## 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]

Item type | Home library | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
---|---|---|---|---|---|---|---|

Libros | Instituto de Matemática, CONICET-UNS | 03 R224 (Browse shelf) | Checked out | 2022-10-12 | A-3931 |

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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