PP-2015-18: The topological theory of belief

PP-2015-18: Baltag, Alexandru and Bezhanishvili, Nick and Smets, Sonja (2015) The topological theory of belief. [Report]

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Stalnaker introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept which captures the `epistemic possibility of knowledge'. In this paper we first provide the most general extensional semantics for this concept of `strong belief', which validates the principles of Stalnaker's epistemic-doxastic logic. We show that this general extensional semantics is a topological semantics, based on so-called extremally disconnected topological spaces. It extends the standard topological interpretation of knowledge (as the interior operator) with a new topological semantics for belief. Formally, our belief modality is interpreted as the `closure of the interior'. We further prove that in this semantics the logic KD45 is sound and complete with respect to the class of extremally disconnected spaces and we compare our approach to a different topological setting in which belief is interpreted in terms of the derived set operator. In the second part of the paper we study (static) belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modalities, respectively. Our investigation of dynamic belief change, is based on hereditarily extremally disconnected spaces. The logic of belief KD45 is sound and complete with respect to the class of hereditarily extremally disconnected spaces (under our proposed semantics), while the logic of knowledge is required to be S4.3. Finally, we provide a complete axiomatization of the logic of conditional belief and knowledge, as well as a complete axiomatization of the corresponding dynamic logic.

Item Type: Report
Report Nr: PP-2015-18
Series Name: Prepublication (PP) Series
Year: 2015
Uncontrolled Keywords: epistemic and doxastic logic, topological semantics, (hereditarily) extremally disconnected spaces, conditional beliefs, and updates, completeness and axiomatization
Subjects: Logic
Depositing User: Nick Bezhanishvili
Date Deposited: 12 Oct 2016 14:37
Last Modified: 12 Oct 2016 14:37
URI: https://eprints.illc.uva.nl/id/eprint/529

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