On the Proof Theory of Conditional Logics Marianna Girlando To cite this version: Marianna Girlando. On the Proof Theory of Conditional Logics. Logic in Computer Science [cs.LO]. Aix-Marseille Universite; Helsinki University, 2019. English. tel-02077109 HAL Id: tel-02077109 https://hal.archives-ouvertes.fr/tel-02077109 Submitted on 22 Mar 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. AIX-MARSEILLE UNIVERSITE´ UNIVERSITY OF HELSINKI ED184 { MATHEMATIQUES´ ET FACULTY OF ARTS INFORMATIQUE DOCTORAL SCHOOL IN LABORATOIRE D'INFORMATI- HUMANITIES AND SOCIAL QUE ET DES SYSTEMES SCIENCES On the Proof Theory of Conditional Logics Marianna GIRLANDO Soutenue le 8/02/2019 { Discussed on 8/02/2019 Aix-Marseille Universit´e Nicola OLIVETTI Aix-Marseille Universit´e Directeur de th`ese Sara NEGRI University of Helsinki Co-directeur de th`ese George METCALFE University of Bern Rapporteur Sonja SMETS University of Amsterdam Rapporteur Didier GALMICHE Universit´ede Lorraine Examinateur Philippe BESNARD Universit´ePaul Sabatier Examinateur Luigi SANTOCANALE Aix-Marseille Universit´e Examinateur Heta PYRHONEN¨ University of Helsinki Examinateur University of Helsinki Sara NEGRI University of Helsinki Supervisor, Custos Nicola OLIVETTI Aix-Marseille Universit´e Supervisor George METCALFE University of Bern Pre-examier Sonja SMETS University of Amsterdam Pre-examiner, Opponent Heta PYRHONEN¨ University of Helsinki Faculty representative Filipe PEREIRA DA SILVA University of Helsinki Faculty representative Abstract This thesis can be ideally placed at the intersection of three research topics: conditional logics, proof theory and neighbourhood semantics. The family of logics under scope stems from the works of Stalnaker and Lewis, and extends classical propositional logic by means of a two-place modal operator, which expresses a fine-grained notion of condi- tionality. The semantics of these logics is modularly defined in terms of neighbourhood models. The research aim is to investigate the proof theory of conditional logics, by defining sequent calculi for them. The proof systems introduced are extensions of Gentzen's sequent calculus; they are either labelled, defined by enriching the language, or internal, which add structural connectives to the sequents. Moreover, the calculi are standard: they are composed of a finite number of rules, each displaying a fixed number of pre- misses. The thesis is organized in six chapters. Chapters 1 contains an axiomatic and seman- tic overview of conditional logics, while Chapter 2 is a short introduction to proof theory. The original contributions to the subject are presented in chapters 3 { 6. Chapter 3 introduces labelled calculi based on neighbourhood models for preferential conditional logics, and Chapter 4 presents different internal proof systems covering counterfactual logics, a subfamily of preferential logics. Chapter 5 analyses the relationship between proof systems by presenting a mapping between a labelled and an internal calculus. Fi- nally, the proof-theoretic methods developed for conditional logics are applied in chapter 6 to a multi-agent epistemic logic. i R´esum´e La th`esese place `al'intersection de trois sujets de recherche : logiques conditionnelles, th´eoriede la d´emonstrationet s´emantique de voisinage. La famille de logiques condition- nelles consid´er´eesprovient d`esouvrages de Stalnaker et Lewis. Elle est une extension de la logique classique propositionnelle avec un op´erateurmodal `adeux places, qui ex- prime une notion affin´eede conditionnalit´e. La s´emantique de ces logiques est d´efinie en termes de mod`elesde voisinage. Le but de la recherche est d'´etudier la th´eoriede la d´emonstrationdes logiques condi- tionnelles, en pr´ecisant leur calculs des sequents. Les calculs d´efinissont des extensions du calcul des sequents de Gentzen ; ils sont ´etiquet´es,c'est `adire d´efinisen enrichissant le langage, ou internes, qui rajoutent des connecteurs structurels aux sequents. La th`eseest organis´eeen six chapitres. Le chapitre 1 pr´esente les axiomes et la s´emantique des logiques conditionnelles et le chapitre 2 introduit la th´eoriede la d´emonstration. Les contributions originelles au sujet sont trait´eesdans les chapitres 3 { 6. Le chapitre 3 introduit des calculs de sequents ´etiquet´esbas´es sur la s´emantique de voisinage pour les logiques conditionnelles pr´ef´erentielles. Le chapitre 4 pr´esente diff´erents syst`emesinternes de calcul pour les logiques counterfactuelles, une sous-famille des logiques pr´ef´erentielles. Le