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Text of Negation Introduction

  • Negation introductionFrom Wikipedia, the free encyclopedia

  • Contents

    1 Absorption (logic) 11.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Proof by truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Formal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    2 Associative property 32.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Generalized associative law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    2.4.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    2.5 Non-associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5.1 Nonassociativity of oating point calculation . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.2 Notation for non-associative operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    2.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    3 Axiom 113.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    3.2.1 Early Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 Modern development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.3 Other sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    3.3 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.1 Logical axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.2 Non-logical axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.3 Role in mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.4 Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    3.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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    3.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    4 Axiom schema 194.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Finite axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Finitely axiomatized theoreies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 In higher-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    5 Axiomatic system 215.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Relative consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4 Axiomatic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    5.4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4.2 Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4.3 Example: The Peano axiomatization of natural numbers . . . . . . . . . . . . . . . . . . . 235.4.4 Axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    5.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    6 Biconditional elimination 246.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    7 Biconditional introduction 267.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    8 Commutative property 278.1 Common uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.2 Mathematical denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    8.3.1 Commutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.3.2 Commutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.3.3 Noncommutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . 298.3.4 Noncommutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 30

    8.4 History and etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.5 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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    8.5.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.5.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    8.6 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.7 Mathematical structures and commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.8 Related properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    8.8.1 Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.8.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

    8.9 Non-commuting operators in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    8.12.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.12.2 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.12.3 Online resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    9 Conjunction elimination 369.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    10 Conjunction introduction 3810.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    11 Constructive dilemma 3911.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2 Variable English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.3 Natural language example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    12 De Morgans laws 4112.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    12.1.1 Substitution form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.1.2 Set theory and Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.1.3 Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.1.4 Text searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

    12.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.3 Informal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    12.3.1 Negation of a disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.3.2 Negation of a conjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    12.4 Formal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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    12.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    13 Destructive dilemma 4913.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.2 Natural language example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.4 Example proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

    14 Disjunction elimination 5114.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    15 Disjunction introduction 5315.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    16 Disjunctive syllogism 5416.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.2 Natural language examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.3 Inclusive and exclusive disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.4 Related argument forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

    17 Distributive property 5717.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5717.2 Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5717.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    17.3.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5817.3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.3.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

    17.4 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.4.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.4.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    17.5 Distributivity and rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017.6 Distributivity in rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017.7 Generalizations of distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    17.7.1 Notions of antidistributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6117.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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    17.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    18 Double negation 6318.1 Double negative elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

    18.1.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.3 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6418.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

    19 Existential generalization 6619.1 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    20 Existential instantiation 6720.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    21 Exportation (logic) 6821.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6821.2 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    21.2.1 Truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6821.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    21.3 Relation to functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6921.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    22 First principle 7022.1 First principles in formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.2 Philosophy in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.3 Aristotles contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.4 Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7122.5 In physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7122.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7122.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7222.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    23 Formal ethics 7323.1 Symbolic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7423.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7523.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    24 Formal proof 7624.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

  • vi CONTENTS

    24.1.1 Formal language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.2 Formal grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.3 Formal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.4 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

    24.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7724.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7724.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

    25 Formal system 7825.1 Related subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    25.1.1 Logical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.1.2 Deductive system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.1.3 Formal proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.1.4 Formal language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.1.5 Formal grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    25.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    26 Hypothetical syllogism 8126.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8126.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8126.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    27 List of formal systems 8327.1 Mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.2 Other formal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    28 Material implication (rule of inference) 8428.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8428.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8428.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    29 Modus ponendo tollens 8629.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

    30 Modus ponens 8730.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8730.2 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8830.3 Justication via truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8830.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

  • CONTENTS vii

    30.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8930.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8930.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

    31 Modus tollens 9031.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9031.2 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9131.3 Relation to modus ponens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9131.4 Justication via truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9231.5 Formal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    31.5.1 Via disjunctive syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9231.5.2 Via reductio ad absurdum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    31.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9231.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9231.8 External link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    32 Negation introduction 9332.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9332.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9332.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    33 Physical symbol system 9433.1 Examples of physical symbol systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9433.2 Arguments in favor of the physical symbol system hypothesis . . . . . . . . . . . . . . . . . . . . 95

    33.2.1 Newell and Simon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9533.2.2 Turing completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

    33.3 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9533.3.1 Dreyfus and the primacy of unconscious skills . . . . . . . . . . . . . . . . . . . . . . . . 9633.3.2 Searle and his Chinese room . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9633.3.3 Brooks and the roboticists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9633.3.4 Connectionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9633.3.5 Embodied philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

    33.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9733.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9733.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

    34 Predicate logic 9934.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9934.2 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9934.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    35 Proof (truth) 10135.1 On proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

  • viii CONTENTS

    35.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10235.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

    36 Rule of inference 10336.1 The standard form of rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10336.2 Axiom schemas and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10436.3 Example: Hilbert systems for two propositional logics . . . . . . . . . . . . . . . . . . . . . . . . 10436.4 Admissibility and derivability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10536.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10536.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

    37 Rule of replacement 10737.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

    38 Tautology (rule of inference) 10838.1 Relation to tautology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10838.2 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10838.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

    39 Transposition (logic) 11039.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11039.2 Traditional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    39.2.1 Form of transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11139.2.2 Sucient condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11139.2.3 Necessary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11139.2.4 Grammatically speaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11139.2.5 Relationship of propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11139.2.6 Transposition and the method of contraposition . . . . . . . . . . . . . . . . . . . . . . . 11239.2.7 Dierences between transposition and contraposition . . . . . . . . . . . . . . . . . . . . . 112

    39.3 Transposition in mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11239.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11239.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11239.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11239.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11339.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

    40 Universal generalization 11440.1 Generalization with hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11440.2 Example of a proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11440.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11440.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

    41 Universal instantiation 11641.1 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

  • CONTENTS ix

    41.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11641.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11741.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 118

    41.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11841.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12241.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

  • Chapter 1

    Absorption (logic)

    Absorption is a valid argument form and rule of inference of propositional logic.[1][2] The rule states that if P impliesQ , then P implies P and Q . The rule makes it possible to introduce conjunctions to proofs. It is called the law ofabsorption because the term Q is absorbed by the term P in the consequent.[3] The rule can be stated:

    P ! Q) P ! (P ^Q)

    where the rule is that wherever an instance of " P ! Q " appears on a line of a proof, " P ! (P ^ Q) " can beplaced on a subsequent line.

