Der Autor vermittelt logisches Grundwissen, fundamentale Beweisprinzipien und Methoden der Mathematik. Dabei geht er u. a. folgenden Fragen nach: Was unterscheidet endliche von unendlichen Mengen? Wie lassen sich die ganzen, rationalen und reellen Zahlen aus den natürlichen Zahlen konstruieren? Welche grundlegenden topologischen Eigenschaften besitzt die Menge der reellen Zahlen? Lassen sich die natürlichen oder reellen Zahlen vollständig axiomatisch beschreiben? Pflichtlektüre für alle Studierenden der Mathematik, Physik und Informatik Preface xi Acknowledgments xv Part I Propositional Logic 1 Chapter 1: Propositions and Arguments 3 1.1 What Is Logic? 3 1.2 Propositions 5 1.3 Arguments 11 1.4 Logical Consequence 14 1.5 Soundness 21 1.6 Connectives 23 Chapter 2: The Language of Propositional Logic 32 2.1 Motivation 32 2.2 Basic Propositions of PL 32 2.3 Connectives of PL 36 2.4 Wff Variables 39 2.5 Syntax of PL 40 Chapter 3: Semantics of Propositional Logic 49 3.1 Truth Tables for the Connectives 49 3.2 Truth Values of Complex Propositions 51 3.3 Truth Tables for Complex Propositions 54 3.4 Truth Tables for Multiple Propositions 58 3.5 Connectives and Truth Functions 59 Chapter 4: Uses of Truth Tables 63 4.1 Arguments 63 4.2 Single Propositions 67 4.3 Two Propositions 69 4.4 Sets of Propositions 74 4.5 More on Validity 75 Chapter 5: Logical Form 79 5.1 Abstracting from Content: From Propositions to Forms 81 5.2 Instances: From Forms to Propositions 82 5.3 Argument Forms 84 5.4 Validity and Form 87 5.5 Invalidity and Form 91 5.6 Notable Argument Forms 94 5.7 Other Logical Properties 95 Chapter 6: Connectives: Translation and Adequacy 97 6.1 Assertibility and Implicature 97 6.2 Conjunction 103 6.3 Conditional and Biconditional 110 6.4 Disjunction 117 6.5 Negation 122 6.6 Functional Completeness 124 7 Trees for Propositional Logic 134 7.1 Tree Rules 136 7.2 Applying the Rules 140 7.3 Uses of Trees 146 7.4 Abbreviations 156 Part II Predicate Logic 161 Chapter 8: The Language of Monadic Predicate Logic 163 8.1 The Limitations of Propositional Logic 164 8.2 MPL, Part I: Names and Predicates 167 8.3 MPL, Part II: Variables and Quantifiers 172 8.4 Syntax of MPL 182 Chapter 9: Semantics of Monadic Predicate Logic 189 9.1 Models; Truth and Falsity of Uncomplicated Propositions 191 9.2 Connectives 196 9.3 Quantified Propositions: The General Case 197 9.4 Semantics of MPL: Summary 204 9.5 Analyses and Methods 206 Chapter 10: Trees for Monadic Predicate Logic 211 10.1 Tree Rules 212 10.2 Using Trees 223 10.3 Infinite Trees 228 Chapter 11: Models, Propositions, and Ways the World Could Be 242 11.1 Translation 243 11.2 Valuation 247 11.3 Axiomatization 251 11.4 Propositions 253 11.5 Logical Consequence and NTP 257 11.6 Postulates 261 Chapter 12: General Predicate Logic 264 12.1 The Language of General Predicate Logic 264 12.2 Semantics of GPL 276 12.3 Trees for General Predicate Logic 282 12.4 Postulates 286 12.5 Moving Quantifiers 293 Chapter 13: Identity 298 13.1 The Identity Relation 299 13.2 The Identity Predicate 303 13.3 Semantics of Identity 306 13.4 Trees for General Predicate Logic with Identity 311 13.5 Numerical Quantifiers 321 13.6 Definite Descriptions 326 13.7 Function Symbols 343 Part III Foundations and Variations 355 14 Metatheory 357 14.1 Soundness and Completeness 358 14.2 Decidability and Undecidability 368 14.3 Other Logical Properties 374 14.4 Expressive Power 382 15 Other Methods of Proof 385 15.1 Axiomatic Systems 386 15.2 Natural Deduction 407 15.3 Sequent Calculus 421 16 Set Theory 438 16.1 Sets 438 16.2 Ordered Pairs and Ordered n-tuples 449 16.3 Relations 453 16.4 Functions 454 16.5 Sequences 458 16.6 Multisets 460 16.7 Syntax 462 Notes 467 References 509 Index 515

