{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T01:02:23Z","timestamp":1781139743006,"version":"3.54.1"},"reference-count":24,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017596","name":"Natural Science Basic Research Program of Shaanxi Province","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100017596","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Approximate Reasoning"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1016\/j.ijar.2026.109742","type":"journal-article","created":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T15:42:18Z","timestamp":1780674138000},"page":"109742","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Randomized truth degrees and inference models in multi-valued modal logics"],"prefix":"10.1016","volume":"197","author":[{"given":"Jingyao","family":"Duan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhaoxin","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dechao","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"9","key":"10.1016\/j.ijar.2026.109742_bib0001","doi-asserted-by":"crossref","first-page":"1106","DOI":"10.1360\/02ys9122","article-title":"Theory of truth degrees of propositions in two-valued logic","volume":"45","author":"Wang","year":"2002","journal-title":"Sci. China A Math."},{"issue":"6","key":"10.1016\/j.ijar.2026.109742_bib0002","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1360\/122005-47","article-title":"Theory of truth degrees of formulas in Lukasiewicz n-valued propositional logic and a limit theorem","volume":"48","author":"Li","year":"2005","journal-title":"Sci. China F Inf. Sci."},{"issue":"3","key":"10.1016\/j.ijar.2026.109742_bib0003","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.ins.2008.09.008","article-title":"Quantitative logic","volume":"179","author":"Wang","year":"2008","journal-title":"Inf. Sci."},{"issue":"9","key":"10.1016\/j.ijar.2026.109742_bib0004","doi-asserted-by":"crossref","first-page":"1843","DOI":"10.1007\/s11432-011-4268-x","article-title":"Borel probabilistic and quantitative logic","volume":"54","author":"Zhou","year":"2011","journal-title":"Sci. China Inf. Sci."},{"key":"10.1016\/j.ijar.2026.109742_bib0005","unstructured":"Y.M. Li, G.J. Wang, B. Zhao, Quantitative Logic and Soft Computing- Proceedings of The Ql & SC 2012, World Scientific Publishing Company, 2012."},{"issue":"1","key":"10.1016\/j.ijar.2026.109742_bib0006","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1360\/jos180033","article-title":"Theory of a-truth degrees in n-valued G\u00f6del propositional logic","volume":"18","author":"Li","year":"2007","journal-title":"J. Softw."},{"issue":"22","key":"10.1016\/j.ijar.2026.109742_bib0007","doi-asserted-by":"crossref","first-page":"2970","DOI":"10.1016\/j.fss.2008.04.005","article-title":"Three and two-valued Lukasiewicz theories in the formal deductive system L* (NM-logic)","volume":"159","author":"Zhou","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"10.1016\/j.ijar.2026.109742_bib0008","series-title":"Modal Logic","author":"Blackburn","year":"2001"},{"issue":"2","key":"10.1016\/j.ijar.2026.109742_bib0009","first-page":"1","article-title":"Valuation semantics for S4","volume":"113","author":"Lopari\u0107","year":"2024","journal-title":"Stud. Log."},{"key":"10.1016\/j.ijar.2026.109742_bib0010","doi-asserted-by":"crossref","DOI":"10.1016\/j.ijar.2024.109167","article-title":"A probabilistic modal logic for context-aware trust based on evidence","volume":"169","author":"Aldini","year":"2024","journal-title":"Int. J. Approx. Reason."},{"issue":"2-3","key":"10.1016\/j.ijar.2026.109742_bib0011","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1080\/11663081.2024.2336385","article-title":"A study of modal logic with semantics based on rough set theory","volume":"34","author":"Khan","year":"2024","journal-title":"J. Appl. Non Class. Log."},{"key":"10.1016\/j.ijar.2026.109742_bib0012","doi-asserted-by":"crossref","DOI":"10.1016\/j.ijar.2024.109305","article-title":"Rough sets, modal logic and approximate reasoning","volume":"176","author":"Chakraborty","year":"2025","journal-title":"Int. J. Approx. Reason."