{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:07:37Z","timestamp":1760058457001,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T00:00:00Z","timestamp":1743638400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article focuses on the relationships between fuzzy logic and classical logic properties using fuzzy t-norms, t-conorms, and fuzzy implications. It aims to contribute to fuzzy set theory by extending the Boolean laws in classical logic to fuzzy logic. We determine the necessary and sufficient conditions for validating the generalizations of the proposed properties from classical to fuzzy logic. Additionally, we provide examples demonstrating the practical applicability of this approach and its advantages over conventional methodologies, reinforcing its effectiveness.<\/jats:p>","DOI":"10.3390\/sym17040548","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T05:51:26Z","timestamp":1743659486000},"page":"548","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some Properties of Boolean-like Laws in Fuzzy Logic"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4615-6863","authenticated-orcid":false,"given":"Sevilay","family":"Demir Sa\u011flam","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Istanbul University, Istanbul 34134, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2258-8266","authenticated-orcid":false,"given":"G\u00fcl","family":"Karadeniz G\u00f6zeri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Istanbul University, Istanbul 34134, T\u00fcrkiye"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,3]]},"reference":[{"key":"ref_1","first-page":"51","article-title":"Bemerkungen zu mehrwertigen Systemen des Aussagenkalk\u00fcls","volume":"23","author":"Lukasiewicz","year":"1930","journal-title":"Comptes Rendus S\u00e9ances Soc. Des Sci. Lettres Vars. Cl. III"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1007\/s00153-004-0267-6","article-title":"Sufficient triangular norms in many-valued logics with standard negation","volume":"44","author":"Butnariu","year":"2005","journal-title":"Arch. Math. Log."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1073\/pnas.28.12.535","article-title":"Statistical metrics","volume":"8","author":"Menger","year":"1942","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"ref_4","first-page":"2092","article-title":"Espaces m\u00e9triques al\u00e9atoires","volume":"247","author":"Schweizer","year":"1958","journal-title":"C. R. Acad. Sci. Paris S\u00e9rie A"},{"key":"ref_5","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":"Inform. Control"},{"key":"ref_6","unstructured":"Klement, E.P. (1980, January 15\u201320). Triangular norms for fuzzy sets. Proceedings of the Second International Seminar on Fuzzy Set Theory, Linz, Austria."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"313","DOI":"10.2140\/pjm.1960.10.313","article-title":"Statistical metric spaces","volume":"10","author":"Schweizer","year":"1960","journal-title":"Pac. J. Math."},{"key":"ref_8","unstructured":"Alsina, C., and Trillas, E. (2005, January 7\u20139). On iterative Boolean-like laws of fuzzy sets. Proceedings of the 4th Conference Fuzzy Logic and Technology, Barcelona, Spain."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1002\/(SICI)1098-111X(200007)15:7<647::AID-INT5>3.0.CO;2-T","article-title":"When QM-operators are implication functions and conditional fuzzy relations","volume":"15","author":"Trillas","year":"2000","journal-title":"Int. J. Intell. 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Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/91.660804","article-title":"Combinatorial rule explosion eliminated by a fuzzy rule configuration","volume":"6","author":"Combs","year":"1998","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1109\/91.983281","article-title":"On the law (p \u2227 q) \u21d2 r \u2261 (p \u21d2 r) \u2228 (q \u21d2 r) in fuzzy logic","volume":"10","author":"Trillas","year":"2002","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_15","first-page":"647","article-title":"Distributivity of ordinal sum implications over overlap and grouping functions","volume":"57","author":"Pan","year":"2021","journal-title":"Kybernetika"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1836","DOI":"10.1016\/j.fss.2007.11.015","article-title":"(S, N)-and R-implications: A state-of-the-art survey","volume":"159","author":"Jayaram","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_17","unstructured":"Baczy\u0144ski, M., and Jayaram, B. (2008). Fuzzy Implications, Springer."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Trillas, E., and Valverde, L. (1993). On implication and indistinguishability in the setting of fuzzy logic. Readings in Fuzzy Sets for Intelligent Systems, Elsevier.","DOI":"10.1016\/B978-1-4832-1450-4.50013-4"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Chen, G., and Tat Pham, T. (2000). Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems, CRC Press.","DOI":"10.1201\/9781420039818"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Pedraza, T., Ramos-Can\u00f3s, J., and Rodr\u00edguez-L\u00f3pez, J. (2021). Aggregation of Weak Fuzzy Norms. Symmetry, 13.","DOI":"10.3390\/sym13101908"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Sarfraz, M., Ullah, K., Akram, M., Pamucar, D., and Bo\u017eani\u0107, D. (2022). Prioritized Aggregation Operators for Intuitionistic Fuzzy Information Based on Aczel\u2013Alsina T-Norm and T-Conorm and Their Applications in Group Decision-Making. Symmetry, 14.","DOI":"10.3390\/sym14122655"},{"key":"ref_22","first-page":"93","article-title":"The Uniqueness of the Decomposition of Distributive Lattices into Sums of Boolean Lattices","volume":"31","author":"Grygiel","year":"1997","journal-title":"Rep. Math. Log."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Klement, E.P., Mesiar, R., and Pap, E. (2000). Triangular Norms, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-9540-7"},{"key":"ref_24","first-page":"125","article-title":"Categorical Abstract Algebraic Logic: Coordinatization is Algebraization","volume":"47","author":"Voutsadakis","year":"2012","journal-title":"Rep. Math. Log."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Fodor, J., and Roubens, M. (1994). Fuzzy Preference Modeling and Multicriteria Decision Support, Kluwer.","DOI":"10.1007\/978-94-017-1648-2"},{"key":"ref_26","unstructured":"Gottwald, S. (2001). A Treatise on Many-Valued Logics, Research Studies Press."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1142\/S0218488514500093","article-title":"On the Boolean-like Law I(x, I(y, x)) = 1","volume":"22","author":"Cruz","year":"2014","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Bustince, H., Fernandez, J., Mesiar, R., and Calvo, T. (2013). Fuzzy Implication Classes Satisfying a Boolean-Like Law. Aggregation Functions in Theory and in Practice, Springer. 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