|Logic, Algebra and Implication
Institute of Computer Science
Artificial Intelligence Research Institute
Abstract Algebraic Logic is a relatively new subfield of Mathematical Logic. It is a natural evolution of Algebraic Logic, a branch of Mathematical Logic studying logical systems by giving them an algebra-based semantics. It can be traced back to George Boole and his study of classical propositional logic by means of a two-element algebra that became its canonical semantics. Other non-classical logics enjoy a strong connection with algebras as well intuitionistic logic and Heyting algebras, substructural logic and residuated lattices, etc.). Abstract Algebraic Logic (AAL) was born as the natural next step to be taken in this evolution: the abstract study of logical systems through their interplay with algebraic semantics.
One of the starting points of AAL is the book by Helena Rasiowa  where she studies logics possessing a reasonable implication connective. Her approach was later gradually generalized into a genuinely abstract theory~[1,4,5]. A crucial technical notion used is this process is the Leibniz operator, which maps any theory of a logic to the congruence relation of the formulae which are provably equivalent in the presence of such theory. Logics were classified by means of properties of Leibniz operator, which gave rise to the two-dimensional Leibniz hierarchy. This classification, starting with the biggest class of protoalgebraic logics, became the core theory of AAL due to its robustness, the characterizations of its classes, and their usefulness for obtaining bridge theorems, i.e. results connecting logical properties to equivalent algebraic properties in the semantics.
The aim of this course is to present a self-contained introduction to AAL. For didactic reasons we present the full Leibniz hierarchy at the end of the tutorial only and, for most of the time, we simplify our account to yet another generalization of Rasiowa approach: the class of weakly implicative logics [2,3]. Although this can be viewed as a rather minor generalization, it provides a relatively simple framework which allows us to describe the arguably more important dimension of Leibniz hierarchy and to demonstrate the strength of existing abstract results.
 Helena Rasiowa. An Algebraic Approach to Non-Classical Logics. North-Holland, Amsterdam, 1974.
This is an introductory course and as such will be highly self-contained. Students will only be assumed to have a basic knowledge on classical propositional logic and elementary set theory, and some rudiments of universal algebra.