Abstract
Two propositions may be regarded as doxastically equivalent if revision of an agent’s beliefs to adopt either has the same effect on the agent’s belief state. We enrich the language of dynamic doxastic logic with formulas expressing this notion of equivalence, and provide it with a formal semantics. A finitary proof system is then defined and shown to be sound and complete for this semantics.
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Notes
- 1.
\(\phi \bowtie \psi \) could conveniently be pronounced ‘\(\phi \) tie \(\psi \)’, from the LaTeX control word \(\backslash \) bowtie for the symbol \(\bowtie \).
- 2.
- 3.
The members of \( Prop \) are sometimes called the admissible propositions of the structure, to distinguish them from other subsets of \(U\). See [5].
- 4.
incl stands for ‘inclusion’, moneys for ‘monotonicity for nonempty segments’, and arrow is named in honour of Kenneth Arrow. See [14], p. 232.
- 5.
Except that in \((\Box )\) and \((\Box \)N), \(\theta \) and \(\omega \) must be pure Boolean when \(\Box \) is \(\mathbf {B}\) or \(\mathbf {K}\).
- 6.
In the models of [14], \( Prop \) is taken to be the set of clopen subsets of a topology on \(U\) that makes it a Stone space, i.e. compact and totally separated. It can be shown that \( Prop _L\) generates a Stone topology on \(U_L\), for which the clopen sets are precisely the members of \( Prop _L\). But we do not make any use of those additional properties.
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Acknowledgments
The \(\bowtie \) notation is due to Tim Stokes, who introduced the study of the concept it denotes as an operation in the algebra of binary relations [2, 3, 8, 15]. I am obliged to him for posing the question of its axiomatisation in the context of mult-modal logics, and for illuminating explanations and discussions about this. This paper also owes much to Krister Segerberg’s innovative contributions to the modelling of dynamic doxastic modalities, and indeed to his approach to the model-theoretic analysis of modalities in general.
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Goldblatt, R. (2014). Equivalent Beliefs in Dynamic Doxastic Logic . In: Trypuz, R. (eds) Krister Segerberg on Logic of Actions. Outstanding Contributions to Logic, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7046-1_9
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