New Fuzzy Modus Ponens and Tollens based on Least Common Multiple

New Fuzzy Modus Ponens and Tollens based on Least Common Multiple | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article New Fuzzy Modus Ponens and Tollens based on Least Common Multiple Kang-Song Jo, Son-Il Kwak, Chol-Jun Hwang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7267543/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 16 You are reading this latest preprint version Abstract The fuzzy approximate reasoning is one of the important research branches in the uncertainty inference of Artificial Intelligence (AI) and Computational Intelligence (CI). This fuzzy approximate reasoning consists of two parts, i.e., fuzzy modus ponens (FMP) and fuzzy modus tollens(FMT). FMP and FMT have an important significance in modeling of the human thinking process, design and development of a lot of several intelligence systems and etc. In this paper we proposed a novel original method of fuzzy approximate reasoning that can open a new direction of research in the uncertainty inference of AI and CI. Firstly, our proposed method is based on distance measure, concretely, an extended distance measure by the least common multiple(LCM). The proposed fuzzy approximate reasoning method LCM based on a least common multiple is an inference one for the SISO fuzzy system with discrete fuzzy set vectors of equal or different dimensions between the antecedent and consequent of fuzzy rule. We call it LCM method. LCM method is consisted of two parts, i.e., FMP–LCM, and FMT–LCM. Secondly, in this paper we proposed and proved four theorems with respect to the information loss. In other words we pointed out that our proposed method LCM has no information loss, and then the previous methods, i.e., Compositional Rule of Inference(CRI), Triple Implication Principle(TIP), Quintuple Implication Principle(QIP) and Approximate Analogical Reasoning Scheme(AARS) have some information loss compared with our proposed method. Thirdly, we compared the reductive properties for the five fuzzy reasoning methods, i.e., CRI, TIP, QIP, AARS, and our proposed LCM with respect to FMP and FMT. The theoretical and experimental results highlight that the proposed fuzzy approximate reasoning method LCM is comparatively more clear and effective, and in accordance with human thinking than the previous fuzzy reasoning methods. Fourthly, in this paper we proposed and proved six theorems with respect to fuzzy controllability of the several fuzzy reasoning methods based on the fuzzy relation and our proposed method based on LCM. We pointed out our proposed method LCM has not only higher reductive property but also higher fuzzy controllability than the previous fuzzy reasoning methods. SISO fuzzy system Fuzzy Approximate Reasoning Least Common Multiple Fuzzy Modus Ponens Fuzzy Modus Tollens Reductive Property Fuzzy Controllability Information Loss Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 01 Oct, 2025 Reviews received at journal 28 Aug, 2025 Reviews received at journal 21 Aug, 2025 Reviews received at journal 08 Aug, 2025 Reviewers agreed at journal 07 Aug, 2025 Reviewers agreed at journal 07 Aug, 2025 Reviews received at journal 07 Aug, 2025 Reviews received at journal 06 Aug, 2025 Reviewers agreed at journal 06 Aug, 2025 Reviewers agreed at journal 06 Aug, 2025 Reviewers agreed at journal 05 Aug, 2025 Reviewers agreed at journal 05 Aug, 2025 Reviewers invited by journal 05 Aug, 2025 Editor assigned by journal 03 Aug, 2025 Submission checks completed at journal 02 Aug, 2025 First submitted to journal 01 Aug, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7267543","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":495776428,"identity":"07a22e84-41bd-4d47-9a1f-69f63a033ce2","order_by":0,"name":"Kang-Song Jo","email":"","orcid":"","institution":"Kim Il Sung University","correspondingAuthor":false,"prefix":"","firstName":"Kang-Song","middleName":"","lastName":"Jo","suffix":""},{"id":495776429,"identity":"e1a71587-5114-44d8-89d6-f1962cf47dd9","order_by":1,"name":"Son-Il Kwak","email":"data:image/png;base64,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","orcid":"","institution":"Kim Il Sung University","correspondingAuthor":true,"prefix":"","firstName":"Son-Il","middleName":"","lastName":"Kwak","suffix":""},{"id":495776430,"identity":"757479f4-d638-4a7b-ac20-bc8bd9071161","order_by":2,"name":"Chol-Jun Hwang","email":"","orcid":"","institution":"Kim Il Sung University","correspondingAuthor":false,"prefix":"","firstName":"Chol-Jun","middleName":"","lastName":"Hwang","suffix":""}],"badges":[],"createdAt":"2025-08-01 05:23:27","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7267543/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7267543/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":88429464,"identity":"8a4e0f62-9f68-48f0-a64d-2b5c11e8f69f","added_by":"auto","created_at":"2025-08-06 10:24:11","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":560184,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7267543/v1_covered_723f9233-f312-4b0e-b030-e7972b9e3b67.