Direct Proof

Direct proofWikipedia

Contents

1 Direct proof 11.1 History and etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 The sum of two even integers equals an even integer . . . . . . . . . . . . . . . . . . . . . 21.2.2 Pythagoras Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 If n is an odd integer, n2 is also an odd integer. . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Analogy 62.1 Usage of the terms source and target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Models and theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Identity of relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Shared abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Special case of induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Shared structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.5 High-level perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.6 Analogy and Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Applications and types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 In language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 In science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 In normative matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.4 In teaching strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Axiom 173.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Early Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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3.2.2 Modern development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.3 Other sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.1 Logical axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 Non-logical axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.3 Role in mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.4 Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Comparison 254.1 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.5 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Contraposition 275.1 Intuitive explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Simple proof by denition of a conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 Simple proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 More rigorous proof of the equivalence of contrapositives . . . . . . . . . . . . . . . . . . . . . . 305.6 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.6.2 Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.7 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 First-order logic 336.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.2.1 Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2.2 Formation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.2.3 Free and bound variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.3.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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6.3.2 Evaluation of truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3.3 Validity, satisability, and logical consequence . . . . . . . . . . . . . . . . . . . . . . . . 406.3.4 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.3.5 First-order theories, models, and elementary classes . . . . . . . . . . . . . . . . . . . . . 416.3.6 Empty domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.4 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.4.1 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.4.2 Hilbert-style systems and natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 426.4.3 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.4.4 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.4.6 Provable identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.5 Equality and its axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.5.1 First-order logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.5.2 Dening equality within a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.6 Metalogical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.6.1 Completeness and undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.6.2 The LwenheimSkolem theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.6.3 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.6.4 Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.7.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.7.2 Formalizing natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.8 Restrictions, extensions, and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.8.1 Restricted languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.8.2 Many-sorted logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.8.3 Additional quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.8.4 Innitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.8.5 Non-classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.8.6 Fixpoint logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.8.7 Higher-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.9 Automated theorem proving and formal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Formal proof 547.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.1.1 Formal language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.1.2 Formal grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.1.3 Formal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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7.1.4 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8 Formation rule 568.1 Formal language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.2 Formal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.3 Propositional and predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9 Law of excluded middle 589.1 Classic laws of thought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.2 Analogous laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

9.3.1 The Law in non-constructive proofs over the innite . . . . . . . . . . . . . . . . . . . . . 599.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9.4.1 Aristotle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609.4.2 Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609.4.3 Bertrand Russell and Principia Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . 61

9.5 Use in computer science proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629.6 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

10 Lemma (mathematics) 6610.1 Comparison with theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.2 Well-known lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

11 Logic 6811.1 The study of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

11.1.1 Logical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6811.1.2 Deductive and inductive reasoning, and abductive inference . . . . . . . . . . . . . . . . . 6911.1.3 Consistency, validity, soundness, and completeness . . . . . . . . . . . . . . . . . . . . . . 7011.1.4 Rival conceptions of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

11.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7011.3 Types of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

11.3.1 Syllogistic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

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11.3.2 Propositional logic (sentential logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7211.3.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7211.3.4 Modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.3.5 Informal reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.3.6 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7411.3.7 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7411.3.8 Computational logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7411.3.9 Bivalence and the law of the excluded middle; non-classical logics . . . . . . . . . . . . . 7511.3.10 Is logic empirical?" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7611.3.11 Implication: strict or material? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7611.3.12 Tolerating the impossible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7611.3.13 Rejection of logical truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

11.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7711.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7811.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8011.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

12 Logical constant 8212.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8212.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8212.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

13 Logical form 8313.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8313.2 Example of argument form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8313.3 Importance of argument form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8413.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8413.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8413.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8513.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

14 Logical truth 8614.1 Logical truths and analytic truths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8614.2 Truth values and tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.3 Logical truth and logical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.4 Logical truth and rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.5 Non-classical logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8814.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

15 Material conditional 8915.1 Denitions of the material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

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15.1.1 As a truth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.1.2 As a formal connective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

15.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.3 Philosophical problems with material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

15.4.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9215.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9215.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9215.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

16 Mathematical induction 9416.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9516.4 Axiom of induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

16.4.1 Characterizing the structure of N by the induction axiom . . . . . . . . . . . . . . . . . . 9616.5 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

16.5.1 Induction basis other than 0 or 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.5.2 Induction basis equal to 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.5.3 Induction on more than one counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9816.5.4 Innite descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9816.5.5 Prex induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9816.5.6 Complete induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

16.6 Equivalence with the well-ordering principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10016.7 Example of error in the inductive step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10116.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10116.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10116.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

16.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10216.10.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

17 Mathematics 10417.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

17.1.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10517.1.2 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

17.2 Denitions of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10817.3 Inspiration, pure and applied mathematics, and aesthetics . . . . . . . . . . . . . . . . . . . . . . . 10917.4 Notation, language, and rigor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11017.5 Fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

17.5.1 Foundations and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11017.5.2 Pure mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11217.5.3 Applied mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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17.6 Mathematical awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11417.7 Mathematics as science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11417.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11617.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11617.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11817.11Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11917.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

18 Modus ponens 12118.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12118.2 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12218.3 Justication via truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12218.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12218.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12318.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12318.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

19 Phenomenology (philosophy) 12419.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12419.2 Historical overview of the use of the term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12619.3 Phenomenological terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

19.3.1 Intentionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12719.3.2 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12719.3.3 Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12719.3.4 Noesis and noema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12719.3.5 Empathy and intersubjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12819.3.6 Lifeworld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

19.4 Husserls Logische Untersuchungen (1900/1901) . . . . . . . . . . . . . . . . . . . . . . . . . . . 12819.5 Transcendental phenomenology after the Ideen (1913) . . . . . . . . . . . . . . . . . . . . . . . . 12819.6 Realist phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12919.7 Existential phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12919.8 Eastern thought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13019.9 Technoethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

