Paraconsistent logic: consistency, contradiction and negation
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2016]
|
| Series: | Logic, epistemology, and the unity of science
Volume 40 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 398 Seiten |
| ISBN: | 9783319332031 |
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Record in the Search Index
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| adam_text | Contents Contradiction and (inconsistency ............................................................ 1.1 Introduction.......................................................................................... 1.2 On the Philosophy of the Logics of Formal Inconsistency............. 1.3 A Historical Sketch: The Forerunners of the Logics of Formal Inconsistency.......................................................... 10 1.4 Paraconsistency and the Nature of Logic......................................... 1.5 Paraconsistency and the Nature of Contradictions .......................... 1.6 Contradiction, Consistency and Negation......................................... 1.6.1 On Contradiction.................................................................... 1.6.2 On Consistency....................................................................... 1.6.3 On Negation............................................................................ 1.7 Varieties of Paraconsistency Involvement........................................ References........................................................................................................ 11 15 18 19 20 21 24 26 A Basic Logic of Formal Inconsistency: mbC........................................ 2.1 Introducing mbC.................................................................................... 2.2 A Valuation Semantics for mbC........................................................ 2.3 Applications of mbC-Valuations........................................................ 2.4 Recovering Classical Logic Inside
mbC........................................... 2.5 Reintroducing mbC as an Expansion of CPL.................................. 2.5.1 The New Presentation mbC1 of mbC............................... 2.5.2 Valuation Semantics for mbC.............................................. 2.5.3 Equivalence Between mbC and mbCJ............................... References......................................................................................................... 29 29 35 39 43 50 51 53 55 61 3 Some Extensions of mbC.............................................................................. 3.1 A Wider Form of Truth-Functionality for Consistency.................. 3.2 A Hidden Consistency Operator........................................................ 3.3 Consistency and Inconsistency as Derived Connectives.................. 3.4 Some Conceptual Differences Between mbC and mbC1............. 3.5 Inconsistency Operators and Double-Negations ............................... 63 63 68 71 91 95 1 2 1 1 4 xxi
xxii Contents 3.6 Propagating Consistency........................................................................ 3.7 da Costa’s Hierarchy and Consistency Propagation......................... 3.8 A Stronger Consistency Propagation................................................. References.......................................................................................................... 104 Ill 115 119 4 Matrices and Algebraizability........................................................................ 4.1 Logical Matrices..................................................................................... 4.2 Uncharacterizability by Finite Matrices............................................... 4.3 The Problem of Algebraizability of LFIs........................................... 4.4 Some З-Valued LFIs................ ........................................................... 4.4.1 Halldén’s Logic of Nonsense (1949)................................. 4.4.2 Segerberg’s Logic of Nonsense (1965)............................... 4.4.3 da Costa and D’Ottaviano’s Logic J3 (1970)..................... 4.4.4 Sette’s Logic Pl (1973)....................................................... 4.4.5 Asenjo-Priest’s Logic LP (1966-1979) ............................... 4.4.6 Ciore and Other Related Systems...................................... 4.4.7 LFI1, MPT and J3............................................................... References.......................................................................................................... 121 121 122 129 136 136 138 140 144 149 151
