{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T15:56:48Z","timestamp":1770825408284,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T00:00:00Z","timestamp":1673222400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T00:00:00Z","timestamp":1673222400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["PID2020-116502GB-I00"],"award-info":[{"award-number":["PID2020-116502GB-I00"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Philos Logic"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Belnap and Dunn\u2019s well-known 4-valued logic FDE is an interesting and useful non-classical logic. FDE is defined by using conjunction, disjunction and negation as the sole propositional connectives. Then the question of expanding FDE with an implication connective is of course of great interest. In this sense, some implicative expansions of FDE have been proposed in the literature, among which Brady\u2019s logic BN4 seems to be the preferred option of relevant logicians. The aim of this paper is to define a class of implicative expansions of FDE in whose elements Boolean negation is definable, whence strong logics such as the paraconsistent and paracomplete logic P\u01414 and BN4 itself are definable, in addition to classical propositional logic.<\/jats:p>","DOI":"10.1007\/s10992-022-09692-2","type":"journal-article","created":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T14:04:46Z","timestamp":1673273086000},"page":"915-938","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Class of Implicative Expansions of Belnap-Dunn Logic in which Boolean Negation is Definable"],"prefix":"10.1007","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6495-0388","authenticated-orcid":false,"given":"Gemma","family":"Robles","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9560-3327","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"M\u00e9ndez","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,9]]},"reference":[{"key":"9692_CR1","volume-title":"Entailment. The Logic of Relevance and Necessity, Vol. I","author":"AR Anderson","year":"1975","unstructured":"Anderson, A. R., & Belnap, N. D. (1975). Entailment. The Logic of Relevance and Necessity Vol. I. Princeton: Princeton University Press."},{"issue":"3","key":"9692_CR2","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1007\/s11787-020-00254-1","volume":"14","author":"A Avron","year":"2020","unstructured":"Avron, A. (2020). The normal and self-extensional extension of Dunn\u2013Belnap Logic. Logica Universalis, 14(3), 281\u2013296. https:\/\/doi.org\/10.1007\/s11787-020-00254-1.","journal-title":"Logica Universalis"},{"key":"9692_CR3","doi-asserted-by":"publisher","unstructured":"Avron, A. (2021). Proof systems for 3-valued logics based on G\u00f6del\u2019s implication. Logic Journal of the IGPL, jzab013. https:\/\/doi.org\/10.1093\/jigpal\/jzab013.","DOI":"10.1093\/jigpal\/jzab013"},{"issue":"1","key":"9692_CR4","doi-asserted-by":"publisher","first-page":"75","DOI":"10.12775\/LLP.2021.010","volume":"31","author":"JM Blanco","year":"2022","unstructured":"Blanco, J. M. (2022). EF4, EF4-M and EF4-\u0141: A companion to BN4 and two modal four-valued systems without strong \u0141ukasiewicz-type modal paradoxes. Logic and Logical Philosophy, 31(1), 75\u2013104. https:\/\/doi.org\/10.12775\/LLP.2021.010.","journal-title":"Logic and Logical Philosophy"},{"key":"9692_CR5","doi-asserted-by":"crossref","unstructured":"Belnap Jr, N. D. (1977a). In Epstein, G., Dunn, J. M., Belnap Jr, N. D., & Reidel D. (Eds.), A useful four-valued logic, (pp. 8\u201337). Dordrecht: Publishing Co.","DOI":"10.1007\/978-94-010-1161-7_2"},{"key":"9692_CR6","unstructured":"Belnap Jr, N. D. (1977b). In Ryle, G., & Belnap Jr, N. D. (Eds.), How a computer should think, (pp. 30\u201355). Stocksfield: Oriel Press Ltd."