{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T11:44:13Z","timestamp":1781869453713,"version":"3.54.5"},"reference-count":36,"publisher":"Cambridge University Press (CUP)","issue":"10","license":[{"start":{"date-parts":[[2024,10,2]],"date-time":"2024-10-02T00:00:00Z","timestamp":1727827200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2024,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (<jats:sc>Set<\/jats:sc>-<jats:sc>Set<\/jats:sc>) and single-conclusion (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000227_inline1.png\"\/>) order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix. We studied these logics extensively in Gomes et al. ((2022). <jats:italic>Electronic Proceedings in Theoretical Computer Science<\/jats:italic><jats:bold>357<\/jats:bold> 56\u201376.) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) <jats:sc>Set<\/jats:sc>-<jats:sc>Set<\/jats:sc> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000227_inline2.png\"\/> order-preserving and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000227_inline3.png\"\/><jats:tex-math>\n$\\top$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil\u2019s \u201csymmetric modal logic.\u201d For the order-preserving <jats:sc>Set<\/jats:sc>-<jats:sc>Set<\/jats:sc> logic, in particular, we obtain a <jats:sc>Set<\/jats:sc>-<jats:sc>Set<\/jats:sc> axiomatization that is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.<\/jats:p>","DOI":"10.1017\/s0960129524000227","type":"journal-article","created":{"date-parts":[[2024,10,2]],"date-time":"2024-10-02T09:16:27Z","timestamp":1727860587000},"page":"1138-1183","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["Adding an implication to logics of perfect paradefinite algebras"],"prefix":"10.1017","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3240-386X","authenticated-orcid":false,"given":"Vitor","family":"Greati","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6941-7555","authenticated-orcid":false,"given":"S\u00e9rgio","family":"Marcelino","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2601-8164","authenticated-orcid":false,"given":"Jo\u00e3o","family":"Marcos","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1364-5003","authenticated-orcid":false,"given":"Umberto","family":"Rivieccio","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2024,10,2]]},"reference":[{"key":"S0960129524000227_ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-006-9003-z"},{"key":"S0960129524000227_ref17","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700023806"},{"key":"S0960129524000227_ref33","doi-asserted-by":"publisher","DOI":"10.1080\/11663081.2012.737154"},{"key":"S0960129524000227_ref21","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511809088"},{"key":"S0960129524000227_ref23","volume-title":"Residuated Lattices: An Algebraic Glimpse at Substructural Logics","volume":"151","author":"Galatos","year":"2007"},{"key":"S0960129524000227_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-010-1161-7_2"},{"key":"S0960129524000227_ref12","unstructured":"Cant\u00fa, L. 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(2005a). Logics of Formal Inconsistency. PhD thesis, Unicamp, Brazil & IST, Portugal."}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129524000227","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T08:44:17Z","timestamp":1740991457000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129524000227\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,2]]},"references-count":36,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2024,11]]}},"alternative-id":["S0960129524000227"],"URL":"https:\/\/doi.org\/10.1017\/s0960129524000227","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,2]]},"assertion":[{"value":"\u00a9 The Author(s), 2024. 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