## CONTENTS

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PREFACE

1. A NOTE ON NOTATION

 Introduction Detachment substitution and 'application' The 'D-notation'

2. GROUNDWORK -- THE PROPOSITIONAL CALCULUS

 Building blocks Implicational systems -- positive implication Deduction Some ICI theses The semisubstitutivity of implication Classical implication Conjunction The full positive logic Classical C-K-A -- the redundancy of A Negation: the full system IC Relationships between IC connectives Classical negation: the full system PC Alternative bases for PC

3. GROUNDWORK--PROPOSITIONAL SEQUENT-LOGIC

 Introduction Formulas, sequences, and sequents Sequent generation Sequent-logic and propositional calculus The 'normal form' theorem The rule of contraction and its elimination Inversion of rules Contraction elimination once again

4. GROUNDWORK -- PROPOSITIONAL LOGIC AND TABLEAUX

 Proof tableaux Proof tableaux -- formal characterization Semantic tableaux

5. THE ABSOLUTELY STRICT SYSTEMS -- S1° AND S1

 Introduction The system S1° The classical PC in S1° Further theses and rules of S1° The semisubstitutivity of strict implication and S1° The system S1; truth-value systems Some work within the system S1

6. THE ABSOLUTELY STRICT SYSTEMS -- S2°, S2, T°, AND T

 The system S2° Semisubstitutivity of strict implication in S2° Some theorems of S2° The system S2 The systems T° and T Modalities in T and its included systems

7. THE ABSOLUTELY STRICT SYSTEMS -- ALTERNATIVE FORMULATIONS

 The 'Lemmon style' bases Other formulations

8. THE ABSOLUTELY STRICT SYSTEMS -- MODAL SEQUENT-LOGIC

 Introduction Sequent-logic versions of S1° and S1 The normal-form theorem in LS1° and LS1 Equivalence of the systems Sequent-logic versions of S2° and S2 Sequent-logic versions of T ° and T Decision procedures and the rule of contraction Modalities in T revisited

9. THE ABSOLUTELY STRICT SYSTEMS -- TABLEAUX

 Modal models Tableaux for modal systems Tableaux for T° and T Tableaux for S2° and S2 Tableaux for S1° and S1

10. THE SYSTEMS OF COMPLETE MODALIZATION -- S3° AND S3

 The system S3° Semisubstitutivity of strict implication in S3° More theorems of S3° Inclusion, containment, and independence with S3° and S3 The full system S3 S3 and complete modalization The 'unreasonableness' of S3 and its included systems

11. THE SYSTEMS OF COMPLETE MODALIZATION -- S4°, S4, AND S5

 The system S4° S4°, S4, and the previously discussed systems The full system S4 The system S5 Complete modalization in S5

12. THE SYSTEMS OF COMPLETE MODALIZATION -- ALTERNATIVE FORMULATIONS

 The Lemmon-style bases Other formulations Finite axiomatizability Formulations without axioms beyond those of the PC The deduction theorem for the systems of this chapter

13. THE SYSTEMS OF COMPLETE MODALIZATION -- SEQUENT-LOGIC, THE BASIC SYSTEMS

 Sequent-logic for S3° and S4° The normal form theorem in LS3° and LS4° The equivalence of the systems Sequent-logics for S3 and S4 The normal form theorem for LS3 and LS4 A relationship between intuitionist logic and S4 Sequent-logic for S5

14. THE SYSTEMS OF COMPLETE MODALIZATION -- BASIC SYSTEMS OF TABLEAUX

 Tableaux for S4° and S4 Tableaux for S3° and S3 Tableaux for S5

15. THE SYSTEMS OF COMPLETE MODALIZATION -- THE S4-S5 SPECTRUM AND RELATED SYSTEMS

 Introduction The system S4.3 The system S4.2 The Diodorean system D S4.4 and some associated systems Sobocinski's family K of 'non-Lewis modal' systems Semantics for S4.9; the 'last stop' before S5 Summary

APPENDIX: BASES FOR KEY SYSTEMS AND STRUCTURES STUDIED IN THIS BOOK

(A) Non-Modal Propositional Calculi

(A.1) Standard ('Hilbert-style') axiomatizations

 (A.1.1) Intuitionistic propositional calculus (IC) (A.1.2) Classical propositional calculus (PC).
(A.2) Non-modal sequent-logic
(A.3) Semantic tableaux for PC (structure MPC)

(B) Modal Calculi

 (B.1) 'Lewis-style' axiomatizations of S1°-S5 (B.2) 'Lemmon-style' bases for S1°-S5 (B.3) Systems between S4 and S5, and Sobocinski's 'non-Lewis modal' systems (B.4) Modal sequent-logic for S1°-S5 (B.5) Sequent-logic for systems between S4 and S5 (B.6) Tableaux for Modal Systems

BIBLIOGRAPHY

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