We present a finitary version of Moss' coalgebraic logic for $T$-coalgebras, where $T$ is a locally monotone endofunctor of the category of posets and monotone maps. The logic uses a single cover modality whose arity is given by the least finitary subfunctor of the dual of the coalgebra functor $T_\omega^\partial$, and the semantics of the modality is given by relation lifting. For the semantics to work, $T$ is required to preserve exact squares. For the finitary setting to work, $T_\omega^\partial$ is required to preserve finite intersections. We develop a notion of a base for subobjects of $T_\omega X$. This in particular allows us to talk about the finite poset of subformulas for a given formula. The notion of a base is introduced generally for a category equipped with a suitable factorisation system. We prove that the resulting logic has the Hennessy-Milner property for the notion of similarity based on the notion of relation lifting. We define a sequent proof system for the logic, and prove its completeness.

• 03B38 - Type theory
• 18M05 - Monoidal categories, symmetric monoidal categories
• 68N15 - Theory of programming languages
• 68N19 - Other programming paradigms (object-oriented, sequential, concurrent, automatic, etc.)
• 68Q12 - Quantum algorithms and complexity in the theory of computing
• 81P68 - Quantum computation