Philosophical and mathematical logic:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cham, Switzerland
Springer
[2018]
|
| Schriftenreihe: | Springer Undergraduate Texts in Philosophy
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | xx, 539 Seiten Illustrationen |
| ISBN: | 9783030032531 |
| ISSN: | 2569-8737 |
Internformat
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| 653 | |a predicate logic | ||
| 653 | |a propositional logic | ||
| 653 | |a set theory | ||
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| 653 | |a SUCO41175: Religion and Philosophy | ||
| 653 | |a SCI16048: Mathematical Logic and Formal Languages | ||
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Datensatz im Suchindex
| _version_ | 1804180754281791488 |
|---|---|
| adam_text | Contents 1 Logic; a First Impression ............................................................................... General................................................................................................... Propositional Logic................................................................................ Sets; Finite and Infinite.......................................................................... Predicate Logic...................................................................................... Arithmetic; Gödel’s Incompleteness Theorem..................................... Modal Logic................................ Philosophy of Language........................................................................ Intuiționism and Intuitionistic Logic.................................................... Applications........................................................................................... 1.9.1 Programming in Logic: Prolog............................................... 1.9.2 Relational Databases............................................................... 1.9.3 Social Choice Theory............................................................. 1.10 Fallacies and Unfair Discussion Methods............................................ 1.11 Solutions ............................................................................................... References...................................................................................................... 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Propositional
Logic........................................................................................... 2.1 Linguistic Considerations...................................................................... 2.2 Semantics; Truth Tables............................................................... 2.2.1 Validity ..................................................................................... 2.3 Semantics; Logical (Valid) Consequence............................................ 2.3.1 Decidability............................................................................... 2.3.2 Sound versus Plausible Arguments; Enthymemes................ 2.4 Semantics: Meta-logical Considerations.............................................. 2.5 About Truthfunctional Connectives. . . ................................................. 2.5.1 Applications in Electrical Engineering and in Jurisdiction .. 2.5.2 Normal Form*; Logic Programming* .................................... 2.5.3 Travelling Salesman Problem (TSP)*; NP-completeness*... 2.6 Syntax: Provability and Deducibility .................................................. 2.7 Syntax: Meta-logical Results........................................................... .. 1 1 2 8 8 12 13 13 14 15 15 16 16 17 19 20 21 21 29 33 37 40 41 44 51 53 55 58 62 71 XV
Contents xvi 2.7.1 Deduction Theorem; Introduction and Elimination Rules ... 73 2.7.2 Natural Deduction* .................................................................. 78 2.8 Tableaux................................................................................................ 82 2.9 Completeness of classical propositional logic................................... 93 2.10 Paradoxes; Historical and Philosophical Remarks............................. 97 2.10.1 Paradoxes ................................................................................. 97 2.10.2 Historical and Philosophical Remarks...................................... 102 2.11 Solutions ......... 112 References.........................................................................................................128 3 Sets: 3.1 3.2 3.3 4 Predicate Logic............................................................................... 181 4.1 Predicate Language................................................................................. 181 4.1.1 Quantifiers, Individual Variables and Constants...................... 182 4.1.2 Translating English into Predicate Logic, Intended and Non-intended Interpretation................................185 4.1.3 Scope, Bound and Free Variables.............................................. 188 4.1.4 Alphabet and Formulas............................................................. 189 4.2 Semantics: Tarski’s Truth Definition; Logical (Valid) Consequence . 194 4.3 Basic Results about Validity and Logical Consequence..................... 204 4.3.1 Quantifiers and
Connectives.................................................... 204 4.3.2 Two different quantifiers.......................................................... 208 4.3.3 About the axioms and rules for V and 3................ 209 4.3.4 Predicate Logic with Function Symbols*.................... 211 4.3.5 Prenex Form* ....................... 212 4.3.6 Skolemization, Clausal Form* ................................................. 213 finite and infinite.................................................................................. 129 Russell’s Paradox . . . ............................................................................ 129 Axioms of Zèrmelo-Fraenkel for Sets...................................................132 Historical·and Philosophical Remarks ................................................. 140 3.3.1 Mathematics and Theology........................................................ 140 3.3.2 Ontology of mathematics........................................................ 140 3.3.3 Analytic-Synthetic...................................................................... 141 3.3.4 Logicism......................................................................................142 3.4 Relations, Functions and Orderings*..................................................... 144 3.4.1 Ordered pairs and Cartesian product . .................................... 144 3.4.2 Relations...................................................................................... 146 3.4.3 Equivalence Relations................................................................ 149 3.4.4
