Algebraic logic for classical conjunction and disjunction (Q1189892)

Relationships between the fragment \(\mathbb{L}\) of classical propositional logic and the variety \(\mathbb{D}\) of distributive lattices are studied from a non-traditional point of view. They are shown to be satisfactorily covered neither by Blok and Pigozzi's algebraizable logics approach nor by use of logical matrices. To make up for the deficiency, the authors introduce a new notion of a model of a sequential calculus \(\mathcal L\) for \(\mathbb{L}\). They find that abstract logics in the sense of Suszko provide a natural tool to convert a distributive lattice into a model of \(\mathcal L\); the specified presentation of \(\mathcal L\) plays here a crucial role. The various results obtained in this paper serve as a basis for a statement that distributive lattices are the right models of \(\mathcal L\).