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Logics of Essence and Accident

In the literature, essence is formalized in two different ways, either de dicto, or de re. Following \cite{Marcos:2005}, we adopt its de dicto formalization: a formula is essential, if once it is true, it is necessarily true; otherwise, it is accidental. In this article, we study the model theory and axiomatization of the logic of essence and accident, i.e. the logic with essence operator (or accident operator) as the only primitive modality. We show that the logic of essence and accident is less expressive than modal logic on non-reflexive models, but the two logics are equally expressive on reflexive models. We prove that some frame properties are undefinable in the logic of essence and accident, while some are. We propose the suitable bisimulation for this logic, based on which we characterize the expressive power of this logic within modal logic and within first-order logic. We axiomatize this logic over various frame classes, among which the symmetric case is missing, and our method is more suitable than those in the literature. We also find a method to compute certain axioms used to axiomatize this logic over special frames in the literature. As a side effect, we answer some open questions raised in \cite{Marcos:2005}.