158 168 LFIs Based on Other Logics........................................................................ 5.1 LFIs Based on Positive Intuitionistic Logic........................................ 5.1.1 Basic Features of Positive Intuitionistic Logic..................... 5.1.2 Johansson’s Minimal Logic.................................................... 5.1.3 Nelson’s Paraconsistent Logic N4......................................... 5.1.4 An Intuitionistic Version of mbC......................................... 5.2 LFIs Based on Fuzzy Logics.............................................................. 5.2.1 Preliminaries on MFL............................................................. 5.2.2 Fuzzy Logics with a Consistency Operator.......................... 5.2.3 Propagation of Consistency and DAT.................................... 5.2.4 Fuzzy Logics with an Inconsistency Operator..................... 5.3 A Modal LFI Based on Belnap and Dunn’s Logic BD................. 5.3.1 The Logic Mim of Tetravalent Modal Algebras................. 5.3.2 Mim as an LFI......................................................................... 5.3.3 Mim as a dC-System................................................................ 5.3.4 The Contrapositive Implication.............................................. 5.3.5 A Hilbert-Style Axiomatization of Mļm.................................. 5.4 Paraconsistent Modalities, Consistency and Determinedness .... References........................................................................................................ 171 171 171 175 179 187
191 191 197 205 208 211 213 218 222 224 226 229 233 6 Semantics of Non-deterministic Character for LFIs.............................. 6.1 Fidel Structures for mbC..................................................................... 6.2 Fidel Structures for Some Extensions of mbC.................................. 6.3 Non-deterministic Matrices................................................................... 6.4 Swap Structures for mbC...................................................................... 6.5 Swap Structures for Some Extensionsof mbC.................................. 6.6 Axiom (cl) and Uncharacterizability by Finite Nmatrices............ 237 238 242 251 253 260 272 5
Contents xxiii Some Remarks on Fidel Structures and Swap Structures............... 279 The Possible-Translations Semantics............................................... 280 6.8.1 Possible-Translations Semantics for Some LFIs............... 280 6.8.2 A З-Valued Possible-TranslationsSemantics for Cila . . . 282 6.8.3Some Remarks on Possible-Translations Semantics .... 288 References................................................................................................... 290 6.7 6.8 7 First-Order LFIs............................................................. 293 7.1 The Logic QmbC................................................................................ 294 7.2 Basic Properties of QmbC................................................................ 296 7.3 Tarskian Paraconsistent Structures..................................................... 302 7.4 Soundness Theorem for QmbC.......................................................... 308 7.5 Completeness Theorem for QmbC..................................................... 310 7.5.1 Henkin Theories................................................................... 311 7.5.2 Canonical Interpretations....................................................... 312 7.6 Compactness and Lowenhëim-Skolem Theorems for QmbC.... 316 7.7 QmbC with Equality........................................................................... 318 7.8 First-Order Characterization of Other Quantified LFIs.................. 322 7.9 First-Order LFI1 and the Logic of Quasi-truth ............................... 324 7.9.1 Semantics of
Partial Structures....................................... 324 7.9.2 The Logic QLFI1O................................................ 332 7.10 First-Order Pl and Partial Structures................................................ 335 References.................................................................................................. 343 8 Paraconsistent Set Theory..................................................................... 345 8.1 Antinomic Sets and Paraconsistency................................................... 346 8.2 LFIs Predicating on Consistency................................. 349 8.3 Some Extensions of ZFmbC............................................................... 354 8.4 Inconsistent Sets and Proper Classes................................................ 360 8.5 On Models............................................................................................. 363 References.................................................................................................. 365 9 Paraconsistency and Philosophy of Science: Foundations and Perspectives..................................................................................... 369 9.1 An Epistemological Understanding of Paraconsistency, and Its Significance for Science.............................................. 369 9.2 Consistency and Contradiction in Scientific Theories.................... 371 9.2.1 The Heritage of Kant............................................................ 371 9.2.2 Some Historical Examples.................................................. 373 9.2.3 The
