The Logic of Secrets, Somebody-Knows and Weak Forms of Conjunctive Distribution in Modal Logics

I will start with discussing formalising the notion of knowing a secret as a modality in modal logic, by combining standard notions of knowledge and ignorance from modal epistemic logic. Roughly speaking, Ann knows a secret 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 \emph{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.