Aristotle's thesis in consistent and inconsistent logics

Abstract

A typical theorem of conaexive logics is Aristotle's Thesis(A), ∼(A→∼A).A cannot be added to classical logic without producing a trivial (Post-inconsistent) logic, so connexive logics typically give up one or more of the classical properties of conjunction, e.g.(A & B)→A, and are thereby able to achieve not only nontriviality, but also (negation) consistency. To date, semantical modellings forA have been unintuitive. One task of this paper is to give a more intuitive modelling forA in consistent logics. In addition, while inconsistent but nontrivial theories, and inconsistent nontrivial logics employing prepositional constants (for which the rule of uniform substitution US fails), have both been studied extensively within the paraconsistent programme, inconsistent nontrivial logics (closed under US) do not seem to have been. This paper gives sufficient conditions for a logic containingA to be inconsistent, and then shows that there is a class of inconsistent nontrivial logics all containingA. A second semantical modelling forA in such logics is given. Finally, some informal remarks about the kind of modellingA seems to require are made.