  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.