19.9.1 Phenomenological approach to technology . . . . . . . . . . . . . . . . . . . . . . . . . . 13019.9.2 Heideggers approach (pre-technological age) . . . . . . . . . . . . . . . . . . . . . . . . . 13019.9.3 The Hubert Dreyfus approach (contemporary society) . . . . . . . . . . . . . . . . . . . . 131

19.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13119.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13119.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

20 Proof (truth) 13420.1 On proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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20.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13520.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

21 Proof by contradiction 13621.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

21.1.1 Irrationality of the square root of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13621.1.2 The length of the hypotenuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13621.1.3 No least positive rational number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13721.1.4 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

21.2 In mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13721.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13721.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13821.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13821.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13821.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

22 Proof by exhaustion 13922.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13922.2 Number of cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13922.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14022.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

23 Proof by innite descent 14123.1 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14123.2 Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

23.2.1 Irrationality of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14223.2.2 Irrationality of k if it is not an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14223.2.3 Non-solvability of r2 + s4 = t4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

23.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14323.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14423.5 Other reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

24 Rule of inference 14524.1 The standard form of rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14524.2 Axiom schemas and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14624.3 Example: Hilbert systems for two propositional logics . . . . . . . . . . . . . . . . . . . . . . . . 14624.4 Admissibility and derivability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14724.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14724.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

25 Satisability 14925.1 Reduction of validity to satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14925.2 Propositional satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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25.3 Satisability in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15025.4 Satisability in model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15025.5 Finite satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15025.6 Numerical constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15125.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15125.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15125.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15125.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

26 Substitution (logic) 15226.1 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

26.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15226.1.2 Tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

26.2 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15326.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15326.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15426.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15426.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

27 Tautology (logic) 15527.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15527.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15627.3 Denition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15627.4 Verifying tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15727.5 Tautological implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15727.6 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15727.7 Ecient verication and the Boolean satisability problem . . . . . . . . . . . . . . . . . . . . . . 15827.8 Tautologies versus validities in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15827.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

27.9.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15927.9.2 Related logical topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

27.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15927.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

28 Tautology (rhetoric) 16028.1 Rhetorical tautology vs. circular reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

28.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16028.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16028.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16128.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

29 Theorem 16229.1 Informal account of theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

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29.2 Provability and theoremhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16329.3 Relation with scientic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16329.4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16329.5 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16429.6 Lore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16529.7 Theorems in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

29.7.1 Syntax and semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16629.7.2 Derivation of a theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16629.7.3 Interpretation of a formal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16729.7.4 Theorems and theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

29.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16729.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16729.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16829.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

30 Transposition (logic) 17330.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17330.2 Traditional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

30.2.1 Form of transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17430.2.2 Sucient condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17430.2.3 Necessary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17430.2.4 Grammatically speaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17430.2.5 Relationship of propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17430.2.6 Transposition and the method of contraposition . . . . . . . . . . . . . . . . . . . . . . . 17530.2.7 Dierences between transposition and contraposition . . . . . . . . . . . . . . . . . . . . . 175

30.3 Transposition in mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17530.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17530.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17530.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17530.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17630.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

31 Truth 17731.1 Denition and etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17731.2 Major theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

31.2.1 Substantive theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17831.2.2 Minimalist (deationary) theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18131.2.3 Pluralist theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18231.2.4 Most believed theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

31.3 Formal theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18231.3.1 Truth in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18331.3.2 Truth in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

CONTENTS xi

31.3.3 Semantic theory of truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18331.3.4 Kripkes theory of truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

31.4 Notable views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18431.4.1 Ancient history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18431.4.2 Middle Ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18531.4.3 Modern age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

31.5 In medicine and psychiatry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18931.6 In religion: omniscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18931.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

31.7.1 Major theorists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19031.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19131.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19431.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

32 Two Dogmas of Empiricism 20432.1 Analyticity and circularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20432.2 Reductionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20532.3 Quines holism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20632.4 Critique and inuence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20632.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20732.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20732.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20732.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

33 Universal instantiation 20833.1 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20833.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20833.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

34 Validity 21034.1 Validity of arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21034.2 Valid formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21134.3 Validity of statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21134.4 Validity and soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21134.5 Satisability and validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21234.6 Preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21234.7 n-Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

34.7.1 -Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21234.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21234.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21234.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21334.11Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 214

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34.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21434.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22334.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Chapter 1

Direct proof

In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by astraightforward combination of established facts, usually axioms, existing lemmas and theorems, without making anyfurther assumptions.[1] In order to directly prove a conditional statement of the form If p, then q", it suces toconsider the situations in which the statement p is true. Logical deduction is employed to reason from assumptionsto conclusion. The type of logic employed is almost invariably rst-order logic, employing the quantiers for all andthere exists. Common proof rules used are modus ponens and universal instantiation.[2]

In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncer-tainties in each of these scenarios until an inescapable conclusion is forced. For example instead of showing directlyp q, one proves its contrapositive ~q ~p (one assumes ~q and shows that it leads to ~p). Since p q and ~q ~p are equivalent by the principle of transposition (see law of excluded middle), p q is indirectly proved. Proofmethods that are not direct include proof by contradiction, including proof by innite descent. Direct proof methodsinclude proof by exhaustion and proof by induction.

