An algebraic approach to non-classical logics / Helena Rasiowa.

Por: Rasiowa, HelenaSeries 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)
Contenidos:
 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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FUNDAMENTOS DE LA MATEMÁTICA


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] --

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