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

TL;DR

This paper introduces a modal logic with adjoint pairs of modalities, providing algebraic semantics and a cut-free sequent calculus, to model and reason about information and misinformation in multi-agent systems.

Contribution

It presents a novel logic with adjoint modalities, an algebraic semantics, and a cut-free sequent calculus, enabling reasoning about information without negation or implication.

Findings

Algebraic semantics using lattices with adjoint operators

Cut-admissibility proven by syntactic methods

Application to reasoning about the muddy children puzzle

Abstract

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.