{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T15:56:52Z","timestamp":1770825412480,"version":"3.50.1"},"reference-count":33,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2023,3,25]],"date-time":"2023-03-25T00:00:00Z","timestamp":1679702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,5,24]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The well-known logic first degree entailment logic (FDE), introduced by Belnap and Dunn, is defined with $\\wedge $, $\\vee $ and $\\sim $ as the sole primitive connectives. The aim of this paper is to establish the lattice formed by the class of all 4-valued C-extending implicative expansions of FDE verifying the axioms and rules of Routley and Meyer\u2019s basic logic B and its useful disjunctive extension B$^{\\textrm {d}}$. It is to be noted that Boolean negation (so, classical propositional logic) is definable in the strongest element in the said class.<\/jats:p>","DOI":"10.1093\/jigpal\/jzad005","type":"journal-article","created":{"date-parts":[[2023,3,25]],"date-time":"2023-03-25T15:16:48Z","timestamp":1679757408000},"page":"493-516","source":"Crossref","is-referenced-by-count":1,"title":["The lattice of all 4-valued implicative expansions of Belnap\u2013Dunn logic containing Routley and Meyer\u2019s basic logic B<i>d<\/i>"],"prefix":"10.1093","volume":"32","author":[{"given":"Gemma","family":"Robles","sequence":"first","affiliation":[{"name":"Departamento de Psicolog\u00eda , Sociolog\u00eda y Filosof\u00eda, Universidad de Le\u00f3n, Campus de Vegazana, s\/n, 24071, Le\u00f3n, Spain , gemma.robles@unileon.es"}]},{"given":"Jos\u00e9 M","family":"M\u00e9ndez","sequence":"additional","affiliation":[{"name":"Universidad de Salamanca , Edificio FES, Campus Unamuno, 37007, Salamanca, Spain , sefus@usal.es"}]}],"member":"286","published-online":{"date-parts":[[2023,3,25]]},"reference":[{"key":"2024061111212139600_ref1","article-title":"Entailment","volume-title":"The Logic of Relevance and Necessity","author":"Anderson","year":"1975"},{"key":"2024061111212139600_ref2","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1007\/978-94-010-1161-7_2","article-title":"A useful four-valued logic","volume-title":"Modern Uses of Multiple-Valued Logic","author":"Belnap Jr","year":"1977"},{"key":"2024061111212139600_ref3","first-page":"30","article-title":"How a computer should think","volume-title":"Contemporary Aspects of Philosophy","author":"Belnap Jr","year":"1977"},{"key":"2024061111212139600_ref4","first-page":"75","article-title":"EF4, EF4-M and EF4-\u0141: a companion to BN4 and two modal four-valued systems without strong \u0141ukasiewicz-type modal paradoxes","volume":"31","author":"Blanco","year":"2022","journal-title":"Logic and Logical Philosophy"},{"key":"2024061111212139600_ref5","first-page":"9","article-title":"Completeness proofs for the systems RM3 and BN4","volume":"25","author":"Brady","year":"1982","journal-title":"Logique et Analyse"},{"key":"2024061111212139600_ref6","volume-title":"Relevant Logics and Their Rivals","author":"Brady, ed","year":"2003"},{"key":"2024061111212139600_ref7","volume-title":"Universal Logic","author":"Brady","year":"2006"},{"key":"2024061111212139600_ref8","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1007\/s11225-014-9595-7","article-title":"Classical negation and expansions of Belnap\u2013Dunn logic","volume":"103","author":"De","year":"2015","journal-title":"Studia Logica"},{"key":"2024061111212139600_ref9","volume-title":"The Algebra of Intensional Logics","author":"Dunn","year":"1966"},{"key":"2024061111212139600_ref10","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/BF00373152","article-title":"Intuitive semantics for first-degree entailments and \u201ccoupled trees.\u201d","volume":"29","author":"Dunn","year":"1976","journal-title":"Philosophical Studies"},{"key":"2024061111212139600_ref11","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1026740726955","article-title":"Partiality and its dual","volume":"66","author":"Dunn","year":"2000","journal-title":"Studia Logica"},{"key":"2024061111212139600_ref12","article-title":"MaTest v.1.3.2a","author":"Gonz\u00e1lez","year":"2011"},{"key":"2024061111212139600_ref13","doi-asserted-by":"crossref","first-page":"51","DOI":"10.2307\/2271336","article-title":"What is strict implication","volume":"28","author":"Hacking","year":"1963","journal-title":"Journal of Symbolic Logic"},{"key":"2024061111212139600_ref14","volume-title":"Estudio Sobre las Variantes de la Matriz Tetravaluada de Brady que Verifican la L\u00f3gica B\u00e1sica de Routley y Meyer","author":"L\u00f3pez Velasco","year":"2020"},{"key":"2024061111212139600_ref15","doi-asserted-by":"crossref","first-page":"27","DOI":"10.18778\/0138-0680.2021.16","article-title":"Ternary relational semantics for the variants of BN4 and E4 which contain Routley and Meyer\u2019s logic B","volume":"51","author":"L\u00f3pez","year":"2021","journal-title":"Bulletin of the Section of Logic, Early View"},{"key":"2024061111212139600_ref16","first-page":"29","article-title":"Belnap\u2013Dunn semantics for the variants of BN4 and E4 which contain Routley and Meyer\u2019s logic B","volume":"31","author":"L\u00f3pez","year":"2022","journal-title":"Logic and Logical Philosophy"},{"key":"2024061111212139600_ref17","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1080\/11663081.2016.1153930","article-title":"The