Beginning of Quantum Theory and Paraconsistency....................................................... 374 9.2.4 Mercury’s Orbit and a Non-existent Planet....................... 374 9.2.5 Contradictions in Phlogiston, the Imponderable............... 376 9.2.6 The Special Theory of Relativity........................................ 377 9.2.7 Mathematics, and the Meaning of Objects that Mean Nothing.................................................. 379
xxiv Contents Quasi-truth and the Reconciliation of Science and Rationality......................................................................... 382 9.4 An Evidence-Based Approach to Paraconsistency.......................... 384 9.5 Summing Up........................................................................................... 385 References........................................................................................................ 387 9.3 Index....................................................................................................................... 391 Index of Logic Systems....................................................................................... 397
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| adam_txt |
Contents Contradiction and (inconsistency . 1.1 Introduction. 1.2 On the Philosophy of the Logics of Formal Inconsistency. 1.3 A Historical Sketch: The Forerunners of the Logics of Formal Inconsistency. 10 1.4 Paraconsistency and the Nature of Logic. 1.5 Paraconsistency and the Nature of Contradictions . 1.6 Contradiction, Consistency and Negation. 1.6.1 On Contradiction. 1.6.2 On Consistency. 1.6.3 On Negation. 1.7 Varieties of Paraconsistency Involvement. References. 11 15 18 19 20 21 24 26 A Basic Logic of Formal Inconsistency: mbC. 2.1 Introducing mbC. 2.2 A Valuation Semantics for mbC. 2.3 Applications of mbC-Valuations. 2.4 Recovering Classical Logic Inside
mbC. 2.5 Reintroducing mbC as an Expansion of CPL. 2.5.1 The New Presentation mbC1 of mbC. 2.5.2 Valuation Semantics for mbC. 2.5.3 Equivalence Between mbC and mbCJ. References. 29 29 35 39 43 50 51 53 55 61 3 Some Extensions of mbC. 3.1 A Wider Form of Truth-Functionality for Consistency. 3.2 A Hidden Consistency Operator. 3.3 Consistency and Inconsistency as Derived Connectives. 3.4 Some Conceptual Differences Between mbC and mbC1. 3.5 Inconsistency Operators and Double-Negations . 63 63 68 71 91 95 1 2 1 1 4 xxi
xxii Contents 3.6 Propagating Consistency. 3.7 da Costa’s Hierarchy and Consistency Propagation. 3.8 A Stronger Consistency Propagation. References. 104 Ill 115 119 4 Matrices and Algebraizability. 4.1 Logical Matrices. 4.2 Uncharacterizability by Finite Matrices. 4.3 The Problem of Algebraizability of LFIs. 4.4 Some З-Valued LFIs. . 4.4.1 Halldén’s Logic of Nonsense (1949). 4.4.2 Segerberg’s Logic of Nonsense (1965). 4.4.3 da Costa and D’Ottaviano’s Logic J3 (1970). 4.4.4 Sette’s Logic Pl (1973). 4.4.5 Asenjo-Priest’s Logic LP (1966-1979) . 4.4.6 Ciore and Other Related Systems. 4.4.7 LFI1, MPT and J3. References. 121 121 122 129 136 136 138 140 144 149 151
158 168 LFIs Based on Other Logics. 5.1 LFIs Based on Positive Intuitionistic Logic. 5.1.1 Basic Features of Positive Intuitionistic Logic. 5.1.2 Johansson’s Minimal Logic. 5.1.3 Nelson’s Paraconsistent Logic N4. 5.1.4 An Intuitionistic Version of mbC. 5.2 LFIs Based on Fuzzy Logics. 5.2.1 Preliminaries on MFL. 5.2.2 Fuzzy Logics with a Consistency Operator. 5.2.3 Propagation of Consistency and DAT. 5.2.4 Fuzzy Logics with an Inconsistency Operator. 5.3 A Modal LFI Based on Belnap and Dunn’s Logic BD. 5.3.1 The Logic Mim of Tetravalent Modal Algebras. 5.3.2 Mim as an LFI. 5.3.3 Mim as a dC-System. 5.3.4 The Contrapositive Implication. 5.3.5 A Hilbert-Style Axiomatization of Mļm. 5.4 Paraconsistent Modalities, Consistency and Determinedness . References. 171 171 171 175 179 187
191 191 197 205 208 211 213 218 222 224 226 229 233 6 Semantics of Non-deterministic Character for LFIs. 6.1 Fidel Structures for mbC. 6.2 Fidel Structures for Some Extensions of mbC. 6.3 Non-deterministic Matrices. 6.4 Swap Structures for mbC. 6.5 Swap Structures for Some Extensionsof mbC. 6.6 Axiom (cl) and Uncharacterizability by Finite Nmatrices. 237 238 242 251 253 260 272 5
Contents xxiii Some Remarks on Fidel Structures and Swap Structures. 279 The Possible-Translations Semantics. 280 6.8.1 Possible-Translations Semantics for Some LFIs. 280 6.8.2 A З-Valued Possible-TranslationsSemantics for Cila . . . 282 6.8.3Some Remarks on Possible-Translations Semantics . 288 References. 290 6.7 6.8 7 First-Order LFIs. 293 7.1 The Logic QmbC. 294 7.2 Basic Properties of QmbC. 296 7.3 Tarskian Paraconsistent Structures. 302 7.4 Soundness Theorem for QmbC. 308 7.5 Completeness Theorem for QmbC. 310 7.5.1 Henkin Theories. 311 7.5.2 Canonical Interpretations. 312 7.6 Compactness and Lowenhëim-Skolem Theorems for QmbC. 316 7.7 QmbC with Equality. 318 7.8 First-Order Characterization of Other Quantified LFIs. 322 7.9 First-Order LFI1 and the Logic of Quasi-truth . 324 7.9.1 Semantics of
