Note: This book was later replaced by "A New Introduction to Modal Logic" (1996).
An earlier book of ours, entitled An Introduction to Modal Logic (IML), was published in 1968. When we wrote it, we were able to give a reasonably comprehensive survey of the state of modal logic at that time. We very much doubt, however, whether any comparable survey would be possible today, for, since 1968, the subject has developed vigorously in a wide variety of directions.
The present book is therefore not an attempt to update IML in the style of that work, but it is in some sense a sequel to it. The bulk of IML was concerned with the description of a range of particular modal systems. We have made no attempt here to survey the very large number of systems found in the recent literature. Good surveys of these will be found in Lemmon and Scott (1977), Segerberg (1971) and Chellas (1980), and we have not wished to duplicate the material found in these works. Our aim has been rather to concentrate on certain recent developments which concern questions about general properties of modal systems and which have, we believe, led to a genuine deepening of our understanding of modal logic. Most of the relevant material is, however, at present available only in journal articles, and then often in a form which is accessible only to a fairly experienced worker in the field. We have tried to make these important developments accessible to all students of modal logic,as we believe they should be.
Author(s): G. E. Hughes, M. J. Cresswell
Publisher: Methuen
Year: 1985
Language: English
Pages: xviii+203
Preface
Note on references
1 Normal propositional modal systems
The propositional calculus
Modal propositional logic
Normal modal systems
Models
Validity
Some extensions of K
Validity-preservingness in a model
Notes
2 Canonical models and completeness proofs
Completeness and consistency
Maximal consistent sets of wff
Canonical models
The completeness of K, T, S4, B and S5
Three further systems
Dead ends
Exercises —2
Notes
3 More results about characterization
General characterization theorems
Conditions not corresponding to any axiom
Exercises —3
Notes
4 Completeness and incompleteness in modal logic
Frames and completeness
An incomplete normal modal system
General frames
What might we understand by incompleteness?
Exercises —4
Notes
5 Frames and models
Equivalent models and equivalent frames
Pseudo-epimorphisms
Distinguishable models
Generated frames
S4.3 reconsidered
Exercises —5
Notes
6 Frames and systems
Frames for T, S4, B and S5
The frames of canonical models
Establishing the rule of disjunction
A complete but non-canonical system
Compactness
Exercises —6
Notes
7 Subordination frames
The canonical subordination frame
Proving completeness by the subordination method
Tree frames
S4.3 and linearity
Systems not containing D
Exercises —7
Notes
8 Finite models
The finite model property
Filtrations
Proving that a system has the finite model property
The completeness of KW
Characterization by classes of finite models
The finite frame property
Decidability
Systems without the finite model property
Exercises —8
Notes
9 Modal predicate logic
Notation and formation rules for modal LPC
Modal predicate systems
Models
Validity and soundness
The V property
Canonical models for S + BF systems
General questions about completeness in modal LPC
Exercises —9
Notes
Bibliography
Glossary
List of axioms for propositional systems
Index
