An algebraic approach to non-classical logics / Helena Rasiowa.
Series Studies in logic and the foundations of mathematics ; v. 78Editor: Amsterdam : New York : North-Holland ; American Elsevier, 1974Descripción: xv, 403 p. ; 23 cmISBN: 0720422647Otra clasificación: 03Gxx (03G10 06Bxx)CONTENTS PART ONE IMPLICATIVE ALGEBRAS AND LATTICES Chapter I. Preliminary set-theoretical, topological and algebraic notions Introduction [3] 1. Sets, mappings [3] 2. Topological spaces [4] 3. Ordered sets and quasi-ordered sets [7] 4. Abstract algebras [9] 5. Exercises [14] Chapter II. Implicative algebras Introduction [15] 1. Definition and elementary properties [16] 2. Positive implication algebras [22] 3. Implicative filters in positive implication algebras [26] 4. Representation theorem for positive implication algebras [28] 5. Implication algebras [30] 6. Implicative filters in implication algebras [32] 7. Representation theorem for implication algebras [34] 8. Exercises [36] Chapter III. Distributive lattices and quasi-Boolean algebras Introduction [38] 1. Lattices [39] 2. Distributive lattices [43] 3. Quasi-Boolean algebras [44] 4. Exercises [48] Chapter IV. Relatively pseudo-complemented lattices, contrapositionally complemented lattices, semi-complemented lattices and pseudo-Boolean algebras Introduction [51] 1. Relatively pseudo-complemented lattices [52] 2. Filters in relatively pseudo-complemented lattices [56] 3. Representation theorem for relatively pseudo-complemented lattices [57] 4. Contrapositionally complemented lattices [58] 5. Semi-complemented lattices [61] 6. Pseudo-Boolean algebras [62] 7. Exercises [65] Chapter V. Quasi-pseudo-Boolean algebras Introduction [67] 1. Definition and elementary properties [68] 2. Equational definability of quasi-pseudo-Boolean algebras [75] 3. Examples of quasi-pseudo-Boolean algebras [81] 4. Filters in quasi-pseudo-Boolean algebras [90] 5. Representation theorem for quasi-pseudo-Boolean algebras [100] 6. Exercises [107] Chapter VI. Boolean algebras and topological Boolean algebras Introduction [110] 1. Definition and elementary properties of Boolean algebras [111] 2. Subalgebras of Boolean algebras [112] 3. Filters and implicative filters in Boolean algebras [113] 4. Representation theorem for Boolean algebras [114] 5. Topological Boolean algebras [115] 6. I-filters in topological Boolean algebras [116] 7. Representation theorem for topological Boolean algebras [120] 8. Strongly compact topological spaces [121] 9. A lemma on imbedding for topological Boolean algebras [122] 10. Connections between topological Boolean algebras, pseudo-Boolean algebras, relatively pseudo-complemented lattices, contrapositionally complemented lattices and semi-complemented lattices [123] 11. Lemmas on imbeddings for pseudo-Boolean algebras, relatively pseudocomplemented lattices, contrapositionally complemented lattices and semi-complemented lattices [127] 12. Exercises [129] Chapter VII. Post algebras Introduction [132] 1. Definition and elementary properties [133] 2. Examples of Post algebras [142] 3. Filters and D-filters in Post algebras [144] 4. Post homomorphisms [150] 5. Post fields of sets [156] 6. Representation theorem for Post algebras [161] 7. Exercises [163] PART TWO NON-CLASSICAL LOGICS Chapter VIII. Implicative extensional propositional calculi Introduction [167] 1. Formalized languages of zero order [170] 2. The algebra of formulas [172] 3. Interpretation of formulas as mappings [174] 4. Consequence operations in formalized languages of zero order [177] 5. The class S of standard systems of implicative extensional propositional calculi [179] 6. L - algebras [181] 7. Completeness theorem [185] 8. Logically equivalent systems [186] 9. L-theories of zero order [189] 10. Standard systems of implicative extensional propositional calculi with semi-negation [192] 11. Theorems on logically