A NEW INTRODUCTION TO MODAL LOGIC G. E. Hughes Late Professor of Philosophy Victoria University of Wellington
M. J. Cresswell Professor of Philosophy...
A NEW INTRODUCTION TO MODAL LOGIC G. E. Hughes Late Professor of Philosophy Victoria University of Wellington
M. J. Cresswell Professor of Philosophy Victoria University of Wellington
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London and New York
CONTENTS
Preface
ix
Part One: Basic Modal Propositional Logic 1 The Basic Notions
3
The language of PC (3) Interpretation (4) Further operators (6) Interpretation of A , D and = (7) Validity (8) Testing for validity: (i) the truth-table method (10) Testing for validity: (ii) the Reductio method (11) Some valid wff of PC (13) Basic modal notions (13) The language of propositional modal logic (16) Validity in propositional modal logic (17) Exercises — 1 (21) Notes (22) 2 The Systems K, T and D
23
Systems of modal logic (23) The system K (24) Proofs of theorems (26) L and M (33) Validity and soundness (36) The system T (41) A definition of validity for T (43) The system D (43) A note on derived rules (45) Consistency (46) Constant wff (47) Exercises — 2 (48) Notes (49) 3 The Systems S4, S5, B, Triv and Ver
51
Iterated modalities (51) The system S4 (53) Modalities in S4 (54) Validity for S4 (56) The system S5 (58) Modalities in S5 (59) Validity for S5 (60) The Brouwerian system (62) Validity for B (63) Some other systems (64) Collapsing into PC (64) Exercises — 3 (68) Notes (70) 4 Testing for validity
72
Semantic diagrams (73) Alternatives in a diagram (80) S4 diagrams (85) S5-diagrams (91) Exercises — 4 (92) Notes (93)
A NEW INTRODUCTION TO MODAL LOGIC
5 Conjunctive Normal Form
94
Equivalence transformations (94) Conjunctive normal form (96) Modal functions and modal degree (97) S5 reduction theorem (98) MCNF theorem (101) Testing formulae in MCNF (103) The completeness of S5 (105) A decision procedure for S5-validity (108) Triv and Ver again (108) Exercises — 5 (110) Notes (110) 6 Completeness
111
Maximal consistent sets of wff (113) Maximal consistent extensions (114) Consistent sets of wff in modal systems (116) Canonical models (117) The completeness of K, T, B, S4 and S5 (119) Triv and Ver again (121) Exercises — 6 (122) Notes (123)
Part Two: Normal Modal Systems 7 Canonical Models
127
Temporal interpretations of modal logic (127) Ending time (131) Convergence (134) The frames of canonical models (136) A non-canonical system (139) Exercises — 7 (141) Notes (142) 8 Finite Models
145
The finite model property (145) Establishing the finite model property (145) The completeness of KW (150) Decidability (152) Systems without the finite model property (153) Exercises — 8 (156) Notes (156) 9 Incompleteness
159
Frames and models (159) An incomplete modal system (160) KH and KW (164) Completeness and the finite model property (165) General frames (166) What might we understand by incompleteness? (168) Exercises — 9 (169) Notes (170) 10
Frames and Systems
172
Frames for T, S4, B and S5 (172) Irreflexiveness (176) Compactness (177) S4.3.1 (179) First-order definability (181) Second-order logic (188) Exercises — 10 (189) Notes (190)
VI
CONTENTS
11
Strict Implication
193
Historical preamble (193) The 'paradoxes of implication' (194) Material and strict implication (195} The 'Lewis' systems (197) The system SI (198) Lemmon's basis for SI (199) The system S2 (200) The system S3 (200) Validity in S2 and S3 (201) Entailment (202) Exercises — 11 (205) Notes (206) 12
Glimpses Beyond
210
Axiomatic PC (210) Natural deduction (211) Multiply modal logics (217) The expressive power of multi-modal logics (219) Propositional symbols (220) Dynamic logic (220) Neighbourhood semantics (221) Intermediate logics (224) 'Syntactical' approaches to modality (225) Probabilistic semantics (227) Algebraic semantics (229) Exercises — 12 (229) Notes (230)
Part Three: Modal Predicate Logic 13
The Lower Predicate Calculus
235
Primitive symbols and formation rules of non-modal LPC (235) Interpretation (237) The Principle of replacement (240) Axiomatization (241) Some theorems of LPC (242) Modal LPC (243) Semantics for modal LPC (243) Systems of modal predicate logic (244) Theorems of modal LPC (244) Validity and soundness (247) De re and de dicto (250) Exercises - 13 (254) Notes (255) 14
The Completeness of Modal LPC
256
Canonical models for Modal LPC (256) Completeness in modal LPC (262) Incompleteness (265) Other incompleteness results (270) The monadic modal LPC (271) Exercises — 14 (272) Notes (272) 15
Expanding Domains
274
Validity without the Barcan Formula (274) Undefined formulae (277) Canonical models without BF (280) Completeness (282) Incompleteness without the Barcan Formula (283) LPC + S4.4 (S4.9) (283) Exercises — 15 (287) Notes (287)
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A NEW INTRODUCTION TO MODAL LOGIC
16
Modality and Existence
289
Changing domains (289) The existence predicate (292) Axiomatization of systems with an existence predicate (293) Completeness for existence predicates (296) Incompleteness (302) Expanding languages (302) Possibilist quantification revisited (303) Kripke-style systems (304) Completeness of Kripke-style systems (306) Exercises — 16 (309) Notes (310) 17
Identity and Descriptions
312
Identity in LPC (312) Soundness and completeness (314) Definite descriptions (318) Descriptions and scope (323) Individual constants and function symbols (327) Exercises — 17 (328) Notes (329) 18
Intensional Objects
330
Contingent identity (330) Contingent identity systems (334) Quantifying over all intensional objects (335) Intensional objects and descriptions (342) Intensional predicates (344) Exercises — 18 (347) Notes (348) 19 Further Issues
349
First-order modal theories (349) Multiple indexing (350) Counterpart theory (353) Counterparts or intensional objects? (357) Notes (358) Axioms, Rules and Systems
359
Axioms for normal systems (359) Some normal systems (361) Nonnormal systems (363) Modal predicate logic (365) Table I: Normal Modal Systems (367) Table II: Non-normal Modal Systems (368) Solutions to Selected Exercises
