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We consider a simple modal logic whose non-modal part has conjunction anddisjunction as connectives and whose modalities come in adjoint pairs, but arenot in general closure operators. Despite absence of negation and implication,and of axioms corresponding to the characteristic axioms of (e.g.) T, S4 andS5, such logics are useful, as shown in previous work by Baltag, Coecke and thefirst author, for encoding and reasoning about information and misinformationin multi-agent systems. For such a logic we present an algebraic semantics,using lattices with agent-indexed families of adjoint pairs of operators, and acut-free sequent calculus. The calculus exploits operators on sequents, in thestyle of "nested" or "tree-sequent" calculi; cut-admissibility is shown byconstructive syntactic methods. The applicability of the logic is illustratedby reasoning about the muddy children puzzle, for which the calculus isaugmented with extra rules to express the facts of the muddy children scenario.

Positive Logic with Adjoint Modalities: Proof Theory, Semantics and Reasoning about Information