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 ( \(\textrm{OL}\) ) proposed and extensively motivated by W.S. Cooper as a more adequate candidate for formalizing everyday reasoning (in English). \(\textrm{OL}\) 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, \(\textrm{OL}\) 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 \(\textrm{OL}\) and its structural companion (that we call \(\textrm{sOL}\) ). We introduce Hilbert-style multiple-conclusion calculi for \(\textrm{OL}\) and \(\textrm{sOL}\) that are both modular and analytic, and easily allow us to obtain single-conclusion axiomatizations. We prove that \(\textrm{sOL}\) 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 \(\textrm{sOL}\) can express the connectives of other three-valued logics, we prove that it is definitionally equivalent to an expansion of the three-valued logic \(\mathcal {J}3\) of D’Ottaviano and da Costa, itself an axiomatic extension of paraconsistent Nelson logic.