Admissible equivalence systems

Abstract

Whenever a logic is the set of theorems of some deductive system, where the latter has an equivalence system, the behavioral theorems of the logic can be determined by means of that equivalence system. In general, this original equivalence system may be too restrictive, because it su ces to check behavioral theorems by means of any admissible equivalence system (that is an equivalence system of the small- est deductive system associated with the given logic). In this paper, we present a range of examples, which show that: 1) there is an admissible equivalence system which is not an equivalence system for the initial deductive system, 2) there is a non- nitely equivalential deductive system with a nite admissible equivalence system, and 3) there is a deductive system with an admissible equivalence sys- tems, such that this deductive system is not even protoalgebraic itself. We use methods and results from algebraic and modal logic.