Functions......................................................................................151 3.4.5 Orderings...................................................................................... 156 3.4.6 Structures and Isomorphisms.................................................... 158 3.5 The Hilbert Hotel; Denumerable Sets................................................... 162 3.6 Non-enumerable Sets............................................................................ 168 3.7 Solutions .................................. 175 References......................................................................................................... 180
Contents xvii Syntax: Provability and Deducibility ................................................... 216 4.4.1 Natural Deduction..................................................................... 221 4.4.2 Tableaux ..................................................................................... 222 4.5 Completeness, Compactness and Löwenheim-Skolem....................... 228 4.5.1 Undecidability ........................................................................... 230 4.5.2 Compactness and Löwenheim-Skolem Theorems .................. 232 4.5.3 Second-order Logic................................................................... 234 4.5.4 Skolėm’s Paradox....................................................................... 235 4.6 Predicate Logic with Equality.............................................................. 237 4.7 About the Relation of Logic with other Disciplines ........................... 242 4.7.1 Logic and Philosophy of Language..........................................242 4.7.2 Logic and Philosophy of Science......................................... 244 4.7.3 Logic and Artificial Intelligence; Prolog.................................. 246 4.7.4 Aristotle’s Organon................................................................... 247 4.8 Solutions ................................................................................................ 248 References........................................................................................................ 260 4.4 5 Arithmetic: Godel’s Incompleteness
Theorems....................................... 261 5.1 Formalization of Elementary Number Theory..................................... 261 5.2 Godel’s first Incompleteness Theorem................................................. 266 5.2.1 Gödel-numbering....................................................................... 268 5.2.2 Provability predicate for ....................................................... 270 5.3 Gödel’s second Incompleteness Theorem............................................. 271 5.3.1 Implications of Gödel’s Incompleteness Theorems ................ 272 5.4 Non-standard Models of Peano’s Arithmetic....................................... 272 5.4.1 Second-order Logic (continued)................................................273 5.5 Solutions ................................... 275 References........................................................................................................ 276 6 Modal Logic.................................................................................................... 277 6.1 Modal Operators................................................................................... 277 6.2 Different systems of Modal Logic........................................................ 279 6.3 Possible World Semantics .................................................................... 282 6.4 Epistemic logic.............................................................................. 287 6.4.1 Muddy Children Puzzle; Reasoning about Knowledge........ 288 6.5 Tableaux for Modal Logics
.................................................................. 290 6.6 Applications of Possible World Semantics.......................................... 296 6.6.1 Direct Reference........................................................................ 296 6.6.2 Rigid Designators...................................................................... 298 6.6.3 De dicto - de re distinction ....................................................... 299 6.6.4 Reasoning about Knowledge..................................................... 300 6.6.5 Common Knowledge.................................................................302 6.7 Completeness of Modal Propositional Logic...................................... 303 6.8 Strict Implication....................................................................................309 6.9 Counterfactuals..................................................................................... 310
xviii Contents 6.10 Weak and Relevant Implication; Entailment*....................................... 313 6.11 Modal Predicate Logic........................................................................... 315 6.11.1 Modal Predicate Logic and Essentialism................................. 316 6.12 The Modal Logic GL ............................................................................. 318 6.13 Solutions ................................................................................................ 320 References........................................................................................................ 327 7 Philosophy of Language..................................................................................329 7.1 Use and Mention..................................................................................... 329 7.2 Frege’s Sinn und Bedeutung (Sense and Reference)........................... 331 7.3 Mannoury (1867-1956), Signifies......................................................... 335 7.4 Speech Acts............. .............................................................................. 340 7.5 Definite Descriptions ............................................................................. 342 7.6 Berry’s add Grelling’s Paradox............................................................. 344 Ί.Ί The Theory of Direct Reference........................................................... 346 7.8 Analytic - Synthetic ..................................................................... 349 7.9 Logicism
................................................................................................ 350 7.10 Logical Positivism...................................................................................351 7.11 Presuppositions...................................................................................... 353 7.12 Wittgenstein on meaning....................................................................... 357 7.13 Syntax - Semantics - Pragnatics............................................................. 363 7.14 Conversational Implicature..................................................................... 365 7.15 Conditionals............................................................................................ 366 7.16 Leibniz.................................................................................................... 367 7.17 De Dicto - De Re.....................................................................................369 7.18 Grammars................................................................................................ 370 7.19 Solutions .................................................................................................374 References........................................... 376 8 Intuiționism and Intuitionistic Logic.......................................................... 379 8.1 Intuiționism vs Platonism; basic ideas ................................. 379 8.1.1 Language....................................................................... 381 8.1.2 First Steps in Intuitionistic
