Abstract algebraic logic

Algberaic Logic is the area of Mathematical Logic that studies the relationship between logical systems, on the one hand, and associated classes of algebras, on the other. The archetypal paradigm of an association of this kind, that lies in the historical origins of Algebraic Logic and at the heart of all subsequently developed subtheories, is the association of the class of Boolean algebras to classical propositional calculus by George Boole in the 1850's.

In classical Algebraic Logic, which comprises all studies in Algebraic Logic until somewhere in the 1950's to 60's, when a transitional period to the modern era emerged primarily in Poland, the focus was on the study of the properties of specific classes of algebras used to "algebraize" specific logical systems of particular interest to specific logical investigations.

Abstract Algebraic Logic is the modern subarea of Algebraic Logic that emerged in the 1950's to 60's with the work of the Polish school of logicians (Rasiowa, Sikorski, Los and Suszko are but a few contributors) but did not reach full maturity until the 1980's, when seminal works by the Polish logician Janusz Czelakowski and the pair consisting of the Dutch logician Willem Blok and the Americal logician Don Pigozzi appeared in print. The focus of Abstract Algebraic Logic, as compared to that of the classical period, is shifted from the study of specific classes of algebras associated to specific logical systems to the study of 1. classes of algebras associated with classes of logical

  systems satisfying collectively some abstract logical
  properties;

2. the process by which a class of algebras is associated

  to a given logical system as its "algebraic counterpart";

3. the relationship between metalogical properties that

  are satisfied by a class of logical systems and 
  corresponding algebraic properties that are satisfied 
  by their algebraic counterparts.

The passage from classical Algebraic Logic to Abstract Algebraic Logic may be compared for a better intuitive understanding to the passage from Modern Algebra (i.e., the study of groups, rings, modules, fields, etc.) to Universal Algebra (the study of classes of algebras of arbitrary similarity types (algebraic signatures)).

The two main motivations for the development of Abstract Algebraic Logic are closely connected to the first and the third directions of study, enumerated above. With respect to the first direction, a critical step in the transition was initiated by the work of Rasiowa, whose goal was to abstract results and methods that were known to hold for classical propositional calculus and Boolean algebras and some other closely related logical systems to a much wider variety of propositional logics. As far as the third direction, Blok and Pigozzi's work was formed with the primary motivation to explore different forms that the well-known Deduction Theorem of classical propositional calculus and of first-order logic assumes in a wide variety of logical systems and to relate it to algebraic properties of the classes of algebras forming the algebraic counterparts of these systems.

Nowadays, Abstract Algebraic Logic is a very well established subfield of Algebraic Logic, increasing in popularity, with many deep and interesting results explaining many properties that arise in different classes of logical systems that were previously explained only in a case by case basis or shrouded mystery. Perhaps its most important achievement has been the classification of propositional logics in a hierarchy, called the Abstract Algebraic Hierarchy or the Leibniz Hierarchy, whose different levels roughly reflect the strength of the ties between a logic in that level and its associated class of algebras and, as a consequence, the extent to which that logic may be studied using algebraic methods and techniques. Once a logic is classified in one of these levels, one may draw upon a powerful arsenal of results that have been accumulated over the last thirty or so years to study many of its logical properties and some of the algebraic properties of its algebraic counterpart.