1.1 History and etymologyA direct proof is the simplest form of proof there is. The word proof comes from the Latin word probare,[3] whichmeans to test. The earliest use of proofs was prominent in legal proceedings. A person with authority, such as anobleman, was said to have probity, which means that the evidence was by his relative authority, which outweighedempirical testimony. In days gone by, mathematics and proof was often intertwined with practical questions withpopulations like the Egyptians and the Greeks showing an interest in surveying land.[4] This lead to a natural curiositywith regards to geometry and trigonometry particularly triangles and rectangles. These were the shapes whichprovided the most questions in terms of practical things, so early geometrical concepts were focused on these shapes,for example, the likes of buildings and pyramids used these shapes in abundance. Another shape which is crucial inthe history of direct proof is the circle, which was crucial for the design of arenas and water tanks. This meant thatancient geometry (and Euclidean Geometry) discussed circles.The earliest form of mathematics was phenomenological. For example, if someone could draw a reasonable picture,or give a convincing description, then that met all the criteria for something to be described as a mathematical fact.On occasion, analogical arguments took place, or even by invoking the gods. The idea that mathematical statementscould be proven had not been developed yet, so these were the earliest forms of the concept of proof, despite notbeing actual proof at all.Proof as we know it came about with one specic question: what is a proof? Traditionally, a proof is a platformwhich convinces someone beyond reasonable doubt that a statement is mathematically true. Naturally, one wouldassume that the best way to prove the truth of something like this (B) would be to draw up a comparison withsomething old (A) that has already been proven as true. Thus was created the concept of deriving a new result froman old result.

1.2 Examples

1

2 CHAPTER 1. DIRECT PROOF

Geometric Constructions

1.2.1 The sum of two even integers equals an even integer

Consider two even integers x and y. Since they are even, they can be written as

x = 2a

y = 2b

respectively for integers a and b. Then the sum can be written as

x+ y = 2a+ 2b = 2(a+ b)

From this it is clear x + y has 2 as a factor and therefore is even, so the sum of any two even integers is even.

1.2.2 Pythagoras Theorem

Observe that we have four right-angled triangles and a square packed into a large square. Each of the triangles hassides a and b and hypotenuse c. The area of a square is dened as the square of the length of its sides - in this case,(a + b)2. However, the area of the large square can also be expressed as the sum of the areas of its components. Inthis case, that would be the sum of the areas of the four triangles and the small square in the middle.[5]

We know that the area of the large square is equal to (a + b)2

The area of a triangle is equal to 12ab

1.2. EXAMPLES 3

New result from an old result

We know that the area of the large square is also equal to the sum of the areas of the triangles, plus the area of thesmall square, and thus the area of the large square equals 4( 12ab) + c2

These are equal, and so:

(a+ b)2 = 4(1/2ab) + c2

After some simplifying:

a2 + 2ab+ b2 = 2ab+ c2

Removing the ab that appears on both sides gives

a2 + b2 = c2

Which proves Pythagoras theorem.

1.2.3 If n is an odd integer, n2 is also an odd integer.By denition, if n is an odd integer, it can be expressed as:

4 CHAPTER 1. DIRECT PROOF

Diagram of Pythagoras Theorem

n = 2k + 1

for some integer k. Thus:

n2 = (2k + 1)2

= (2k + 1)(2k + 1)

= 4k2 + 2k + 2k + 1

= 4k2 + 4k + 1

= 2(2k2 + 2k) + 1

As (2k2 + 2k) is an integer, our answer can be expressed as:

2k + 1

And hence we have shown that n2 is odd.

1.3. REFERENCES 5

1.3 References[1] Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001. Page 3.

[2] C. Gupta, S. Singh, S. Kumar Advanced Discrete Structure. I.K. International Publishing House Pvt. Ltd., 2010. Page 127.

[3] New Shorter Oxford English Dictionary

[4] Krantz, Steven G. The History and Concept of Mathematical Proof. February 5, 2007.

[5] Krantz, Steven G. The Proof is the Pudding. Springer, 2010. Page 43.

1.4 Sources Franklin, J.; A. Daoud (2011). Proof in Mathematics: An Introduction. Sydney: Kew Books. ISBN 0-646-

54509-4. (Ch. 1.)

1.5 External links Direct Proof from Larry W. Cusicks How To Write Proofs. Direct Proofs from Patrick Keef and David Guichards Introduction to Higher Mathematics. Direct Proof section of Richard Hammacks Book of Proof.

Chapter 2

Analogy

Analogy (from Greek , analogia, proportion[1][2]) is a cognitive process of transferring information ormeaning from a particular subject (the analogue or source) to another particular subject (the target), or a linguisticexpression corresponding to such a process. In a narrower sense, analogy is an inference or an argument from oneparticular to another particular, as opposed to deduction, induction, and abduction, where at least one of the premisesor the conclusion is general. The word analogy can also refer to the relation between the source and the targetthemselves, which is often, though not necessarily, a similarity, as in the biological notion of analogy.Analogy plays a signicant role in problem solving such as, decision making, perception, memory, creativity, emotion,explanation and communication. It lies behind basic tasks such as the identication of places, objects and people,for example, in face perception and facial recognition systems. It has been argued that analogy is the core ofcognition.[3] Specic analogical language comprises exemplication, comparisons, metaphors, similes, allegories,and parables, but not metonymy. Phrases like and so on, and the like, as if, and the very word like also rely on an ana-logical understanding by the receiver of a message including them. Analogy is important not only in ordinary languageand common sense (where proverbs and idioms give many examples of its application) but also in science, philosophyand the humanities. The concepts of association, comparison, correspondence, mathematical and morphological ho-mology, homomorphism, iconicity, isomorphism, metaphor, resemblance, and similarity are closely related to anal-ogy. In cognitive linguistics, the notion of conceptual metaphor may be equivalent to that of analogy.Analogy has been studied and discussed since classical antiquity by philosophers, scientists and lawyers. The last fewdecades have shown a renewed interest in analogy, most notably in cognitive science.

2.1 Usage of the terms source and targetWith respect to the terms source and target there are two distinct traditions of usage:

The logical and cultures and economics tradition speaks of an arrow, homomorphism, mapping, or morphismfrom what is typically the more complex domain or source to what is typically the less complex codomain ortarget, using all of these words in the sense of mathematical category theory.

The tradition in cognitive psychology, in literary theory, and in specializations within philosophy outside oflogic, speaks of a mapping from what is typically the more familiar area of experience, the source, to what istypically the more problematic area of experience, the target.