equivalent systems in S [193] 12. Deductive filters [199] 13. The connection between L-theories and deductive filters [203] 14. Exercises [208] Chapter IX. Positive implicative logic and classical implicative logic Introduction [210] 1. Propositional calculus L πl of positive implicative logic [212] 2. algebras [213] 3. Positive implicative logic Lπl [215] 4. LK(-theories of zero order [217] 5. The connection between Lπl-theories and implicative filters [218] 6. Propositional calculus Lϰl of classical implicative logic [221] 7. Lϰl-algebras [222] 8. Classical implicative logic [223] 9. Lϰl- theories of zero order [226] 10. The connection between Lϰl-theories of zero order and implicative filters [228] 11. Exercises [232] Chapter X. Positive logic Introduction [234] 1. Propositional calculus Lπ of positive logic [235] 2. Lπ-algebras [237] 3. Positive logic Lπ [238] 4. On disjunctions derivable in the propositional calculi of positive logic [243] 5. Ln-theories of zero order [244] 6. The connection between Lπ-theories and filters [245] 7. Exercises [248] Chapter XL Minimal logic, positive logic with semi-negation and intuitionistic logic Introduction [250] 1. Propositional calculus Lμ of minimal logic [252] 2. Minimal logic Lμ [253] 3. Lp-theories of zero order and their connection with filters [255] 4. Propositional calculus Lv of positive logic with semi-negation [258] 5. Positive logic with semi-negation Lv [259] 6. Ly- theories of zero order and their connection with filters [261] 7. Propositional calculus Lx of intuitionistic logic [263] 8. Intuitionistic logic Lx [265] 9. Lx-theories of zero order and their connection with filters [267] 10. Prime Lx-theories [271] 11. Exercises [274] Chapter XII. Constructive logic with strong negation Introduction [276] 1. Propositional calculus LN of constructive logic with strong negation [279] 2. L N-algebras [282] 3. Constructive logic with strong negation LN [283] 4. Connections between constructive logic with strong negation and intuitionistic logic [286] 5. A topological characterization of formulas derivable in propositional calculi of constructive logic with strong negation [293] 6. LN-theories of zero order [295] 7. The connection between LN-theories of zero order and special filters of the first kind [298] 8. Prime LN-theories [302] 9. Exercises [309] Chapter Xm. Classical logic and modal logic Introduction [311] 1. Propositional calculus Lx of classical logic [313] 2. Classical logic Lx [314] 3. Lx-theories of zero order and their connection with filters [315] 4. Propositional calculus Lλ of modal logic [318] 5. Modal logic Lλ [319] 6. Lλ-theories of zero order and their connection with I-filters [323] 7. I-prime Lλ-theories [328] 8. Exercises [331] Chapter XIV. Many-valued logics Introduction [333] 1. Propositional calculus of Lm of m-valued logic [335] 2. Lm-algebras [337] 3. m-valued logic Lm [338] 4. Lm-theories of zero order and their connection with D-filters [340] 5. Exercises [346] SUPPLEMENT First order predicate calculi of non-classical logics Introduction [347] 1. Formalized languages of first order [351] 2. First order predicate calculi of a logic L [353] 3. Elementary L-theories [357] 4. The algebra of terms [358] 5. Realizations of terms [359] 6. Implicative algebras with generalized joins and meets [361] 7. The algebra of an elementary L-theory [363] 8. Realizations of first order formalized languages associated with a logic L [367] 9. Canonical realizations for elementary L-theories [369] 10. L-models [371] 11. The completeness theorem for the first order predicate calculi of a logic L [374] 12. The existence of L-models for consistent elementary L-theories [375] 13. Exercises [377] Bibliography [380] List of symbols [391] Author index [393] Subject index [396]
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Bibliografía: p. [380]-390.
