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Develops a special form of tense definite descriptions and provides a constructive proof of the interpolation theorem for the calculus, extending results by Blackburn and Marx.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/definite-descriptions-and-hybrid-tense-logic/381352/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/definite-descriptions-and-hybrid-tense-logic/381352.png","ImageObject",442,249,{"name":42,"@type":43},"eBook King","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-27","2026-09-24",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",7,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What is the main logical extension proposed in the document?","Question",{"text":63,"@type":64},"It provides a version of first-order hybrid tense logic with predicate abstracts and definite descriptions as the only non-rigid terms, formalized through a tableau calculus over satformulas.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How are definite descriptions handled in the approach?",{"text":68,"@type":64},"Descriptions are treated as genuine terms in a Russell-inspired framework, while retaining reductionist aspects in several ways and also introducing tensal definite descriptions using tense variables.",{"name":70,"@type":61,"acceptedAnswer":71},"What is proven about interpolation?",{"text":72,"@type":64},"A constructive proof of the interpolation theorem is given for the tableau calculus, extending interpolation results previously developed by Blackburn and Marx.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},381352,1791746479,{"code":4,"msg":82,"data":83},"success",[84,88,93,98,103,108,113,118,123],{"id":85,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":86,"slug":87},11,90,"presentations",{"id":89,"doc_module":22,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},12,"Resumes",80,"resumes",{"id":94,"doc_module":22,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},14,"Invoices",70,"invoices",{"id":99,"doc_module":22,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},15,"Posters",60,"posters",{"id":104,"doc_module":22,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},16,"Social Media",50,"social-media",{"id":109,"doc_module":22,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},17,"Forms",40,"forms",{"id":114,"doc_module":22,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},18,"Letters",30,"letters",{"id":119,"doc_module":22,"doc_module_name":25,"category_name":120,"show_sort_weight":121,"slug":122},21,"Paper Templates",5,"papers-templates",{"id":124,"doc_module":22,"doc_module_name":25,"category_name":125,"show_sort_weight":4,"slug":126},158,"General","general-158",{"code":4,"msg":82,"data":128},{"doc_id":79,"user_id":129,"nickname":42,"user_avatar":130,"doc_module":22,"category_id":85,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":116,"language":136,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":12,"update_tm":140,"read_time":85},962088006270,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","Springer Nature 2021 LATEX template  \narXiv :2412 .0 1484v 1 [ cs .LO] 2 Dec 2024  \nDe􀀜nite descriptions and hybrid tense logic  \nAndrzej Indrzejczak ORCID: 0000-0003-4063-1651†  \nand Michał Zawidzki ORCID: 0000-0002-2394-6056†  \nDepartment of Logic, University of Lodz, Lindleya 3/5, Łódź ,  \n90-131, Poland.  \n*Corresponding author(s) . E-mail(s): michal.zawidzki@fi[lhist.uni.lodz.pl](lhist.uni.lodz.pl) ; Contributing authors: andrzej.indrzejczak@fi[lhist.uni.lodz.pl](lhist.uni.lodz.pl) ;  \n†These authors contributed equally to this work.  \nAbstract  \nWe provide a version of 􀀜rst-order hybrid tense logic with predicate abstracts and de􀀜nite descriptions as the only non-rigid terms.  \nIt is formalised by means of a tableau calculus working on satformulas. A particular theory of DD exploited here is essentially based on the approach of Russell, but with descriptions treated as genuine terms. However, the reductionist aspect of the Russellian approach is retained in several ways. Moreover, a special form of tense de􀀜nite descriptions is formally developed. A constructive proof of the interpolation theorem for this calculus is given, which is an extension of the result provided by Blackburn and Marx.  \nKeywords: 􀀜rst-order tense logic, hybrid tense logic, de􀀜nite descriptions,  \ntableau calculus, interpolation  \n1 Introduction  \nHybrid logic (HL) is an important augmentation of standard modal logic with rich syntactic resources. The basic language of HL is obtained by adding a second sort of propositional atoms, called nominals, each of which holds true at exactly one state of a model and serves as a name of this state. Additionally, one can introduce several extra operators; the most important one is the satisfaction, or @-, operator which takes as its first argument a nominal j and  \nSpringer Nature 2021 LATEX template  \n2 De􀀜nite descriptions and hybrid tense logic  \nas the second one an arbitrary HL-formula ϕ . A formula @j ϕ indicates that ϕ is satisfied at the state denoted by j. This allows us to internalise an essential part of the semantics in the language. Another specific operator is the downarrow binder (↓) which binds the value of a state variable to the current state. What is nice about HL is that the additional hybrid machinery does not seriously affect the modal logic core it is based on. In particular, modifications in the relational semantics are minimal. The concept of frame remains intact. Only at the level of models we have some changes. Moreover, adding a binderfree hybrid toolkit typically does not increase the computational complexity of the underlying modal logic. These relatively small modifications of standard modal languages give us many advantages: 1 . a more expressive language, 2 . abetter behaviour in completeness theory, 3 . a more natural and simpler proof theory. In particular, defining frame conditions such as irreflexivity, asymmetry, trichotomy, and others, impossible in standard modal languages, becomes possible in HL. This machinery and results are easily extendable to multimodal logics, in particular to tense and temporal logic (Blackburn & Jørgensen, 2012; Blackburn & Tzakova, 1999) . Proof theory of HL offers an even more general approach than applying labels popular in proof theory for standard modal logic, namely it allows for internalising those labels as part of standard hybrid formulas (Braüner, 2011; Indrzejczak, 2010) .  \nHL offers considerable benefits pertaining to the interpolation property. It is well known that for many modal logics in standard languages this property fails. The situation is particularly bad for the first-order case; Fine (1979) showed that the first-order variant of S5 does not enjoy interpolation, and also all modal logics from the modal cube with constant domains fail to satisfy it. On the other hand, HL offers resources which significantly improve the situation. In this case the ↓-binder turns out particularly useful. The uniform interpolation theorem for all propositional","cbCaigXzwK5JwMYi","https://ap.wps.com/l/cbCaigXzwK5JwMYi","pdf",437176,"English","# Abstract\n# 1 Introduction\n## Hybrid logic basics\n## Interpolation property and prior work\n## Extensions with definite descriptions and tense definite descriptions","[{\"question\":\"What is the main logical extension proposed in the document?\",\"answer\":\"It provides a version of first-order hybrid tense logic with predicate abstracts and definite descriptions as the only non-rigid terms, formalized through a tableau calculus over satformulas.\"},{\"question\":\"How are definite descriptions handled in the approach?\",\"answer\":\"Descriptions are treated as genuine terms in a Russell-inspired framework, while retaining reductionist aspects in several ways and also introducing tensal definite descriptions using tense variables.\"},{\"question\":\"What is proven about interpolation?\",\"answer\":\"A constructive proof of the interpolation theorem is given for the tableau calculus, extending interpolation results previously developed by Blackburn and Marx.\"}]","De􀀜nite descriptions and hybrid tense logic | PDF",1790244966]