Reasoning..................................... 382 8.2 Intuitionistic Propositional Logic: Syntax........................................... 385 8.3 Tableaux for Intuitionistic Propositional Logic............................... 387 8.4 Intuitionistic Propositional Logic: Semantics....................................... 393 8.5 Completeness of Intuitionistic Propositional Logic............................. 397 8.6 Quantifiers in Intuiționism; Intuitionistic Predicate Logic................. 402 8.6.1 Deducibility for Intuitionistic Predicate Logic....................... 404 8.6.2 Tableaux for Intuitionistic Predicate Logic ..............................406 8.6.3 Kripke Semantics for Intuitionistic Predicate Logic.............. 407 8.6.4 Soundness and Completeness.......... ........................................ 409 8.7 Sets in Intuiționism: Construction Projects and Spreads...................... 411 8.8 The Brouwer Kripke axiom....................................................................416 8.9 Solutions ................................................................................................. 417
Contents xix References...................................................................................................... 426 9 Applications: Prolog; Relational Databases and SQL; Social Choice Theory.............................................................................. 427 9.1 Programming in Logic.......................................................................... 427 9.1.1 Recursion ................................................................................. 430 9.1.2 Declarative versus Procedural Programming.......................... 432 9.1.3 Syntax....................................................................................... 434 9.1.4 Matching versus Unification................................................... 436 9.1.5 Lists, Arithmetic....................................................................... 438 9.1.6 Cut.............................................................................................440 9.1.7 Negation as Failure................................................................... 443 9.1.8 Applications: Deductive Databases and Artificial Intelligence444 9.1.9 Pitfalls....................................................................................... 447 9.2 Relational Databases and SQL............................................................... 451 9.2.1 SQL...........................................................................................459 9.3 Social Choice Theory; Majority Judgment............................................462 9.3.1
Introduction........................................................... 463 9.3.2 Plurality Rule (PR): most votes count.................................... 464 9.3.3 Majority Rule (MR): pairwise comparison............................ 466 9.3.4 Borda Rule (BR)....................................................................... 467 9.3.5 Outcome depends on the Voting Rule .................................... 468 9.3.6 Arrow’s Impossibility Theorem.............................................. 468 9.3.7 Domination............................................................................... 470 9.3.8 Majority Judgment (MJ)......................................................... 471 9.3.9 Properties of Majority Judgment ............................................ 473 9.3.10 Point Summing and Approval Voting...................................... 474 9.3.11 Majority Judgment with many Voters .................................... 475 9.3.12 Presidential Elections in the USA............................................ 475 9.3.13 Presidential Elections in France.............................................. 477 9.3.14 Elections for Parliament in the Netherlands .......................... 479 9.4 Solutions ................................................................................................ 482 References....................................................................................................... 487 10 Fallacies and Unfair Discussion Methods ................................................. 489 10.1 Introduction
...................................................................................... 489 10.2 Fallacies.................................................................................................. 491 10.2.1 Clichés and Killers.................................................................... 491 10.2.2 Improper or hasty Generalizations........................................... 493 10.2.3 Thinking simplistically ............. .............................................. 494 10.2.4 Appeal to ignorance.................................................................. 496 10.2.5 Speculative Thinking.................................................................497 10.2.6 Incredulity.................................................................................. 499 10.2.7 The use of Terms with a vague Meaning................................. 500 10.2.8 The Danger of Words with more than one Meaning.............. 502
XX Contents 10.2.9 Aprioristic Reasoning............................................................... 504 10.2.10 Circular Reasoning............................................... 504 10.2.11 Applying double Standards....................................................... 505 10.2.12Rationalizing...............................................................................506 10.2.13 After this, therefore because of this....................508 10.3 Unfair Discussion Methods................................................................... 510 10.3.1 Pushing someone into an extreme corner................................. 510 10.3.2 Straw man argument ................................................................. 511 10.3.3 Diversion maneuvers................................................................. 512 10.3.4 Suggestive Methods................................................................... 515 10.3.5 Either/Or Fallacy ....................................................................... 524 10.3.6 The treacherous paradox........................................................... 524 10.3.7 Ad Hominem Arguments........................................................... 525 10.3.8 Argumentum ad baculum ......................................................... 527 10.3.9 Secrecy.......................................................................................527 Ю.З.ЮТҺе Retirement Home’s Discussion......................................... 528 10.4 Summary................................................................................................
