Counterfactuals as modal conditionals, and their probability

In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal...

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Authors: Rosella, Giuliano, Flaminio, Tommaso, Bonzio, Stefano
Format: article
Status:Published version
Publication Date:2023
Country:España
Institution:Consejo Superior de Investigaciones Científicas (CSIC)
Repository:DIGITAL.CSIC. Repositorio Institucional del CSIC
OAI Identifier:oai:digital.csic.es:10261/404527
Online Access:http://hdl.handle.net/10261/404527
https://api.elsevier.com/content/abstract/scopus_id/85165233634
Access Level:Open access
Keyword:Belief functions
Boolean algebras
Conditionals
Counterfactuals
Imaging rule
Modal logic
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spelling Counterfactuals as modal conditionals, and their probabilityRosella, GiulianoFlaminio, TommasoBonzio, StefanoBelief functionsBoolean algebrasConditionalsCounterfactualsImaging ruleModal logicIn this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update.The authors thank the anonymous referees and journal editors for their comments, suggestions, and constructive criticism. The authors are also indebted to Lluis Godo and Jan Sprenger for their help. Flaminio acknowledges partial support by the Spanish project PID2019-111544GB-C21 funded by MCIN/AEI/10.13039/501100011033 and Bonzio by the INdAM, GNSAGA - Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni. Flaminio and Bonzio acknowledge partial support by the MOSAIC project EU H2020-MSCA-RISE-2020 Project 101007627.Peer reviewedElsevier BVAgencia Estatal de Investigación (España)Ministerio de Ciencia, Innovación y Universidades (España)Istituto Nazionale di Alta Matematica Francesco SeveriEuropean CommissionRosella, Giuliano [0000-0002-3148-6125]Flaminio, Tommaso [0000-0002-9180-7808]Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]202520252023info:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501Publisher's versioninfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/10261/404527https://api.elsevier.com/content/abstract/scopus_id/85165233634reponame:DIGITAL.CSIC. Repositorio Institucional del CSICinstname:Consejo Superior de Investigaciones Científicas (CSIC)Inglés#PLACEHOLDER_PARENT_METADATA_VALUE##PLACEHOLDER_PARENT_METADATA_VALUE#info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2019-111544GB-C21info:eu-repo/grantAgreement/EC/H2020/101007627https://doi.org/10.1016/j.artint.2023.103970Síinfo:eu-repo/semantics/openAccessoai:digital.csic.es:10261/4045272026-05-22T06:33:51Z
dc.title.none.fl_str_mv Counterfactuals as modal conditionals, and their probability
title Counterfactuals as modal conditionals, and their probability
spellingShingle Counterfactuals as modal conditionals, and their probability
Rosella, Giuliano
Belief functions
Boolean algebras
Conditionals
Counterfactuals
Imaging rule
Modal logic
title_short Counterfactuals as modal conditionals, and their probability
title_full Counterfactuals as modal conditionals, and their probability
title_fullStr Counterfactuals as modal conditionals, and their probability
title_full_unstemmed Counterfactuals as modal conditionals, and their probability
title_sort Counterfactuals as modal conditionals, and their probability
dc.creator.none.fl_str_mv Rosella, Giuliano
Flaminio, Tommaso
Bonzio, Stefano
author Rosella, Giuliano
author_facet Rosella, Giuliano
Flaminio, Tommaso
Bonzio, Stefano
author_role author
author2 Flaminio, Tommaso
Bonzio, Stefano
author2_role author
author
dc.contributor.none.fl_str_mv Agencia Estatal de Investigación (España)
Ministerio de Ciencia, Innovación y Universidades (España)
Istituto Nazionale di Alta Matematica Francesco Severi
European Commission
Rosella, Giuliano [0000-0002-3148-6125]
Flaminio, Tommaso [0000-0002-9180-7808]
Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]
dc.subject.none.fl_str_mv Belief functions
Boolean algebras
Conditionals
Counterfactuals
Imaging rule
Modal logic
topic Belief functions
Boolean algebras
Conditionals
Counterfactuals
Imaging rule
Modal logic
description In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update.
publishDate 2023
dc.date.none.fl_str_mv 2023
2025
2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
http://purl.org/coar/resource_type/c_6501
Publisher's version
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10261/404527
https://api.elsevier.com/content/abstract/scopus_id/85165233634
url http://hdl.handle.net/10261/404527
https://api.elsevier.com/content/abstract/scopus_id/85165233634
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
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info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2019-111544GB-C21
info:eu-repo/grantAgreement/EC/H2020/101007627
https://doi.org/10.1016/j.artint.2023.103970
Sí
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
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dc.publisher.none.fl_str_mv Elsevier BV
publisher.none.fl_str_mv Elsevier BV
dc.source.none.fl_str_mv reponame:DIGITAL.CSIC. Repositorio Institucional del CSIC
instname:Consejo Superior de Investigaciones Científicas (CSIC)
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