Abstract
We introduce a multi-agent topological semantics for evidence-based belief and knowledge, which extends the dense interior semantics developed in [2]. We provide the complete logic of this multi-agent framework together with generic models for a fragment of the language. We also define a new notion of group knowledge which differs conceptually from previous approaches.
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Notes
- 1.
This paper is based on Saúl Fernández González’s Master’s thesis [10].
- 2.
A set \(U\subseteq X\) is dense whenever \({{\,\mathrm{Cl}\,}}U=X\) or equivalently whenever \(U\cap V\ne \varnothing \) for all nonempty open sets V.
- 3.
We recall that \(\mathsf {S4}\) is the least normal modal logic containing the axioms (T) \(\square \phi \rightarrow \phi \) and (4) \(\square \phi \rightarrow \square \square \phi \); that \(\mathsf {S5}\) is \(\mathsf {S4}\) plus the axiom (5) \(\lnot \square \phi \rightarrow \square \lnot \square \phi \), and that \(\mathsf {S4.2}\) is \(\mathsf {S4}\) plus the axiom (.2) \(\Diamond \square \phi \rightarrow \square \Diamond \phi \).
- 4.
\(K\phi \equiv \square \phi \wedge [\forall ]\square \Diamond \phi \) and \(B\phi \equiv \lnot K \lnot K\phi \).
- 5.
In [10] this is discussed in more depth and an example is provided.
References
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Acknowledgement
We wish to thank Guram Bezhanishvili for very fruitful discussions on these topics. Special thanks go to the reviewers for their very valuable comments, which helped us improve the paper.
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Baltag, A., Bezhanishvili, N., Fernández González, S. (2022). Topological Evidence Logics: Multi-agent Setting. In: Özgün, A., Zinova, Y. (eds) Language, Logic, and Computation. TbiLLC 2019. Lecture Notes in Computer Science, vol 13206. Springer, Cham. https://doi.org/10.1007/978-3-030-98479-3_12
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