},{"issue":"1","key":"10.1016\/j.ijar.2026.109742_bib0013","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1007\/s11432-009-0008-x","article-title":"Theory of (n) truth degrees of formulas in modal logic and a consistency theorem","volume":"52","author":"Wang","year":"2009","journal-title":"Sci. China F Inf. Sci."},{"issue":"12","key":"10.1016\/j.ijar.2026.109742_bib0014","doi-asserted-by":"crossref","first-page":"3074","DOI":"10.3724\/SP.J.1001.2012.04212","article-title":"Quantitative method for multi-value modal logics","volume":"23","author":"Shi","year":"2012","journal-title":"J. Softw."},{"issue":"03","key":"10.1016\/j.ijar.2026.109742_bib0015","first-page":"40","article-title":"Theory of truth degree of modal logic","volume":"49","author":"Li","year":"2013","journal-title":"Comput. Eng. Appl."},{"key":"10.1016\/j.ijar.2026.109742_bib0016","series-title":"Quantitative Research on Modal Logic and Its Application in Model Checking","author":"Shi","year":"2014"},{"issue":"03","key":"10.1016\/j.ijar.2026.109742_bib0017","first-page":"1","article-title":"Theory of probability truth degree for modal logic formulas","volume":"42","author":"Li","year":"2022","journal-title":"J. Baoji Univ. Arts Sci."},{"issue":"6","key":"10.1016\/j.ijar.2026.109742_bib0018","doi-asserted-by":"crossref","first-page":"1635","DOI":"10.1111\/theo.12371","article-title":"The uniqueness of necessary truth and the status of S4 and S5","volume":"87","author":"Marco","year":"2021","journal-title":"Theoria"},{"issue":"4","key":"10.1016\/j.ijar.2026.109742_bib0019","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1080\/11663081.2024.2366752","article-title":"Truth diagrams for some non-classical and modal logics","volume":"34","author":"Ba\u015fkent","year":"2024","journal-title":"J. Appl. Non Class. Log."},{"issue":"7","key":"10.1016\/j.ijar.2026.109742_bib0020","doi-asserted-by":"crossref","first-page":"341","DOI":"10.3390\/axioms11070341","article-title":"Graded many-valued modal logic and its graded rough truth","volume":"11","author":"Li","year":"2022","journal-title":"Axioms"},{"issue":"6","key":"10.1016\/j.ijar.2026.109742_bib0021","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1007\/s11432-007-0067-9","article-title":"Randomization of classical inference patterns and its application","volume":"50","author":"Wang","year":"2007","journal-title":"Sci. China F Inf. Sci."},{"issue":"03","key":"10.1016\/j.ijar.2026.109742_bib0022","first-page":"21","article-title":"Randomization of classical inference patterns and its application(II)","author":"Shi","year":"2008","journal-title":"Fuzzy Syst. Math."},{"issue":"02","key":"10.1016\/j.ijar.2026.109742_bib0023","first-page":"209","article-title":"The randomized truth degree of formulas an approximate reasoning in n-valued lukasiewicz propositional logic system","volume":"35","author":"Cui","year":"2012","journal-title":"Acta Math. Appl. Sin."},{"key":"10.1016\/j.ijar.2026.109742_bib0024","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1016\/j.ijar.2013.09.023","article-title":"On robustness of the full implication triple I inference method with respect to finner measurements","volume":"55","author":"Wang","year":"2014","journal-title":"Int. J. Approx. Reason."}],"container-title":["International Journal of Approximate Reasoning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0888613X26001179?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0888613X26001179?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T00:37:13Z","timestamp":1781138233000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0888613X26001179"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10]]},"references-count":24,"alternative-id":["S0888613X26001179"],"URL":"https:\/\/doi.org\/10.1016\/j.ijar.2026.109742","relation":{},"ISSN":["0888-613X"],"issn-type":[{"value":"0888-613X","type":"print"}],"subject":[],"published":{"date-parts":[[2026,10]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Randomized truth degrees and inference models in multi-valued modal logics","name":"articletitle","label":"Article Title"},{"value":"International Journal of Approximate Reasoning","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.ijar.2026.109742","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"109742"}}