Equational Coalgebraic Logic

Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible one-step behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for T-coalgebras. We compare two existing proposals, Moss's coalgebraic logic and the logic of all predicate liftings, by providing one-step translations between them, extending the results in [Raul Andres Leal, Predicate liftings versus nabla modalities, Electr. Notes Theor. Comput. Sci. 203 (2008), no. 5, 195–220] by making systematic use of Stone duality. Our main contribution then is a novel coalgebraic logic, which can be seen as an equational axiomatization of Moss's logic. The three logics are equivalent for a natural but restricted class of functors. We give examples showing that the logics fall apart in general. Finally, we argue that the quest for a generic logic for T-coalgebras is still open in the general case.