    1.1 Formal notation

    The absorption rule may be expressed as a sequent:

    P ! Q ` P ! (P ^Q)

    where ` is a metalogical symbol meaning that P ! (P ^Q) is a syntactic consequences of (P ! Q) in some logicalsystem;and expressed as a truth-functional tautology or theorem of propositional logic. The principle was stated as a theoremof propositional logic by Russell and Whitehead in Principia Mathematica as:

    (P ! Q)$ (P ! (P ^Q))

    where P , and Q are propositions expressed in some formal system.

    1.2 Examples

    If it will rain, then I will wear my coat.Therefore, if it will rain then it will rain and I will wear my coat.

    1

  • 2 CHAPTER 1. ABSORPTION (LOGIC)

    1.3 Proof by truth table

    1.4 Formal proof

    1.5 References[1] Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 362.

    [2] http://www.philosophypages.com/lg/e11a.htm

    [3] Russell and Whitehead, Principia Mathematica

  • Chapter 2

    Associative property

    This article is about associativity in mathematics. For associativity in the central processing unit memory cache, seeCPU cache. For associativity in programming languages, see operator associativity.Associative and non-associative redirect here. For associative and non-associative learning, see Learning#Types.

    In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associa-tivity is a valid rule of replacement for expressions in logical proofs.Within an expression containing two or more occurrences in a row of the same associative operator, the order inwhich the operations are performed does not matter as long as the sequence of the operands is not changed. That is,rearranging the parentheses in such an expression will not change its value. Consider the following equations:

    (2 + 3) + 4 = 2 + (3 + 4) = 9

    2 (3 4) = (2 3) 4 = 24:Even though the parentheses were rearranged, the values of the expressions were not altered. Since this holds truewhen performing addition and multiplication on any real numbers, it can be said that addition and multiplication ofreal numbers are associative operations.Associativity is not to be confused with commutativity, which addresses whether a b = b a.Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups andcategories) explicitly require their binary operations to be associative.However, many important and interesting operations are non-associative; some examples include subtraction, exponentiationand the vector cross product. In contrast to the theoretical counterpart, the addition of oating point numbers in com-puter science is not associative, and is an important source of rounding error.

    2.1 DenitionFormally, a binary operation on a set S is called associative if it satises the associative law:

    (x y) z = x (y z) for all x, y, z in S.

    Here, is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbollike for the multiplication.

    (xy)z = x(yz) = xyz for all x, y, z in S.

    The associative law can also be expressed in functional notation thus: f(f(x, y), z) = f(x, f(y, z)).

    3

  • 4 CHAPTER 2. ASSOCIATIVE PROPERTY

    Associative binary operation on the set S.

    2.2 Generalized associative lawIf a binary operation is associative, repeated application of the operation produces the same result regardless how validpairs of parenthesis are inserted in the expression.[2] This is called the generalized associative law. For instance, aproduct of four elements may be written in ve possible ways:

    1. ((ab)c)d

    2. (ab)(cd)

    3. (a(bc))d

    4. a((bc)d)

    5. a(b(cd))

    If the product operation is associative, the generalized associative law says that all these formulas will yield the sameresult, making the parenthesis unnecessary. Thus the product can be written unambiguously as

    abcd.

    As the number of elements increases, the number of possible ways to insert parentheses grows quickly, but theyremain unnecessary for disambiguation.

    2.3 ExamplesSome examples of associative operations include the following.

    The concatenation of the three strings hello, " ", world can be computed by concatenating the rst twostrings (giving hello ") and appending the third string (world), or by joining the second and third string(giving " world) and concatenating the rst string (hello) with the result. The two methods produce thesame result; string concatenation is associative (but not commutative).

    In arithmetic, addition and multiplication of real numbers are associative; i.e.,

  • 2.3. EXAMPLES 5

    (((ab)c)d)e

    ((ab)c)(de)

    ((ab)(cd))e

    ((a(bc))d)e

    (ab)(c(de))

    (a(bc))(de)

    (ab)((cd)e)

    (a(b(cd)))e

    a(b(c(de)))

    a((bc)(de))

    a(b((cd)e))

    a(((bc)d)e)

    a((b(cd))e)

    (a((bc)d))e

    In the absence of the associative property, ve factors a, b, c, d, e result in a Tamari lattice of order four, possibly dierent products.

    (x+ y) + z = x+ (y + z) = x+ y + z(x y)z = x(y z) = x y z

    for all x; y; z 2 R:

    Because of associativity, the grouping parentheses can be omitted without ambiguity.

  • 6 CHAPTER 2. ASSOCIATIVE PROPERTY

    (x + z+ y)

    x + z)+ (y=

    The addition of real numbers is associative.

    Addition and multiplication of complex numbers and quaternions are associative. Addition of octonions is alsoassociative, but multiplication of octonions is non-associative.

    The greatest common divisor and least common multiple functions act associatively.

    gcd(gcd(x; y); z) = gcd(x; gcd(y; z)) = gcd(x; y; z)lcm(lcm(x; y); z) = lcm(x; lcm(y; z)) = lcm(x; y; z)

    for all x; y; z 2 Z:

    Taking the intersection or the union of sets:

    (A \B) \ C = A \ (B \ C) = A \B \ C(A [B) [ C = A [ (B [ C) = A [B [ C

    for all sets A;B;C:

    IfM is some set and S denotes the set of all functions fromM toM, then the operation of functional compositionon S is associative:

    (f g) h = f (g h) = f g h for all f; g; h 2 S:

    Slightly more generally, given four sets M, N, P and Q, with h: M to N, g: N to P, and f: P to Q, then

    (f g) h = f (g h) = f g h

    as before. In short, composition of maps is always associative.

    Consider a set with three elements, A, B, and C. The following operation:

    is associative. Thus, for example, A(BC)=(AB)C = A. This mapping is not commutative.

    Because matrices represent linear transformation functions, with matrix multiplication representing functionalcomposition, one can immediately conclude that matrix multiplication is associative.