},{"issue":"97","key":"9692_CR7","first-page":"9","volume":"25","author":"RT Brady","year":"1982","unstructured":"Brady, R. T. (1982). Completeness proofs for the systems RM3 and BN4. Logique et Analyse, 25(97), 9\u201332.","journal-title":"Logique et Analyse"},{"key":"9692_CR8","unstructured":"Brady, R. T. (Ed.) (2003). Relevant logics and their rivals, vol II. Aldershot: Ashgate."},{"key":"9692_CR9","unstructured":"Brady, R. T. (2006). Universal logic CSLI. Stanford, CA."},{"key":"9692_CR10","doi-asserted-by":"publisher","first-page":"115","DOI":"10.12775\/LLP.2000.008","volume":"8","author":"W Carnielli","year":"2000","unstructured":"Carnielli, W., Marcos, J., & Amo, S. D. E. (2000). Formal inconsistency and evolutionary databases. Logic and Logical Philosophy, 8, 115\u2013152. https:\/\/doi.org\/10.12775\/LLP.2000.008.","journal-title":"Logic and Logical Philosophy"},{"issue":"4","key":"9692_CR11","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1007\/s11225-014-9595-7","volume":"103","author":"M De","year":"2015","unstructured":"De, M., & Omori, H. (2015). Classical negation and expansions of Belnap\u2013Dunn logic. Studia Logica, 103(4), 825\u2013851. https:\/\/doi.org\/10.1007\/s11225-014-9595-7.","journal-title":"Studia Logica"},{"key":"9692_CR12","unstructured":"Dunn, J. M. (1966). The algebra of intensional logics. In PhD Thesis. University of Pittsburgh. UMI, Ann Arbor, MI. (Published as vol. 2 in the Logic PhDs series by College Publications, London, UK, 2019."},{"key":"9692_CR13","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/BF00373152","volume":"29","author":"JM Dunn","year":"1976","unstructured":"Dunn, J. M. (1976). Intuitive semantics for first-degree entailments and \u201ccoupled trees.\u201d Philosophical Studies, 29, 149\u2013168.","journal-title":"Philosophical Studies"},{"issue":"1","key":"9692_CR14","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1023\/A:1026740726955","volume":"66","author":"JM Dunn","year":"2000","unstructured":"Dunn, J. M. (2000). Partiality and its dual. Studia Logica, 66(1), 5\u201340.  https:\/\/doi.org\/10.1023\/A:1026740726955.","journal-title":"Studia Logica"},{"key":"9692_CR15","unstructured":"Gonz\u00e1lez, C. (2011). MaTest, v. 1.3.2a. https:\/\/sites.google.com\/site\/sefusmendez\/matest. Last Accessed 29 Apr 2022."},{"key":"9692_CR16","doi-asserted-by":"publisher","unstructured":"Kamide, N., & Omori, H. (2017). An extended first-order Belnap-Dunn logic with classical negation. In A. Baltag, J. Seligman, & Yamada T. (Eds.) Logic, Rationality, and Interaction: Lecture Notes in Computer Science, vol. 10455 (pp. 79\u201393). Springer, Berlin, Heidelbereg. https:\/\/doi.org\/10.1007\/978-3-662-55665-8_6.","DOI":"10.1007\/978-3-662-55665-8_6"},{"key":"9692_CR17","doi-asserted-by":"publisher","first-page":"81","DOI":"10.12775\/LLP.1999.006","volume":"7","author":"AS Karpenko","year":"1999","unstructured":"Karpenko, A. S. (1999). Ja\u015bkowski\u2019s criterion and three-valued paraconsistent logics. Logic and Logical Philosophy, 7, 81\u201386. https:\/\/doi.org\/10.12775\/LLP.1999.006.","journal-title":"Logic and Logical Philosophy"},{"key":"9692_CR18","unstructured":"L\u00f3pez Velasco, S. M. (2020). Estudio sobre las variantes de la matriz tetravaluada de Brady que verifican la l\u00f3gica b\u00e1sica de Routley y Meyer. PhD thesis Universidad de Salamanca, Salamanca, Spain."},{"issue":"1","key":"9692_CR19","doi-asserted-by":"publisher","first-page":"29","DOI":"10.12775\/LLP.2021.004","volume":"31","author":"SM L\u00f3pez","year":"2022","unstructured":"L\u00f3pez, S. M. (2022). Belnap-Dunn semantics for the variants of BN4 and E4 which contain Routley and Meyer\u2019s logic B. Logic and Logical Philosophy, 31(1), 29\u201356. https:\/\/doi.org\/10.12775\/LLP.2021.004.","journal-title":"Logic and Logical Philosophy"},{"issue":"23","key":"9692_CR20","first-page":"30","volume":"III","author":"J \u0141ukasiewicz","year":"1930","unstructured":"\u0141ukasiewicz, J., & Tarski, A. (1930). Untersuchungen \u00fcber den Aussagenkalk\u00fcl. Comptes Rendus des