[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85393-en":3,"doc-seo-85393-105":30,"detail-sidebar-cat-0-en-105":91},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85393,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Simply-typed Constant-domain Modal Lambda Calculus I: Distanced Beta Reduction and Combinatory Logic","A system λθ is developed that combines modal logic with simply-typed lambda calculus, generalizing the Montague–Gallin approach while using today’s standard typed base theory. A parameter θ controls state types and state variables more flexibly than in the original work. The paper establishes metatheoretic results: an Andrews-like characterization via combinatory logic with a BCKW-like basis, and semantic conservation/expressibility theorems relating λθ to the maximal system λω. Comparable relations also connect λω to ordinary simply-typed lambda calculus, addressing a Zimmermann question.","SIMPLY-TYPED CONSTANT-DOMAIN MODAL LAMBDA CALCULUS I: DISTANCED BETA REDUCTION AND COMBINATORY  \nLOGIC  \nSEAN WALSH  a  \narXiv :2410 . 17463v 5 [ cs .LO] 13 Jul 2026  \nUCLA Department of Philosophy, 390 Portola Plaza, 300 Dodd Hall, Box 951451, Los Angeles, CA 90095-1451  \ne-mail address: [walsh@ucla.edu](walsh@ucla.edu)  \nAbstract. A system λθ is developed that combines modal logic and simply-typed lambda calculus, and that generalizes the system studied by Montague and Gallin. Whereas Montague and Gallin worked with Church’s simple theory of types, the system λθ is developed in the typed base theory most commonly used today, namely the simply-typed lambda calculus. Further, the system λθ is controlled by a parameter θ which allows more options for state types and state variables than is present in Montague and Gallin. A main goal of the paper is to establish some basic metatheory of λθ : (i) an Andrews-like characterization of its models in terms of combinatory logic is given, and this combinatory logic involves a BCKW-like basis rather than an SKI-like basis and (ii) semantic conservation and expressibility results relating λθ to the maximal system λω are proven. Similar results are proven for the relation between λω and λ, the corresponding ordinary simply-typed lambda calculus. This answers a question of Zimmermann in the semantics of the simply typed setting. In a companion paper this is extended to Church’s simple theory of types.  \nWe further develop a partial correspondence between a pure combinatory logic centered on the BCKW-like basis and the weak deductive system for λω wherein β-reduction is not allowed under a lambda abstract, and we use this to show partial deductive conservation between the maximal system λω and the intermediary systems λθ .  \n1. Introduction  \nTwo of the great achievements of modern logic are modal logic and typed lambda calculus. Atthe advent of formal semantics in linguistics, Montague developed a system that integrated the two.1 However, by contemporary lights, Montague’s theory is both too strong and too weak. It is too strong in that he worked only with Church’s simple theory of types,  \nKey words and phrases: Type theory, Modal and temporal logics, Higher order logic.  \nThanks to Thomas Ede Zimmermann, Dana Scott, and the reviewers for comments and feedback.  \n1Montague’s work [Mon74] is discussed at length in standard semantics textbooks, such as [DWP81],[Gam91], and [CMG00] . Montague’s work was made well-known in part through the work of Partee; see [PH97] for some of the history. The theory is sometimes divided into the intensional theory of types and Montague grammar (e.g. [Gam91, Chapters 5-6]) . This paper focuses on the intensional theory of types, as did Chapters 1-2 of Gallin’s book [Gal75] . In recent decades, textbook treatments of semantics focus foremost on Montague grammar in extensional contexts. That is the topic of [HK98], with its anticipated sequel [VFH23] being devoted to intensional matters.  \nPreprint submitted to  \nLogical Methods in Computer Science  \n© S. Walsh  \n⃝CC Creative Commons  \nreplete with the resources of quantification and identity.2 But modern typed lambda calculi work with a weaker base system, and have many different extensions besides Church’s simple theory of types.3 Further, Montague’s theory is too weak in that it does not have many of the features of modern modal logics, such as two-dimensionality and actuality operators and other devices for referring to many distinct states, and binding many distinct variables of state type, within one and the same expression.4 (Following common usage in modal logic, “state” is a term of art which, depending on application, covers worlds, times, machine-configurations, etc.) A chief aim of this paper is to remedy this deficit, and to begin the development of a thoroughly modern version of Montague’s simply-typed modal lambda calculus. This should be of interest wherever modal logic and simply-typed lambda c","cbCaihJSWYQKUicu","https://ap.wps.com/l/cbCaihJSWYQKUicu","pdf",736816,3,1,61,"English","en",105,"# Introduction\n## System λθ and goals\n## Metatheory and conservation results\n## Motivation from Montague’s original system","[{\"question\":\"What is the main contribution of the system λθ in this paper?\",\"answer\":\"It integrates modal logic with simply-typed lambda calculus, generalizing the Montague–Gallin framework while using the modern simply-typed lambda calculus as the typed base theory and introducing parameter θ for additional control over state types and variables.\"},{\"question\":\"How does the paper characterize models using combinatory logic?\",\"answer\":\"It provides an Andrews-like characterization of λθ models in terms of combinatory logic, using a BCKW-like basis rather than an SKI-like basis.\"},{\"question\":\"What conservation and expressibility results are proven?\",\"answer\":\"Semantic conservation and expressibility theorems relate λθ to the maximal system λω, and similar results relate λω to the corresponding ordinary simply-typed lambda calculus, answering a question posed by Zimmermann.\"}]",1784203102,154,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"simply-typed-constant-domain-modal-lambda-calculus-i-distanced-beta-reduction-and-combinatory-logic","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/simply-typed-constant-domain-modal-lambda-calculus-i-distanced-beta-reduction-and-combinatory-logic/85393/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-21","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main contribution of the system λθ in this paper?","Question",{"text":75,"@type":76},"It integrates modal logic with simply-typed lambda calculus, generalizing the Montague–Gallin framework while using the modern simply-typed lambda calculus as the typed base theory and introducing parameter θ for additional control over state types and variables.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper characterize models using combinatory logic?",{"text":80,"@type":76},"It provides an Andrews-like characterization of λθ models in terms of combinatory logic, using a BCKW-like basis rather than an SKI-like basis.",{"name":82,"@type":73,"acceptedAnswer":83},"What conservation and expressibility results are proven?",{"text":84,"@type":76},"Semantic conservation and expressibility theorems relate λθ to the maximal system λω, and similar results relate λω to the corresponding ordinary simply-typed lambda calculus, answering a question posed by 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