[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82157-en":3,"doc-seo-82157-105":29,"detail-sidebar-cat-0-en-105":83},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82157,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Quantum Logic as the Logic of Contexts","Quantum logic is developed for a finite, fully computable setting where the usual explanation order is reversed: classical Boolean logic arises after forgetting experimental context rather than serving as the starting point. Using the free orthomodular lattice on two generators, the work decomposes it into a register-of-contexts factor and a Boolean content factor, yielding context–bit-vector calculus. Three results classify context layers, show rigid layer duality under orthocomplementation, and prove context-forgetting forms a surjective homomorphism whose quotient is classical logic.","arXiv :2607 .09032v1 [ quant-ph] 10 Jul 2026  \nQuantum Logic as the Logic of Contexts  \nHaruki Emori  1, 2, 3, ∗ Atsushi Iriki  3, 2, 4,† Andrei Khrennikov  5,‡ and Kazunori Kondo 6, §  \n, , ,  \n1 Graduate School of Information Science and Technology, Hokkaido University,  \nKita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido 060-0814, Japan  \n2 RIKEN Center for Interdisciplinary Theoretical and Mathematical  \nSciences (iTHEMS), 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan  \n3 Teikyo University Advanced Comprehensive Research Organization (ACRO),  \n2-21-1 Kaga, Itabashi-ku, Tokyo, 173-0003, Japan  \n4 Brain Mind and Consciousness Program, Canadian Institute for Advanced Research (CIFAR),  \n661 University Avenue, Suite 505, Toronto, ON M5G 1M1 Canada  \n5 Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, V¨axj¨o, SE-351 95, Sweden  \n6 Graduate School of Human Sciences, Department of Human Sciences,  \nThe University of Osaka, 1-2 Yamadaoka, Suita, Osaka, 565-0871, Japan  \n(Dated: July 13, 2026)  \nQuantum logic is usually presented as a non-classical departure from ordinary reasoning forcedon us by quantum mechanics, with classical logic kept as the secure starting point. We argue for the opposite order of explanation in a finite and fully computable setting. The free orthomodular lattice on two generators has ninety-six elements, the direct product of a six-element non-distributive factor and a sixteen-element Boolean factor. Reading the first factor as a register of contexts and the second as Boolean content, we obtain a calculus whose elements are context–bit-vector pairs and whose operations act component by component. With this calculus we establish three results. First, we classify the six layers by commutativity, identifying the central kernel of context-neutral propositions together with a dual central layer in which all complementary contexts are present. Second, we show that orthocomplementation rearranges the layers exactly as the complementation of the small factor rearranges its elements, which makes the duality among the layers rigid rather than accidental. Third, we prove that the operation forgetting the context is a surjective homomorphism of orthocomplemented lattices whose quotient is the classical Boolean algebra, so that classical logic is a six-to-one, information-losing image of the contextual calculus.  \nI. INTRODUCTION  \nQuantum logic was introduced by Birkhoff and von Neumann [1] as the calculus of experimental propositions about a quantum system, with the lattice of projections of a Hilbert space taking the place of a Boolean algebra and with the distributive law no longer valid. Since then it has most often been read as a non-classical departure from ordinary reasoning, one made necessary by the peculiarities of quantum mechanics, while classical Boolean logic is kept as the secure foundation against which such departures are measured. The present paper argues, in a finite and completely computable setting, that this order of explanation can be reversed. Classical logic is better understood as a quotient of a contextual quantum logic, obtained by forgetting which context a proposition belongs to.  \nMany cognitive and psychological experiments share a feature that ordinary logic does not record, namely that an answer depends on the question asked, on the order of questions, and on the context in which a question is posed. Classical Boolean logic can store the final answer, but it does not store the context-dependent process by which that answer became determinate. The order of questions, the task demand, and the measurement basis selected by an experiment jointly shape which proposition becomes determinate, which is the reason classical Boolean logic is insufficient as a generative logic even though it remains adequate as a record of final answers. The present paper isolates, in a finite and fully computable model, the logical operation that turns such a contextual process i","cbCaimFJ4AqXBxK1","https://ap.wps.com/l/cbCaimFJ4AqXBxK1","pdf",399267,1,18,"English","en",105,"# Abstract\n# Introduction\n## Background: Birkhoff–von Neumann quantum logic\n## Contextuality in cognition and experiments\n## Complementarity and contextuality theorems","[{\"question\":\"What does the context-forgetting operation prove about classical logic?\",\"answer\":\"Forgetting context is a surjective homomorphism of orthocomplemented lattices. Its quotient is the classical Boolean algebra, so classical logic becomes a six-to-one, information-losing image of the contextual calculus.\"}]",1784178504,45,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":27},"quantum-logic-as-the-logic-of-contexts","",{"@graph":35,"@context":77},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/quantum-logic-as-the-logic-of-contexts/82157/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"What does the context-forgetting operation prove about classical logic?","Question",{"text":75,"@type":76},"Forgetting context is a surjective homomorphism of orthocomplemented lattices. Its quotient is the classical Boolean algebra, so classical logic becomes a six-to-one, information-losing image of the contextual calculus.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]