Π2-rule systems and inductive classes of Gödel algebras

Abstract

In this paper we present a general theory of ${\Pi }_2$-rules for systems of intuitionistic and modal logic. We introduce the notions of ${\Pi }_2$-rule system and of an inductive class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools. As an illustration of the general theory, we analyse the structure of inductive classes of Gödel algebras, from a structure theoretic and logical point of view. We show that unlike other well-studied settings (such as logics, or single-conclusion rule systems), there are continuum many ${\Pi }_2$-rule systems extending $\mathrm{LC}=\mathrm{IPC}+(p\to q)\vee (q\to p)$, and show how our methods allow easy proofs of the admissibility of the well-known Takeuti-Titani rule. Our final results concern general questions admissibility in $\mathrm{LC}$: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all ${\Pi }_2$-rules which are admissible are derivable, and (2) show that the problem of admissibility of ${\Pi }_2$-rules over $\mathrm{LC}$ is decidable.