Partial Structures. 324 7.9.2 The Logic QLFI1O. 332 7.10 First-Order Pl and Partial Structures. 335 References. 343 8 Paraconsistent Set Theory. 345 8.1 Antinomic Sets and Paraconsistency. 346 8.2 LFIs Predicating on Consistency. 349 8.3 Some Extensions of ZFmbC. 354 8.4 Inconsistent Sets and Proper Classes. 360 8.5 On Models. 363 References. 365 9 Paraconsistency and Philosophy of Science: Foundations and Perspectives. 369 9.1 An Epistemological Understanding of Paraconsistency, and Its Significance for Science. 369 9.2 Consistency and Contradiction in Scientific Theories. 371 9.2.1 The Heritage of Kant. 371 9.2.2 Some Historical Examples. 373 9.2.3 The
Beginning of Quantum Theory and Paraconsistency. 374 9.2.4 Mercury’s Orbit and a Non-existent Planet. 374 9.2.5 Contradictions in Phlogiston, the Imponderable. 376 9.2.6 The Special Theory of Relativity. 377 9.2.7 Mathematics, and the Meaning of Objects that Mean Nothing. 379
xxiv Contents Quasi-truth and the Reconciliation of Science and Rationality. 382 9.4 An Evidence-Based Approach to Paraconsistency. 384 9.5 Summing Up. 385 References. 387 9.3 Index. 391 Index of Logic Systems. 397 |
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| id | DE-604.BV048557053 |
| illustrated | Not Illustrated |
| index_date | 2024-07-03T20:58:56Z |
| indexdate | 2024-07-10T09:41:24Z |
| institution | BVB |
| isbn | 9783319332031 |
| language | English |
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| physical | XXIV, 398 Seiten |
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| publisher | Springer |
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| series | Logic, epistemology, and the unity of science |
| series2 | Logic, epistemology, and the unity of science |
| spelling | Carnielli, Walter Alexandre 1952- Verfasser (DE-588)1014998530 aut Paraconsistent logic consistency, contradiction and negation Walter Carnielli, Marcelo Esteban Coniglio Cham Springer [2016] © 2016 XXIV, 398 Seiten txt rdacontent n rdamedia nc rdacarrier Logic, epistemology, and the unity of science Volume 40 Negation (DE-588)4137553-1 gnd rswk-swf Paradoxon (DE-588)4044593-8 gnd rswk-swf Parakonsistente Logik (DE-588)4226190-9 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Inkonsistenz (DE-588)4226191-0 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 s Inkonsistenz (DE-588)4226191-0 s Parakonsistente Logik (DE-588)4226190-9 s Paradoxon (DE-588)4044593-8 s Negation (DE-588)4137553-1 s DE-604 Coniglio, Marcelo Esteban Verfasser (DE-588)1274662141 aut Erscheint auch als Online-Ausgabe 978-3-319-33205-5 Logic, epistemology, and the unity of science Volume 40 (DE-604)BV019709027 40 Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033933316&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Carnielli, Walter Alexandre 1952- Coniglio, Marcelo Esteban Paraconsistent logic consistency, contradiction and negation Logic, epistemology, and the unity of science Negation (DE-588)4137553-1 gnd Paradoxon (DE-588)4044593-8 gnd Parakonsistente Logik (DE-588)4226190-9 gnd Mathematische Logik (DE-588)4037951-6 gnd Inkonsistenz (DE-588)4226191-0 gnd |
| subject_GND | (DE-588)4137553-1 (DE-588)4044593-8 (DE-588)4226190-9 (DE-588)4037951-6 (DE-588)4226191-0 |
| title | Paraconsistent logic consistency, contradiction and negation |
| title_auth | Paraconsistent logic consistency, contradiction and negation |
| title_exact_search | Paraconsistent logic consistency, contradiction and negation |
| title_exact_search_txtP | Paraconsistent logic consistency, contradiction and negation |
| title_full | Paraconsistent logic consistency, contradiction and negation Walter Carnielli, Marcelo Esteban Coniglio |
| title_fullStr | Paraconsistent logic consistency, contradiction and negation Walter Carnielli, Marcelo Esteban Coniglio |
| title_full_unstemmed | Paraconsistent logic consistency, contradiction and negation Walter Carnielli, Marcelo Esteban Coniglio |
| title_short | Paraconsistent logic |
| title_sort | paraconsistent logic consistency contradiction and negation |
| title_sub | consistency, contradiction and negation |
| topic | Negation (DE-588)4137553-1 gnd Paradoxon (DE-588)4044593-8 gnd Parakonsistente Logik (DE-588)4226190-9 gnd Mathematische Logik (DE-588)4037951-6 gnd Inkonsistenz (DE-588)4226191-0 gnd |
| topic_facet | Negation Paradoxon Parakonsistente Logik Mathematische Logik Inkonsistenz |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033933316&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV019709027 |
| work_keys_str_mv | AT carnielliwalteralexandre paraconsistentlogicconsistencycontradictionandnegation AT conigliomarceloesteban paraconsistentlogicconsistencycontradictionandnegation |