Lectures on Propositional Logic

Title......Page 1
Contents......Page 3
Preface......Page 7
1. Propositional languages......Page 9
2. Logical calculi......Page 10
3. Closure bases......Page 14
4. Consequence operations from a complete lattice......Page 15
5. A bit more about the lattice of structural consequence......Page 16
6. Structural completeness......Page 23
7. Rules of inference and inferential bases......Page 24
8. More on structural completeness......Page 26
9. Some examples......Page 28
10. More on inferential bases. Proofs......Page 33
11. Basic definitions and theorems......Page 37
12. Deductive systems and well-determined logics......Page 39
13. Are Łukasiewicz logics well-determined?......Page 40
14. Well determined modal logics......Page 42
15. Surprisingly enough, relevant logics are not well-determined......Page 44
16. Theories vs truth-valuations......Page 45
17. Epistemic valuations for L......Page 47
18. Epistemic valuation for L~......Page 48
19. Neighborhood valuations......Page 50
20. Relational valuations......Page 51
21. Completeness lemma......Page 53
22. Consequence with Lindenbaum property......Page 54
23. Inferential bases for K, H, Jmin, J, and some useful theorems......Page 57
24. An inferential base for N and some adequacy theorems. The method of canonical frames......Page 60
25. Canonical frames for modal logics......Page 64
26. Are all classical modal systems natural? System K4.3W......Page 66
27. The problem of completeness......Page 68
28. Some preparatory results......Page 71
29. Conditions for a consequence to be finitary......Page 73
30. All Łn, n є ω are standard: an example of application of theorem 29.1.......Page 75
31. Matrices and matrix semantics......Page 77
32. First two completeness theorems......Page 79
33. Simple matrices and two more completeness theorems......Page 80
34. Łoś-Suszco's theorem......Page 81
35. A few comments on Łoś-Suszco's theorem......Page 84
36. Ramified matrices and ramified logics......Page 86
37. Implicative logics......Page 89
38. More on algebraic semantics......Page 91
39. Properly l-algebraic logics......Page 93
40. A bit of philosophy......Page 95
41. Some operations on matrices......Page 99
42. Reduced products of matrices......Page 101
43. Czelakowski's theorems......Page 102
44. The proof of Czelakowski's theorems......Page 103
45. Some more conditions for a logic to be standard......Page 107
46. Some corollaries to theorem 43.1......Page 108
47. Matr*(C) for equivalential logics......Page 110
48. Subdirectly irreducible matrices......Page 113
52. K--standard referential matrices......Page 117
53. Referential matrices vs neighborhood frames......Page 119
54. Referential matrices vrs relational frames......Page 121
55. Comparing the relative strength of different semantics......Page 122
49. Referential algebras......Page 125
50. Selfextensional logics......Page 128
51. An useful lemma......Page 130
56. A syntactical test for strong finiteness......Page 133
57. The lattices of strengthenings of a strongly finite consequences......Page 135
58. Hereditary properties......Page 137
59. Degree of maximality......Page 141
60. Some applications of Theorem 59.3......Page 143
61. Two algebraic lattices......Page 145
62. Finitely axiomatizable theories and finitely based logics......Page 146
63. Axiomatizable theories and parafinitely based logics......Page 147
64. A generalized version of Herrop's theorem and some problems concerning decidability......Page 148
65. Finite approximability and finite model property......Page 151
66. Definitional extensions......Page 153
67. Definability......Page 156
68. Definitional variants......Page 157
Bibliography......Page 161
A-C......Page 173
D-E-F......Page 174
H-I-K-L......Page 175
M......Page 176
N-P-Q-R......Page 177
S......Page 178
T-U-V......Page 179