[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117515-en":3,"doc-seo-117515-105":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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},117515,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","A Modal Logic for Uncertainty - a Completeness Theorem","The paper axiomatizes a modal logic for reasoning about probability functions, belief functions, and lower probabilities, including their dual notions. It combines an S5 necessity operator acting on formulas of the infinite-valued Łukasiewicz logic with a unary modality describing probability-function behaviour. The resulting language is expressive enough to characterize probability, belief, and lower-probability theories. Soundness and completeness are proved after restricting to suitable sublanguages: with belief-function models for belief-function formulas, and with lower-probability evaluations for lower-probability formulas.","A Modal Logic for Uncertainty: a Completeness Theorem  \nEsther Anna Corsi  \nDepartment of Philosophy, University of Milan, Italy  \nTommaso Flaminio  \nLluís Godo  \nArtificial Intelligence Research Institute (IIIA - CSIC), Campus UAB, Spain  \nHykel Hosni  \n[esther.corsi@unimi.it](esther.corsi@unimi.it)  \n[tommaso@iiia.csic.es](tommaso@iiia.csic.es)[ ](tommaso@iiia.csic.es)[godo@iiia.csic.es](godo@iiia.csic.es)  \n[hykel.hosni@unimi.it](hykel.hosni@unimi.it)  \nDepartment of Philosophy, University of Milan, Italy  \nAbstract  \nIn the present paper, we axiomatize a logic that allows a general approach for reasoning about probability functions, belief functions, lower probabilities and their corresponding duals. The formal setting we consider arises from combining a modal S5 necessity operator  \n􀀃 that applies to the formulas of the infinite-valued Łukasiewicz logic with the unary modality 􀀥 that describes the behaviour of probability functions. The modality 􀀥 together with an S5 modality 􀀃 provides a language rich enough to characterise probability, belief and lower probability theories. For this logic, we provide an axiomatization and we prove that, once we restrict to suitable sublanguages, it turns out to be sound and complete with respect to belief functions and lower probability models.  \nKeywords: fuzzy logic, Dempster-Shafer belief functions, probability functions, imprecise probabilities, modal logic  \n1. Introduction  \nThe relationship between modal logics, fuzzy logics and uncertainty measures is not new. In [21, 17, 19], see also [14] for a survey, probability functions are defined via a fuzzy modal operator 􀀥 applied on classical propositional formulas. Thus, the probability of a boolean formula 􀁩 is taken to be the truth degree of the fuzzy proposition 􀀥􀁩 =“􀁩 is probable”. Remarkably, this modal fuzzy approach to probability has been proved in [2] to be equivalent to the possibly better known setting proposed and studied by Fagin, Halpern and Megiddo in [11]. The same approach has been then generalized to represent Dempster-Shafer belief functions and in [18] the belief degree of classical boolean 􀁩 is the truth degree of the modal formula 􀀗􀁩 = 􀀥 􀀃􀁩, where 􀀃 is an S5 modality. In [24, 25], lower and upper probabilities have been formalized in a similar way. Furthermore, these setting have been also generalized to deal with nonclassical events in [13, 15, 12] .  \nIn the recent short paper [9], the authors propose an approach to deal with several uncertainty theories within a unique and general logical language that, in addition to the previously recalled modality 􀀥, also contains an additional S5 modal operator 􀀃 . In the same paper [9], the problem of determining an axiomatization for that general logic was left as open. In the present paper we approach that issue showing an axiomatization for our logic. More precisely, the language proposed in the aforementioned paper, in addition to the probability formulas of the form 􀀥 (􀁩) , was claimed to allow expressing “belief function formulas”by combining 􀀥 and 􀀃 as 􀀥 ( 􀀃􀁩) and “lower probability formulas” as 􀀃􀀥 (􀁩) . Although belief function formulas asthe above were already considered in the literature (see [18] for instance), the models presented in [9] are slightly more general as they also allow to interpret lower probability formulas.  \nIn the present paper we show that if we restrict to belief function formulas, our logic is sound and complete with respect to belief function models, while if we restrict to lower probability formulas, the same logic is complete with respect to lower probability evaluations. As the former will be a direct consequence of the completeness theorem shown in [18], the latter is entirely new. Having a unique logic to deal with uncertainty theories, and with belief functions and lower probability in particular, is interesting also in light of the result presented in [8] showing that in some nontrivial situations, these two uncertainty measures ca","cbCaivbIFkNbmeql","https://ap.wps.com/l/cbCaivbIFkNbmeql","pdf",361751,1,11,"English","en",105,"# Introduction\n# Modal Logics and Uncertainty Measures\n## Logical Preliminaries\n## Algebraic Semantics and MV-Algebras\n# Axiomatization and Completeness Theorems\n# Comparative Analysis and Future Work","[{\"question\":\"What kinds of uncertainty measures does the proposed modal logic target?\",\"answer\":\"It targets probability functions, belief functions, and lower probabilities, together with the corresponding dual perspectives expressed in the same logical framework.\"},{\"question\":\"How are the modal operators combined in the logic?\",\"answer\":\"The logic combines an S5 necessity operator applied to infinite-valued Łukasiewicz formulas with a unary modality that describes the behaviour of probability functions.\"},{\"question\":\"What completeness results are proved in the paper?\",\"answer\":\"When restricted to belief-function formulas, the logic is sound and complete with respect to belief-function models; when restricted to lower-probability formulas, it is complete with respect to lower-probability evaluations.\"}]","A Modal Logic for Uncertainty - a Completeness Theorem | PDF",1785676509,28,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-modal-logic-for-uncertainty-a-completeness-theorem","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/a-modal-logic-for-uncertainty-a-completeness-theorem/117515/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What kinds of uncertainty measures does the proposed modal logic target?","Question",{"text":75,"@type":76},"It targets probability functions, belief functions, and lower probabilities, together with the corresponding dual perspectives expressed in the same logical framework.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the modal operators combined in the logic?",{"text":80,"@type":76},"The logic combines an S5 necessity operator applied to infinite-valued Łukasiewicz formulas with a unary modality that describes the behaviour of probability functions.",{"name":82,"@type":73,"acceptedAnswer":83},"What completeness results are proved in the paper?",{"text":84,"@type":76},"When restricted to belief-function formulas, the logic is sound and complete with respect to belief-function models; 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