Abstract
We introduce a new topological semantics for belief logics in which the belief modality is interpreted as the interior of the closure of the interior operator. We show that the system wKD45, a weakened version of KD45, is sound and complete with respect to the class of all topological spaces. While generalizing the topological belief semantics proposed in [1, 2] to all spaces, we model conditional beliefs and updates and give complete axiomatizations of the corresponding logics with respect to the class of all topological spaces.
A. Özgün—Acknowledges support from European Research Council grant EPS 313360.
S. Smets—Contribution to this paper has received funding from the European Research Council under the European Community’s 7th Framework Programme/ERC Grant agreement no. 283963.
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Notes
- 1.
- 2.
For a more detailed discussion on Stalnaker’s approach, we refer the reader to [2].
- 3.
\(\langle K\rangle \) denotes the dual of K, i.e., \(\lnot K\lnot \varphi :=\langle K\rangle \varphi \).
- 4.
Originally, McKinsey and Tarski [23] introduce the interior semantics for the basic modal language. Since we talk about this semantics in the context of knowledge, we use the basic epistemic language.
- 5.
- 6.
A set \(A\subseteq X\) is called an upset of (X, R) if for each \(x, y\in X\), xRy and \(x\in A\) imply \(y\in A\).
- 7.
A topological property is said to be hereditary if for any topological space \((X, \tau )\) that has the property, every subspace of \((X, \tau )\) also has it [14, p. 68].
- 8.
A subset \(A\subseteq X\) is called nowhere dense in \((X, \tau )\) if \(\mathrm {Int}(\mathrm {Cl}(A))=\emptyset \).
- 9.
A subset \(A\subseteq X\) of a topological space \((X, \tau )\) satisfying the condition \(A=\mathrm {Int}(\mathrm {Cl}(A)\) is called regular open [14].
- 10.
In fact, for any \(A\subseteq X\), the set \(\mathrm {Int}(\mathrm {Cl}(A))\) is regular open, however, it is not always the case that \(A\subseteq \mathrm {Int}(\mathrm {Cl}(A))\).
- 11.
Brushes and pins are introduced in [25] and a similar terminology is used in this paper.
- 12.
In [2], we propose topological semantics for conditional beliefs based on hereditarily extremally disconnected spaces.
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We thank the anonymous referees for their valuable comments that help us improve the presentation of the paper significantly.
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Baltag, A., Bezhanishvili, N., Özgün, A., Smets, S. (2017). The Topology of Full and Weak Belief. In: Hansen, H., Murray, S., Sadrzadeh, M., Zeevat, H. (eds) Logic, Language, and Computation. TbiLLC 2015. Lecture Notes in Computer Science(), vol 10148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54332-0_12
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