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Inquisitive Conditional-Doxastic Logic

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Inquisitive Conditional-Doxastic Logic Inquisitive Conditional-Doxastic Logic MSc Thesis (Afstudeerscriptie) written by Benjamin Sparkes (born 31/05/1989 in Bristol, United Kingdom) under the supervision of Dr. Ivano Ciardelli and Dr. Floris Roelofsen, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: August 30 2016 Dr. Ivano Ciardelli Prof. Dr. Benedikt Löwe (chair) Dr. Soroush Rafiee Rad Dr. Floris Roelofsen Dr. Katrin Schulz Prof. Dr. Frank Veltman Abstract This thesis develops inquisitive conditional-doxastic logic, obtained by enriching classi- cal multi-agent plausibility models with issues for each agent. The aim of these models is to allow for a finer approach to modelling inquiry. Issues capture informative and interrogative content, and so by associating an issue to an agent we are able to capture both their information state and their inquisitive state, while a plausibility map on worlds captures their doxastic state. Moreover, inquisitive plausibility models allow for conditionalisation on both informative and interrogative content. Two conditional-doxastic modalities are introduced and axiomatised; a considers modality unique to inquisitive conditional-doxastic logic, which conditionalises on is- sues with respect to both an agent’s doxastic and inquisitive state, and a generalisation of (conditional) belief, which conditionalises solely on an agent’s doxastic state. We show that inquisitive conditional-doxastic logic encodes the same assumptions concerning conditionalisation and conditional-doxastic logic. And, just as conditional- doxastic logic may be taken as the static counterpart to a dynamic logic of belief revi- sion, inquisitive conditional-doxastic logic can be taken as the static counterpart to a dy- namic logic of belief revision within an enriched setting that includes formal resources to model interrogatives. Inquisitive conditional-doxastic logic is shown to be sound and complete with re- spect to inquisitive plausibility models, and it is shown that both (conditional) belief and knowledge modalities can be defined in terms of the considers modality. However, we also show that the entertains modality of inquisitive epistemic logic cannot be de- fined by the considers modality. Therefore, as the basic inquisitive conditional-doxastic logic is axiomatised solely by the considers modality over the base inquisitive seman- tics it cannot be used to reason about an agent’s inquisitive state independently of their doxastic state. For this reason, we also axiomatise inquisitive plausibility logic and show it is sound and complete with respect to the same class of inquisitive plausibility models. This logic introduces modalities which restrict the issue associated to each agent to the worlds considered at least as plausible as the current world or state of evaluation, and those strictly less plausible. The entertains modality of inquisitive epistemic logic is taken as basic in the axiomatisation of inquisitive plausibility logic, and we show the considers modality of basic inquisitive conditional-doxastic logic is definable. Therefore, inquis- itive plausibility logic allows for a full study of the interaction between epistemic and conditional-doxastic modalities within the framework of inquisitive semantics. Acknowledgements First and foremost, I would like to thank my supervisors Floris Roelofsen and Ivano Ciardelli for their support in writing this thesis. The many meetings we had, and emails exchanged, were never less than inspirational. I had not anticipated how challenging the project would be, and working through many of the problems that presented themselves with Ivano has improved my understanding of logic immensely, and without Ivano’s ex- cellent guidance this thesis would not have been completed. Ivano, it has been a privilege working with you. I would also like to thank the members of my thesis committee, Soroush Rafiee Rad, Katrin Schulz, and Frank Veltman for their questions and comments. Thanks also to all the people at the ILLC who have taught me so much, there are too many of you to name individually! And thanks in particular to Maria Aloni, who has been a superb and incredibly supportive mentor. Finally a special thanks to Maša Močnik, you made my time at the ILLC wonderful. Contents Introduction 3 1 Inquisitive Semantics 5 1.1 Inquisitiveness, informally ......................... 5 1.2 InqD ..................................... 6 2 Inquisitive Epistemic Logic 15 2.1 Epistemic Logic ............................... 15 2.2 Inquisitive Epistemic Logic ........................ 16 3 Inquisitive Conditional-Doxastic Logic 21 3.1 Conditional-Doxastic Logic ........................ 