FLAP Journal 2016 Journal Article
In this paper, we reflect on some themes related to the formulation of math- ematics against the backdrop of a connexive logic. From a positive perspective, we will consider some remarks of the Kneales concerning Aristotle’s position on connexive implication and suggest that common themes between the Kneales’ Aristotle and the hyper-constructive arithmetic of David Nelson may provide a philosophical basis for connexive mathematics. We will also consider some his- torical points, including Łukasiewicz’ argument that connexive principles may be refuted by appeal to number-theoretic intuitions. Finally, we will take more concrete steps towards the implementation of connexive mathematics by ex- amining how weak subtheories of arithmetic fare when formulated in modest first-order extensions of three connexive logics: Richard Angell’s PA1 and PA2 and Graham Priest’s PN. Unfortunately, we will observe that severe patholo- gies emerge when even extraordinarily weak subsystems of Peano arithmetic are evaluated in these logics, suggesting that Angell and Priest’s systems constitute strained, if not unserviceable, bases for arithmetic. 1 The Allure of Connexive Mathematics Frequently, non-classical logics are presented as formalizations of correct deductive reasoning and mathematical reasoning, as an a priori discipline, is uniquely sensitive to the adoption or rejection of logical principles. The analysis of how various mathematical theories fare under enriched or restrained theories of inference makes up one of the most salient applications of a non-classical theory of deduction. Historically, this is most evident in the case of intuitionism, insofar as the I am very grateful for some helpful comments due to Maarten McKubre-Jordens and Can Bas ˛kent when the material was presented at the Workshop on Connexive Logic at the Fifth World Congress and School on Universal Logic. I also appreciate the very helpful comments of two anonymous referees.