An Extension of Connexive Logic C

The connexive logic $\mathsf{C}$ is a simple variant of Nelson's Logic $\mathsf{N4}$, obtained by making a small change in the falsification clause for the conditional. This was an important step marked in the field of connexive logic since $\mathsf{C}$ can be seen as the first system of connexive logic with an intuitively plausible semantics. The aim of the present paper is to consider an extension of $\mathsf{C}$ obtained by adding the law of excluded middle with respect to the strong negation. The extension of $\mathsf{C}$ is motivated by three questions. The first question comes from a system $\mathsf{CN}$ devised by John Cantwell. The second question concerns how many more connexive theses, beside the basic theses of Aristotle and Boethius, can be captured within the framework suggested in the above paper. The third question addresses the relation between constructivity and the law of excluded middle. We will show that the quantified version of our extension of $\mathsf{C}$ satisfies the Existence Property and its dual, but fails to satisfy the Disjunction Property and its dual when the law of excluded middle is restricted to atomic formulas.


Based on joint work with Hitoshi Omori.