The Review of Symbolic Logic

Research Article

POSITIVE LOGIC WITH ADJOINT MODALITIES: PROOF THEORY, SEMANTICS, AND REASONING ABOUT INFORMATION

MEHRNOOSH SADRZADEHa1 c1 and ROY DYCKHOFFa2 c2

a1 Computing Laboratory, University of Oxford

a2 School of Computer Science, St Andrews University

Abstract

We consider a simple modal logic whose nonmodal 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 multiagent systems. For the propositional-only fragment of such a dynamic epistemic 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.

(Received May 20 2009)

Correspondence:

c1 OXFORD UNIVERSITY COMPUTING LABORATORY, OXFORD, UK. E-mail: mehrs@comlab.ox.ac.uk

c2 SCHOOL OF COMPUTER SCIENCE, ST ANDREWS UNIVERSITY, ST ANDREWS, SCOTLAND, UK. E-mail: rd@st-andrews.ac.uk