Axiomatizing the Logic of Ordinary Discourse

TL;DR

This paper analyzes the Logic of Ordinary Discourse (OL), a non-classical, three-valued logic intended to formalize ordinary discourse using a gap value $\rac{1}{2}$ and designated values $\rac{1}{2}$ and $1$. OL challenges classical logic by rejecting certain principles while embracing non-classically valid theses such as Boethius' and Aristotle's theses, and it is non-structural, connexive, paraconsistent, and contradictory. The authors introduce a structural companion, sOL, and develop modular, analytic Hilbert-style multi- and single-conclusion calculi for OL and sOL, along with an algebraic semantics. They show that sOL is algebraizable with an equivalent semantics given by a discriminator variety generated by the three-element algebra $O_3$, and they establish a definitional equivalence between sOL and an expansion of the three-valued logic $\,J3$ (itself an extension of paraconsistent Nelson logic). The work thus links OL/sOL to well-studied non-classical logics, provides complete axiomatizations, and offers a rich algebraic perspective on the logic of ordinary discourse.

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.

Figures (5)

• Figure 1: Truth tables for $\mathrm{OL}$Cooper1968.
• Figure 2: Truth tables of term-definable connectives of $\mathrm{OL}$/$\mathrm{sOL}$.
• Figure 3: Multiple-conclusion rules for the connectives of $\mathrm{sOL}$.
• Figure 4: Derivations of Boethius' thesis and Aristotle's thesis in $\mathrm{sOL}$ (see Figure \ref{['fig:mc-rules-connectives']}).
• Figure 5: Single-conclusion calculus for single-conclusion $\mathrm{sOL}$ produced via Definition \ref{['def:trans-disj-mc-to-sc']}. By Theorem \ref{['the:no-imp-frags-set-fmla']}, one can modularly add suitable rules to $\mathsf{R}_{\curlyvee} \cup \mathsf{R}_\neg \cup \mathsf{R}_\lor$ to axiomatize fragments/expansions of $\mathrm{sOL}$ over signatures $\Sigma \supseteq \{ \neg,\lor \}$ ($\lor$ may be replaced by $\land$).

Theorems & Definitions (19)

• Lemma 1
• Theorem 2
• Theorem 3
• proof
• Definition 1
• Definition 2
• Definition 3
• Theorem 4: ss1978
• Theorem 5
• Theorem 6