Considerations on Everett J. Nelson's connexive logic

TL;DR

This paper analyzes Everett J. Nelson's connexive logic, providing algebraic-relational semantics, exploring extensions to better align with Nelson's philosophical ideas, and relating it to modern ordered structures.

Contribution

It introduces algebraic-relational semantics for Nelson's connexive logic and investigates extensions that incorporate Nelson's philosophical concepts more accurately.

Findings

Established algebraic-relational models for Nelson's logic.

Explored extensions with irreflexive incompatibility relations.

Linked Nelson's ideas to modern ordered algebraic structures.

Abstract

This work explores Everett John Nelson's connexive logic, outlined in his PhD thesis and partially summarized in his 1930 paper \emph{Intensional Relations}, which is obtained by extending the system $\mathsf{NL}$ (reconstructed by E. Mares and F. Paoli) with a weak conjunction elimination rule explicitly assumed in the former but not in the latter. After a preliminary analysis of Nelson's philosophical ideas, we provide an algebraic-relational semantics for his logic and we investigate possible extensions thereof which are able to cope with Nelson's ideas with much more accuracy than the original system. For example, we will inquire into extensions whose algebraic-relational models are endowed with irreflexive incompatibility relations, or determine a ``weakly'' transitive entailment. Such an investigation will allow us to establish relationships between some of the trademarks of Nelson's thought and concepts of prominent importance for connexive logic, as e.g. Kapsner's strong connexivity and superconnexivity, as well as between the algebraic-relational semantics of Nelsonian logics and ordered structures that have gained great attention over the past years, namely partially ordered involutive residuate groupoids and (non-orthomodular) orthoposets.