Π₂-rule systems and inductive classes of Gödel algebras

In this paper we present a general theory of Π₂-rules for systems of intuitionistic and modal logic. We introduce the notions of Π₂-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 Π₂-rule systems extending LC=IPC+(p→q)∨(q→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 LC: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all Π₂-rules which are admissible are derivable, and (2) show that the problem of admissibility of Π₂-rules over LC is decidable.