2.2 Models and theories

2.2.1 Identity of relationIn ancient Greek the word (analogia) originally meant proportionality, in the mathematical sense, and itwas indeed sometimes translated to Latin as proportio. From there analogy was understood as identity of relationbetween any two ordered pairs, whether of mathematical nature or not. Kants Critique of Judgment held to this

6

2.2. MODELS AND THEORIES 7

n = 1

n = 2

n = 3Increasing energy

of orbits

A photon is emittedwith energy E = hf

Rutherfords model of the atom (modied by Niels Bohr) made an analogy between the atom and the solar system.

notion. Kant argued that there can be exactly the same relation between two completely dierent objects. The samenotion of analogy was used in the US-based SAT tests, that included analogy questions in the form A is to Bas C is to what?" For example, Hand is to palm as foot is to ____?" These questions were usually given in theAristotelian format: HAND : PALM : : FOOT : ____ While most competent English speakers will immediatelygive the right answer to the analogy question (sole), it is more dicult to identify and describe the exact relation thatholds both between pairs such as hand and palm, and between foot and sole. This relation is not apparent in somelexical denitions of palm and sole, where the former is dened as the inner surface of the hand, and the latter as theunderside of the foot. Analogy and abstraction are dierent cognitive processes, and analogy is often an easier one.This analogy is not comparing all the properties between a hand and a foot, but rather comparing the relationshipbetween a hand and its palm to a foot and its sole.[4] While a hand and a foot have many dissimilarities, the analogyfocuses on their similarity in having an inner surface. A computer algorithm has achieved human-level performanceon multiple-choice analogy questions from the SAT test. The algorithm measures the similarity of relations betweenpairs of words (e.g., the similarity between the pairs HAND:PALM and FOOT:SOLE) by statistical analysis of alarge collection of text. It answers SAT questions by selecting the choice with the highest relational similarity.[5]

2.2.2 Shared abstractionGreek philosophers such as Plato and Aristotle actually used a wider notion of analogy. They saw analogy as a sharedabstraction.[6] Analogous objects did not share necessarily a relation, but also an idea, a pattern, a regularity, anattribute, an eect or a philosophy. These authors also accepted that comparisons, metaphors and images (alle-

8 CHAPTER 2. ANALOGY

In several cultures, the sun is the source of an analogy to God.

gories) could be used as arguments, and sometimes they called them analogies. Analogies should also make thoseabstractions easier to understand and give condence to the ones using them.The Middle Age saw an increased use and theorization of analogy. Roman lawyers had already used analogicalreasoning and the Greek word analogia. Medieval lawyers distinguished analogia legis and analogia iuris (see below).In Islamic logic, analogical reasoning was used for the process of qiyas in Islamic sharia law and qh jurisprudence.In Christian theology, analogical arguments were accepted in order to explain the attributes of God. Aquinas made adistinction between equivocal, univocal and analogical terms, the last being those like healthy that have dierent butrelated meanings. Not only a person can be healthy, but also the food that is good for health (see the contemporarydistinction between polysemy and homonymy). Thomas Cajetan wrote an inuential treatise on analogy. In all ofthese cases, the wide Platonic and Aristotelian notion of analogy was preserved. James Francis Ross in PortrayingAnalogy (1982), the rst substantive examination of the topic since Cajetans De Nominum Analogia, demonstratedthat analogy is a systematic and universal feature of natural languages, with identiable and law-like characteristicswhich explain how the meanings of words in a sentence are interdependent.

2.2.3 Special case of induction

On the contrary, Ibn Taymiyya,[7][8][9] Francis Bacon and later John Stuart Mill argued that analogy is simply aspecial case of induction.[6] In their view analogy is an inductive inference from common known attributes to anotherprobable common attribute, which is known only about the source of the analogy, in the following form:

Premises a is C, D, E, F, G

b is C, D, E, F

Conclusion b is probably G.

This view does not accept analogy as an autonomous mode of thought or inference, reducing it to induction. However,autonomous analogical arguments are still useful in science, philosophy and the humanities (see below), which makesthis reduction philosophically uninteresting. Moreover, induction tries to achieve general conclusions, while analogylooks for particular ones.

2.2. MODELS AND THEORIES 9

2.2.4 Shared structure

According to Shelley (2003), the study of the coelacanth drew heavily on analogies from other sh.

Contemporary cognitive scientists use a wide notion of analogy, extensionally close to that of Plato and Aristotle, butframed by Gentners (1983) structure mapping theory.[10] The same idea of mapping between source and target is usedby conceptual metaphor and conceptual blending theorists. Structure mapping theory concerns both psychology andcomputer science. According to this view, analogy depends on the mapping or alignment of the elements of sourceand target. The mapping takes place not only between objects, but also between relations of objects and betweenrelations of relations. The whole mapping yields the assignment of a predicate or a relation to the target. Structuremapping theory has been applied and has found considerable conrmation in psychology. It has had reasonablesuccess in computer science and articial intelligence (see below). Some studies extended the approach to specicsubjects, such as metaphor and similarity.[11]

Keith Holyoak and Paul Thagard (1997) developed their multiconstraint theory within structure mapping theory.They defend that the "coherence" of an analogy depends on structural consistency, semantic similarity and purpose.Structural consistency is maximal when the analogy is an isomorphism, although lower levels are admitted. Similaritydemands that the mapping connects similar elements and relations of source and target, at any level of abstraction.It is maximal when there are identical relations and when connected elements have many identical attributes. Ananalogy achieves its purpose insofar as it helps solve the problem at hand. The multiconstraint theory faces somediculties when there are multiple sources, but these can be overcome.[6] Hummel and Holyoak (2005) recast themulticonstraint theory within a neural network architecture. A problem for the multiconstraint theory arises fromits concept of similarity, which, in this respect, is not obviously dierent from analogy itself. Computer applicationsdemand that there are some identical attributes or relations at some level of abstraction. The model was extended(Doumas, Hummel, and Sandhofer, 2008) to learn relations from unstructured examples (providing the only currentaccount of how symbolic representations can be learned from examples).[12]

Mark Keane and Brayshaw (1988) developed their Incremental Analogy Machine (IAM) to include working memoryconstraints as well as structural, semantic and pragmatic constraints, so that a subset of the base analog is selectedand mapping from base to target occurs in a serial manner.[13][14] Empirical evidence shows that human analogicalmapping performance is inuenced by information presentation order.[15]

