Abstract
In this article we formalise the notion of knowing a secret as a modality, by combining standard notions of knowledge and ignorance from modal epistemic logic. Roughly speaking, Ann knows a secreet if and only if she knows it and she knows that everyone else does not know it. The main aim is to study the properties of these secretly knowing modalities. It turns out that the modalities are non-normal, and are characterised by a derivation rule we call Interpolation that is stronger than Equivalence but weaker than Monotonicity. We study the Interpolation rule and position it in the landscape of non-normal modal logics. We show that it, in combination with basic axioms, gives us a complete characterisation of the properties of the secretly knowing modalities under weak assumptions about the properties of individual knowledge, in the form of a sound and complete axiomatisation. This characterisation gives us the most basic and fundamental principles of secretly knowing.
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05 April 2023
A Correction to this paper has been published: https://doi.org/10.1007/s10472-022-09825-y
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Acknowledgements
We are indebted to two anonymous AMAI reviewers, who helped us improve the paper significantly. We are grateful to Frederik Van De Putte for discussions about non-normal logics with (K) and for sharing the manuscript [23] with us. We also thank anonymous reviewers of the LAMAS 2020 workshop for valuable suggestions, and participants of the workshop for their comments.
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Partial financial support was received by the first author by MOE Liberal arts and Social Sciences Foundation (project no. 20YJC7204002).
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Xiong, Z., Ågotnes, T. The logic of secrets and the interpolation rule. Ann Math Artif Intell 91, 375–407 (2023). https://doi.org/10.1007/s10472-022-09815-0
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DOI: https://doi.org/10.1007/s10472-022-09815-0