Axiomatizing the Logic of Ordinary Discourse

TL;DR

This paper formalizes the three-valued Logic of Ordinary Discourse (OL), exploring its unusual features and providing modular axiomatizations, showing its algebraizability and relation to other three-valued logics.

Contribution

It introduces Hilbert-style calculi for OL and sOL, proves sOL's algebraizability, and establishes its equivalence to an extension of J3 logic, advancing formal understanding of everyday reasoning.

Findings

sOL is algebraizable with a discriminator variety semantics

sOL is definitionally equivalent to an extension of J3 logic

The calculi for OL and sOL are modular and analytic

Abstract

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.