2.2.5 High-level perceptionDouglas Hofstadter and his team[16] challenged the shared structure theory and mostly its applications in computerscience. They argue that there is no line between perception, including high-level perception, and analogical thought.In fact, analogy occurs not only after, but also before and at the same time as high-level perception. In high-levelperception, humans make representations by selecting relevant information from low-level stimuli. Perception isnecessary for analogy, but analogy is also necessary for high-level perception. Chalmers et al. conclude that analogyactually is high-level perception. Forbus et al. (1998) claim that this is only a metaphor.[17] It has been argued


(Morrison and Dietrich 1995) that Hofstadters and Gentners groups do not defend opposite views, but are insteaddealing with dierent aspects of analogy.[18]

2.2.6 Analogy and Complexity

Antoine Cornujols[19] has presented analogy as a principle of economy and computational complexity.Reasoning by analogy is a process of, from a given pair (x,f(x)), extrapolating the function f. In the standard modeling,analogical reasoning involves two objects": the source and the target. The target is supposed to be incomplete andin need for a complete description using the source. The target has an existing part St and a missing part Rt. Weassume that we can isolate a situation of the source Ss, which corresponds to a situation of target St, and the result ofthe source Rs, which correspond to the result of the target Rt. With Bs, the relation between Ss and Rs, we want Bt,the relation between St and Rt.If the source and target are completely known:Using Kolmogorov complexity K(x), dened as the size of the smallest description of x and Solomono's approachto induction, Rissanen (89),[20] Wallace & Boulton (68) proposed the principle of Minimum description length. Thisprinciple leads to minimize the complexity K(target| Source) of producing the target from the source.This is unattractive in Articial Intelligence, as it requires a computation over abstract Turing machines. Supposethat Ms and Mt are local theories of the source and the target, available to the observer. The best analogy between asource case and a target case is the analogy that minimizes:

K(Ms) + K(Ss|Ms) + K(Bs|Ms) + K(Mt|Ms) + K(St|Mt) + K(Bt|Mt) (1).

If the target is completely unknown:All models and descriptions Ms, Mt, Bs, Ss, and St leading to the minimization of:

K(Ms) + K(Ss|Ms) + K(Bs|Ms) + K(Mt|Ms) + K(St|Mt) (2)

are also those who allow to obtain the relationship Bt, and thus the most satisfactory Rt for formula (1).The analogical hypothesis, which solves an analogy between a source case and a target case, has two parts:

Analogy, like induction, is a principle of economy. The best analogy between two cases is the one whichminimizes the amount of information necessary for the derivation of the source from the target (1). Its mostfundamental measure is the computational complexity theory.

When solving or completing a target case with a source case, the parameters which minimize (2) are postulatedto minimize (1), and thus, produce the best response.

However, a cognitive agent may simply reduce the amount of information necessary for the interpretation of the sourceand the target, without taking into account the cost of data replication. So, it may prefer to the minimization of (2)the minimization of the following simplied formula:

K(Ms) + K(Bs|Ms) + K(Mt|Ms)

2.3 Applications and types

2.3.1 In language

Logic

Logicians analyze how analogical reasoning is used in arguments from analogy.

2.3. APPLICATIONS AND TYPES 11

Rhetoric

An analogy can be a spoken or textual comparison between two words (or sets of words) to highlight someform of semantic similarity between them. Such analogies can be used to strengthen political and philosophicalarguments, even when the semantic similarity is weak or non-existent (if crafted carefully for the audience).Analogies are sometimes used to persuade those that cannot detect the awed or non-existent arguments.

Linguistics

An analogy can be the linguistic process that reduces word forms perceived as irregular by remaking them inthe shape of more common forms that are governed by rules. For example, the English verb help once had thepreterite holp and the past participle holpen. These obsolete forms have been discarded and replaced by helpedby the power of analogy (or by widened application of the productive Verb-ed rule.) This is called leveling.However, irregular forms can sometimes be created by analogy; one example is the American English pasttense form of dive: dove, formed on analogy with words such as drive: drove.

Neologisms can also be formed by analogy with existing words. A good example is software, formed by analogywith hardware; other analogous neologisms such as rmware and vaporware have followed. Another exampleis the humorous[21] term underwhelm, formed by analogy with overwhelm.

Analogy is often presented as an alternative mechanism to generative rules for explaining productive formationof structures such as words. Others argue that in fact they are the same mechanism, that rules are analogiesthat have become entrenched as standard parts of the linguistic system, whereas clearer cases of analogy havesimply not (yet) done so (e.g. Langacker 1987.445447). This view has obvious resonances with the currentviews of analogy in cognitive science which are discussed above.

2.3.2 In science

Analogues are often used in theoretical and applied sciences in the form of models or simulations which can beconsidered as strong analogies. Other much weaker analogies assist in understanding and describing functional be-haviours of similar systems. For instance, an analogy commonly used in electronics textbooks compares electricalcircuits to hydraulics. Another example is the analog ear based on electrical, electronic or mechanical devices.

Mathematics

Some types of analogies can have a precise mathematical formulation through the concept of isomorphism. In detail,this means that given two mathematical structures of the same type, an analogy between them can be thought of as abijection between them which preserves some or all of the relevant structure. For example, R2 and C are isomorphicas vector spaces, but the complex numbers, C , have more structure than R2 does: C is a eld as well as a vectorspace.Category theory takes the idea of mathematical analogy much further with the concept of functors. Given twocategories C and D, a functor F from C to D can be thought of as an analogy between C and D, because F has tomap objects of C to objects of D and arrows of C to arrows of D in such a way that the compositional structure ofthe two categories is preserved. This is similar to the structure mapping theory of analogy of Dedre Gentner, in thatit formalizes the idea of analogy as a function which satises certain conditions.