  • 2.4. PROPOSITIONAL LOGIC 7

    2.4 Propositional logic

    2.4.1 Rule of replacementIn standard truth-functional propositional logic, association,[3][4] or associativity[5] are two valid rules of replacement.The rules allow one to move parentheses in logical expressions in logical proofs. The rules are:

    (P _ (Q _R)), ((P _Q) _R)

    and

    (P ^ (Q ^R)), ((P ^Q) ^R);

    where ", " is a metalogical symbol representing can be replaced in a proof with.

    2.4.2 Truth functional connectivesAssociativity is a property of some logical connectives of truth-functional propositional logic. The following logicalequivalences demonstrate that associativity is a property of particular connectives. The following are truth-functionaltautologies.Associativity of disjunction:

    (P _ (Q _R))$ ((P _Q) _R)

    ((P _Q) _R)$ (P _ (Q _R))Associativity of conjunction:

    ((P ^Q) ^R)$ (P ^ (Q ^R))

    (P ^ (Q ^R))$ ((P ^Q) ^R)Associativity of equivalence:

    ((P $ Q)$ R)$ (P $ (Q$ R))

    (P $ (Q$ R))$ ((P $ Q)$ R)

    2.5 Non-associativityA binary operation on a set S that does not satisfy the associative law is called non-associative. Symbolically,

    (x y) z 6= x (y z) for some x; y; z 2 S:

    For such an operation the order of evaluation does matter. For example:

    Subtraction

    (5 3) 2 6= 5 (3 2)

  • 8 CHAPTER 2. ASSOCIATIVE PROPERTY

    Division

    (4/2)/2 6= 4/(2/2)

    Exponentiation

    2(12) 6= (21)2

    Also note that innite sums are not generally associative, for example:

    (1 1) + (1 1) + (1 1) + (1 1) + (1 1) + (1 1) + : : : = 0

    whereas

    1 + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + : : : = 1

    The study of non-associative structures arises from reasons somewhat dierent from the mainstream of classicalalgebra. One area within non-associative algebra that has grown very large is that of Lie algebras. There the associativelaw is replaced by the Jacobi identity. Lie algebras abstract the essential nature of innitesimal transformations, andhave become ubiquitous in mathematics.There are other specic types of non-associative structures that have been studied in depth; these tend to come fromsome specic applications or areas such as combinatorial mathematics. Other examples are Quasigroup, Quasield,Non-associative ring, Non-associative algebra and Commutative non-associative magmas.

    2.5.1 Nonassociativity of oating point calculation

    In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, theaddition and multiplication of oating point numbers is not associative, as rounding errors are introduced whendissimilar-sized values are joined together.[6]

    To illustrate this, consider a oating point representation with a 4-bit mantissa:(1.000220 + 1.000220) + 1.000224 = 1.000221 + 1.000224 = 1.0012241.000220 + (1.000220 + 1.000224) = 1.000220 + 1.000224 = 1.000224

    Even though most computers compute with a 24 or 53 bits of mantissa,[7] this is an important source of roundingerror, and approaches such as the Kahan Summation Algorithm are ways to minimise the errors. It can be especiallyproblematic in parallel computing.[8] [9]

    2.5.2 Notation for non-associative operations

    Main article: Operator associativity

    In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears morethan once in an expression. However, mathematicians agree on a particular order of evaluation for several commonnon-associative operations. This is simply a notational convention to avoid parentheses.A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,

    x y z = (x y) zw x y z = ((w x) y) zetc.

    9=; for all w; x; y; z 2 Swhile a right-associative operation is conventionally evaluated from right to left:

  • 2.5. NON-ASSOCIATIVITY 9

    x y z = x (y z)w x y z = w (x (y z))etc.

    9=; for all w; x; y; z 2 SBoth left-associative and right-associative operations occur. Left-associative operations include the following:

    Subtraction and division of real numbers:

    x y z = (x y) z for all x; y; z 2 R;x/y/z = (x/y)/z for all x; y; z 2 R with y 6= 0; z 6= 0:

    Function application:

    (f x y) = ((f x) y)

    This notation can be motivated by the currying isomorphism.

    Right-associative operations include the following:

    Exponentiation of real numbers:

    xyz

    = x(yz):

    The reason exponentiation is right-associative is that a repeated left-associative exponentiation operationwould be less useful. Multiple appearances could (and would) be rewritten with multiplication:

    (xy)z = x(yz):

    Function denition

    Z! Z! Z = Z! (Z! Z)x 7! y 7! x y = x 7! (y 7! x y)

    Using right-associative notation for these operations can be motivated by the Curry-Howard correspon-dence and by the currying isomorphism.

    Non-associative operations for which no conventional evaluation order is dened include the following.

    Taking the Cross product of three vectors:

    ~a (~b ~c) 6= (~a~b) ~c for some ~a;~b;~c 2 R3

    Taking the pairwise average of real numbers:

    (x+ y)/2 + z

    26= x+ (y + z)/2

    2for all x; y; z 2 R with x 6= z:

    Taking the relative complement of sets (AnB)nC is not the same as An(BnC) . (Compare material nonim-plication in logic.)

  • 10 CHAPTER 2. ASSOCIATIVE PROPERTY

    2.6 See also Lights associativity test A semigroup is a set with a closed associative binary operation. Commutativity and distributivity are two other frequently discussed properties of binary operations. Power associativity, alternativity and N-ary associativity are weak forms of associativity.

    2.7 References[1] Thomas W. Hungerford (1974). Algebra (1st ed.). Springer. p. 24. ISBN 0387905189. Denition 1.1 (i) a(bc) = (ab)c

    for all a, b, c in G.

    [2] Durbin, John R. (1992). Modern Algebra: an Introduction (3rd ed.). New York: Wiley. p. 78. ISBN 0-471-51001-7. Ifa1; a2; : : : ; an (n 2) are elements of a set with an associative operation, then the product a1a2 : : : an is unambiguous;this is, the same element will be obtained regardless of how parentheses are inserted in the product

    [3] Moore and Parker

    [4] Copi and Cohen

    [5] Hurley

    [6] Knuth, Donald, The Art of Computer Programming, Volume 3, section 4.2.2

    [7] IEEEComputer Society (August 29, 2008). IEEE Standard for Floating-Point Arithmetic. IEEE. doi:10.1109/IEEESTD.2008.4610935.ISBN 978-0-7381-5753-5. IEEE Std 754-2008.

