Abstract
In the previous paper with a similar title (see Shtakser in Stud Log 106(2):311–344, 2018), we presented a family of propositional epistemic logics whose languages are extended by two ingredients: (a) by quantification over modal (epistemic) operators or over agents of knowledge and (b) by predicate symbols that take modal (epistemic) operators (or agents) as arguments. We denoted this family by \({\mathcal {P}\mathcal {E}\mathcal {L}}_{(QK)}\). The family \({\mathcal {P}\mathcal {E}\mathcal {L}}_{(QK)}\) is defined on the basis of a decidable higher-order generalization of the loosely guarded fragment (HO-LGF) of first-order logic. And since HO-LGF is decidable, we obtain the decidability of logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{(QK)}\). In this paper we construct an alternative family of decidable propositional epistemic logics whose languages include ingredients (a) and (b). Denote this family by \({\mathcal {P}\mathcal {E}\mathcal {L}}^{alt}_{(QK)}\). Now we will use another decidable fragment of first-order logic: the two variable fragment of first-order logic with two equivalence relations (FO\(^2\)+2E) [the decidability of FO\(^2\)+2E was proved in Kieroński and Otto (J Symb Log 77(3):729–765, 2012)]. The families \({\mathcal {P}\mathcal {E}\mathcal {L}}^{alt}_{(QK)}\) and \({\mathcal {P}\mathcal {E}\mathcal {L}}_{(QK)}\) differ in the expressive power. In particular, we exhibit classes of epistemic sentences considered in works on first-order modal logic demonstrating this difference.
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Acknowledgements
I have dedicated the paper [24] to the memory of my teacher Prof. Alexander Chagrov. The content of this article was also discussed with him. I am grateful to Prof. Alexander Chagrov for his support, which I still feel. I would also like to express my appreciation for the time and effort of two referees, whose comments and criticism were extremely helpful. Referee 1 pointed out to me the very important reference [17] that contains more precise properties of the fragment FO\(^2\)+2E. I am especially grateful to referee 1 for this comment which greatly strengthened the results of the paper.
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Presented by Heinrich Wansing; Received December 12, 2017.
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Shtakser, G. Propositional Epistemic Logics with Quantification Over Agents of Knowledge (An Alternative Approach). Stud Logica 107, 753–780 (2019). https://doi.org/10.1007/s11225-018-9824-6
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DOI: https://doi.org/10.1007/s11225-018-9824-6