CONTENTS --
PART ONE --
IMPLICATIVE ALGEBRAS AND LATTICES --
Chapter I. Preliminary set-theoretical, topological and algebraic notions --
Introduction [3] --
1. Sets, mappings [3] --
2. Topological spaces [4] --
3. Ordered sets and quasi-ordered sets [7] --
4. Abstract algebras [9] --
5. Exercises [14] --
Chapter II. Implicative algebras --
Introduction [15] --
1. Definition and elementary properties [16] --
2. Positive implication algebras [22] --
3. Implicative filters in positive implication algebras [26] --
4. Representation theorem for positive implication algebras [28] --
5. Implication algebras [30] --
6. Implicative filters in implication algebras [32] --
7. Representation theorem for implication algebras [34] --
8. Exercises [36] --
Chapter III. Distributive lattices and quasi-Boolean algebras --
Introduction [38] --
1. Lattices [39] --
2. Distributive lattices [43] --
3. Quasi-Boolean algebras [44] --
4. Exercises [48] --
Chapter IV. Relatively pseudo-complemented lattices, contrapositionally complemented lattices, semi-complemented lattices and pseudo-Boolean algebras --
Introduction [51] --
1. Relatively pseudo-complemented lattices [52] --
2. Filters in relatively pseudo-complemented lattices [56] --
3. Representation theorem for relatively pseudo-complemented lattices [57] --
4. Contrapositionally complemented lattices [58] --
5. Semi-complemented lattices [61] --
6. Pseudo-Boolean algebras [62] --
7. Exercises [65] --
Chapter V. Quasi-pseudo-Boolean algebras --
Introduction [67] --
1. Definition and elementary properties [68] --
2. Equational definability of quasi-pseudo-Boolean algebras [75] --
3. Examples of quasi-pseudo-Boolean algebras [81] --
4. Filters in quasi-pseudo-Boolean algebras [90] --
5. Representation theorem for quasi-pseudo-Boolean algebras [100] --
6. Exercises [107] --
Chapter VI. Boolean algebras and topological Boolean algebras --
Introduction [110] --
1. Definition and elementary properties of Boolean algebras [111] --
2. Subalgebras of Boolean algebras [112] --
3. Filters and implicative filters in Boolean algebras [113] --
4. Representation theorem for Boolean algebras [114] --
5. Topological Boolean algebras [115] --
6. I-filters in topological Boolean algebras [116] --
7. Representation theorem for topological Boolean algebras [120] --
8. Strongly compact topological spaces [121] --
9. A lemma on imbedding for topological Boolean algebras [122] --
10. Connections between topological Boolean algebras, pseudo-Boolean --
algebras, relatively pseudo-complemented lattices, contrapositionally complemented lattices and semi-complemented lattices [123] --
11. Lemmas on imbeddings for pseudo-Boolean algebras, relatively pseudocomplemented lattices, contrapositionally complemented lattices and semi-complemented lattices [127] --
12. Exercises [129] --
Chapter VII. Post algebras --
Introduction [132] --
1. Definition and elementary properties [133] --
2. Examples of Post algebras [142] --
3. Filters and D-filters in Post algebras [144] --
4. Post homomorphisms [150] --
5. Post fields of sets [156] --
6. Representation theorem for Post algebras [161] --
7. Exercises [163] --
PART TWO --
NON-CLASSICAL LOGICS --
Chapter VIII. Implicative extensional propositional calculi --
Introduction [167] --
1. Formalized languages of zero order [170] --
2. The algebra of formulas [172] --
3. Interpretation of formulas as mappings [174] --
4. Consequence operations in formalized languages of zero order [177] --
5. The class S of standard systems of implicative extensional propositional calculi [179] --
6. L - algebras [181] --
7. Completeness theorem [185] --
8. Logically equivalent systems [186] --
9. L-theories of zero order [189] --
10. Standard systems of implicative extensional propositional calculi with semi-negation [192] --
11. Theorems on logically equivalent systems in S [193] --
12. Deductive filters [199] --
13. The connection between L-theories and deductive filters [203] --