    [8] Villa, Oreste; Chavarra-mir, Daniel; Gurumoorthi, Vidhya; Mrquez, Andrs; Krishnamoorthy, Sriram, Eects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems (PDF), retrieved 2014-04-08

    [9] Goldberg, David, What Every Computer Scientist ShouldKnowAbout Floating Point Arithmetic (PDF),ACMComputingSurveys 23 (1): 548, doi:10.1145/103162.103163, retrieved 2014-04-08

  • 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 deemworthy, 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.

    11

  • 12 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 which were accepted without proof. Sucha hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of eachparticular science were dierent. Their validity had to be established by means of real-world experience. Indeed,Aristotle warns that the content of a science cannot be successfully communicated, if the learner is in doubt aboutthe truth of the postulates.[7]

    The classical approach is well-illustrated by Euclids Elements, where a list of postulates is given (common-sensicalgeometric facts drawn from our experience), followed by a list of common notions (very basic, self-evident asser-tions).

    Postulates

    1. It is possible to draw a straight line from any point to any other point.2. It is possible to extend a line segment continuously in both directions.3. It is possible to describe a circle with any center and any radius.4. It is true that all right angles are equal to one another.5. ("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior

    angles on the same side less than two right angles, the two straight lines, if produced indenitely,intersect on that side on which are the angles less than the two right angles.

    Common notions

    1. Things which are equal to the same thing are also equal to one another.2. If equals are added to equals, the wholes are equal.3. If equals are subtracted from equals, the remainders are equal.4. Things which coincide with one another are equal to one another.5. The whole is greater than the part.

    3.2.2 Modern developmentA lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathemat-ical assertions (axioms, postulates, propositions, theorems) and denitions. One must concede the need for primitive

  • 3.2. HISTORICAL DEVELOPMENT 13

    notions, or undened terms or concepts, in any study. Such abstraction or formalization makes mathematical knowl-edge more general, capable of multiple dierent meanings, and therefore useful in multiple contexts. AlessandroPadoa, Mario Pieri, and Giuseppe Peano were pioneers in this movement.Structuralist mathematics goes further, and develops theories and axioms (e.g. eld theory, group theory, topology,vector spaces) without any particular application in mind. The distinction between an axiom and a postulatedisappears. The postulates of Euclid are protably motivated by saying that they lead to a great wealth of geometricfacts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwingout Euclids fth postulate we get theories that have meaning in wider contexts, hyperbolic geometry for example.We must simply be prepared to use labels like line and parallel with greater exibility. The development ofhyperbolic geometry taught mathematicians that postulates should be regarded as purely formal statements, and notas facts based on experience.When mathematicians employ the eld axioms, the intentions are even more abstract. The propositions of eld theorydo not concern any one particular application; the mathematician now works in complete abstraction. There are manyexamples of elds; eld theory gives correct knowledge about them all.It is not correct to say that the axioms of eld theory are propositions that are regarded as true without proof. Rather,the eld axioms are a set of constraints. If any given system of addition and multiplication satises these constraints,then one is in a position to instantly know a great deal of extra information about this system.Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded asmathematical objects, and mathematics itself can be regarded as a branch of logic. Frege, Russell, Poincar, Hilbert,and Gdel are some of the key gures in this development.In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formallystated assertions follow by the application of certain well-dened rules. In this view, logic becomes just another formalsystem. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom. A setof axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regardedas an axiom.It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, couldbe derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbertsformalization of Euclidean geometry, and the related demonstration of the consistency of those axioms.In a wider context, there was an attempt to base all of mathematics on Cantors set theory. Here the emergence ofRussells paradox, and similar antinomies of nave set theory raised the possibility that any such system could turnout to be inconsistent.The formalist project suered a decisive setback, when in 1931 Gdel showed that it is possible, for any sucientlylarge set of axioms (Peanos axioms, for example) to construct a statement whose truth is independent of that set ofaxioms. As a corollary, Gdel proved that the consistency of a theory like Peano arithmetic is an unprovable assertionwithin the scope of that theory.It is reasonable to believe in the consistency of Peano arithmetic because it is satised by the system of natural num-bers, an innite but intuitively accessible formal system. However, at present, there is no known way of demonstratingthe consistency of the modern ZermeloFraenkel axioms for set theory. Furthermore, using techniques of forcing(Cohen) one can show that the continuum hypothesis (Cantor) is independent of the ZermeloFraenkel axioms. Thus,even this very general set of axioms cannot be regarded as the denitive foundation for mathematics.

    3.2.3 Other sciences

    Axioms play a key role not only in mathematics, but also in other sciences, notably in theoretical physics. In particular,the monumental work of Isaac Newton is essentially based on Euclid's axioms, augmented by a postulate on the non-relation of spacetime and the physics taking place in it at any moment.In 1905, Newtons axioms were replaced by those of Albert Einstein's special relativity, and later on by those ofgeneral relativity.Another paper of Albert Einstein and coworkers (see EPR paradox), almost immediately contradicted by Niels Bohr,concerned the interpretation of quantum mechanics. This was in 1935. According to Bohr, this new theory should beprobabilistic, whereas according to Einstein it should be deterministic. Notably, the underlying quantum mechanicaltheory, i.e. the set of theorems derived by it, seemed to be identical. Einstein even assumed that it would besucient to add to quantum mechanics hidden variables to enforce determinism. However, thirty years later, in

  • 14 CHAPTER 3. AXIOM

    1964, John Bell found a theorem, involving complicated optical correlations (see Bell inequalities), which yieldedmeasurably dierent results using Einsteins axioms compared to using Bohrs axioms. And it took roughly anothertwenty years until an experiment of Alain Aspect got results in favour of Bohrs axioms, not Einsteins. (Bohrs axiomsare simply: The theory should be probabilistic in the sense of the Copenhagen interpretation.)As a consequence, it is not necessary to explicitly cite Einsteins axioms, the more so since they concern subtle pointson the reality and locality of experiments.Regardless, the role of axioms in mathematics and in the above-mentioned sciences is dierent. In mathematics oneneither proves nor disproves an axiom for a set of theorems; the point is simply that in the conceptual realmidentied by the axioms, the theorems logically follow. In contrast, in physics a comparison with experiments alwaysmakes sense, since a falsied physical theory needs modication.

