{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T17:50:00Z","timestamp":1769536200417,"version":"3.49.0"},"reference-count":26,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T00:00:00Z","timestamp":1768953600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","award":["PID2022-139248NB-I00"],"award-info":[{"award-number":["PID2022-139248NB-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study focuses on the Modus Tollens (MT) property induced by discrete uninorms. Specifically, we identify the set of necessary and sufficient criteria for a discrete implication function to comply with this logical property. This rule of inference is studied by using discrete residual implication functions derived from uninorms of two of the most important families of these discrete operators (Umin, idempotents), exploring which properties these operators must satisfy, as well as providing some characterizations of the Modus Tollens in this domain of definition. Our findings contribute to a deeper understanding of reasoning mechanisms in fuzzy logic, particularly in discrete settings.<\/jats:p>","DOI":"10.3390\/axioms15010077","type":"journal-article","created":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T13:59:54Z","timestamp":1769003994000},"page":"77","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modus Tollens in the Setting of Discrete Uninorms"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4434-3203","authenticated-orcid":false,"given":"Isabel","family":"Aguil\u00f3","sequence":"first","affiliation":[{"name":"Soft Computing, Image Processing and Aggregation (SCOPIA) Research Group, Department of Mathematics and Computer Science, University of the Balearic Islands, 07122 Palma, Spain"},{"name":"Health Research Institute of the Balearic Islands (IdISBa), 07120 Palma, Spain"},{"name":"Artificial Intelligence Research Institute of the Balearic Islands (IAIB), 07122 Palma, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2929-1625","authenticated-orcid":false,"given":"Pilar","family":"Fuster-Parra","sequence":"additional","affiliation":[{"name":"Soft Computing, Image Processing and Aggregation (SCOPIA) Research Group, Department of Mathematics and Computer Science, University of the Balearic Islands, 07122 Palma, Spain"},{"name":"Health Research Institute of the Balearic Islands (IdISBa), 07120 Palma, Spain"},{"name":"Artificial Intelligence Research Institute of the Balearic Islands (IAIB), 07122 Palma, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0878-8301","authenticated-orcid":false,"given":"Juan Vicente","family":"Riera","sequence":"additional","affiliation":[{"name":"Soft Computing, Image Processing and Aggregation (SCOPIA) Research Group, Department of Mathematics and Computer Science, University of the Balearic Islands, 07122 Palma, Spain"},{"name":"Health Research Institute of the Balearic Islands (IdISBa), 07120 Palma, Spain"},{"name":"Artificial Intelligence Research Institute of the Balearic Islands (IAIB), 07122 Palma, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy Sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. 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The Relationship between Fuzzy Reasoning Methods Based on Intuitionistic Fuzzy Sets and Interval-Valued Fuzzy Sets. Axioms, 11.","DOI":"10.3390\/axioms11080419"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"120212","DOI":"10.1016\/j.ins.2024.120212","article-title":"Fuzzy inference system with interpretable fuzzy rules: Advancing explainable artificial intelligence for disease diagnosis\u2014A comprehensive review","volume":"662","author":"Cao","year":"2024","journal-title":"Inf. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"108474","DOI":"10.1016\/j.fss.2023.01.009","article-title":"The f-index of inclusion as optimal adjoint pair for fuzzy modus ponens","volume":"466","author":"Madrid","year":"2023","journal-title":"Fuzzy Sets Syst."},{"key":"ref_13","unstructured":"Kawaguchi, M.F., Ohno, T., Tachibana, H., Miyakoshi, M., and Da-Te, T. (1996, January 11). Modus ponens and modus tollens under the compositional rule of inference with triangular norms. Proceedings of the IEEE 5th International Fuzzy Systems Conference, New Orleans, LA, USA."},{"key":"ref_14","first-page":"259","article-title":"On MPT-implication functions for Fuzzy Logic","volume":"98","author":"Trillas","year":"2004","journal-title":"Rev. Real Acad. Cienc. Ser. A Mat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.ijar.2020.10.003","article-title":"Modus Tollens with respect to uninorms: U-Modus Tollens","volume":"127","author":"Riera","year":"2020","journal-title":"Int. J. Approx. Reason."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Mas, M., Monreal, J., Monserrat, M., Riera, J.V., and Torrens, J. (2016). Modus Tollens on Fuzzy Implication Functions Derived from Uninorms. Fuzzy Logic and Information Fusion, Springer.","DOI":"10.1007\/978-3-319-30421-2_5"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.knosys.2014.11.001","article-title":"On multi-granular fuzzy linguistic modeling in group decision making problems: A systematic review and future trends","volume":"74","year":"2015","journal-title":"Knowl.-Based Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S021848850900570X","article-title":"Idempotent uninorms on finite ordinal scales","volume":"17","author":"Baets","year":"2009","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1002\/(SICI)1098-111X(199909)14:9<909::AID-INT4>3.0.CO;2-B","article-title":"t-Operators and uninorms on a finite totally ordered sets","volume":"14","author":"Mas","year":"1999","journal-title":"Int. J. Intell. Syst."},{"key":"ref_20","first-page":"3","article-title":"S-implications and R-implications on a finite chain","volume":"40","author":"Mas","year":"2004","journal-title":"Kybernetika"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.ijar.2005.05.001","article-title":"On two types of discrete implications","volume":"40","author":"Mas","year":"2005","journal-title":"Int. J. Approx. Reason."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1016\/j.ijar.2008.04.002","article-title":"Modus ponens and modus tollens in discrete implications","volume":"49","author":"Mas","year":"2008","journal-title":"Int. J. Approx. Reason."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.fss.2014.10.020","article-title":"A characterization of discrete uninorms having smooth underlying operators","volume":"359","author":"Ruiz","year":"2015","journal-title":"Fuzzy Sets Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"109023","DOI":"10.1016\/j.fss.2024.109023","article-title":"A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results","volume":"490","author":"Munar","year":"2024","journal-title":"Fuzzy Sets Syst."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Magdalena, L., Verdegay, J., and Esteva, F. (2015). Residual Implications from Discrete Uninorms. A Characterization. Enric Trillas: A Passion for Fuzzy Sets, Springer.","DOI":"10.1007\/978-3-319-16235-5"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Klement, E.P., and Mesiar, R. (2005). 7\u2014Triangular norms on discrete settings. Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, Elsevier Science.","DOI":"10.1016\/B978-044451814-9\/50004-5"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/77\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T05:11:11Z","timestamp":1769490671000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/77"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,21]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["axioms15010077"],"URL":"https:\/\/doi.org\/10.3390\/axioms15010077","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,21]]}}}