Ricardo Arturo Nicolás-Francisco

This paper discusses the approaches to paraconsistency proposed by Jaśkowski in 1948 and independently by da Costa in 1963, which are historically important and highly influential in the context of non-classical logics. In their seminal works, they initiated extensive investigations into logics with non-explosive negations, later baptized paraconsistent logics, thus distinguishing the concepts of triviality and inconsistency, which are classically equivalent. The focus will be on Jaśkowski’s logic $$\textit{D}_\textit{2}$$ and da Costa’s $$C_n$$ hierarchy. We will discuss the properties of these formal systems and their further developments, as well as the main motivations and the historical context.

Seen from the point of view of evaluation conditions, a usual way to obtain a connexive logic is to take a well-known negation, for example, Boolean negation or de Morgan negation, and then assign special properties to the conditional to validate Aristotle’s and Boethius’ Theses. Nonetheless, another theoretical possibility is to have the extensional or the material conditional and then assign special properties to the negation to validate the theses. In this paper we examine that possibility, not sufficiently explored in the connexive literature yet.We offer a characterization of connexive negation disentangled from the cancellation account of negation, a previous attempt to define connexivity on top of a distinctive negation. We also discuss an ancient view on connexive logics, according to which a valid implication is one where the negation of the consequent is incompatible with the antecedent, and discuss the role of our idea of connexive negation for this kind of view.

In this paper I explore whether three different approaches to connexivity – namely, Lewis and Langford’s, Wansing’s and Egré and Politzer’s – can be non-trivially related. Lewis and Langford’s approach consists of the definition of the conditional in terms of consistency via the definition of possibility in terms of consistency and Lewis’ definition of the (strict) conditional in terms of possibility; Wansing’s approach is based on his definition of the falsity condition for the conditional as “if A is true then B is false”; Egré and Politzer’s approach relies on the modalization of the falsity condition for the conditional in its conditional form (A→◇~B). With this background in mind, I build upon Estrada-González’s work relating connexivism and possibilism, the thesis according to which everything is possible. Estrada-González’s work relates Lewis and Langford’s connexive approach with Nelson’s idea that every proposition is self-consistent. He does so using a fragment of LP augmented with a special conditional. This relation has resulted in the study and postulation of several truth-functional modalities, some of which have already been discussed elsewhere. I employ several truth-functional modalities to evaluate Egré and Politzer’s falsity conditions for the conditional. I show that in using Béziau’s truth-functional modality ◇B, definable in Estrada-González’s framework, one can retain in a single framework the three different approaches. More specifically, distinguishing the notion of logical equivalence from the notion of interderivability and these two from the notion of coimplication, it is possible to relate ~(A→B) and (A→◇~B) as interderivable schemas and in a valid biconditional. I also show some consequences of this for Béziau’s negative appraisal of three-valued truthfunctional modalities.