A NEW INTRODUCTION
TO MODAL LOGIC
Preface ix
Part One: Basic Modal Propositional Logic
1 The Basic Notions 3
The language of PC C) Interpretation D) Further operators F)
Interpretation of A , D and s G) Validity (8) Testing for validity: (i) the
truth-table method A0) Testing for validity: (ii) the Reductio method A1)
Some valid wff of PC A3) Basic modal notions A3) The language of
propositional modal logic A6) Validity in propositional modal logic A7)
Exercises — 1 B1) Notes B2)
2 The Systems K, T and D 23
Systems of modal logic B3) The system K B4) Proofs of theorems B6)
L and M C3) Validity and soundness C6) The system T D1) A definition
of validity for T D3) The system D D3) A note on derived rules D5)
Consistency D6) Constant wff D7) Exercises — 2 D8) Notes D9)
3 The Systems S4, S5, B, Triv and Ver 51
Iterated modalities E1) The system S4 E3) Modalities in S4 E4) Validity
for S4 E6) The system S5 E8) Modalities in S5 E9) Validity for S5 F0)
The Brouwerian system F2) Validity for B F3) Some other systems F4)
Collapsing into PC F4) Exercises — 3 F8) Notes G0)
4 Testing for validity 72
Semantic diagrams G3) Alternatives in a diagram (80) S4 diagrams (85)
S5-diagrams (91) Exercises — 4 (92) Notes (93)
5 Conjunctive Normal Form 94
Equivalence transformations (94) Conjunctive normal form (96) Modal
functions and modal degree (97) S5 reduction theorem (98) MCNF
theorem A01) Testing formulae in MCNF A03) The completeness of S5
A05) A decision procedure for S5-validity A08) Triv and Ver again A08)
Exercises — 5 A10) Notes A10)
6 Completeness 111
Maximal consistent sets of wff A13) Maximal consistent extensions A14)
Consistent sets of wff in modal systems A16) Canonical models A17) The
completeness of K, T, B, S4 and S5 A19) Triv and Ver again A21)
Exercises — 6 A22) Notes A23)
Part Two: Normal Modal Systems
7 Canonical Models 127
Temporal interpretations of modal logic A27) Ending time A31)
Convergence A34) The frames of canonical models A36) A non-canonical
system A39) Exercises — 7 A41) Notes A42)
8 Finite Models 145
The finite model property A45) Establishing the finite model property
A45) The completeness of KW A50) Decidability A52) Systems without
the finite model property A53) Exercises — 8 A56) Notes A56)
9 Incompleteness 159
Frames and models A59) An incomplete modal system A60) KH and KW
A64) Completeness and the finite model property A65) General frames
A66) What might we understand by incompleteness? A68) Exercises —
9 A69) Notes A70)
10 Frames and Systems 172
Frames for T, S4, B and S5 A72) Irreflexiveness A76) Compactness
A77) S4.3.1 A79) First-order definability A81) Second-order logic A88)
Exercises — 10 A89) Notes A90)
11 Strict Implication 193
Historical preamble A93) The 'paradoxes of implication' A94) Material
and strict implication A95) The 'Lewis' systems A97) The system SI
A98) Lemmon's basis for SI A99) The system S2 B00) The system S3
B00) Validity in S2 and S3 B01) Entailment B02) Exercises — 11 B05)
Notes B06)
12 Glimpses Beyond 210
Axiomatic PC B10) Natural deduction B11) Multiply modal logics B17)
The expressive power of multi-modal logics B19) Propositional symbols
B20) Dynamic logic B20) Neighbourhood semantics B21) Intermediate
logics B24) 'Syntactical' approaches to modality B25) Probabilistic
semantics B27) Algebraic semantics B29) Exercises — 12 B29) Notes
B30)
Part Three: Modal Predicate Logic
13 The Lower Predicate Calculus 235
Primitive symbols and formation rules of non-modal LPC B35)
Interpretation B37) The Principle of replacement B40) Axiomatization
B41) Some theorems of LPC B42) Modal LPC B43) Semantics for
modal LPC B43) Systems of modal predicate logic B44) Theorems of
modal LPC B44) Validity and soundness B47) De re and de dicto B50)
Exercises — 13 B54) Notes B55)
14 The Completeness of Modal LPC 256
Canonical models for Modal LPC B56) Completeness in modal LPC
B62) Incompleteness B65) Other incompleteness results B70) The
monadic modal LPC B71) Exercises — 14 B72) Notes B72)
15 Expanding Domains 274
Validity without the Barcan Formula B74) Undefined formulae B77)
Canonical models without BF B80) Completeness B82) Incompleteness
without the Barcan Formula B83) LPC + S4.4 (S4.9) B83) Exercises —
15 B87) Notes B87)
16 Modality and Existence 289
Changing domains B89) The existence predicate B92) Axiomatization of
systems with an existence predicate B93) Completeness for existence
predicates B96) Incompleteness C02) Expanding languages C02)
Possibilist quantification revisited C03) Kripke-style systems C04)
Completeness of Kripke-style systems C06) Exercises — 16 C09) Notes
C10)
17 Identity and Descriptions 312
Identity in LPC C12) Soundness and completeness C14) Definite
descriptions C18) Descriptions and scope C23) Individual constants and
function symbols C27) Exercises — 17 C28) Notes C29)
18 Intensional Objects 330
Contingent identity C30) Contingent identity systems C34) Quantifying
over all intensional objects C35) Intensional objects and descriptions C42)
Intensional predicates C44) Exercises — 18 C47) Notes C48)
19 Further Issues 349
First-order modal theories C49) Multiple indexing C50) Counterpart
theory C53) Counterparts or intensional objects? C57) Notes C58)
Axioms, Rules and Systems 359
Axioms for normal systems C59) Some normal systems C61) Non-
normal systems C63) Modal predicate logic C65) Table I: Normal Modal
Systems C67) Table II: Non-normal Modal Systems C68)
Solutions to Selected Exercises 369
Bibliography 384
Index 398
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