    3.3 Mathematical logicIn the eld of mathematical logic, a clear distinction is made between two notions of axioms: logical and non-logical(somewhat similar to the ancient distinction between axioms and postulates respectively).

    3.3.1 Logical axiomsThese are certain formulas in a formal language that are universally valid, that is, formulas that are satised by everyassignment of values. Usually one takes as logical axioms at least some minimal set of tautologies that is sucientfor proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required,in order to prove logical truths that are not tautologies in the strict sense.

    Examples

    Propositional logic In propositional logic it is common to take as logical axioms all formulae of the followingforms, where , , and can be any formulae of the language and where the included primitive connectives areonly " : " for negation of the immediately following proposition and " ! " for implication from antecedent toconsequent propositions:

    1. ! ( ! )2. (! ( ! ))! ((! )! (! ))3. (:! : )! ( ! ):

    Each of these patterns is an axiom schema, a rule for generating an innite number of axioms. For example, if A , B, and C are propositional variables, then A! (B ! A) and (A! :B)! (C ! (A! :B)) are both instancesof axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modusponens, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schematais sucient for proving all tautologies with modus ponens.Other axiom schemas involving the same or dierent sets of primitive connectives can be alternatively constructed.[8]

    These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed to include aquantier in the calculus.[9]

    First-order logic Axiom of Equality. Let L be a rst-order language. For each variable x , the formula

    x = x

    is universally valid.This means that, for any variable symbol x ; the formula x = x can be regarded as an axiom. Also, in this example,for this not to fall into vagueness and a never-ending series of primitive notions, either a precise notion of what wemean by x = x (or, for that matter, to be equal) has to be well established rst, or a purely formal and syntactical

  • 3.3. MATHEMATICAL LOGIC 15

    usage of the symbol= has to be enforced, only regarding it as a string and only a string of symbols, and mathematicallogic does indeed do that.Another, more interesting example axiom scheme, is that which provides us with what is known as Universal In-stantiation:Axiom scheme for Universal Instantiation. Given a formula in a rst-order language L , a variable x and aterm t that is substitutable for x in , the formula

    8x! xt

    is universally valid.Where the symbol xt stands for the formula with the term t substituted for x . (See Substitution of variables.) Ininformal terms, this example allows us to state that, if we know that a certain property P holds for every x and thatt stands for a particular object in our structure, then we should be able to claim P (t) . Again, we are claiming thatthe formula 8x ! xt is valid, that is, we must be able to give a proof of this fact, or more properly speaking, ametaproof. Actually, these examples are metatheorems of our theory of mathematical logic since we are dealing withthe very concept of proof itself. Aside from this, we can also have Existential Generalization:Axiom scheme for Existential Generalization. Given a formula in a rst-order language L , a variable x and aterm t that is substitutable for x in , the formula

    xt ! 9x

    is universally valid.

    3.3.2 Non-logical axiomsNon-logical axioms are formulas that play the role of theory-specic assumptions. Reasoning about two dierentstructures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logicalaxioms aim to capture what is special about a particular structure (or set of structures, such as groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate.[10]

    Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that inprinciple every theory could be axiomatized in this way and formalized down to the bare language of logical formulas.Non-logical axioms are often simply referred to as axioms in mathematical discourse. This does not mean that it isclaimed that they are true in some absolute sense. For example, in some groups, the group operation is commutative,and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite welldeveloping (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference dene adeductive system.

    Examples

    This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms(axioms, henceforth). A rigorous treatment of any of these topics begins with a specication of these axioms.Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, butimplicitly or explicitly there is generally an assumption that the axioms being used are the axioms of ZermeloFraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like VonNeumannBernaysGdel set theory, a conservative extension of ZFC. Sometimes slightly stronger theories suchas Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieckuniverse are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, suchas second-order arithmetic.The study of topology in mathematics extends all over through point set topology, algebraic topology, dierentialtopology, and all the related paraphernalia, such as homology theory, homotopy theory. The development of abstractalgebra brought with itself group theory, rings, elds, and Galois theory.

  • 16 CHAPTER 3. AXIOM

    This list could be expanded to includemost elds ofmathematics, includingmeasure theory, ergodic theory, probability,representation theory, and dierential geometry.Combinatorics is an example of a eld of mathematics which does not, in general, follow the axiomatic method.

    Arithmetic The Peano axioms are the most widely used axiomatization of rst-order arithmetic. They are a set ofaxioms strong enough to prove many important facts about number theory and they allowed Gdel to establish hisfamous second incompleteness theorem.[11]

    We have a languageLNT = f0; Sg where 0 is a constant symbol and S is a unary function and the following axioms:

    1. 8x::(Sx = 0)2. 8x:8y:(Sx = Sy ! x = y)3. (((0) ^ 8x: ((x)! (Sx)))! 8x:(x) for any LNT formula with one free variable.

    The standard structure isN = hN; 0; Si where N is the set of natural numbers, S is the successor function and 0 isnaturally interpreted as the number 0.

    Euclidean geometry Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclids postulates ofplane geometry. The axioms are referred to as 4 + 1 because for nearly two millennia the fth (parallel) postulate(through a point outside a line there is exactly one parallel) was suspected of being derivable from the rst four.Ultimately, the fth postulate was found to be independent of the rst four. Indeed, one can assume that exactly oneparallel through a point outside a line exists, or that innitely many exist. This choice gives us two alternative formsof geometry in which the interior angles of a triangle add up to exactly 180 degrees or less, respectively, and areknown as Euclidean and hyperbolic geometries. If one also removes the second postulate (a line can be extendedindenitely) then elliptic geometry arises, where there is no parallel through a point outside a line, and in which theinterior angles of a triangle add up to more than 180 degrees.

    Real analysis The object of study is the real numbers. The real numbers are uniquely picked out (up to isomorphism)by the properties of a Dedekind complete ordered eld, meaning that any nonempty set of real numbers with an upperbound has a least upper bound. However, expressing these properties as axioms requires use of second-order logic.The Lwenheim-Skolem theorems tell us that if we restrict ourselves to rst-order logic, any axiom system for thereals admits other models, including both models that are smaller than the reals and models that are larger. Some ofthe latter are studied in non-standard analysis.

