Rewriting the History of Connexive Logic

Wolfgang Lenzen

doi:10.1007/s10992-021-09640-6

Abstract

The “official” history of connexive logic was written in 2012 by Storrs McCall who argued that connexive logic was founded by ancient logicians like Aristotle, Chrysippus, and Boethius; that it was further developed by medieval logicians like Abelard, Kilwardby, and Paul of Venice; and that it was rediscovered in the 19th and twentieth century by Lewis Carroll, Hugh MacColl, Frank P. Ramsey, and Everett J. Nelson. From 1960 onwards, connexive logic was finally transformed into non-classical calculi which partly concur with systems of relevance logic and paraconsistent logic. In this paper it will be argued that McCall’s historical analysis is fundamentally mistaken since it doesn’t take into account two versions of connexivism. While “humble” connexivism maintains that connexive properties (like the condition that no proposition implies its own negation) only apply to “normal” (e.g., self-consistent) antecedents, “hardcore” connexivism insists that they also hold for “abnormal” propositions. It is shown that the overwhelming majority of the forerunners of connexive logic were only “humble” connexivists. Their ideas concerning (“humbly”) connexive implication don’t give rise, however, to anything like a non-classical logic.

Highlights

What is connexive logic? According to [57]: Systems of connexive logic are contra-classical in the sense that they are neither subsystems nor extensions of classical logic
Lenzen language gives rise to the trivial system, so that any non-trivial system of connexive logic will have to leave out some theorems of classical logic
Our investigation of the first 2200 years has shown, that the vast majority of the logicians either understood their claims only in the sense of “humble” connexivism; or, if they originally believed in “hardcore” connexivism, they were eventually convinced by other logicians to give up this belief since it is incompatible with the validity of other, better entrenched laws of logic

Summary

Introduction

What is connexive logic? According to [57]: Systems of connexive logic are contra-classical in the sense that they are neither subsystems nor extensions of classical logic. Negation remains an indispensable ingredient of term logic, and it seems safe to conclude that McCall’s propositional “transliteration” (‘If p, if q r; if q not-r; not-p’) correctly reflects Boethius’ term-logical principle ‘Si est A, cum sit B, est C; atqui cum sit B, non est C, non est igitur A’ In this sense McCall’s conclusion ([39], 417) “that, for Boethius, ¬(p → ¬q) follows from p → q” is entirely correct, and Boethius may rightly be considered as an advocate of principle BOETH. One may reasonably assume that the intuitions, which guided Aristotle and Boethius in putting forward their laws of connexive implication, were based on the tacit assumption that the antecedents are “normal” This hypothesis, again, appears much more likely than to assume that they would have been willing to defend their laws for “abnormal”, self-inconsistent propositions. The most important examples shall be discussed in the subsequent section

Some Twelfth Century Debates

11 Conclusion