Preface ......................................................... 7
1. Introduction ................................................. 9
1.1. Syntactic Refutations ................................... 9
1.2. The Classical Logic .................................... 11
1.3. The Intuitionistic Logic ............................... 13
1.4. Scott's Method ......................................... 14
1.5. Validity ............................................... 16
1.6. Modal Logics ........................................... 17
2. Syntactic Refutations in INT ................................ 20
2.1. Preliminaries .......................................... 20
2.2. Normal Forms ........................................... 21
2.3. The Rank of a Normal Form .............................. 23
2.4. Syntactic Reductions ................................... 23
2.5. Special Normal Forms of Rank 0 ......................... 24
2.6. A Refutation Procedure ................................. 25
3. Models for INT .............................................. 27
3.1. Preliminaries .......................................... 27
3.2. Syntactic Refutation Trees ............................. 27
3.3. Constructing Models .................................... 28
3.4. Validity ............................................... 29
3.5. The Jaskowski Trees .................................... 31
4. Algebraic Characterizations of INT .......................... 33
4.1. Preliminaries .......................................... 33
4.2. Finite Heyting Algebras ................................ 35
4.3. Jaskowski's Construction + ............................. 35
4.4. Jaskowski's Sequence ................................... 37
5. Syntactic Characterizations of INT .......................... 39
5.1. Formulas Characterizing Algebras ....................... 39
5.2. Formulas Corresponding to + ............................ 41
5.3. A Kind of Disjunction Property ......................... 43
5.4. Kleene's Property ...................................... 44
6. Syntactic Refutations in S4 ................................. 46
6.1. Preliminaries .......................................... 46
6.2. Normal Forms ........................................... 47
6.3. Syntactic Reductions ................................... 48
6.4. A Refutation Procedure ................................. 50
7. Models for S4 ............................................... 52
7.1. Preliminaries .......................................... 52
7.2. Constructing Models .................................... 52
7.3. Validity ............................................... 53
7.4. More Valid Rules ....................................... 54
8. Algebraic Characterizations of S4 ........................... 59
8.1. Preliminaries .......................................... 59
8.2. Finite Interior Algebras ............................... 60
8.3. Two Algebraic Constructions ............................ 61
8.4. An Algebraic Property .................................. 63
9. Syntactic Characterizations of S4 ........................... 66
9.1. Yankov-style Formulas .................................. 66
9.2. Sequential Refutation Systems ........................ 66
9.3. Formulas Corresponding to u ............................ 68
9.4. A Characteristic Rule ................................. 69
10.Grzegorczyk's Logic ......................................... 72
10.1.Normal Forms ........................................... 72
10.2.Syntactic Reductions ................................... 73
10.3.A Refutation Procedure ................................. 74
10.4.Constructing Models ................................... 76
10.5.A Kind of Modal Disjunction Property ................... 77
Bibliography ................................................... 83
Subject Index .................................................. 85
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