The algebra of ordinary discourse. On the semantics of Cooper’s logic

We develop an algebraic study of W.S. Cooper’s three-valued propositional logic of ordinary discourse (\(\mathcal{O}\mathcal{L}\)). This logic displays a number of unusual features: \(\mathcal{O}\mathcal{L}\) is not weaker but incomparable with classical logic, it is connexive, paraconsistent and contradictory. As a non-structural logic, \(\mathcal{O}\mathcal{L}\) cannot be algebraized by the standard methods. However, we show that \(\mathcal{O}\mathcal{L}\) has an algebraizable structural companion, and determine its equivalent semantics, which turns out to be a finitely-generated discriminator variety. We provide an equational and a twist presentation for this class of algebras, which allow us to compare it with other well-known algebras of non-classical logics. In this way we establish that \(\mathcal{O}\mathcal{L}\) is definitionally equivalent to an expansion of the three-valued logic \({\mathcal {J}}3\) of D’Ottaviano and da Costa, itself a schematic extension of paraconsistent Nelson logic.