Rewriting the History of Connexive Logic

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.

Connexivity in the Logic of Reasons

“…This is essentially the view advocated in Iacona[12]. Restricted forms of connexivity are discussed in Unterhuber[32], Kapsner[14], and Lenzen[15].…”

“…However, there is another kind of defeaters which is no less important for the purposes of a theory of reasons, namely, undercutting defeaters: a defeater r for p → q is undercutting if it questions the connection between p and q, thus providing a reason against p → q. 15 Boethius Thesis and Boethius Left Thesis may be regarded as higher-order principles about undercutting defeaters, where r is itself a proposition of the form p → q. 14 An interesting exception is the trivalent theory mentioned in footnote 11.…”

“…This theory allows embeddings, however it invalidates Boethius Left Thesis. 15 The distinction between rebutting and undercutting defeaters goes back to Pollock [21], pp. 73-74.…”

Bilateral Connexive Logic

“…For the history of connexive logics see [4,5]. The implicit assumptions in much 2 of the literature on connexive logics are as follows:…”

“…A precursor to the distinction in the meaning of the two occurrences of negation in the connexive axioms can be found already in Boethius himself, who uses the inner occurrences as a classifier of conditionals into affirmative or negative, as shown by the following quotation of Boethius in [5]: Some hypothetical propositions, however, are affirmative and others are negative [. .…”