    3.3.3 Role in mathematical logicDeductive systems and completeness

    A deductive system consists of a set of logical axioms, a set of non-logical axioms, and a set f(; )g of rulesof inference. A desirable property of a deductive system is that it be complete. A system is said to be complete if,for all formulas ,

    if j= then `

    that is, for any statement that is a logical consequence of there actually exists a deduction of the statement from .This is sometimes expressed as everything that is true is provable, but it must be understood that true here meansmade true by the set of axioms, and not, for example, true in the intended interpretation. Gdels completenesstheorem establishes the completeness of a certain commonly used type of deductive system.Note that completeness has a dierent meaning here than it does in the context of Gdels rst incompletenesstheorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete,in the sense that there will always exist an arithmetic statement such that neither nor : can be proved from thegiven set of axioms.There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that ofcompleteness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despitetheir names, do not contradict one another.

  • 3.4. SEE ALSO 17

    3.3.4 Further discussionEarly mathematicians regarded axiomatic geometry as a model of physical space, and obviously there could only beone such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians ofthe 19th century and the developers of systems such as Boolean algebra made elaborate eorts to derive them fromtraditional arithmetic. Galois showed just before his untimely death that these eorts were largely wasted. Ultimately,the abstract parallels between algebraic systems were seen to be more important than the details and modern algebrawas born. In the modern view axioms may be any set of formulas, as long as they are not known to be inconsistent.

    3.4 See also Axiomatic system Dogma List of axioms Model theory Regul Juris Theorem

    3.5 References[1] A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim,

    rule, law axiom, n., denition 1a. Oxford English Dictionary Online, accessed 2012-04-28. Cf. Aristotle, PosteriorAnalytics I.2.72a18-b4.

    [2] Cf. axiom, n., etymology. Oxford English Dictionary, accessed 2012-04-28.

    [3] Oxford American College Dictionary: n. a statement or proposition that is regarded as being established, accepted, orself-evidently true. ORIGIN: late 15th cent.: ultimately from Greek axima 'what is thought tting,' from axios 'worthy.'http://www.highbeam.com/doc/1O997-axiom.html (subscription required)

    [4] A proposition (whether true or false)" axiom, n., denition 2. Oxford English Dictionary Online, accessed 2012-04-28.

    [5] Wol, P. Breakthroughs in Mathematics, 1963, New York: New American Library, pp 478

    [6] Heath, T. 1956. The Thirteen Books of Euclids Elements. New York: Dover. p200

    [7] Aristotle, Metaphysics Bk IV, Chapter 3, 1005b Physics also is a kind of Wisdom, but it is not the rst kind. And theattempts of some of those who discuss the terms on which truth should be accepted, are due to want of training in logic;for they should know these things already when they come to a special study, and not be inquiring into them while they arelistening to lectures on it. W.D. Ross translation, in The Basic Works of Aristotle, ed. Richard McKeon, (Random House,New York, 1941)|date=June 2011

    [8] Mendelson, 6. Other Axiomatizations of Ch. 1

    [9] Mendelson, 3. First-Order Theories of Ch. 2

    [10] Mendelson, 3. First-Order Theories: Proper Axioms of Ch. 2

    [11] Mendelson, 5. The Fixed Point Theorem. Gdels Incompleteness Theorem of Ch. 2

    3.6 Further reading Mendelson, Elliot (1987). Introduction to mathematical logic. Belmont, California: Wadsworth & Brooks.ISBN 0-534-06624-0

  • 18 CHAPTER 3. AXIOM

    3.7 External links Axiom at PhilPapers Axiom at PlanetMath.org. Metamath axioms page

  • Chapter 4

    Axiom schema

    In mathematical logic, an axiom schema (plural: axiom schemata) generalizes the notion of axiom.

    4.1 Formal denitionAn axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variablesappear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, whichmay or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free,or that certain variables not appear in the subformula or term.

    4.2 Finite axiomatizationGiven that the number of possible subformulas or terms that can be inserted in place of a schematic variable iscountably innite, an axiom schema stands for a countably innite set of axioms. This set can usually be denedrecursively. A theory that can be axiomatized without schemata is said to be nitely axiomatized. Theories that canbe nitely axiomatized are seen as a bit more metamathematically elegant, even if they are less practical for deductivework.

    4.3 ExamplesTwo very well known instances of axiom schemata are the:

    induction schema that is part of Peanos axioms for the arithmetic of the natural numbers;

    axiom schema of replacement that is part of the standard ZFC axiomatization of set theory.

    It has been proved (rst by Richard Montague) that these schemata cannot be eliminated. Hence Peano arithmeticand ZFC cannot be nitely axiomatized. This is also the case for quite a few other axiomatic theories in mathematics,philosophy, linguistics, etc.

    4.4 Finitely axiomatized theoreiesAll theorems of ZFC are also theorems of von NeumannBernaysGdel set theory, but the latter is, quite surpris-ingly, nitely axiomatized. The set theory New Foundations can be nitely axiomatized, but only with some loss ofelegance.

    19

  • 20 CHAPTER 4. AXIOM SCHEMA

    4.5 In higher-order logicSchematic variables in rst-order logic are usually trivially eliminable in second-order logic, because a schematicvariable is often a placeholder for any property or relation over the individuals of the theory. This is the case withthe schemata of Induction and Replacement mentioned above. Higher-order logic allows quantied variables to rangeover all possible properties or relations.

    4.6 See also Axiom schema of predicative separation Axiom schema of replacement Axiom schema of specication

    4.7 References Schema entry by John Corcoran in the Stanford Encyclopedia of Philosophy, 2008-09-21 Corcoran, J. 2006. Schemata: the Concept of Schema in the History of Logic. Bulletin of Symbolic Logic 12:219-40.

    Mendelson, Elliot, 1997. Introduction to Mathematical Logic, 4th ed. Chapman & Hall. Potter, Michael, 2004. Set Theory and its Philosophy. Oxford Univ. Press.

  • Chapter 5

    Axiomatic system

    In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunctionto logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems. Anaxiomatic system that is completely described is a special kind of formal system; usually though, the eort towardscomplete formalisation brings diminishing returns in certainty, and a lack of readability for humans. A formal theorytypically means an axiomatic system, for example formulated within model theory. A formal proof is a completerendition of a mathematical proof within a formal system.

