Partially-ordered Modalities

Modal logic is extended by partially ordering the modalities. The modalities are normal, i.e., commute with either conjunctions or disjunctions and preserve either Truth or Falsity (respectively). The partial order does not conflict with type of modality (K, S4, etc.) although this paper will concentrate on S4 since partially ordered S4 systems appear to be numerous. The partially-ordered normal modal systems considered are both sound and complete. Hilbert and Gentzen systems are given. A cut-elimination theorem holds (for partially ordered S4), and the Hilbert and Gentzen systems present the same logic. The partial order induces a 2-category structure on a coalgebraic formulation of descriptive frames. Channel theory is used to ‘move’ modal logics when the source and target languages may be different. A particular partially ordered modal system is shown to be applicable to security properties.