Articial intelligence

Steven Phillips and William H. Wilson [22][23] use category theory to mathematically demonstrate how the analogicalreasoning in the human mind, that is free of the spurious inferences that plague conventional articial intelligencemodels, (called systematicity), could arise naturally from the use of relationships between the internal arrows that keepthe internal structures of the categories rather than the mere relationships between the objects (called representationalstates). Thus, the mind may use analogies between domains whose internal structures t according with a naturaltransformation and reject those that do not.See also case-based reasoning.


Anatomy

See also: Analogy (biology)

In anatomy, two anatomical structures are considered to be analogous when they serve similar functions but are notevolutionarily related, such as the legs of vertebrates and the legs of insects. Analogous structures are the result ofconvergent evolution and should be contrasted with homologous structures.

Engineering

Often a physical prototype is built to model and represent some other physical object. For example, wind tunnels areused to test scale models of wings and aircraft, which act as an analog to full-size wings and aircraft.For example, the MONIAC (an analog computer) used the ow of water in its pipes as an analog to the ow of moneyin an economy.

Cybernetics

Where there is dependence and hence interaction between a pair or more of biological or physical participants com-munication occurs and the stresses produced describe internal models inside the participants. Pask in his ConversationTheory asserts there exists an analogy exhibiting both similarities and dierences between any pair of the participantsinternal models or concepts.

2.3.3 In normative matters

Morality

Analogical reasoning plays a very important part in morality. This may be in part because morality is supposed tobe impartial and fair. If it is wrong to do something in a situation A, and situation B is analogous to A in all relevantfeatures, then it is also wrong to perform that action in situation B. Moral particularism accepts analogical moralreasoning, rejecting both deduction and induction, since only the former can do without moral principles.

Law

In law, analogy is used to resolve issues on which there is no previous authority. A distinction has to be made betweenanalogous reasoning from written law and analogy to precedent case law.

Analogies from codes and statutes In civil law systems, where the preeminent source of law is legal codes andstatutes, a lacuna (a gap) arises when a specic issue is not explicitly dealt with in written law. Judges will try toidentify a provision whose purpose applies to the case at hand. That process can reach a high degree of sophistication,as judges sometimes not only look at a specic provision to ll lacunae (gaps), but at several provisions (from whichan underlying purpose can be inferred) or at general principles of the law to identify the legislator's value judgementfrom which the analogy is drawn. Besides the not very frequent lling of lacunae, analogy is very commonly usedbetween dierent provisions in order to achieve substantial coherence. Analogy from previous judicial decisions isalso common, although these decisions are not binding authorities.

Analogies from precedent case law By contrast, in common law systems, where precedent cases are the primarysource of law, analogies to codes and statutes are rare (since those are not seen as a coherent system, but as incursionsinto the common law). Analogies are thus usually drawn from precedent cases: The judge nds that the facts ofanother case are similar to the one at hand to an extent that the analogous application of the rule established in theprevious case is justied.

2.4. SEE ALSO 13

In gender science

In the 19th century, there was increased attention to dierences in gender. Scientists started to use an analogybetween race and gender to explain gender dierences. In gender, the female represents a lower race than the male.Researchers record the data of womens bodies for analysis. Nancy Leys Stepan believes that the analogy is so crucialin science that it shapes and inuences scientic study. In her article Race and Gender: The Role of Analogyin Science,[24] she states The analogy guided research, generated new hypotheses, and helped disseminate new,usually technical vocabularies. The analogy dened what was problematic about these social groups, what aspects ofthem needed further investigation, and which kinds of measurements and what data would be signicant for scienticinquiry.

2.3.4 In teaching strategiesAnalogies as dened in rhetoric, are a comparison between words, but an analogy can be used in teaching as well. Ananalogy as used in teaching would be comparing a topic that students are already familiar with, with a new topic thatis being introduced so that students can get a better understanding of the topic and relate back to previous knowledge.Shawn Glynn, a professor in the department of educational psychology and instructional technology at the Universityof Georgia,[25] developed a theory on teaching with analogies and developed steps to explain the process of teachingwith this method. The steps for teaching with analogies are as follows: Step one is introducing the new topic thatis about to be taught and giving some general knowledge on the subject. Step two is reviewing the concept that thestudents already know to ensure they have the proper knowledge to assess the similarities between the two concepts.Step three is nding relevant features within the analogy of the two concepts. Step four is nding similarities betweenthe two concepts so students are able to compare and contrast them in order to understand. Step ve is indicatingwhere the analogy breaks down between the two concepts. And nally, step six is drawing a conclusion about theanalogy and comparison of the new material with the already learned material. Typically this method is used to learntopics in science.[26]

In 1989, Kerry Ruef, a teacher began an entire program, which she titled The Private Eye Project. It is a methodof teaching that revolves around using analogies in the classroom to better explain topics. She thought of the ideato use analogies as a part of curriculum because she was observing objects once and she said, my mind was notingwhat else each object reminded me of... This led her to teach with the question, what does [the subject or topic]remind you of?" The idea of comparing subjects and concepts led to the development of The Private Eye Project asa method of teaching.[27] The program is designed to build critical thinking skills with analogies as one of the mainthemes revolving around it. While Glynn focuses on using analogies to teach science, The Private Eye Project canbe used for any subject including writing, math, art, social studies, and invention. It is now used by thousands ofschools around the country.[28] There are also various pedagogic innovations now emerging that use visual analogiesfor cross-disciplinary teaching and research, for instance between science and the humanities.[29]

2.4 See also Conceptual metaphor Conceptual blending Argument from analogy False analogy Metaphor Simile Hypocatastasis Allegory Argumentum e contrario Parable


2.5 Notes[1] , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus Digital Library

[2] analogy, Online Etymology Dictionary

[3] Hofstadter in Gentner et al. 2001.

[4] , Michael A. Martin, The Use of Analogies and Heuristics in Teaching Introductory Statistical Methods

[5] Turney 2006

[6] Shelley 2003

[7] Hallaq, Wael B. (19851986). The Logic of Legal Reasoning in Religious and Non-Religious Cultures: The Case ofIslamic Law and the Common Law. Cleveland State Law Review 34: 7996 [935]

[8] Ruth Mas (1998). Qiyas: A Study in Islamic Logic (PDF). Folia Orientalia 34: 113128. ISSN 0015-5675.

