## A New Introduction to Modal LogicThis entirely new work guides the reader through the most basic systems of modal propositional logic up to systems of modal predicate with identity, dealing with both technical developments and discussing philosophical applications.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. |

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### Contents

The language of PC 3 Interpretation 4 Further operators | 6 |

Some valid wff of PC 13 Basic modal notions 13 The language | 17 |

The Systems K T and D | 23 |

and M 33 Validity and soundness 36 The system T 41 A definition | 45 |

The Systems S4 S5 B Triv and Ver | 51 |

Iterated modalities 51 The system S4 53 Modalities in S4 54 Validity | 60 |

Collapsing into PC 64 Exercises 3 68 Notes 70 | 70 |

Semantic diagrams 73 Alternatives in a diagram 80 S4 diagrams 85 | 85 |

Strict Implication | 193 |

200 Validity in S2 and S3 201 Entailment 202 Exercises 11 205 | 205 |

Axiomatic PC 210 Natural deduction 211 Multiply modal logics 217 | 217 |

logics 224 Syntactical approaches to modality 225 Probabilistic | 230 |

Interpretation 237 The Principle of replacement 240 Axiomatization | 241 |

modal LPC 243 Systems of modal predicate logic 244 Theorems | 250 |

The Completeness of Modal LPC | 256 |

Canonical models for Modal LPC 256 Completeness in modal | 262 |

S5diagrams 91 Exercises 4 92 Notes 93 | 93 |

Conjunctive Normal Form | 94 |

Equivalence transformations 94 Conjunctive normal form 96 Modal | 108 |

Maximal consistent sets of wff 113 Maximal consistent extensions 114 | 114 |

Consistent sets of wff in modal systems 116 Canonical models 117 | 121 |

Canonical Models | 127 |

Convergence 134 The frames of canonical models 136 A noncanonical | 142 |

the finite model property 153 Exercises 8 156 Notes 156 | 156 |

Frames and models 159 An incomplete modal system 160 KH and | 164 |

Frames and Systems | 172 |

Frames for T S4 B and S5 172 Irreflexiveness 176 Compactness | 188 |

monadic modal LPC 271 Exercises 14 272 Notes 272 | 272 |

Canonical models without BF 280 Completeness 282 Incompleteness | 287 |

Modality and Existence | 289 |

Changing domains 289 The existence predicate 292 Axiomatization | 302 |

Identity in LPC 312 Soundness and completeness 314 Definite | 329 |

Contingent identity 330 Contingent identity systems 334 Quantifying | 342 |

Firstorder modal theories 349 Multiple indexing 350 Counterpart | 358 |

Bibliography | 384 |

398 | |

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### Common terms and phrases

assignment axiom axiomatic Barcan Formula called canonical model Chapter characterized class of frames conjunction conjunctive normal form contains counterpart theory D L(p defined definition diagram disjunction equivalent fact false falsifying finite model property finite number frame W,R G Dw Hughes and Cresswell inconsistent individual variables induction infinitely intensional objects interpretation irreflexive K-valid L(Lp L(p D q lemma Lemmon Lp A Lq Lp D LLp maximal consistent sets MCNF means modal LPC modal operators modal predicate logic model based natural deduction non-modal normal modal system obtained PC wff PC-valid player propositional logic propositional variables prove quantifier rectangle reflexive frames replaced result S-consistent satisfies second-order logic semantics set of wff subset substitution-instance Suppose symbols tense logic transformation rules Triv true truth-functional truth-value V-property valid wff value-assignment values wff of modal wff of PC