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: 223
Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 8
Preface......Page 12
Note on references......Page 18
The propositional calculus......Page 20
Modal propositional logic......Page 22
Normal modal systems......Page 23
Models......Page 26
Validity......Page 28
Some extensions of K......Page 29
Validity-preservingness in a model......Page 31
Notes......Page 33
Completeness and consistency......Page 35
Maximal consistent sets of wff......Page 37
Canonical models......Page 41
The completeness of K, T, S4, B and S5......Page 46
Three further systems......Page 48
Dead ends......Page 52
Exercises......Page 57
Notes......Page 58
General characterization theorems......Page 60
Conditions not corresponding to any axiom......Page 66
Notes......Page 70
4
Completeness and incompleteness in modal logic......Page 71
Frames and completeness......Page 72
An incomplete normal modal system......Page 76
General frames......Page 81
What might we understand by incompleteness?......Page 84
Notes......Page 85
Equivalent models and equivalent frames......Page 87
Pseudo-epimorphisms......Page 89
Distinguishable models......Page 94
Generated frames......Page 96
S4.3 reconsidered......Page 100
Exercises......Page 105
Notes......Page 106
6
Frames and systems......Page 108
Frames for T, S4, B and S5......Page 109
The frames of canonical models......Page 113
Establishing the rule of disjunction......Page 117
A complete but non-canonical system......Page 119
Compactness......Page 122
Notes......Page 129
7
Subordination frames......Page 131
The canonical subordination frame......Page 132
Proving completeness by the subordination method......Page 134
Tree frames......Page 137
S4.3 and linearity......Page 142
Exercises......Page 152
Notes......Page 153
The finite model property......Page 154
Filtrations......Page 155
Proving that a system has the finite model property......Page 160
The completeness of KW......Page 164
Characterization by classes of finite models......Page 167
The finite frame property......Page 168
Decidability......Page 171
Systems without the finite model property......Page 173
Exercises......Page 180
Notes......Page 181
Notation and formation rules for modal LPC......Page 183
Modal predicate systems......Page 184
Models......Page 186
Validity and soundness......Page 187
The for-all property......Page 191
Canonical models for S + BF systems......Page 195
General questions about completeness in modal LPC......Page 198
Notes......Page 203
B-C......Page 205
F-G-H-L-M......Page 206
P-R-S-T-U-Z......Page 207
A-B-C......Page 208
D......Page 209
E-F-G......Page 210
I-L-M-N-P-R-S......Page 211
T-U-V......Page 212
W......Page 213
List of axioms for propositional systems......Page 214
A-B-C......Page 216
D-E-F......Page 217
H-I-J-K......Page 218
L-M-N-O-P......Page 219
R-S......Page 220
T-U-V-W-Z......Page 221