[9] John F. Sowa; Arun K. Majumdar (2003). Analogical reasoning. Conceptual Structures for Knowledge Creation andCommunication, Proceedings of ICCS 2003. Berlin: Springer-Verlag., pp. 1636

[10] See Dedre Gentner et al. 2001

[11] See Gentner et al. 2001 and Gentners publication page.

[12] Doumas, Hummel, and Sandhofer, 2008

[13] Keane, M.T. and Brayshaw, M. (1988). The Incremental Analogical Machine: a computational model of analogy. In D.Sleeman (Ed). European working session on learning. (pp.5362). London: Pitman.

[14] Keane, M.T. Ledgeway; Du, S (1994). Constraints on analogical mapping: a comparison of three models. CognitiveScience 18: 287334.

[15] Keane, M.T. (1997). What makes an analogy dicult? The eects of order and causal structure in analogical mapping.Journal of Experimental Psychology: Learning, Memory and Cognition 123: 946967.

[16] See Chalmers et al. 1991

[17] Forbus et al., 1998

[18] Morrison and Dietrich, 1995

[19] Cornujols, A. (1996). Analogie, principe dconomie et complexit algorithmique. InActes des 11mes Journes Franaisesde lApprentissage. Ste.

[20] Rissanen J. (1989) : Stochastical Complexity and Statistical Inquiry. World Scientic Publishing Company, 1989.

[21] http://www.oxforddictionaries.com/us/definition/american_english/underwhelm

[22] Phillips, Steven; Wilson, William H. (July 2010). Categorial Compositionality: A Category Theory Explanation for theSystematicity of Human Cognition. PLoS Computational Biology 6 (7). doi:10.1371/journal.pcbi.1000858.

[23] Phillips, Steven; Wilson, William H. (August 2011). Categorial Compositionality II: Universal Constructions and a Gen-eral Theory of (Quasi-)Systematicity in Human Cognition. PLoS Computational Biology 7 (8). doi:10.1371/journal.pcbi.1002102.

[24] Stepan, Nancy Leys (Jun 1986). Race and Gender: The Role of Analogy in Science. Isis 77 (2): 261277. doi:10.1086/354130.JSTOR 232652. Retrieved 2012-12-30.

[25] University of Georgia. Curriculum Vitae of Shawn M. Glynn. 2012. 16 October 2013

[26] Glynn, Shawn M. Teaching with Analogies. 2008.

[27] Johnson, Katie. Educational Leadership: Exploring the World with the Private Eye. September 1995. 16 October 2013 .

[28] The Private Eye Project. The Private Eye Project. 2013.

[29] Mario Petrucci. Crosstalk, Mutation, Chaos: bridge-building between the sciences and literary studies using Visual Anal-ogy.

2.6. REFERENCES 15

2.6 References Cajetan, Tommaso De Vio, (1498), De Nominum Analogia, P.N. Zammit (ed.), 1934, The Analogy of Names,

Koren, Henry J. and Bushinski, Edward A (trans.), 1953, Pittsburgh: Duquesne University Press.

Chalmers, D.J. et al. (1991). Chalmers, D.J., French, R.M., Hofstadter, D., High-Level Perception, Repre-sentation, and Analogy.

Coelho, Ivo (2010). Analogy. ACPI Encyclopedia of Philosophy. Ed. Johnson J. Puthenpurackal. Banga-lore: ATC. 1:64-68.

Cornujols, A. (1996). Analogie, principe dconomie et complexit algorithmique. In Actes des 11mesJournes Franaises de lApprentissage. Ste.

Doumas, L. A. A., Hummel, J.E., and Sandhofer, C. (2008). A Theory of the Discovery and Predication ofRelational Concepts. Psychological Review, 115, 1-43.

Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy. Cognitive Science, 7, 155170.(Reprinted in A. Collins & E. E. Smith (Eds.), Readings in cognitive science: A perspective from psychologyand articial intelligence. Palo Alto, CA: Kaufmann).

Forbus, K. et al. (1998). Analogy just looks like high-level perception. Gentner, D., Holyoak, K.J., Kokinov, B. (Eds.) (2001). The Analogical Mind: Perspectives from Cognitive

Science. Cambridge, MA, MIT Press, ISBN 0-262-57139-0

Hofstadter, D. (2001). Analogy as the Core of Cognition, in Dedre Gentner, Keith Holyoak, and BoichoKokinov (eds.) The Analogical Mind: Perspectives from Cognitive Science, Cambridge, MA: The MITPress/Bradford Book, 2001, pp. 499538.

Holland, J.H., Holyoak, K.J., Nisbett, R.E., and Thagard, P. (1986). Induction: Processes of Inference, Learn-ing, and Discovery. Cambridge, MA, MIT Press, ISBN 0-262-58096-9.

Holyoak, K.J., and Thagard, P. (1995). Mental Leaps: Analogy in Creative Thought. Cambridge, MA, MITPress, ISBN 0-262-58144-2.

Holyoak, K.J., and Thagard, P. (1997). The Analogical Mind. Hummel, J.E., and Holyoak, K.J. (2005). Relational Reasoning in a Neurally Plausible Cognitive Architecture Itkonen, E. (2005). Analogy as Structure and Process. Amsterdam/Philadelphia: John Benjamins Publishing

Company.

Juthe, A. (2005). Argument by Analogy, in Argumentation (2005) 19: 127. Keane, M.T. Ledgeway; Du, S (1994). Constraints on analogical mapping: a comparison of three models.Cognitive Science 18: 287334.

Keane, M.T. (1997). What makes an analogy dicult? The eects of order and causal structure in analogicalmapping. Journal of Experimental Psychology: Learning, Memory and Cognition 123: 946967.

