Introduction

The propositional modal language is an extension of the pure propositional language formed by adding a battery of new 1-ary connectives (known informally as box connectives). Originally there was just one new connective □, however for many purposes it is necessary to add several (possibly infinitely many) such connectives [i], one for each element i of an index set I. Thus there are many possible modal languages, one for each index set I. The syntax, semantics, and proof systems associated with modal languages are designed to subsume those of the proposition language, in fact, propositional logic can be regarded as the extreme version of modal logic where I = ∅.

The element i of I are called labels and I itself is called the signature of the modal language. Thus two languages are identical precisely when they have the same signature. (We are never going to consider how one language may be be translated into another, so we need not worry about comparison of signatures.)

Unlike the propositional connectives ¬, →, ∧, ∨, ⊤, and ⊥, the box connectives [i] do not have a fixed interpretation. For each formula φ (of the modal language) we may use [i] to obtain a new formula

[i]φ

This may be read in several ways, and different readings suggest different semantics and proof systems.

This book is an introduction to modal logic, more precisely, to classically based propositional modal logic. There are few books on this subject and even fewer books worth looking at. None of these give an acceptable mathematically correct account of the subject. This book is a first attempt to fill that gap.

Apart from its mathematical clarity, some other features of the book are:

• The central concept of the book is that of a labelled transition structure, and polymodal languages are used from the beginning.

• Modal languages are viewed as a tool for analysing the properties of transition structures, not the other way round.

• There is not an overemphasis on syntactic (proof theoretic) matters.

• Nevertheless, a detailed explanation is given of the differences between the weak completeness and Kripke completeness of formal systems.

• Correspondence properties (the expressibility properties of modal languages) are stressed as an important tool.

• Bisimulations are used as a method of comparing transition structures.

• Each chapter has a decent selection of exercises and over one sixth of the book consists of a comprehensive set of solutions to these exercises.

The book is aimed primarily at a computer science readership. However there is no computer science in the book and very little material which is directly attributable to a computer science motivation. Thus the reader of the book may be interested in modal logic in its own right or because of one or several of its applications in computer science.

Introduction

This chapter is one of the first parts of this book that you should read. You might think this is a little strange since it is also one of the last things in the book. However, all of the material in this chapter should appear somewhere, and I believe that it is better if it is all in one place rather than scattered throughout the book. This position is as good as any.

Beginning

Put yourself in the position of a complete beginner to modal logic; someone who already knows the basics of propositional and predicate logic and who now wants to learn something of modal logic. (You may actually be such a person.) There are many reasons why you may want to do this, from mere curiosity to an eventual use in a particular application. What should you do to acquire this knowledge?

One thing you could do is attend a course on modal logic, but let us assume that this is not an option. The other thing to do is read various text books on the subject. Which ones? There are, in fact, only a few possibilities.

The first possibility is [29] by Hughes and Cress well, first published in 1968. This, at the time it was written, was the most comprehensive and accessible introduction to the subject. It contains descriptions of many of the systems that were, and to some extent still are, of interest.

This part includes all the proof theoretic machinery and results presented in this book. First, in the short motivating Chapter 7, various semantic consequence relations are introduced. Then, in Chapter 8, the notion of a standard formal system is developed. Such a system is given by a set of axioms and has a proof structure based on modus ponens and necessitation (a rule designed to cope with the box connectives). Once developed, this proof theoretic machinery has to be justified (in the sense that it has to be shown to be correct and powerful enough). This is done by proving a completeness theorem. Chapter 9 contains a completeness result which is applicable to all standard systems, and hence can be regarded as rather superficial. The proof of this result is important for the method used is applicable in many other situations. Chapter 10 contains a more refined completeness result which is widely, but not generally, applicable. This kind of completeness was first developed by Kripke and it was this advancement which brought modal logic out of the dark ages.

We have isolated three important properties of standard formal systems: being canonical, and having the fmp, both of which imply the third property of being Kripke-complete. We have also seen may examples of systems with these properties. In this part we look more closely at the connection between these properties.

In each of Chapters 14 and 15 we look at a system which has the fmp but is not canonical. Both of these systems have independent interest. In Chapter 16 we consider a system which is canonical but does not have the fmp. This system is custom built to have these properties but may, in time, be found to have interest in its own right. Finally, in Chapter 17, we look at two systems, one of which has all three properties and the other having none of the properties. Furthermore, these two systems have precisely the same class of unadorned models.

Taken as a whole these four chapters hint at some of the complexities that can arise in modal logic.