Fernando Cano-Jorge

The idea of a paraconsistent computability theory has been proposed as a way to work effectively with inconsistent sets of numbers. The viability of such a theory, though—the very coherence of the idea of an ‘inconsistent recursive relation’—has been called into doubt, most recently in (Choi, Synthese 200(5):418, 2022). In this paper we remove some doubt, by setting out a simple model of (naïve) set theory in LP, showing how to compute inconsistent sets in terms of extensions and antiextensions, and establishing further representability results. This suggests a way that a longstanding and apparently impossible-to-answer question—how can inconsistency be computed?—can be answered.

Mortensen introduced a connexive logic commonly known as 'M3V'. M3V is obtained by adding a special conditional to LP. Among its most notable features, besides its being connexive, M3V is negation-inconsistent and it validates the negation of every conditional. But Mortensen has also studied and applied extensively other non-connexive logics, for example, closed set logic, CSL, and a variant of Sette's logic, identified and called 'P2' by Marcos. In this paper, we analyze and compare systematically the connexive variants of CSL and P2, obtained by adding the M3V conditional to them. Our main observations are two. First, that the inconsistency of M3V is exacerbated in the connexive variant of closed set logic, while it is attenuated in the connexive variant of the Sette-like P2. Second, that the M3V conditional is, unlike other conditionals, "connexively stable", meaning that it remains connexive when combined with the main paraconsistent negations.

One tradition in relevant and paraconsistent logics has been to develop sys- tems intended for applications to arithmetic and computability theory. The aspiration, as in Meyer [38] and others, is to recover enough working mathe- maticsforrealcomputation, butwithoutthelimitativeresultsofTuring, Gödel, etc.; or more cautiously, as in Dunn [22], to respect relevance and with that be insulated against the possibility of a genuine inconsistency. We distill these goals into guiding questions, and study the options for logics within a range of relevant systems. We focus on strong truth functional logics RM3and PAC [6] and their expansions, with application to inconsistent arithmetics [62, 63]. We argue that this approach, while having many virtues, does not fully answer our guiding questions. This points to weak relevant logics like Routley/Sylvan’s DKQ [54], Brady’s MCQ [14], and Logan and Boccuni’s DL2Qt,f c [31]. The recurring theme is that paraconsistent computability struggles with functional- ity [17, 41, 43]. A method for advancing on the ‘function problem’ is sketched with Kleene’s theorem as a worked example.

Computer hardware is heavily reliant on classical logic, but could we use some non- classical logic instead? In this paper I suggest an alternative model of implementation of logic gates in electronics. In particular, I propose a model that takes descriptions of logical connectives through means of Dunn semantics and implements them as logic gates by using a double current system: one for truth and one for falsity, instead of the classical use of a single current for both values. The outcome of this proposal should pave the way to new models of non-classical and even contra-classical computation based on paraconsistent, paracomplete and paranormal logics.

Projects directed at a universal logic (notably, Brady's [Brady 2006]) have long struggled with paradoxes. In just the domain of computability, Hilbert's call for a general decision procedure was scuttled by Turing, using a diagonal argument. Indeed, Turing's halting problem can be straightforwardly viewed as a paradox, of the same type as others like the Liar and Curry's. This is confirmed here by reconstructing a halting predicate in a sequent calculus setting, thereby fitting a ``recipe' for paradox in [Ahmad, AJL 2022]. The usual range of possible solutions then apply, including especially substructural approaches. Bringing this work as a response to Brady's ``"Starting the Dismantling of Classical Mathematics" [Brady, AJL 2018], then, we assess his proposals to drop the law of excluded middle and other logical principles---asking whether this strategy avoids all versions of the halting paradox. Brady has well begun, in his idiom, the re-construction of mathematics; there is more yet to do.

Following the strategy in [15] to develop inconsistent models for relevant arithmetics, we formulate a connexive variant of arithmetic by replacing the conditional of RM3 with the Belikov–Loginov conditional. We obtain thus the connexive logic cRM3 which serves as a base logic for arithmetics cRM3$^{i}$, cRM3$^{i\sharp }$, cRM$^{\sharp }$, cRMn$^{i}$, and cRM$^\omega $. We compare these with their counterparts RM3$^{i\sharp }$, RM$^{\sharp }$ and $\mathbf{RM}^\omega$ that extend relevant arithmetic $\mathbf{R}^\sharp$.