### Axiomatizing the Logic of Ordinary Discourse

Most non-classical logics are subclassical, that is, every inference/theorem they validate is also valid classically. A notable exception is the three-valued propositional Logic of Ordinary Discourse (OL) proposed and extensively motivated by W.S. Cooper as a more adequate candidate for formalizing everyday reasoning (in English). OL challenges classical logic not only by rejecting some theses, but also by accepting non-classically valid principles, such as so-called Aristotle’s and Boethius’ theses. Formally, OL shows a number of unusual features – it is non-structural, connexive, paraconsistent and contradictory – making it all the more interesting for the mathematical logician. We present our recent findings on OL and its structural companion (that we call sOL). We introduce Hilbert-style multiple-conclusion calculi for OL and sOL that are both modular and analytic, and easily allow us to obtain single-conclusion axiomatizations. We prove that sOL is algebraizable and single out its equivalent semantics, which turns out to be a discriminator variety generated by a three-element algebra. Having observed that sOL can express the connectives of other three-valued logics, we prove that it is definitionally equivalent to an expansion of the three- valued logic J3 of D’Ottaviano and da Costa, itself an axiomatic extension of paraconsistent Nelson logic.