S\u00e9ances de la Soci\u00e9t\u00e9 des Sciences et des Lettres de Varsovie, Classe, III(23), 30\u201350.","journal-title":"Comptes Rendus des S\u00e9ances de la Soci\u00e9t\u00e9 des Sciences et des Lettres de Varsovie, Classe"},{"issue":"4","key":"9692_CR21","doi-asserted-by":"publisher","first-page":"501\u2014522","DOI":"10.1007\/s11787-015-0130-z","volume":"9","author":"JM M\u00e9ndez","year":"2015","unstructured":"M\u00e9ndez, J. M., & Robles, G. (2015). A strong and rich 4-valued modal logic without \u0141ukasiewicz-type paradoxes. Logica Universalis, 9(4), 501\u2014522. https:\/\/doi.org\/10.1007\/s11787-015-0130-z.","journal-title":"Logica Universalis"},{"issue":"1","key":"9692_CR22","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1080\/11663081.2016.1153930","volume":"26","author":"JM M\u00e9ndez","year":"2016","unstructured":"M\u00e9ndez, J. M., & Robles, G. (2016a). The logic determined by Smiley\u2019s matrix for Anderson and Belnap\u2019s first-degree entailment logic. Journal of Applied Non-Classical Logics, 26(1), 47\u201368. https:\/\/doi.org\/10.1080\/11663081.2016.1153930.","journal-title":"Journal of Applied Non-Classical Logics"},{"issue":"2","key":"9692_CR23","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/s10849-016-9237-8","volume":"25","author":"JM M\u00e9ndez","year":"2016","unstructured":"M\u00e9ndez, J. M., & Robles, G. (2016b). Strengthening Brady\u2019s paraconsistent 4-valued logic BN4 with truth-functional modal operators. Journal of Logic Language and Information, 25(2), 163\u2013189. https:\/\/doi.org\/10.1007\/s10849-016-9237-8.","journal-title":"Journal of Logic Language and Information"},{"key":"9692_CR24","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1007\/BF02429841","volume":"43","author":"RK Meyer","year":"1984","unstructured":"Meyer, R. K., Giambrone, S., & Brady, R. T. (1984). Where gamma fails. Studia Logica, 43, 247\u2013256. https:\/\/doi.org\/10.1007\/BF02429841.","journal-title":"Studia Logica"},{"issue":"6","key":"9692_CR25","doi-asserted-by":"publisher","first-page":"1021","DOI":"10.1007\/s11225-017-9748-6","volume":"10","author":"H Omori","year":"2017","unstructured":"Omori, H., & Wansing, H. (2017). 40 years of FDE: An introductory overview. Studia Logica, 10(6), 1021\u20131049. https:\/\/doi.org\/10.1007\/s11225-017-9748-6.","journal-title":"Studia Logica"},{"key":"9692_CR26","doi-asserted-by":"publisher","unstructured":"Omori, H. & Wansing H. (Eds.) (2019). New essays on Belnap-Dunn logic. Synthese Library (Stud ies in Epistemology, Logic, Methodology, and Philosophy of Science). (Vol. 418). Cham: Springer. https:\/\/doi.org\/10.1007\/978-3-030-31136-0.","DOI":"10.1007\/978-3-030-31136-0"},{"issue":"4","key":"9692_CR27","doi-asserted-by":"publisher","first-page":"1452","DOI":"10.1007\/s40879-019-00344-5","volume":"6","author":"Y Petrukhin","year":"2020","unstructured":"Petrukhin, Y., & Shangin, V. (2020). Correspondence analysis and automated proof-searching for first degree entailment. European Journal of Mathematics, 6(4), 1452\u20131495. https:\/\/doi.org\/10.1007\/s40879-019-00344-5.","journal-title":"European Journal of Mathematics"},{"issue":"5","key":"9692_CR28","doi-asserted-by":"publisher","first-page":"838","DOI":"10.1093\/jigpal\/jzw011","volume":"24","author":"G Robles","year":"2016","unstructured":"Robles, G., & M\u00e9ndez, J. M. (2016). A companion to Brady\u2019s 4-valued relevant logic BN4: The 4-valued logic of entailment E4. Logic Journal of the IGPL, 24 (5), 838\u2013858. https:\/\/doi.org\/10.1093\/jigpal\/jzw011.","journal-title":"Logic Journal of the IGPL"},{"issue":"1","key":"9692_CR29","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1080\/11663081.2018.1534487","volume":"29","author":"G Robles","year":"2019","unstructured":"Robles, G., & M\u00e9ndez, J. M. (2019). Belnap-Dunn semantics for natural implicative expansions of Kleene\u2019s strong three-valued matrix with two designated