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Nicholas Jeremy Josef Smith

Princeton University, Department of Art & Archaeology

Princeton Electronic

United States, United States of America

Princeton, N.J, New Jersey, 2012

Princeton, cop. 2012

4, 2012

kolxoz -- 10

lg872610

producers:
Acrobat Distiller 9.0.0 (Windows); modified using iTextSharp 5.0.0 (c) 1T3XT BVBA

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Includes bibliographical references and index.

Cover 1
Contents 8
Preface 12
Acknowledgments 16
PART I: Propositional Logic 18
1 Propositions and Arguments 20
1.1 What Is Logic? 20
1.2 Propositions 22
1.3 Arguments 28
1.4 Logical Consequence 31
1.5 Soundness 38
1.6 Connectives 40
2 The Language of Propositional Logic 49
2.1 Motivation 49
2.2 Basic Propositions of PL 49
2.3 Connectives of PL 53
2.4 Wff Variables 56
2.5 Syntax of PL 57
3 Semantics of Propositional Logic 66
3.1 Truth Tables for the Connectives 66
3.2 Truth Values of Complex Propositions 68
3.3 Truth Tables for Complex Propositions 71
3.4 Truth Tables for Multiple Propositions 75
3.5 Connectives and Truth Functions 76
4 Uses of Truth Tables 80
4.1 Arguments 80
4.2 Single Propositions 84
4.3 Two Propositions 86
4.4 Sets of Propositions 91
4.5 More on Validity 92
5 Logical Form 96
5.1 Abstracting from Content: From Propositions to Forms 98
5.2 Instances: From Forms to Propositions 99
5.3 Argument Forms 101
5.4 Validity and Form 104
5.5 Invalidity and Form 108
5.6 Notable Argument Forms 111
5.7 Other Logical Properties 112
6 Connectives: Translation and Adequacy 114
6.1 Assertibility and Implicature 114
6.2 Conjunction 120
6.3 Conditional and Biconditional 127
6.4 Disjunction 134
6.5 Negation 139
6.6 Functional Completeness 141
7 Trees for Propositional Logic 151
7.1 Tree Rules 153
7.2 Applying the Rules 157
7.3 Uses of Trees 163
7.4 Abbreviations 173
PART II: Predicate Logic 178
8 The Language of Monadic Predicate Logic 180
8.1 The Limitations of Propositional Logic 181
8.2 MPL, Part I: Names and Predicates 184
8.3 MPL, Part II: Variables and Quantifiers 189
8.4 Syntax of MPL 199
9 Semantics of Monadic Predicate Logic 206
9.1 Models; Truth and Falsity of Uncomplicated Propositions 208
9.2 Connectives 213
9.3 Quantified Propositions: The General Case 214
9.4 Semantics of MPL: Summary 221
9.5 Analyses and Methods 223
10 Trees for Monadic Predicate Logic 228
10.1 Tree Rules 229
10.2 Using Trees 240
10.3 Infinite Trees 245
11 Models, Propositions, and Ways the World Could Be 259
11.1 Translation 260
11.2 Valuation 264
11.3 Axiomatization 268
11.4 Propositions 270
11.5 Logical Consequence and NTP 274
11.6 Postulates 278
12 General Predicate Logic 281
12.1 The Language of General Predicate Logic 281
12.2 Semantics of GPL 293
12.3 Trees for General Predicate Logic 299
12.4 Postulates 303
12.5 Moving Quantifiers 310
13 Identity 315
13.1 The Identity Relation 316
13.2 The Identity Predicate 320
13.3 Semantics of Identity 323
13.4 Trees for General Predicate Logic with Identity 328
13.5 Numerical Quantifiers 338
13.6 Definite Descriptions 343
13.7 Function Symbols 360
PART III: Foundations and Variations 372
14 Metatheory 374
14.1 Soundness and Completeness 375
14.2 Decidability and Undecidability 385
14.3 Other Logical Properties 391
14.4 Expressive Power 399
15 Other Methods of Proof 402
15.1 Axiomatic Systems 403
15.2 Natural Deduction 424
15.3 Sequent Calculus 438
16 Set Theory 455
16.1 Sets 455
16.2 Ordered Pairs and Ordered n-tuples 466
16.3 Relations 470
16.4 Functions 471
16.5 Sequences 475
16.6 Multisets 477
16.7 Syntax 479
Notes 484
References 526
Index 532
A 532
B 532
C 533
D 534
E 535
F 535
G 536
H 536
I 536
J 537
K 537
L 537
M 537
N 538
O 539
P 539
Q 540
R 541
S 541
T 543
U 544
V 544
W 545
Z 545

