A Few Historical Glimpses into the Interplay Between Algebra and Logic and Investigations into Gautama Algebras

Abstract

This chapter consists of two parts. PART I presents a few historical glimpses into the fascinating interplay between algebra and logic that essentially started in the middle of the nineteenth century with Boole’s work. It is mostly non-technical and highlights a few significant examples of this interplay. PART I includes a brief discussion of Boole’s algebra of logic, Frege’s mathematical logic, and the eventual meeting of Frege’s tradition with Boole’s tradition in Tarski’s papers (in the early 1930s). It also discusses Rasiowa’s implicative logics and Blok and Pigozzi’s algebraizable logics, and provides examples – some known and some new. It concludes with a discussion of the impact, on logic, of algebra arising from rough set theory.

PART II further illustrates the interplay between algebra and logic. More precisely, we define a new equational class of algebras called “Gautama algebras.” They are named in honor and memory of the two founders of Indian Logic – Medhatithi Gautama and Akshapada Gautama. The variety $\mathbbm{G}$ of Gautama algebras is a common generalization of the varieties of regular double Stone algebras and regular Kleene Stone algebras, both of which grew out of Boolean algebras, which, in turn, arose from Boole’s algebra of logic. We provide an explicit description of subdirectly irreducible Gautama algebras and the lattice of subvarieties of the variety of Gautama algebras. We also introduce another variety $\mathbbm{GH}$ of Gautama Heyting algebras and show that it is term-equivalent to the variety of Gautama algebras. We then define new propositional logics called GAUTAMA, RDBLSt, and RKLSt and show them to be algebraizable (in the sense of Blok and Pigozzi), with the varieties of Gautama algebras, regular double Stone algebras and regular Kleene Stone algebras, respectively, as their equivalent algebraic semantics. The chapter concludes with some open problems for further research and also with an extensive bibliography.

2010 Mathematics Subject Classification: Primary: 03G10, 03G27, 03B50, 03G25, 06D20, 06D15; Secondary: 08B26, 08B15, 06D30