Algebraic Tools for Modal Logic

Mai Gehrke and Yde Venema
Section: Logic
Level: Advanced


There is a long and strong tradition in logic research of applying algebraic techniques in order to deepen our understanding of logic. Such applications are possible because many logics correspond to classes of algebras; typically, the consequence relation of the logic translates into the equational theory of the corresponding class of algebras.

This correspondence between logic and algebra allows one, on a first level, to study the algebras in order to understand the deductive system. But metalogical properties also often end up having algebraic counterparts. In modal logic, a striking example of this phenomenon can be found using the duality theory between Kripke structures and Boolean Algebras with Operators. For instance, a modal logic is complete if and only if its corresponding algebraic variety is generated by the class of algebras that are dual to the Kripke frames of the logic.

A central tool in proving completeness for modal logics is the notion of canonicity, which has both a logical and an algebraic expression. Another problem related to the completeness problem is the translation of axioms for a logic into properties of the Kripke frames. This is the concern of so called correspondence theory.

Apart from giving a general introduction to the fundamental ideas and methods of applying algebra in logic, the purpose of the course is to present recent developments from algebra as well as modal logic, in an integrated format. Our intention is to illuminate and generalize existing results concerning the issues of completeness, canonicity and correspondence for Kripke style semantics for modal and generalized modal logics.

In the first part of the course we give a general introduction to the algebraic perspective on logic. As our running examples we take classical propositional logic and modal logic. We show how the technical notion of a modal logic corresponds to the algebraic one of a variety of Boolean algebras with operators, and we discuss the connection between logical and algebraic properties.

In the last three lectures we concentrate on canonicity and correspondence. The notion of canonicity plays an equally fundamental role in the theory of modal logic as in the algebraic theory of Boolean algebras with operators. For a long time the algebraic and the logical strand of research have been carried out in relative isolation. The aim of this part of the course is threefold: (i) to introduce the notion of canonicity, both from the logical and from the algebraic perspective, (ii) to survey some of the connections between the two areas, and (iii) to present some recent results in the field that generalize and illuminate the classical results.


We assume that the students have some basic knowledge of modal logic and exposure to lattice theory (to be specified in due time).


PDFlecture notes


Mai Gehrke
Department of Mathematical Sciences
New Mexico State University
Las Cruces
New Mexico
88003 USA
(tel) ++ 1 505 646 4218
(fax) ++ 1 505 646 1064

Yde Venema
Institute for Logic, Language and Computation
University of Amsterdam
Plantage Muidergracht 24
1018 TV Amsterdam
(tel) ++ 31 20 525 5299
(fax) ++ 31 20 525 5206