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 nonstructural, 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 singleconclusion 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 . J 3
of D’Ottaviano and da Costa, itself an axiomatic extension of paraconsistent Nelson logic.