-rules. Let A and B be formulas variables. ⟨A⟩ and ⟨B⟩ are names for A and B, respectively. The so-called naive validity principles are the following: A ⊢ B VP ⊢ V al(⟨A⟩, ⟨B⟩) VD A, V al(⟨A⟩, ⟨B)⟩⊢ B ⊢ V al(⟨A⟩, ⟨B⟩) ⊢ A MetaVD ⊢ B If the theory achieves self-reference through something like strong diagonaliza- tion, there will be a sentence A definitionally equivalent to Val(⟨A⟩, ⟨⟩), usually 1