Logic: A History of its Central Concepts

Cover......Page 1
Title Page......Page 5
Contents......Page 7
Preface......Page 9
List of Authors......Page 12
1 Introduction......Page 13
1.1 Necessity and Counterexamples......Page 16
1.2 Formality and Structure......Page 17
1.3 A Priori and Giving Reasons......Page 18
2 Aristotle [384 BCE-322 BCE]......Page 19
3 Stoics [300 BCE-200 CE]......Page 22
4 Medievals [476 CE-1453 CE]......Page 23
4.1 Consequentiæ......Page 24
4.2 Self-Reference and Insolubilia......Page 27
4.3 Obligationes......Page 29
5 Leibniz [1646-1716]......Page 30
6 Kant [1724-1804]......Page 31
7 Bolzano [1781-1848]......Page 33
9 Frege [1848-1925]......Page 35
10 Russell [1872-1970]......Page 37
11 Carnap [1891-1970]......Page 40
12 Gentzen [1909-1945]......Page 43
13 Tarski [1902-1983]......Page 44
14 Gödel [1906-1978]......Page 47
15 Modal Logics......Page 49
16 Nonmonotonic Options......Page 51
17 The Substructural Landscape......Page 53
18 Monism or Pluralism......Page 55
Bibliography......Page 59
1 Aristotle\'s Quantification Theory......Page 65
1.1 Validity as a Matter of Form......Page 66
1.2 Aristotle\'s Completeness Proof......Page 68
1.3 The Square of Opposition......Page 74
2.1 The Old Logic......Page 75
2.2 The New Logic......Page 79
2.3 The Mature Logic of Terms......Page 82
3.1 The Port-Royal Logic......Page 87
3.2 The English Textbook Tradition......Page 91
3.3 Quantifiers in the Predicate......Page 95
4 The Rise of Modern Logic......Page 98
4.1 Algebraic Approaches to Quantification......Page 100
4.2 Peirce\'s Quantification Theory......Page 103
4.3 Frege\'s Begriffschrift......Page 106
5 Contemporary Quantification Theory......Page 110
5.1 Limitations of Classical First-Order Logic......Page 111
5.2 Generalized Quantifiers......Page 114
5.3 Quantification and Anaphora......Page 119
Bibliography......Page 124
Introduction: Grice as a catalyst......Page 129
The background: Negation and opposition in Aristotelian logic......Page 132
Modernists and Neo-Traditionalists in post-Gricean hindsight......Page 134
Modernism......Page 135
Syntax for negation......Page 136
Semantics for negation......Page 141
Neo-Traditionalism......Page 142
Robbing Peter to pay Paul......Page 144
What the eye no longer sees the heart no longer grieves for......Page 146
Negation as otherness......Page 149
From contradiction to contrariety: Bosanquet, Anselm et al.......Page 152
Asymmetry revisited: the rise and fall of the Neo-Idealists......Page 155
Negation, presupposition, and the bracketing device......Page 157
Implicature and negation: scales, scopes, and metalinguistic negation......Page 159
Implicature and negation: Subcontrariety and the three-cornered square......Page 164
Negation and denial......Page 165
Negation and falsity......Page 167
Acknowledgments......Page 170
Bibliography......Page 171
1 Aristotelian Foundations......Page 177
2 Stoic Logic......Page 179
2.1 The Nature of the Conditional......Page 180
2.2 Stoic Theories of Conjunction, Disjunction, and Negation......Page 183
2.3 The Stoic Deduction System......Page 186
3 Hypothetical Syllogisms......Page 190
4 Early Medieval Theories......Page 194
5 Later Medieval Theories......Page 197
6 Leibniz\'s Logic......Page 203
7 Standard Modern-Era Logic......Page 207
8 Bolzano......Page 210
9 Boole......Page 216
10 Frege......Page 224
11 Peirce and Peano......Page 227
12 On to the Twentieth Century......Page 230
Bibliography......Page 232
1.1 The whole story......Page 237
1.2 What are we talking about?......Page 238
1.3 The main character of the story: MTV......Page 240
2.1 Tarski and the origin of logical matrices......Page 241
2.2 Logical matrix vs truth table......Page 243
2.3 Suszko and abstract nonsense......Page 246
3.1 Boole......Page 249
3.2 Peirce......Page 253
3.3 Frege......Page 258
4.1 Russell and Wittgenstein......Page 263
4.2 Bernays and Post......Page 270
4.3 Quine and Gödel......Page 274
5.1 Indeterminacy......Page 276
5.2 Gaps and Gluts......Page 280
5.3 Order......Page 282
6.1 Truth, Models, and Truth-values......Page 285
6.2 Truth, Worlds and Truth-values......Page 291
7.1 Back to Suszko......Page 294
7.2 Newton da Costa\'s theory of valuation......Page 296
Bibliography......Page 300
1 Extensional Modal Conceptions in Ancient and Medieval Philosophy......Page 311
2 Modality as Alternativeness......Page 319
Ancient and Medieval Modal Logic and Modal Syllogistic......Page 323
Modalities in Early Modern Philosophy......Page 331
Primary Literature......Page 335
Secondary Literature......Page 337
2 Object Language Natural Deduction......Page 343
2.1 The Wider Notion of Natural Deduction......Page 344
2.2 Different Proof Systems......Page 346
2.3 The Beginnings of Natural Deduction: Jaśkowski and Gentzen (and Suppes) on Representing Natural Deduction Proofs......Page 349
2.4 Natural Deduction in Elementary Textbooks......Page 354
2.5 More Features of the Prototype of Natural Deduction......Page 356
2.6 Natural Deduction Quantificational Rules......Page 360
