An Algebraic Study of Admissible Rules

Abstract

In this thesis we shall study admissible rules within the general framework of Abstract Algebraic Logic (AAL). Following Lorenzen, we say that a rule is admissible for a logic S whenever it does not add new theorems to S. Despite the seemingly natural definition, the determination of admissible rules in particular logics is usually a difficult problem and requires a deep understanding of the structural properties of the logic. Our purpose is not to study particular cases but instead, we intent to present algebraic conditions of the admissibility of a rule for a logic both in the general case and also depending on its classification in the Leibniz hierarchy. Particular cases will be presented as examples or counter-examples, whenever it is necessary.