values. Journal of Applied Non-Classical Logics, 29(1), 37\u201363. https:\/\/doi.org\/10.1080\/11663081.2018.1534487.","journal-title":"Journal of Applied Non-Classical Logics"},{"issue":"3","key":"9692_CR30","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s10849-019-09306-2","volume":"29","author":"G Robles","year":"2020","unstructured":"Robles, G., & M\u00e9ndez, J. M. (2020). The class of all natural implicative expansions of Kleene\u2019s strong logic functionally equivalent to \u0141ukasiewicz\u2019s 3-valued logic \u01413. Journal of Logic Language and Information, 29(3), 349\u2013374. https:\/\/doi.org\/10.1007\/s10849-019-09306-2.","journal-title":"Journal of Logic Language and Information"},{"issue":"1","key":"9692_CR31","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1093\/jigpal\/jzaa028","volume":"30","author":"G Robles","year":"2022","unstructured":"Robles, G., & M\u00e9ndez, J. M. (2022a). A remark on functional completeness of binary expansions of Kleene\u2019s strong 3-valued logic. Logic Journal of the IGPL, 30(1), 21\u201333. https:\/\/doi.org\/10.1093\/jigpal\/jzaa028.","journal-title":"Logic Journal of the IGPL"},{"issue":"10","key":"9692_CR32","first-page":"2435","volume":"8","author":"G Robles","year":"2022","unstructured":"Robles, G., & M\u00e9ndez J. M. (2022b). A 2 set-up Routley-Meyer semantics for the 4-valued logic P\u01414. Journal of Applied Logics \u2014 IfCoLog Journal of Logics and their Applications, 8(10), 2435\u20132446.","journal-title":"Journal of Applied Logics \u2014 IfCoLog Journal of Logics and their Applications"},{"key":"9692_CR33","doi-asserted-by":"publisher","unstructured":"Robles, G., & M\u00e9ndez, J. M. (2022c). A note on functional relations in a certain class of implicative expansions of FDE related to Brady\u2019s 4-valued logic BN4. Logic Journal of the IGPL, jzac045, 1\u20138. https:\/\/doi.org\/10.1093\/jigpal\/jzac045.","DOI":"10.1093\/jigpal\/jzac045"},{"key":"9692_CR34","unstructured":"Routley, R., Meyer, R. K., Plumwood, V., & Brady R. T. (1982). Relevant logics and their rivals, vol. 1. Atascadero, CA, Ridgeview Publishing Co."},{"key":"9692_CR35","doi-asserted-by":"publisher","unstructured":"Slaney, J. (2005). Relevant logic and paraconsistency. In L. Bertossi, A. Hunter, & T. Schaub (Eds.) Inconsistency Tolerance, Lecture Notes in Computer Science, vol. 3300 (pp. 270\u2013293). Berlin: Springer, DOI https:\/\/doi.org\/10.1007\/978-3-540-30597-2_9, (to appear in print).","DOI":"10.1007\/978-3-540-30597-2_9"}],"container-title":["Journal of Philosophical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10992-022-09692-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10992-022-09692-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10992-022-09692-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T15:04:35Z","timestamp":1684163075000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10992-022-09692-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,9]]},"references-count":35,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["9692"],"URL":"https:\/\/doi.org\/10.1007\/s10992-022-09692-2","relation":{},"ISSN":["0022-3611","1573-0433"],"issn-type":[{"value":"0022-3611","type":"print"},{"value":"1573-0433","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,9]]},"assertion":[{"value":"5 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 November 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 January 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare there are no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Competing interests"}},{"value":"The authors declare that they have no financial interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Financial interests"}},{"value":"The authors declare that they have no financial interests. The authors declare that they have no non-financial interests.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Non-financial interests"}}]}}