    5.1 PropertiesAn axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement andits denial from the systems axioms.In an axiomatic system, an axiom is called independent if it is not a theorem that can be derived from other ax-ioms in the system. A system will be called independent if each of its underlying axioms is independent. Althoughindependence is not a necessary requirement for a system, consistency is.An axiomatic system will be called complete if for every statement, either itself or its negation is derivable.

    5.2 Relative consistencyBeyond consistency, relative consistency is also the mark of a worthwhile axiom system. This is when the undenedterms of a rst axiom system are provided denitions from a second, such that the axioms of the rst are theoremsof the second.A good example is the relative consistency of neutral geometry or absolute geometry with respect to the theory of thereal number system. Lines and points are undened terms in absolute geometry, but assigned meanings in the theoryof real numbers in a way that is consistent with both axiom systems.

    5.3 ModelsA model for an axiomatic system is a well-dened set, which assigns meaning for the undened terms presented inthe system, in a manner that is correct with the relations dened in the system. The existence of a concrete modelproves the consistency of a system. A model is called concrete if the meanings assigned are objects and relationsfrom the real world, as opposed to an abstract model which is based on other axiomatic systems.Models can also be used to show the independence of an axiom in the system. By constructing a valid model fora subsystem without a specic axiom, we show that the omitted axiom is independent if its correctness does notnecessarily follow from the subsystem.Two models are said to be isomorphic if a one-to-one correspondence can be found between their elements, in amanner that preserves their relationship. An axiomatic system for which every model is isomorphic to another is

    21

  • 22 CHAPTER 5. AXIOMATIC SYSTEM

    called categorial (sometimes categorical), and the property of categoriality (categoricity) ensures the completenessof a system.

    5.4 Axiomatic methodStating denitions and propositions in a way such that each new term can be formally eliminated by the priorlyintroduced terms requires primitive notions (axioms) to avoid innite regress. This way of doing mathematics iscalled the axiomatic method.[1]

    A common attitude towards the axiomatic method is logicism. In their book Principia Mathematica, Alfred NorthWhitehead and Bertrand Russell attempted to show that all mathematical theory could be reduced to some collectionof axioms. More generally, the reduction of a body of propositions to a particular collection of axioms underliesthe mathematicians research program. This was very prominent in the mathematics of the twentieth century, inparticular in subjects based around homological algebra.The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that themathematician would like to work with. For example, mathematicians opted that rings need not be commutative,which diered from Emmy Noether's original formulation. Mathematicians decided to consider topological spacesmore generally without the separation axiom which Felix Hausdor originally formulated.The Zermelo-Fraenkel axioms, the result of the axiomatic method applied to set theory, allowed the proper for-mulation of set-theory problems and helped to avoid the paradoxes of nave set theory. One such problem was theContinuum hypothesis.

    5.4.1 History

    Mathematical methods developed to some degree of sophistication in ancient Egypt, Babylon, India, and China,apparently without employing the axiomatic method.Euclid of Alexandria authored the earliest extant axiomatic presentation of Euclidean geometry and number theory.Many axiomatic systems were developed in the nineteenth century, including non-Euclidean geometry, the founda-tions of real analysis, Cantor's set theory, Frege's work on foundations, and Hilbert's 'new' use of axiomatic methodas a research tool. For example, group theory was rst put on an axiomatic basis towards the end of that century.Once the axioms were claried (that inverse elements should be required, for example), the subject could proceedautonomously, without reference to the transformation group origins of those studies.

    5.4.2 Issues

    Not every consistent body of propositions can be captured by a describable collection of axioms. Call a collectionof axioms recursive if a computer program can recognize whether a given proposition in the language is an axiom.Gdels First Incompleteness Theorem then tells us that there are certain consistent bodies of propositions with norecursive axiomatization. Typically, the computer can recognize the axioms and logical rules for deriving theorems,and the computer can recognize whether a proof is valid, but to determine whether a proof exists for a statementis only soluble by waiting for the proof or disproof to be generated. The result is that one will not know whichpropositions are theorems and the axiomatic method breaks down. An example of such a body of propositions is thetheory of the natural numbers. The Peano Axioms (described below) thus only partially axiomatize this theory.In practice, not every proof is traced back to the axioms. At times, it is not clear which collection of axioms a proofappeals to. For example, a number-theoretic statement might be expressible in the language of arithmetic (i.e. thelanguage of the Peano Axioms) and a proof might be given that appeals to topology or complex analysis. It might notbe immediately clear whether another proof can be found that derives itself solely from the Peano Axioms.Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitraryaxiomatic systemwill not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything.Philosophers of mathematics sometimes assert that mathematicians choose axioms arbitrarily, but the truth is thatalthough they may appear arbitrary when viewed only from the point of view of the canons of deductive logic, that ismerely a limitation on the purposes that deductive logic serves.

  • 5.5. SEE ALSO 23

    5.4.3 Example: The Peano axiomatization of natural numbersThe mathematical system of natural numbers 0, 1, 2, 3, 4, ... is based on an axiomatic system rst written down by themathematician Peano in 1889. He chose the axioms (see Peano axioms), in the language of a single unary functionsymbol S (short for successor), for the set of natural numbers to be:

    There is a natural number 0. Every natural number a has a successor, denoted by Sa. There is no natural number whose successor is 0. Distinct natural numbers have distinct successors: if a b, then Sa Sb. If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it ispossessed by all natural numbers ("Induction axiom").

    5.4.4 AxiomatizationIn mathematics, axiomatization is the formulation of a system of statements (i.e. axioms) that relate a number ofprimitive terms in order that a consistent body of propositions may be derived deductively from these statements.Thereafter, the proof of any proposition should be, in principle, traceable back to these axioms.

    5.5 See also Axiom schema Gdels incompleteness theorem Hilbert-style deduction system Logicism ZermeloFraenkel set theory, an axiomatic system for set theory and todays most common foundation formathematics.

