An Algebraic Study of Bilattice-based Logics

Abstract

The aim of this work is to develop a study from the perspective of Abstract Algebraic Logic of some bilattice-based logical systems introduced in the nineties by Ofer Arieli and Arnon Avron. The motivation for such an investigation has two main roots. On the one hand there is an interest in bilattices as an elegant formalism that gave rise in the last two decades to a variety of applications, especially in the field of Theoretical Computer Science and Artificial Intelligence. In this respect, the present study aims to be a contribution to a better understanding of the mathematical and logical framework that underlie these applications. On the other hand, our interest in bilattice-based logics comes from Abstract Algebraic Logic. In very general terms, algebraic logic can be described as the study of the connections between algebra and logic. One of the main reasons that motivate this study is the possibility to treat logical problems with algebraic methods and viceversa: this is accomplished by associating to a logical system a class of algebraic models that can be regarded as the algebraic counterpart of that logic. Starting from the work of Tarski and his collaborators, the method of algebraizing logics has been increasingly developed and generalized. In the last two decades, algebraic logicians have focused their attention on the process of algebraization itself: this kind of investigation forms now a subfield of algebraic logic known as Abstract Algebraic Logic (which we abbreviate AAL).