On monotone modalities and adjointness

We fix a logical connection (Stone Pred : Set op −→ BA given by 2 as a schizophrenic object) and study coalgebraic modal logic that is induced by a functor T : Set −→ Set which is finitary, standard, preserves weak pullbacks and finite sets. We prove that for any such T , the cover modality nabla is a left (and its dual delta is a right) adjoint relative to Pω. We then consider monotone unary modalities arising from the logical connection and show that they all are left (or right) adjoints relative to Pω. We are going to study universal properties of modalities in coalgebraic modal logic, considered as monotone operations on the set of modal formulas, preordered by the semantic consequence relation. In coalgebraic logic, there are essentially two approaches to modalities: modalities are given by predicate liftings (which can be viewed as modalities described in 2.B below), (Pattinson (2003)), or, in case of set-coalgebras, by cover modalities that identify the modalities with the coalgebra functor, (Moss (1999)). In any case one is naturally interested in adjointness properties of the modalities that would entail their " nice " behaviour, e.g., with respect to suprema/infima in the consequence preorder. However, it can be almost immediately seen that a monotone modality can rarely be a left or right adjoint, since, typically, a modality preserves only some and not all suprema/infima. In this paper, we show that all cover modalities and all monotone unary modalities indeed enjoy adjointness properties in a weaker sense: the desired left/right adjoints do 2 exist if we require the adjointness property to hold only relative to the doctrine P ω of free join-semilattices, see Definition 2.11 below for details. In fact, as we argue below, this weak notion of adjointness is, when proper adjunction is not available, the " second best " one can hope for in case of finitary languages. Moreover, such adjointness has a proof-theoretic significance: proper adjointness is closely related to a possibility to formulate a sound and invertible rule for the operator in question. The above weaker adjointness property indicates a possibility of formulating a weakly invertible rule. The rule, read backwards, gives finitely many possible continuations of the proof search — the situation well-known in proof theory of modal logics. Adjointness properties of modalities only make sense, of course, for monotone ones. Monotone modalities yield expressive languages for coalgebraic functors preserving …