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"New Fuzzy Modus Ponens and Tollens based on Least Common Multiple","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"international-journal-of-machine-learning-and-cybernetics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jmlc","sideBox":"Learn more about [International Journal of Machine Learning and Cybernetics](http://actavetscand.biomedcentral.com/)","snPcode":"13042","submissionUrl":"https://submission.nature.com/new-submission/13042/3","title":"International Journal of Machine Learning and Cybernetics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"SISO fuzzy system, Fuzzy Approximate Reasoning, Least Common Multiple, Fuzzy Modus Ponens, Fuzzy Modus Tollens, Reductive Property, Fuzzy Controllability, Information Loss","lastPublishedDoi":"10.21203/rs.3.rs-7267543/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7267543/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The fuzzy approximate reasoning is one of the important research branches in the uncertainty inference of Artificial Intelligence (AI) and Computational Intelligence (CI). This fuzzy approximate reasoning consists of two parts, i.e., fuzzy modus ponens (FMP) and fuzzy modus tollens(FMT). FMP and FMT have an important significance in modeling of the human thinking process, design and development of a lot of several intelligence systems and etc. In this paper we proposed a novel original method of fuzzy approximate reasoning that can open a new direction of research in the uncertainty inference of AI and CI. Firstly, our proposed method is based on distance measure, concretely, an extended distance measure by the least common multiple(LCM). The proposed fuzzy approximate reasoning method LCM based on a least common multiple is an inference one for the SISO fuzzy system with discrete fuzzy set vectors of equal or different dimensions between the antecedent and consequent of fuzzy rule. We call it LCM method. LCM method is consisted of two parts, i.e., FMP–LCM, and FMT–LCM. Secondly, in this paper we proposed and proved four theorems with respect to the information loss. In other words we pointed out that our proposed method LCM has no information loss, and then the previous methods, i.e., Compositional Rule of Inference(CRI), Triple Implication Principle(TIP), Quintuple Implication Principle(QIP) and Approximate Analogical Reasoning Scheme(AARS) have some information loss compared with our proposed method. Thirdly, we compared the reductive properties for the five fuzzy reasoning methods, i.e., CRI, TIP, QIP, AARS, and our proposed LCM with respect to FMP and FMT. The theoretical and experimental results highlight that the proposed fuzzy approximate reasoning method LCM is comparatively more clear and effective, and in accordance with human thinking than the previous fuzzy reasoning methods. Fourthly, in this paper we proposed and proved six theorems with respect to fuzzy controllability of the several fuzzy reasoning methods based on the fuzzy relation and our proposed method based on LCM. We pointed out our proposed method LCM has not only higher reductive property but also higher fuzzy controllability than the previous fuzzy reasoning methods.","manuscriptTitle":"New Fuzzy Modus Ponens and Tollens based on Least Common Multiple","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-06 10:23:36","doi":"10.21203/rs.3.rs-7267543/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-10-02T01:24:37+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-29T03:16:56+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-21T21:17:29+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-08T16:19:30+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"100607892568198323397624591922713879258","date":"2025-08-07T16:14:31+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"261551536343211956234015677636969157901","date":"2025-08-07T13:09:05+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-07T10:14:31+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-06T11:09:59+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"266278411176592864743370509628446147934","date":"2025-08-06T10:33:57+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"10031411445699339458969340063579949473","date":"2025-08-06T10:28:46+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"281153624437248849016009526111283255439","date":"2025-08-05T04:45:02+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"160405747821596141916830502396752589905","date":"2025-08-05T04:12:12+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-08-05T04:05:12+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-08-04T02:16:26+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-08-02T05:46:35+00:00","index":"","fulltext":""},{"type":"submitted","content":"International Journal of Machine Learning and Cybernetics","date":"2025-08-01T05:15:28+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"international-journal-of-machine-learning-and-cybernetics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jmlc","sideBox":"Learn more about [International Journal of Machine Learning and Cybernetics](http://actavetscand.biomedcentral.com/)","snPcode":"13042","submissionUrl":"https://submission.nature.com/new-submission/13042/3","title":"International Journal of Machine Learning and Cybernetics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"5d6bc9e3-709e-4818-ad63-7b85bce52c43","owner":[],"postedDate":"August 6th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-08T05:54:34+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-06 10:23:36","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7267543","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7267543","identity":"rs-7267543","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}