logic determined by Smiley\u2019s matrix for Anderson and Belnap\u2019s first-degree entailment logic","volume":"26","author":"M\u00e9ndez","year":"2016","journal-title":"Journal of Applied Non-Classical Logics"},{"key":"2024061111212139600_ref18","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s10849-016-9237-8","article-title":"Strengthening Brady\u2019s paraconsistent 4-valued logic BN4 with truth-functional modal operators","volume":"25","author":"M\u00e9ndez","year":"2016","journal-title":"Journal of Logic, Language and Information"},{"key":"2024061111212139600_ref19","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/BF02429841","article-title":"Where gamma fails","volume":"43","author":"Meyer","year":"1984","journal-title":"Studia Logica"},{"key":"2024061111212139600_ref20","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1007\/s11225-017-9748-6","article-title":"40 years of FDE: an introductory overview","volume":"105","author":"Omori","year":"2017","journal-title":"Studia Logica"},{"key":"2024061111212139600_ref21","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-31136-0","article-title":"New essays on Belnap\u2013Dunn logic","volume-title":"Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science)","author":"Omori","year":"2019"},{"key":"2024061111212139600_ref22","doi-asserted-by":"crossref","first-page":"1452","DOI":"10.1007\/s40879-019-00344-5","article-title":"Correspondence analysis and automated proof-searching for first degree entailment","volume":"6","author":"Petrukhin","year":"2020","journal-title":"European Journal of Mathematics"},{"key":"2024061111212139600_ref23","volume-title":"An Algebraic Approach to Non-Classical Logics","author":"Rasiowa","year":"1974"},{"key":"2024061111212139600_ref24","first-page":"173","article-title":"Relational semantics for the 4-valued relevant logics BN4 and E4","volume":"25","author":"Robles","year":"2016","journal-title":"Logic and Logical Philosophy"},{"key":"2024061111212139600_ref25","doi-asserted-by":"crossref","first-page":"93","DOI":"10.18778\/0138-0680.45.2.03","article-title":"A 2-set-up Routley\u2013Meyer semantics for the 4-valued relevant logic E4","volume":"45","author":"Robles","year":"2016","journal-title":"Bulletin of the Section of Logic"},{"key":"2024061111212139600_ref26","doi-asserted-by":"crossref","first-page":"838","DOI":"10.1093\/jigpal\/jzw011","article-title":"A companion to Brady\u2019s 4-valued relevant logic BN4: the 4-valued logic of entailment E4","volume":"24","author":"Robles","year":"2016","journal-title":"Logic Journal of the IGPL"},{"key":"2024061111212139600_ref27","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s10849-019-09306-2","article-title":"The class of all natural implicative expansions of Kleene\u2019s strong logic functionally equivalent to \u0141ukasiewicz\u2019s 3-valued logic \u01413","volume":"29","author":"Robles","year":"2020","journal-title":"Journal of Logic, Language and Information"},{"key":"2024061111212139600_ref28","first-page":"2435","article-title":"A 2 set-up Routley\u2013Meyer semantics for the 4-valued logic P\u01414","volume":"8","author":"Robles","year":"2022","journal-title":"Journal of Applied Logics\u2014IfCoLog Journal of Logics and their Applications"},{"key":"2024061111212139600_ref29","first-page":"1","article-title":"A note on functional relations in a certain class of implicative expansions of FDE related to Brady\u2019s 4-valued logic BN4","volume":"jzac045","author":"Robles","year":"2022","journal-title":"Logic Journal of the IGPL"},{"key":"2024061111212139600_ref30","volume-title":"Relevant Logics and Their Rivals","author":"Routley","year":"1982"},{"key":"2024061111212139600_ref31","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1007\/978-3-540-30597-2_9","article-title":"Relevant logic and paraconsistency","volume-title":"Inconsistency Tolerance","author":"Slaney","year":"2005"},{"key":"2024061111212139600_ref32","doi-asserted-by":"crossref","first-page":"138","DOI":"10.21146\/2074-1472-2015-21-1-138-143","article-title":"Natural implication and Modus Ponens principle","volume":"21","author":"Tomova","year":"2015","journal-title":"Logical Investigations"},{"key":"2024061111212139600_ref33","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-015-6942-2","volume-title":"Theory of Logical Calculi: Basic Theory of Consequence Operations","author":"W\u00f3jcicki","year":"1988"}],"container-title":["Logic Journal of the IGPL"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jigpal\/article-pdf\/32\/3\/493\/58195742\/jzad005.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jigpal\/article-pdf\/32\/3\/493\/58195742\/jzad005.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T21:40:37Z","timestamp":1718142037000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jigpal\/article\/32\/3\/493\/7079130"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,25]]},"references-count":33,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3,25]]},"published-print":{"date-parts":[[2024,5,24]]}},"URL":"https:\/\/doi.org\/10.1093\/jigpal\/jzad005","relation":{},"ISSN":["1367-0751","1368-9894"],"issn-type":[{"value":"1367-0751","type":"print"},{"value":"1368-9894","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2024,6]]},"published":{"date-parts":[[2023,3,25]]}}}