FLAP Journal 2025 Journal Article
On Strong and Weak Logics for Paraconsistent Computability
- Zach Weber
- Fernando Cano-Jorge
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- matics 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 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. 2020 Mathematics Subject Classification. Primary: 03B47, Secondary: 03B53, 03B80. Thanks to audiences at Logic Day in Dunedin 2023 and the 2nd Third Workshop at the University of Alberta, 2024, and to Jack Copeland. Thanks to the referees for helpful comments that improved the paper, and to the editor of this special issue. ∗ Research supported by the Marsden Fund, Royal Society of New Zealand; the PAPIIT project IN406225; and the CONAHCYT project CBF2023-2024-55.