Kokinov, B. (1994). A hybrid model of reasoning by analogy. Kokinov, B. and Petrov, A. (2001). Integration of Memory and Reasoning in Analogy-Making. Lamond, G. (2006). Precedent and Analogy in Legal Reasoning, in Stanford Encyclopedia of Philosophy. Langacker, Ronald W. (1987). Foundations of Cognitive grammar. Vol. I, Theoretical prerequisites. Stanford:

Stanford University Press.

Little, J. (2000). Analogy in Science: Where Do We Go From Here? Rhetoric Society Quarterly, 30, 6992. Little, J. (2008). The Role of Analogy in George Gamows Derivation of Drop Energy. Technical Communi-cation Quarterly, 17, 119.

Morrison, C., and Dietrich, E. (1995). Structure-Mapping vs. High-level Perception.


Ross, J.F., (1982), Portraying Analogy. Cambridge: Cambridge University Press. Ross, J.F. (October 1970). Analogy and The Resolution of Some Cognitivity Problems. The Journal ofPhilosophy 67 (20): 725746. doi:10.2307/2024008. JSTOR 2024008.

Ross, J.F. (September 1961). Analogy as a Rule of Meaning for Religious Language. International Philo-sophical Quarterly 1 (3): 468502. doi:10.5840/ipq19611356.

Ross, J.F., (1958), A Critical Analysis of the Theory of Analogy of St Thomas Aquinas, (Ann Arbor, MI:University Microlms Inc).

Shelley, C. (2003). Multiple analogies in Science and Philosophy. Amsterdam/Philadelphia: John BenjaminsPublishing Company.

Turney, P.D., and Littman, M.L. (2005). Corpus-based learning of analogies and semantic relations. MachineLearning, 60 (13), 251278.

Turney, P.D. (2006). Similarity of semantic relations. Computational Linguistics, 32 (3), 379416. Cornujols, A. (1996). Analogy, principle of economy and computational complexity.

2.7 External links Stanford Encyclopedia of Philosophy Analogy and Analogical Reasoning, by Paul Bartha. Stanford Encyclopedia of Philosophy Medieval Theories of Analogy, by E. Jennifer Ashworth. Stanford Encyclopedia of Philosophy Precedent and Analogy in Legal Reasoning, by Grant Lamond. Dictionary of the History of Ideas Analogy in Early Greek Thought. Dictionary of the History of Ideas Analogy in Patristic and Medieval Thought. Dedre Gentners publications page, most of them on analogy and available for download. Shawn Glynns publications page, all on teaching with analogies and some available for download. Keith Holyoaks publications page, many on analogy and available for download. Boicho Kokinovs publications page, most of them on analogy and available for download. The Private Eye Projects publications page, all on teaching with analogies (and thinking by analogy) and some

available for download.

jMapper Java Library for Analogy/Metaphor Generation Analogy games Analogy used as the basis for a cultural game

Chapter 3

Axiom

This article is about logical propositions. For other uses, see Axiom (disambiguation).Axiomatic redirects here. For other uses, see Axiomatic (disambiguation).Postulation redirects here. For the term in algebraic geometry, see Postulation (algebraic geometry).

An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premiseso evident as to be accepted as true without controversy.[1] The word comes from the Greek axma () 'thatwhich is thought worthy or t' or 'that which commends itself as evident.'[2][3] As used in modern logic, an axiom issimply a premise or starting point for reasoning.[4] What it means for an axiom, or any mathematical statement, tobe true is a central question in the philosophy of mathematics, with modern mathematicians holding a multitude ofdierent opinions.In mathematics, the term axiom is used in two related but distinguishable senses: logical axioms and non-logicalaxioms. Logical axioms are usually statements that are taken to be true within the system of logic they dene(e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions aboutthe elements of the domain of a specic mathematical theory (such as arithmetic). When used in the latter sense,axiom, postulate, and assumption may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. As modernmathematics admits multiple, equally true systems of logic, precisely the same thing must be said for logical axioms- they both dene and are specic to the particular system of logic that is being invoked. To axiomatize a systemof knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms).There are typically multiple ways to axiomatize a given mathematical domain.In both senses, an axiom is any mathematical statement that serves as a starting point from which other statementsare logically derived. Within the system they dene, axioms (unless redundant) cannot be derived by principlesof deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there isnothing else from which they logically follow otherwise they would be classied as theorems. However, an axiom inone system may be a theorem in another, and vice versa.

3.1 EtymologyThe word axiom comes from the Greek word (axioma), a verbal noun from the verb (axioein),meaning to deem worthy, but also to require, which in turn comes from (axios), meaning being in balance,and hence having (the same) value (as)", worthy, proper. Among the ancient Greek philosophers an axiom wasa claim which could be seen to be true without any need for proof.The root meaning of the word 'postulate' is to 'demand'; for instance, Euclid demands of us that we agree that somethings can be done, e.g. any two points can be joined by a straight line, etc.[5]

Ancient geometers maintained some distinction between axioms and postulates. While commenting Euclids booksProclus remarks that "Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom,since it does not, like the rst three Postulates, assert the possibility of some construction but expresses an essentialproperty.[6] Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscriptsthis usage was not always strictly kept.

17

18 CHAPTER 3. AXIOM

3.2 Historical development

3.2.1 Early GreeksThe logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) throughthe application of sound arguments (syllogisms, rules of inference), was developed by the ancient Greeks, and hasbecome the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing isassumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. Theyare accepted without demonstration. All other assertions (theorems, if we are talking about mathematics) must beproven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changedfrom ancient times to the modern, and consequently the terms axiom and postulate hold a slightly dierent meaningfor the present day mathematician, than they did for Aristotle and Euclid.The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on parwith scientic facts. As such, they developed and used the logico-deductive method as a means of avoiding error, andfor structuring and communicating knowledge. Aristotles posterior analytics is a denitive exposition of the classicalview.An axiom, in classical terminology, referred to a self-evident assumption common to many branches of science. Agood example would be the assertion that

When an equal amount is taken from equals, an equal amount results.

At the foundation of the various sciences lay certain additional hypotheses whi

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