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

We consider a simple modal logic whose non-modal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.) T, S4 and S5, such logics are useful, as shown in previous work by Baltag, Coecke and the first author, for encoding and reasoning about information and misinformation in multi-agent systems. For such a logic we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of “nested” or “tree-sequent” calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario.

💡 Research Summary

The paper introduces a novel positive modal logic that deliberately omits negation and implication, yet equips the language with a pair of adjoint modal operators for each agent. The authors motivate this design by pointing out that many multi‑agent scenarios involve information that is not closed under classical necessity or possibility; agents may possess partial, uncertain, or even contradictory knowledge. To capture such phenomena they replace the usual closure‑based modalities with an adjoint pair ⟨a⟩ and