2.7 A Summary of Elementary Natural Deduction Textbooks......Page 363
2.8 Exploiting Natural Deduction Techniques: Modal Logic......Page 365
3.1 Normalizing Natural Deduction......Page 371
3.3 Sequent Natural Deduction......Page 376
3.4 From Sequent Natural Deduction to Sequent Calculus......Page 378
3.5 The Sequent Calculus LK......Page 380
3.6 Variants of Sequent Calculi......Page 384
3.7 Sequent Calculi and Tableaux......Page 386
3.8 Natural Deduction with Sequences......Page 391
3.9 Size of Normal Proofs......Page 392
4.1 The Concept of Natural Deduction: Further Informal Thoughts......Page 394
4.2 Natural Deduction and Computers......Page 396
4.3 Natural Deduction and Semantics......Page 399
4.4 The One True Logic? Some Philosophical Reflections......Page 405
Bibliography......Page 408
Elementary Logic Textbooks Described in Table 1......Page 415
1 Two Thousand Three Hundred Years of Connexive Implication......Page 417
2 Connexive Conditionals: An Empirical Approach......Page 423
3 Paradoxes of Implication......Page 426
4 The Avoidance of Paradox......Page 428
5 A Consistent System of Connexive Logic......Page 429
6 Connexive Logic in Subproof Form......Page 432
7 Connexive Logic and the Syllogism......Page 435
8 Connexive Class Logic......Page 436
9 First-Degree Connexive Formulae......Page 437
10 Causal Implication......Page 439
11 Contemporary Work on Connexive Implication: Meyer, Routley, Mortensen, Priest, Lowe, Pizzi, Wansing, Rahman and Rückert......Page 445
Bibliography......Page 449
1 Introduction......Page 453
2 Prehistory of types......Page 458
2.1 Paradox threats......Page 459
2.2 Paradox threats in formal systems......Page 461
3 Type theory in Principia Mathematica......Page 467
3.1 Principia\'s propositional functions......Page 469
3.2 Principia\'s propositional functions as λ-terms......Page 471
3.3 Remarks on Principia\'s pfs and their translation in λ-calculus......Page 473
3.4 Principia\'s substitution and related notions......Page 476
3.5 Principia\'s Ramified Theory of Types......Page 480
3.6 A Formalisation RTT of the Ramified Theory of Types......Page 484
3.7 Discussion and examples......Page 487
3.8 Properties of RTT......Page 492
3.9 Legal propositional functions in RTT......Page 494
4.1 The problematic character of RTT......Page 498
4.2 The Axiom of Reducibility......Page 500
4.3 Deramification......Page 501
5.1 Constructing the Simple Theory of Types STT from RTT......Page 503
5.2 Church\'s Simply typed λ-calculus λ_{→C}......Page 504
5.3 Comparing RTT and λ_{→C}......Page 506
5.4 Comparison of STT with Church\'s λ_{→C}......Page 507
6 Conclusion......Page 508
Bibliography......Page 510
1 Introductory remarks......Page 515
1.1 The No-Theory problem......Page 516
2 Aristotle (384-322 BC)......Page 517
2.1 Aristotle\'s importance......Page 536
3 The Hellenistic and mediaeval periods......Page 540
3.1.1 Modes of supposition: Fallacies thereof......Page 546
3.2 The importance of this period......Page 547
4.1 Background remarks......Page 549
4.2 Idols of the mind......Page 550
5 Antoine Arnauld (1612-1694) and Pierre Nicole (1625-1695)......Page 554
5.1 Logic......Page 555
5.2 Sophisms......Page 557
5.3 Scientific sophisms......Page 560
5.4 Public and everyday sophisms......Page 561
5.5 The Port Royalists\' importance......Page 565
6 Isaac Watts: An interlude......Page 566
7 John Locke......Page 567
7.1 Locke\'s Essay......Page 568
7.2 Wrong assent, or error......Page 569
7.3 Critique of logic......Page 570
7.4 The imperfection and abuse of words......Page 572
7.5 Arguments ad......Page 573
7.6 Locke\'s Importance......Page 577
8.1 Logic......Page 578
9 John Stuart Mill (1806-1873)......Page 582
9.1 Deduction and inference......Page 583
9.2 Fallacy Theory......Page 588
9.3 Fallacies Classified......Page 589
9.3.1 Fallacies of simple inspection......Page 590
9.3.3 Fallacy of Generalization......Page 592
9.3.4 Fallacies of Ratiocination......Page 596
9.3.5 Fallacies of Confusion......Page 597
9.3.6 Mill\'s Importance......Page 598
10.1 On Fallacies......Page 599
10.2 Does the syllogism beg the question?......Page 601
10.3 DeMorgan\'s importance......Page 602
11 The Great Depression: 1848-1970......Page 603
12 Now......Page 606
Bibliography......Page 607
1 Introduction......Page 613
2.1 Euler\'s circles......Page 618
2.2 Venn\'s diagrams......Page 620
2.3 Peirce\'s improvements......Page 623
3 Representing information with diagrams......Page 625
3.1 Division and dichotomy......Page 626
3.2 Linear diagrams......Page 631
3.3 Tabular diagrams......Page 634
4.1 Working syllogisms......Page 640
4.2 Beyond syllogisms......Page 646
5 The Frege-Peirce affair......Page 651
5.1 Frege\'s two-dimensional notation......Page 654
5.2 Peirce\'s chef-d\'oeuvre......Page 663
5.3 The Frege-Peirce Rendez-vous......Page 671
6.1 Accuracy pursued......Page 672
6.2 Efficiency revisited......Page 675
6.3 Multi-modal reasoning and logic diagrams......Page 676
Bibliography......Page 679
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