21 3.2 Inquisitive conditional-doxastic Logic .................. 26 3.3 Inquisitive Plausibility Models ....................... 26 3.3.1 The Considers Modality ...................... 29 3.3.2 Conditional Belief ......................... 33 3.4 Axioms & Rules .............................. 36 3.4.1 Axioms for ICDL Enriched with Operators for Conditional Be- lief and Knowledge ........................ 36 3.4.2 Observations ............................ 37 3.5 The Road Ahead .............................. 38 4 Inquisitive Conditional-Doxastic Models 39 4.1 Inquisitive Conditional-Doxastic Models ................. 39 4.2 Observations ................................ 41 4.3 Soundness of ICDL with Respect to icdms ................. 42 4.4 Some Rules and Theorems of ICDL ..................... 46 5 Connexions 48 5.1 From icdms to Plausibility Models .................... 48 5.2 From Inquisitive Plausibility Models to icdms .............. 53 5.3 The Entertains Modality Is Not Definable in ICDL ............. 55 6 Completeness of ICDL 57 6.1 Introduction ................................ 57 6.2 Foundations for the Canonical Model ................... 58 6.2.1 Syntactic Characterisation of Atoms, States, and Propositions. 64 6.3 The Canonical Model ........................... 67 6.4 Results ................................... 75 1 7 Inquisitive Plausibility Logic 77 7.1 Inquisitive Plausibility Logic ........................ 77 7.1.1 Binary Operators ......................... 82 7.2 Inquisitive Plausibility Neighbourhood Models .............. 86 7.3 Axioms and Rules of IPL .......................... 87 7.4 Completeness ................................ 90 7.4.1 Preliminaries ............................ 90 7.4.2 Constructing the Canonical Model ................ 91 7.4.3 The Support Lemma ........................ 91 7.5 Connexions Between ipms and ipnms ................... 94 7.5.1 From ipnms to ipms ........................ 94 7.6 From ipms to ipnms ............................ 97 7.7 Results ................................... 99 7.8 Additional Observations .......................... 99 8 Conclusion 102 A Dynamics 104 A.1 Updates on ipms .............................. 104 A.2 Inquisitive Dynamic Conditional-Doxastic Logic and Inquisitive Dy- namic Plausibility Logic .......................... 105 A.3 Results ................................... 106 Cheat Sheet 109 List of Theorems, Lemmas, etc. 116 2 Introduction Inquisitive semantics is a semantic framework based on a notion of meaning intended to capture both informative and inquisitive content, formally modelled via issues, in order to analyse information exchange—seen as process of raising and resolving issues. We restrict ourselves to a propositional language in this thesis, which allows us to characterise issues as downward closed sets of sets of possible worlds. The logic InqD (Ciardelli, Groenendijk, and Roelofsen 2015) captures the relation of entailment be- tween issues characterised this way. InqD is introduced and outlined in chapter 1. Inquisitive epistemic logic (Ciardelli 2014b) extends InqD to an epistemic setting, by associating to each of a set of agents an issue, intended to capture the information available to an agent and the issues they entertain. In this way the relationship between inquisitive epistemic logic and InqD parallels the relationship between epistemic logic and propositional logic (cf. Ciardelli 2015, §7.2) For, in epistemic logic an information state—a set of possible worlds—is associated to each agent in order to capture the information available to them. Using the information states associated to agent’s in epistemic logic the propositional attitude of knowledge can be formally modelled. Following this parallel, inquisitive epistemic logic models the propositional attitudes of both knowledge and of entertaining an issue, with the attitude of entertaining standing in the same relation to issues as knowledge does to classical proposition. Chapter 2 introduces epistemic logic in section 1 and inquisitive epistemic logic in section 2. However, epistemic logic does not have sufficient expressive power to capture the im- portant propositional attitude of belief. This is captured by doxastic logic, or Moreover, an epistemic-doxastic logic may be used to capture to interaction between the proposi- tional attitudes of knowledge and belief (cf. Stalnaker 2006). So, a natural progression of inquisitive epistemic logic may be to investigate a logic capturing the proposition at- titude with respect to issues which parallels belief, and its interaction with the attitude of entertaining. Alternatively, one may study the attitude of conditional belief—intending to capture what an agent believes conditional upon certain information being
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