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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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    On Strong and Weak Logics for Paraconsistent Computability
    (College Publications, 2025)
    Weber, Zach
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    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.