DS-2017-07: Evidence in Epistemic Logic: A Topological Perspective

DS-2017-07: Aybüke Özgün (2017) Evidence in Epistemic Logic: A Topological Perspective. Doctoral thesis, University of Amsterdam.

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Abstract

This dissertation studies logics of knowledge, belief and information dynamics using topological spaces as models. It is concerned with the formal representation of evidence and its link to justification, justified belief, knowledge, and evidence-based information dynamics. Topological spaces emerge naturally as mathematically elegant and epistemically rich information structures to formalize these epistemic notions. In the following, we give an overview of the content of this thesis.

Part I investigates the role of evidence in forming justified belief and knowledge of a rational idealized agent, where evidence is represented semantically as sets of possible worlds, as well as syntactically via evidence modalities.

Chapter 3 provides background material and motivation for Part I. It introduces the interior-based topological semantics for the basic modal language, focusing on its epistemic interpretation. In this chapter, we motivate the use of topological spaces as models for knowledge, and discuss the status quo of the use of topological spaces as belief models.

Chapter 4 focuses primarily on a topological interpretation of belief: how topological models can contribute to the semantics of existing epistemic/doxastic logics. In particular, we study a notion of belief defined as epistemic possibility of knowledge, whose topological semantics is derived from the interior semantics for knowledge presented in Chapter 3. We provide soundness and completeness results for the belief logic KD45_B with respect to the class of extremally and hereditarily extremally disconnected spaces, and study public announcements based on topological models in the latter class. The notion of evidence in this setting is described at a purely semantic level as the corresponding syntax does not have any components representing evidence.

Chapter 5 presents the main contribution of Part I. We propose a topological semantics for various notions of evidence, evidence-based justification, belief, and knowledge, and explore the connections between these epistemic notions. The corresponding syntax bears evidence modalities, making various notions of evidence an explicit part of the logic. Our investigations in this chapter are not limited to a static setting. We discuss evidence-based actions such as evidence addition, upgrade, and feasible evidence combination as well as receiving information from infallible truthful sources via public announcements. Our main technical results are concerned with completeness, decidability and the finite model property for the associated logics. These investigations have philosophical consequences, as they allow us to formalize some post-Gettier debates surrounding justified belief and knowledge.

Part II focuses on knowledge and knowledge change. More precisely, it studies the notions of absolutely certain knowledge and knowability as potential knowledge, as well as the interplay between the notion of epistemic effort encompassing any method of evidence acquisition and the well-studied dynamic attitudes such as public and arbitrary public announcements.

Chapter 6 provides the background material of Part II, introducing subset space semantics and a topological version of public announcements.

Chapter 7 designs a formal framework elucidating the relationship between three dynamic notions of interest: effort, public announcements, and arbitrary announcements. While the established link between effort and public announcements makes the meaning of the intended notion of effort more transparent, our technical results concerning expressivity and completeness simplify, and in a sense, improve on some of the earlier approaches.

Finally, in Chapter 8, we generalize the single-agent setting presented in Chapter 7 to a multi-agent setting. We present a multi-agent logic of knowledge and knowability, as well as its extensions with public and arbitrary announcements, interpreted on topological spaces. We provide soundness and completeness results for the corresponding systems.

To sum up, this dissertation on one hand re-interprets some existing epistemic and doxastic logics and their dynamic extensions from a topological perspective, providing an evidence-based interpretation. On the other hand, it uses topological tools to refine and extend earlier analysis, leading to novel logics of evidence and information dynamics.

Item Type:	Thesis (Doctoral)
Report Nr:	DS-2017-07
Series Name:	ILLC Dissertation (DS) Series
Year:	2017
Subjects:	Language Logic
Depositing User:	Dr Marco Vervoort
Date Deposited:	14 Jun 2022 15:17
Last Modified:	14 Jun 2022 15:17
URI:	https://eprints.illc.uva.nl/id/eprint/2147

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