<p>Logic is essential to correct reasoning and also has important theoretical applications in philosophy, computer science, linguistics, and mathematics. This book provides an exceptionally clear introduction to classical logic, with a unique approach that emphasizes both the hows and whys of logic. Here Nicholas Smith thoroughly covers the formal tools and techniques of logic while also imparting a deeper understanding of their underlying rationales and broader philosophical significance. In addition, this is the only introduction to logic available today that presents all the major forms of proof—trees, natural deduction in all its major variants, axiomatic proofs, and sequent calculus. The book also features numerous exercises, with solutions available on an accompanying website.</p>
<p><i>Logic</i> is the ideal textbook for undergraduates and graduate students seeking a comprehensive and accessible introduction to the subject.</p>
<ul>
<li>Provides an essential introduction to classical logic</li>
<li>Emphasizes the how and why of logic</li>
<li>Covers both formal and philosophical issues</li>
<li>Presents all the major forms of proof—from trees to sequent calculus</li>
<li>Features numerous exercises, with solutions available at personal.usyd.edu.au/~njjsmith/lawsoftruth</li>
<li>The ideal textbook for undergraduates and graduate students</li>
</ul>

Logic is essential to correct reasoning and also has important theoretical applications in philosophy, computer science, linguistics, and mathematics. This book provides an exceptionally clear introduction to classical logic, with a unique approach that emphasizes both the hows and whys of logic. Here Nicholas Smith thoroughly covers the formal tools and techniques of logic while also imparting a deeper understanding of their underlying rationales and broader philosophical significance. In addition, this is the only introduction to logic available today that presents all the major forms of proof--trees, natural deduction in all its major variants, axiomatic proofs, and sequent calculus. The book also features numerous exercises, with solutions available on an accompanying website. Logic is the ideal textbook for undergraduates and graduate students seeking a comprehensive and accessible introduction to the subject. Provides an essential introduction to classical logic Emphasizes the how and why of logic Covers both formal and philosophical issues Presents all the major forms of proof--from trees to sequent calculus Features numerous exercises, with solutions available at http://njjsmith.com/philosophy/lawsoftruth/ The ideal textbook for undergraduates and graduate students

Logic is essential to correct reasoning and also has important theoretical applications in philosophy, computer science, linguistics, and mathematics. This book provides an exceptionally clear introduction to classical logic, with a unique approach that emphasizes both the hows and whys of logic. Here Nicholas Smith thoroughly covers the formal tools and techniques of logic while also imparting a deeper understanding of their underlying rationales and broader philosophical significance. In addition, this is the only introduction to logic available today that presents all the major forms of proof - trees, natural deduction in all its major variants, axiomatic proofs, and sequent calculus. The book also features numerous exercises, with solutions available on an accompanying website. "Logic" is the ideal textbook for undergraduates and graduate students seeking a comprehensive and accessible introduction to the subject. This title provides an essential introduction to classical logic. It emphasizes the how and why of logic. It covers both formal and philosophical issues. It presents all the major forms of proof - from trees to sequent calculus. It features numerous exercises. This is the ideal textbook for undergraduates and graduate students

2012-12-29