Algebraic analysis of Visser's formal propositional logic
by
Majid Alizadeh
Institute for Studies in Theoretical Physics and Mathematics. University of Tehran

In 1981 Albert Visser characterized a propositional logic that is embedded into GL by the Gödel translation and called it formal propositional logic, FPL. In this talk we introduce the variety of Löb algebras, the algebraic structures associated with formal propositional logic. So Löb algebras take the role for formal propositional logic that Heyting algebras play for intuitionistic propositional logic, IPL, and Boolean algebras play for classical propositional logic, CPL.

It is well-known that there are only eight sub-varieties of Heyting algebras which have the amalgamation property. Therefore there are only eight intermediate logics over IPL which have the interpolation property. But the lattice of subvarieties of the variety of basic algebras is more complicated than the lattice of subvarieties of the variety of Heyting algebras. Here we first prove some partial results about the variety of basic algebras, applying them to study the lattice of subvarieties of the variety of Löb algebras. We show that there are infinitely many subvarieties of basic algebras ( Löb algebras) which have the amalgamation property. We also investigate minimality in the class of all basic algebras. In fact we show that the variety of basic algebras has two minimal subvarieties. The variety of Boolean algebras and a subvariety of the variety of Löb algebras called, L₁-algebras.

Date received: July 22, 2007