On strong and weak logics for paraconsistent computability

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.