{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T22:32:32Z","timestamp":1768429952825,"version":"3.49.0"},"reference-count":28,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2023,12,13]],"date-time":"2023-12-13T00:00:00Z","timestamp":1702425600000},"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":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2024,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present a novel treatment of set theory in a four-valued <jats:italic>paraconsistent<\/jats:italic> and <jats:italic>paracomplete<\/jats:italic> logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell\u2019s paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be used to reason informally about incomplete and inconsistent phenomena, and is sufficiently similar to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000382_inline1.png\"\/><jats:tex-math>\n${\\mathrm {ZFC}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> to enable the development of interesting mathematics.<\/jats:p><jats:p>We propose an axiomatic system <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000382_inline2.png\"\/><jats:tex-math>\n${\\mathrm {BZFC}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, obtained by analysing the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000382_inline3.png\"\/><jats:tex-math>\n${\\mathrm {ZFC}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-axioms and translating them to a four-valued setting in a careful manner, avoiding many of the obstacles encountered by other attempted formalizations. We introduce the <jats:italic>anti-classicality axiom<\/jats:italic> postulating the existence of non-classical sets, and prove a surprising results stating that the existence of a single non-classical set is sufficient to produce any other type of non-classical set.<\/jats:p><jats:p>Our theory is naturally bi-interpretable with <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020323000382_inline4.png\"\/><jats:tex-math>\n${\\mathrm {ZFC}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and provides a philosophically satisfying view in which non-classical sets can be seen as a natural extension of classical ones, in a similar way to the non-well-founded sets of Peter Aczel [1].<\/jats:p><jats:p>Finally, we provide an interesting application concerning Tarski semantics, showing that the classical definition of the satisfaction relation yields a logic precisely reflecting the non-classicality in the meta-theory.<\/jats:p>","DOI":"10.1017\/s1755020323000382","type":"journal-article","created":{"date-parts":[[2023,12,13]],"date-time":"2023-12-13T08:15:22Z","timestamp":1702455322000},"page":"965-995","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["PARACONSISTENT AND PARACOMPLETE ZERMELO\u2013FRAENKEL SET THEORY"],"prefix":"10.1017","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1763-876X","authenticated-orcid":false,"given":"YURII","family":"KHOMSKII","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1594-340X","authenticated-orcid":false,"given":"HRAFN VALT\u00ddR","family":"ODDSSON","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2023,12,13]]},"reference":[{"key":"S1755020323000382_r4","doi-asserted-by":"crossref","first-page":"276","DOI":"10.2307\/2274919","article-title":"Natural \n\n\n\n$3$\n\n\n-valued logics\u2014Characterization and proof theory","volume":"56","author":"Avron","year":"1991","journal-title":"The Journal of Symbolic Logic"},{"key":"S1755020323000382_r13","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/978-3-030-31136-0_11","volume-title":"New Essays on Belnap\u2013Dunn Logic","author":"Hazen","year":"2019"},{"key":"S1755020323000382_r19","doi-asserted-by":"crossref","unstructured":"[20] Omori, H. , & Waragai, T. 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On the logic of logical pluralism","volume":"63","author":"Sereni","year":"2017","journal-title":"Inquiry"},{"key":"S1755020323000382_r2","doi-asserted-by":"crossref","first-page":"231","DOI":"10.2307\/2274105","article-title":"Constructible falsity and inexact predicates","volume":"49","author":"Almukdad","year":"1984","journal-title":"The Journal of Symbolic Logic"},{"key":"S1755020323000382_r20","doi-asserted-by":"crossref","first-page":"12663","DOI":"10.1007\/s11229-021-03348-5","article-title":"Should pluralists be pluralists about pluralism?","volume":"199","author":"Passmann","year":"2021","journal-title":"Synthese"},{"key":"S1755020323000382_r26","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780199696529.001.0001","volume-title":"Varieties of Logic","author":"Shapiro","year":"2014"},{"key":"S1755020323000382_r9","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/logcom\/ext020","article-title":"Paraconsistent set theory by predicating on consistency","volume":"26","author":"Carnielli","year":"2016","journal-title":"Journal of Logic and Computation"},{"key":"S1755020323000382_r18","unstructured":"[19] Oddsson, H. 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Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}