14. Exercises [208] --
Chapter IX. Positive implicative logic and classical implicative logic --
Introduction [210] --
1. Propositional calculus L πl of positive implicative logic [212] --
2. algebras [213] --
3. Positive implicative logic Lπl [215] --
4. LK(-theories of zero order [217] --
5. The connection between Lπl-theories and implicative filters [218] --
6. Propositional calculus Lϰl of classical implicative logic [221] --
7. Lϰl-algebras [222] --
8. Classical implicative logic [223] --
9. Lϰl- theories of zero order [226] --
10. The connection between Lϰl-theories of zero order and implicative filters [228] --
11. Exercises [232] --
Chapter X. Positive logic --
Introduction [234] --
1. Propositional calculus Lπ of positive logic [235] --
2. Lπ-algebras [237] --
3. Positive logic Lπ [238] --
4. On disjunctions derivable in the propositional calculi of positive logic [243] --
5. Ln-theories of zero order [244] --
6. The connection between Lπ-theories and filters [245] --
7. Exercises [248] --
Chapter XL Minimal logic, positive logic with semi-negation and intuitionistic logic --
Introduction [250] --
1. Propositional calculus Lμ of minimal logic [252] --
2. Minimal logic Lμ [253] --
3. Lp-theories of zero order and their connection with filters [255] --
4. Propositional calculus Lv of positive logic with semi-negation [258] --
5. Positive logic with semi-negation Lv [259] --
6. Ly- theories of zero order and their connection with filters [261] --
7. Propositional calculus Lx of intuitionistic logic [263] --
8. Intuitionistic logic Lx [265] --
9. Lx-theories of zero order and their connection with filters [267] --
10. Prime Lx-theories [271] --
11. Exercises [274] --
Chapter XII. Constructive logic with strong negation --
Introduction [276] --
1. Propositional calculus LN of constructive logic with strong negation [279] --
2. L N-algebras [282] --
3. Constructive logic with strong negation LN [283] --
4. Connections between constructive logic with strong negation and intuitionistic logic [286] --
5. A topological characterization of formulas derivable in propositional calculi of constructive logic with strong negation [293] --
6. LN-theories of zero order [295] --
7. The connection between LN-theories of zero order and special filters of the first kind [298] --
8. Prime LN-theories [302] --
9. Exercises [309] --
Chapter Xm. Classical logic and modal logic --
Introduction [311] --
1. Propositional calculus Lx of classical logic [313] --
2. Classical logic Lx [314] --
3. Lx-theories of zero order and their connection with filters [315] --
4. Propositional calculus Lλ of modal logic [318] --
5. Modal logic Lλ [319] --
6. Lλ-theories of zero order and their connection with I-filters [323] --
7. I-prime Lλ-theories [328] --
8. Exercises [331] --
Chapter XIV. Many-valued logics --
Introduction [333] --
1. Propositional calculus of Lm of m-valued logic [335] --
2. Lm-algebras [337] --
3. m-valued logic Lm [338] --
4. Lm-theories of zero order and their connection with D-filters [340] --
5. Exercises [346] --
SUPPLEMENT --
First order predicate calculi of non-classical logics --
Introduction [347] --
1. Formalized languages of first order [351] --
2. First order predicate calculi of a logic L [353] --
3. Elementary L-theories [357] --
4. The algebra of terms [358] --
5. Realizations of terms [359] --
6. Implicative algebras with generalized joins and meets [361] --
7. The algebra of an elementary L-theory [363] --
8. Realizations of first order formalized languages associated with a logic L [367] --
9. Canonical realizations for elementary L-theories [369] --
10. L-models [371] --
11. The completeness theorem for the first order predicate calculi of a logic L [374] --
12. The existence of L-models for consistent elementary L-theories [375] --
13. Exercises [377] --
Bibliography [380] --
List of symbols [391] --
Author index [393] --
Subject index [396] --
MR, 56 #5285
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