529 References........................................................................................................ 530 Index.......................................................... 531
Springer Undergraduate Texts in Philosophy Harrie de SwartPhilosophical and Mathematical Logic This book was written to serve as an introduction to logic, with in each chapter – if applicable – special emphasis on the interplay between logic and philosophy, mathematics, language and (theoretical) computer science. The reader will not only be provided with an introduction to classical logic, but to philosophical (modal, epistemic, deontic, temporal) and intuitionistic logic as well. The first chapter is an easy to read non-technical Introduction to the topics in the book. The next chapters are consecutively about Propositional Logic, Sets (finite and infinite), Predicate Logic, Arithmetic and Gödel’s Incompleteness Theorems, Modal Logic, Philosophy of Language, Intuitionism and Intuitionistic Logic, Applications (Prolog; Relational Databases and SQL; Social Choice Theory, in particular Majority Judgment) and finally, Fallacies and Unfair Discussion Methods. Throughout the text, the author provides some impressions of the historical development of logic: Stoic and Aristotelian logic, logic in the Middle Ages and Frege s Begriffsschrift, together with the works of George Boole (1815-1864) and August De Morgan (1806-1871), the origin of modern logic. Since if …, then … can be considered to be the heart of logic, throughout this book much attention is paid to conditionals: material, strict and relevant implication, entailment, counterfactuals and conversational implicature are treated and many references for further reading are given. Each chapter is concluded with answers to
the exercises.
|
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| classification_rvk | CC 2400 SK 130 SG 700 |
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| discipline | Mathematik Philosophie |
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| id | DE-604.BV046292392 |
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| indexdate | 2024-07-10T08:40:47Z |
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| isbn | 9783030032531 |
| issn | 2569-8737 |
| language | English |
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| record_format | marc |
| series2 | Springer Undergraduate Texts in Philosophy |
| spelling | De Swart, Harrie 1944- Verfasser (DE-588)120024691 aut Philosophical and mathematical logic Harrie de Swart Cham, Switzerland Springer [2018] © 2018 xx, 539 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Springer Undergraduate Texts in Philosophy 2569-8737 Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Logik (DE-588)4036202-4 gnd rswk-swf PHI004000 HPK arithmetic and Gödel’s incompleteness theorem modal logic philosophical logic predicate logic propositional logic set theory SCE13000: Epistemology SUCO41175: Religion and Philosophy SCI16048: Mathematical Logic and Formal Languages SCM24005: Mathematical Logic and Foundations 1521: Hardcover, Softcover / Philosophie/Allgemeines, Lexika Logik (DE-588)4036202-4 s Mathematische Logik (DE-588)4037951-6 s DE-604 Erscheint auch als Online-Ausgabe 978-3-030-03255-5 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031669835&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031669835&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | De Swart, Harrie 1944- Philosophical and mathematical logic Mathematische Logik (DE-588)4037951-6 gnd Logik (DE-588)4036202-4 gnd |
| subject_GND | (DE-588)4037951-6 (DE-588)4036202-4 |
| title | Philosophical and mathematical logic |
| title_auth | Philosophical and mathematical logic |
| title_exact_search | Philosophical and mathematical logic |
| title_full | Philosophical and mathematical logic Harrie de Swart |
| title_fullStr | Philosophical and mathematical logic Harrie de Swart |
| title_full_unstemmed | Philosophical and mathematical logic Harrie de Swart |
| title_short | Philosophical and mathematical logic |
| title_sort | philosophical and mathematical logic |
| topic | Mathematische Logik (DE-588)4037951-6 gnd Logik (DE-588)4036202-4 gnd |
| topic_facet | Mathematische Logik Logik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031669835&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031669835&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT deswartharrie philosophicalandmathematicallogic |