The logic of secrets and the interpolation rule

• Published in: Annals of mathematics and artificial intelligence Vol. 91; no. 4; pp. 375 - 407
• Main Authors: Xiong, Zuojun, Ågotnes, Thomas
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2023; Springer; Springer Nature B.V
• Subjects: Access control; Artificial Intelligence; Axioms; Complex Systems; Computer Science; Dictionaries; Epistemology; Interpolation; Knowledge; Logic; Mathematics; Semantics; Non-normal modal logic; Doxastic logic; MCS Primary 03B42; Secondary 03B45; Epistemic logic; Completeness
• ISSN: 1012-2443; 1573-7470
• DOI: 10.1007/s10472-022-09815-0

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.