# A New Introduction to Modal Logic

## Synopsis

This long-awaited book replaces Hughes and Cresswell's two classic studies of modal logic: An Introduction to Modal Logic and A Companion to Modal Logic . A New Introduction to Modal Logic is an entirely new work, completely re-written by the authors. They have incorporated all the new developments that have taken place since 1968 in both modal propositional logic and modal predicate logic, without sacrificing the clarity of exposition and approachability that were essential features of their earlier works.The book takes readers from the most basic systems of modal propositional logic right up to systems of modal predicate with identity. It covers both technical developments such as completeness and incompleteness, and finite and infinite models, and their philosophical applications, especially in the area of modal predicate logic.

## Excerpt

Modal logic is the logic of necessity and possibility, of 'must be' and 'may be'. By this is meant that it considers not only truth and falsity applied to what is or is not so as things actually stand, but considers what would be so if things were different. If we think of how things are as the actual world then we may think of how things might have been as how things are in an alternative, non-actual but possible, state of affairs—or possible world. Logic is concerned with truth and falsity. In modal logic we are concerned with truth or falsity in other possible worlds as well as the real one. In this sense a proposition will be necessary in a world if it is true in all worlds which are possible relative to that world, and possible in a world if it is true in at least one world possible relative to that world. All this is explained in the first chapter of this book.

Our aim in this book is to introduce readers to modal logic, and we assume that to begin with the reader knows nothing of modal logic. We have attempted to make the book self contained so that it could even be tackled by someone who had not studied any logic at all. However, we anticipate that most readers will already know a little about the (non-modal) propositional and predicate calculi, and will be able to use this knowledge as a foundation for understanding modal logic.

This book is intended as a replacement for our earlier two books An Introduction to Modal Logic (Hughes and Cresswell, 1968, IML) and A Companion to Modal Logic (Hughes and Cresswell, 1984, CML) and we shall here say a little about the relation between it and the earlier books. Part I covers most of the ground covered in IML with two important changes. First, as in CML, we take the system K as basic rather than T. Second, as also in CML, we have (in Chapter 6) used the method of canonical models to prove completeness. We have retained (in Chapter 5) the method of modal conjunctive normal forms to prove the completeness of S5, but while (in Chapter 4) we have retained from IML the method of semantic diagrams for testing formulae, we have omitted the completeness proofs based on this method.

Part II covers a range of topics in modal propositional logic, most of which are also discussed in CML. In the present work we have attempted to be particularly sensitive to its role as an introduction. Thus, to take one . . .

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