chapitre 5 analyse la relation parmi les syst`emesde preuve en pr´esentant les deux c^ot´esd'une traduction entre un calcul ´etiquet´eet un cal- cul interne. Finalement, au chapitre 6, les m´ethodes de la th´eoriede la d´emonstration conditionnelle sont appliqu´ees`aune logique ´epist´emiquemulti-agente. ii Yhteenveto T¨am¨av¨ait¨oskirjasijoittuu ideaalisesti kolmen tutkimusaiheen risteykseen: konditiona- alien logiikka, todistusteoria, ja ymp¨arist¨osemantiikka. Katettujen logiikoiden perhe periytyy Stalnakein ja Lewisin t¨oist¨aja laajentaa klassillista lauselogiikkaa kaksipaikkai- sen modaalioperaattorin kautta. Se ilmaisee hienojakoisen konditionaalisuuden k¨asit- teen. N¨aidenlogiikoiden semantiikka m¨a¨aritell¨a¨anmodulaarisesti ymp¨arist¨osemantiikan termein. Tutkimuksen tavoitteena on selvitt¨a¨akonditionaalien logiikan todistusteoriaa m¨a¨arit- telem¨all¨aniille sekvenssikalkyyleita. Esitellyt todistussysteemit ovat Gentzenin sekvens- sikalkyylin laajennuksia; ne ovat joko merkeill¨avarustettuja, joihin p¨a¨adyt¨a¨ankielen rikastuksella, tai sis¨aisi¨a,joihin p¨a¨adyt¨a¨anlis¨a¨am¨all¨asekvensseihin rakenteellisia kon- nektiiveja. N¨am¨alaajennukset ovat my¨osstandardikalkyyleja: ne muodostuvat ¨a¨arellis- est¨am¨a¨ar¨ast¨as¨a¨ant¨oj¨a,joissa jokaisessa on kiinte¨am¨a¨ar¨apremissej¨a. V¨ait¨oskirjamuodostuu kuudesta kappaleesta. Kappale 1 sis¨alt¨a¨aaksiomattisen ja semanttisen katsauksen konditionaalien logiikkaan, ja kappale 2 on taas lyhyt johda- tus todistusteoriaan. Uudet tutkimuspanokset aiheeseen on esitetty kappaleissa 3-6. Kappale 3 esitt¨a¨ans. merkityt sekvenssikalkyylit joiden perustana on preferentiaalis- ten konditionaalien logiikan ymp¨arist¨omallit,kappale 4 esitt¨a¨aerilaisia sis¨aisi¨atodis- tussysteemej¨ajotka kattavat kontrafaktuaalien logiikat, jotka ovat preferentiaalisten logiikoiden alaperhe. Kappale 5 analysoi eri todistussysteemien v¨alisi¨asuhteita merkki- kalkyyylien ja sis¨aistenkalkyylien v¨alisen kuvauksen kautta. Kuudennessa ja viimeisess¨a kappaleessa sovelletaan konditionaalien logiikalle kehitettyj¨atodistusteoreettisia metode- ja useamman agentin episteemiseen logiikkaan. iii iv Acknowledgements This thesis was written in the course of three years within an interdisciplinary agreement between the University of Aix-Marseille and the University of Helsinki. I am deeply grateful to both my supervisors, Nicola Olivetti and Sara Negri, for their expertise, advice and support throughout the thesis. It goes without saying that their contribution has been fundamental to this work, and to the general welfare of my Ph.D. A special thanks goes to the people whom I have worked with, most notably Bj¨orn Lellmann, Gian Luca Pozzato and Vincent Risch. A significant part of the results presented in this thesis comes from their ideas and initiative. A particular thanks to Bj¨orn,for his proof-theoretic skills and TEX abilities. I wish to thank the reviewers of this thesis, Sonja Smets and George Metcalfe, who kindly accepted to go through the two-hundred-pages manuscript. Their accurate re- ports were extremely useful in the final stages of the revision process. A cotutelle agreement has to deal with a great amount of bureaucracy. The guidance and patience of doctoral schools and administrative offices were essential to find my way in the maze. For Marseille, thanks to Nadia Creignou and Sonia Asseum; for Helsinki, my thanks to Anniina Sj¨oblomand Jutta Kajander. I should also mention that the Ph.D. grant has been founded by the LabEx Archim`ede, and that the research work has been partially supported by the Project TICAMORE ANR-16-CE91-0002-01 and the Suomen Akatemia Project Modalities and conditionals: Systematic and historical studies. Next, a great thanks to my friends from the proof-theory community, for the logic- related and logic-unrelated discussions and their precious advice. I am especially grateful to Elaine, Alberto, Charles, Andreas, to the people in Vienna and to the members of the P.T. Erasmus. A special thanks to Tiziano, my brother-in-Ph.D. I probably wouldn't have survived without my friends in Marseille and Helsinki. Thanks to Rasa and Andrea for the Vieux-Port aperitifs, to Paolo, for the Saint-Charles sessions, to Federico and Belen, for the Cours Julien company, and to Elena, Anna, Guillaume and the Luminy people for π, pies and parties. Thanks to Martina, for the moral discussions, to Fra and Paolo, my first Marseille friends, and to Giulio, for the dinners. Thanks to Niko, for the coffees, and to Hannu, for the apartment.