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Extending computational trinitarianism
(Oxford University Press, 2026)
Cano-Jorge, Fernando
Computational trinitarianism is the view that a single notion of computation has three different manifestations: in logic as proofs, in typed-calculus as programs, and in category theory as morphisms. This idea has traditionally been closely associated with intuitionistic logic but here we argue that the connection is not exclusive. We provide a logician friendly, self-contained introduction to this topic by presenting the trinities for linear, affine, and relevant logic. The ground we set for that goal is then used to show how to obtain paraconsistent trinities by including the De Morgan negation. ©The author © Oxford University Press.
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Inconsistent sets and how to compute them
(Springer Science and Business Media LLC, 2026)
Cano-Jorge, Fernando
;
Weber, Zach
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. ©The author ©Springer.
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Connexive arithmetic formulated relevantly
(Oxford University Press, 2025)
Cano-Jorge, Fernando
;
Luis Estrada-González
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, cRM3⁠, cRM⁠, cRMn⁠, and cRM⁠. We compare these with their counterparts RM3⁠, RM and that extend relevant arithmetic. ©The authors ©Oxford University Press.
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On Strong and Weak Logics for Paraconsistent Computability
(College Publications, 2025)
Weber, Zach
;
Cano-Jorge, Fernando
One tradition in relevant and paraconsistent logics has been to develop systems intended for applications to arithmetic and computability theory. The aspiration, as in Meyer [38] and others, is to recover enough working mathematics for real computation, but without the limitative results of Turing, 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 RM3 and PAC [6] and their expansions, with application to inconsistent arithmetics [61, 62]. 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 DL2Q*f [31]. The recurring theme is that paraconsistent computability struggles with functionality [17, 41, 43]. A method for advancing on the ‘function problem' is sketched with Kleene's theorem as a worked example. ©The authors ©College Publications © Journal of Applied Logics.
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Gates and circuits via Dunn semantics
(Oxford University Press (OUP), 2025)
Cano-Jorge, Fernando
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. © The author © Oxford University Press.

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