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

Abstract We develop an algebraic study of W.S. Cooper’s three-valued propositional logic of ordinary discourse ( $$\mathcal{O}\mathcal{L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> ). This logic displays a number of unusual features: $$\mathcal{O}\mathcal{L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> is not weaker but incomparable with classical logic, it is connexive, paraconsistent and contradictory. As a non-structural logic, $$\mathcal{O}\mathcal{L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> cannot be algebraized by the standard methods. However, we show that $$\mathcal{O}\mathcal{L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> 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 $$\mathcal{O}\mathcal{L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> is definitionally equivalent to an expansion of the three-valued logic $${\mathcal {J}}3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> of D’Ottaviano and da Costa, itself a schematic extension of paraconsistent Nelson logic.