Characterization of the reduced matrices for the {∧,∨}-fragment of classical logic

Abstract

In this note we will deal with the {^,_}-fragment of classical logic, i.e. the deductive system L = hF,‘i where F = hForm,^,_i is the absolutely free algebra of type (2,2) and ‘ is semantically determined by the matrix h2,{1}i, where 2 = h2,^,_i is the two-element distributive lattice. We completely characterize the class of reduced matrices for this deductive system, disproving the common belief that their algebraic reducts are all distributive lattices. In fact, that class is a proper subclass of distributive lattices with unit and is not even a quasivariety.