    5.6 References[1] "Set Theory and its Philosophy, a Critical Introduction S.6; Michael Potter, Oxford, 2004

    Hazewinkel, Michiel, ed. (2001), Axiomatic method, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

    EricW.Weisstein,Axiomatic System, FromMathWorldAWolframWebResource. Mathworld.wolfram.com& Answers.com

  • Chapter 6

    Biconditional elimination

    Biconditional elimination is the name of two valid rules of inference of propositional logic. It allows for one toinfer a conditional from a biconditional. If (P $ Q) is true, then one may infer that (P ! Q) is true, and alsothat (Q ! P ) is true.[1] For example, if its true that I'm breathing if and only if I'm alive, then its true that if I'mbreathing, I'm alive; likewise, its true that if I'm alive, I'm breathing. The rules can be stated formally as:

    (P $ Q)) (P ! Q)and

    (P $ Q)) (Q! P )

    where the rule is that wherever an instance of " (P $ Q) " appears on a line of a proof, either " (P ! Q) " or "(Q! P ) " can be placed on a subsequent line;

    6.1 Formal notationThe biconditional elimination rule may be written in sequent notation:

    (P $ Q) ` (P ! Q)

    and

    (P $ Q) ` (Q! P )

    where ` is a metalogical symbol meaning that (P ! Q) , in the rst case, and (Q ! P ) in the other are syntacticconsequences of (P $ Q) in some logical system;or as the statement of a truth-functional tautology or theorem of propositional logic:

    (P $ Q)! (P ! Q)

    (P $ Q)! (Q! P )where P , and Q are propositions expressed in some formal system.

    24

  • 6.2. SEE ALSO 25

    6.2 See also Logical biconditional

    6.3 References[1] Cohen, S. Marc. Chapter 8: The Logic of Conditionals (PDF). University of Washington. Retrieved 8 October 2013.

  • Chapter 7

    Biconditional introduction

    In propositional logic, biconditional introduction[1][2][3] is a valid rule of inference. It allows for one to infer abiconditional from two conditional statements. The rule makes it possible to introduce a biconditional statement intoa logical proof. If P ! Q is true, and if Q! P is true, then one may infer that P $ Q is true. For example, fromthe statements if I'm breathing, then I'm alive and if I'm alive, then I'm breathing, it can be inferred that I'mbreathing if and only if I'm alive. Biconditional introduction is the converse of biconditional elimination. The rulecan be stated formally as:

    P ! Q;Q! P) P $ Q

    where the rule is that wherever instances of " P ! Q " and " Q ! P " appear on lines of a proof, " P $ Q " canvalidly be placed on a subsequent line.

    7.1 Formal notationThe biconditional introduction rule may be written in sequent notation:

    (P ! Q); (Q! P ) ` (P $ Q)

    where ` is a metalogical symbol meaning that P $ Q is a syntactic consequence when P ! Q and Q ! P areboth in a proof;or as the statement of a truth-functional tautology or theorem of propositional logic:

    ((P ! Q) ^ (Q! P ))! (P $ Q)

    where P , and Q are propositions expressed in some formal system.

    7.2 References[1] Hurley

    [2] Moore and Parker

    [3] Copi and Cohen

    26

  • Chapter 8

    Commutative property

    For other uses, see Commute (disambiguation).In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

    =

    ==

    This image illustrates that addition is commutative.

    It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar asthe name of the property that says 3 + 4 = 4 + 3 or 2 5 = 5 2, the property can also be used in more advancedsettings. The name is needed because there are operations, such as division and subtraction that do not have it (forexample, 3 5 5 3), such operations are not commutative, or noncommutative operations. The idea that simpleoperations, such as multiplication and addition of numbers, are commutative was for many years implicitly assumedand the property was not named until the 19th century when mathematics started to become formalized.

    8.1 Common uses

    The commutative property (or commutative law) is a property generally associatedwith binary operations and functions.If the commutative property holds for a pair of elements under a certain binary operation then the two elements aresaid to commute under that operation.

    27

  • 28 CHAPTER 8. COMMUTATIVE PROPERTY

    8.2 Mathematical denitionsFurther information: Symmetric function

    The term commutative is used in several related senses.[1][2]

    1. A binary operation on a set S is called commutative if:

    x y = y x for all x; y 2 S

    An operation that does not satisfy the above property is called noncommutative.2. One says that x commutes with y under if:

    x y = y x

    3. A binary function f : AA! B is called commutative if:

    f(x; y) = f(y; x) for all x; y 2 A

    8.3 Examples

    8.3.1 Commutative operations in everyday life Putting on socks resembles a commutative operation, since which sock is put on rst is unimportant. Eitherway, the result (having both socks on), is the same.

    The commutativity of addition is observed when paying for an item with cash. Regardless of the order the billsare handed over in, they always give the same total.

    8.3.2 Commutative operations in mathematicsTwo well-known examples of commutative binary operations:[1]

    The addition of real numbers is commutative, since

    y + z = z + y for all y; z 2 R

    For example 4 + 5 = 5 + 4, since both expressions equal 9.

    The multiplication of real numbers is commutative, since

    yz = zy for all y; z 2 R

    For example, 3 5 = 5 3, since both expressions equal 15.

    Some binary truth functions are also commutative, since the truth tables for the functions are the same whenone changes the order of the operands.

  • 8.3. EXAMPLES 29

    b

    b

    aaa+

    b

    The addition of vectors is commutative, because ~a+~b = ~b+ ~a .

    For example, the logical biconditional function p q is equivalent to q p. This function is also writtenas p IFF q, or as p q, or as Epq.The last form is an example of the most concise notation in the article on truth functions, which lists thesixteen possible binary truth functions of which eight are commutative: Vpq = Vqp; Apq (OR) = Aqp;Dpq (NAND) = Dqp; Epq (IFF) = Eqp; Jpq = Jqp; Kpq (AND) = Kqp; Xpq (NOR) = Xqp; Opq = Oqp.

    Further examples of commutative binary operations include addition and multiplication of complex numbers,addition and scalar multiplication of vectors, and intersection and union of sets.

    8.3.3 Noncommutative operations in everyday life Concatenation, the act of joining character strings together, is a noncommutative operation. For example

    EA+ T = EAT 6= TEA = T + EA

    Washing and drying clothes resembles a noncommutative operation; washing and then drying produces amarkedly dierent result to drying and then washing.

  • 30 CHAPTER 8. COMMUTATIVE PROPERTY

    Rotating a book 90 around a vertical axis then 90 around a