Introduction

In Chapter 3 I claimed that modal logic should be viewed as a tool for describing and analysing properties of structures (that is, of labelled transition structures). How can this be, and how effective is this tool? The kind of simple things we might want to know about a relation are whether it is reflexive, symmetric, transitive, confluent, etc. We may want to know whether one relation is included in another, or is the converse of another, or whether one relation can be decomposed as the composite of two others, etc. We may want to know more complicated things like whether a relation is well-founded, or whether one relation is the ⋆-closure of another.

Remarkably, these and many other properties are characterized by quite simple modal formulas. It is this characterizing ability which makes modal logic such a powerful tool. Once it is understood, it can be seen that modal logic is a quite extensive part of full second order logic, and it is the ability to capture second order properties which gives it is power.

Some examples

As an illustration of the kind of thing we are going to do we begin with a quite simple example of a correspondence result. In this result we focus on one particular label with its associated relation ≺ and connective □.

PROPOSITION. For each structure A the conditions

• (i) The distinguished relation ≺ is reflexive.

• (ii) For each formula φ, A ⊩u □φ → φ.

• (iii) For some variable P, A ⊩u □P → P.