Some Remarks on the Model Theory of Epistemic Plausibility Models

Classical logics of knowledge and belief are usually interpreted on Kripke models, for which a mathematically well-developed model theory is available. However, such models are inadequate to capture dynamic phenomena. Therefore, epistemic plausibility models have been introduced. Because these are much richer structures than Kripke models, they do not straightforwardly inherit the model-theoretical results of modal logic. Therefore, while epistemic plausibility structures are well-suited for modeling purposes, an extensive investigation of their model theory has been lacking so far. The aim of the present paper is to fill exactly this gap, by initiating a systematic exploration of the model theory of epistemic plausibility models. Like in 'ordinary' modal logic, the focus will be on the notion of bisimulation. We define various notions of bisimulations (parametrized by a language L) and show that L-bisimilarity implies L-equivalence. We prove a Hennesy-Milner type result, and also two undefinability results. However, our main point is a negative one, viz. that bisimulations cannot straightforwardly be generalized to epistemic plausibility models if conditional belief is taken into account. We present two ways of coping with this issue: (i) adding a modality to the language, and (ii) putting extra constraints on the models. Finally, we make some remarks about the interaction between bisimulation and dynamic model changes.Comment: 19 pages, 3 figure

The Dynamics of Surprise

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Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic

Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams

This paper studies the logical context-sensitivity of Aristotelian diagrams. I propose a new account of measuring this type of context-sensitivity, and illustrate it by means of a small-scale example. Next, I turn toward a more large-scale case study, based on Aristotelian diagrams for the categorical statements with subject negation. On the practical side, I describe an interactive application that can help to explain and illustrate the phenomenon of context-sensitivity in this particular case study. On the theoretical side, I show that applying the proposed measure of context-sensitivity leads to a number of precise yet highly intuitive results.status: publishe

Een geünificeerde theorie van bepaalde en onbepaalde beschrijvingen

A recent topic in the study of descriptions is the possibility of dropping uniqueness from the semantic content of definite descriptions, thus attaining a unified quantificational account of the semantics of definite and indefinite descriptions. In this article, this unified account is presented and placed within a broader historical context. We argue that uniqueness is not as central to Russell's account of definite descriptions as is traditionally supposed and that there are good reasons to drop it. We provide an overview of the meaning differences between definite and indefinite descriptions and present an (independently motivated) pragmatic account of these differences.status: publishe

Aristotelian Diagrams in the Debate on Future Contingents

© 2018 Springer Science+Business Media B.V., part of Springer Nature In the recent debate on future contingents and the nature of the future, authors such as G. A. Boyd, W. L. Craig, and E. Hess have made use of various logical notions, such as (the difference between) the Aristotelian relations of contradiction and contrariety, and the ‘open future square of opposition.’ My aim in this paper is not to enter into this philosophical debate itself, but rather to highlight, at a more abstract methodological level, the important role that Aristotelian diagrams (such as the open future square of opposition, but also others) can play in organizing and clarifying the debate. After providing a brief survey of the specific ways in which Boyd and Hess make use of Aristotelian relations and diagrams in the debate on the nature of the future, I argue that the position of open theism is best represented by means of a hexagon of opposition (rather than a square of opposition). Next, I show that on the classical theist account, this hexagon of opposition ‘collapses’ into a single pair of contradictory statements. This collapse from a hexagon into a pair has several aspects, which can all be seen as different manifestations of a single underlying change (viz., the move from a tripartition to a bipartition of logical space).status: publishe

Agreeing to Disagree in Probabilistic Dynamic Epistemic Logic

This paper studies Aumann’s agreeing to disagree theorem from the perspective of dynamic epistemic logic. This was first done by Dégremont and Roy (J Phil Log 41:735–764, 2012) in the qualitative framework of plausibility models. The current paper uses a probabilistic framework, and thus stays closer to Aumann’s original formulation. The paper first introduces enriched probabilistic Kripke frames and models, and various ways of updating them. This framework is then used to prove several agreement theorems, which are natural formalizations of Aumann’s original result. Furthermore, a sound and complete axiomatization of a dynamic agreement logic is provided, in which one of these agreement theorems can be derived syntactically. These technical results are used to show the importance of explicitly representing the dynamics behind the agreement theorem, and lead to a clarification of some conceptual issues surrounding the agreement theorem, in particular concerning the role of common knowledge. The formalization of the agreement theorem thus constitutes a concrete example of the so-called dynamic turn in logic.status: publishe

De dynamische wending in de epistemische logica

© 2016 by Tijdschrift voor Filosofie. All rights reserved. This article describes the historical development of epistemic logic, focusing on the dynamic turn that has taken place in the last few decades. Although this dynamic turn was mainly motivated by technical considerations in computer science and game theory, it is argued that it can also be relevant from a more philosophical perspective: the application of dynamic epistemic logics to analyze prima facie static notions, theorems, etc. fits perfectly in a Wittgensteinian approach to philosophy as conceptual elucidation. Furthermore, it turns out that these conceptual elucidations often lead to a number of other advantages, such as a higher degree of empirical adequacy. In order to illustrate this line of argumentation, the system of public announcement logic is presented, and it is shown how this system can be used to analyze the psychological phenomenon of surprise in a conceptually and empirically fruitful way.status: publishe

Aristotelian Diagrams for Semantic and Syntactic Consequence

© 2018, Springer Nature B.V. Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation.status: Published onlin