Introduction .................................................. vii
1 Basic notions ................................................. 1
1.1 Propositional languages ................................... 1
1.2 Abstract algebras ......................................... 3
1.3 Preliminary lattice-theoretical notions ................... 6
1.4 Propositional logics ..................................... 19
1.5 Brief exposition of the most important propositional
logics ....................................................... 31
2 Semantic methods in propositional logic ...................... 41
2.1 Preordered sets .......................................... 41
2.2 Preordered algebras ...................................... 50
2.3 Logical matrices ......................................... 55
2.4 Adequacy ................................................. 63
2.5 Propositional logic and lattice theory ................... 75
3 Completeness of propositional logics ......................... 91
3.1 Generalized completeness ................................. 91
3.2 Post-completeness ........................................ 97
3.3 The problem of uniqueness of Lindenbaum extensions ...... 105
3.4 Structural completeness ................................. 112
3.5 Some related concepts ................................... 120
4 Characterizations of propositional connectives .............. 131
4.1 Cn-definitions .......................................... 131
4.2 The system (D) .......................................... 133
4.3 Variants ................................................ 137
4.4 The system (I) .......................................... 141
4.5 Classical logic ......................................... 149
Appendix A
The fundamental metatheorem for the classical
propositional logic ......................................... 153
Appendix В
A proof system for the classical logic ...................... 157
Bibliography .................................................. 165
Notation ...................................................... 175
Index ......................................................... 177
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