Universal quanticationFrom Wikipedia, the free encyclopedia

Contents

1 Adjoint functors 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Spelling (or morphology) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Solutions to optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Symmetry of optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Universal morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 Counit-unit adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.4 Hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Adjunctions in full . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Universal morphisms induce hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Counit-unit adjunction induces hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . 61.4.3 Hom-set adjunction induces all of the above . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.1 Ubiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.2 Problems formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.3 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6.1 Free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6.2 Free constructions and forgetful functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6.3 Diagonal functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6.4 Colimits and diagonal functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7.3 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7.4 Limit preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7.5 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.8 Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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1.8.1 Universal constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8.2 Equivalences of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8.3 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Atomic formula 172.1 Atomic formula in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Binary relation 193.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Category theory 294.1 An abstraction of other mathematical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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4.6 Other concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.6.1 Universal constructions, limits, and colimits . . . . . . . . . . . . . . . . . . . . . . . . . 334.6.2 Equivalent categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.6.3 Further concepts and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.6.4 Higher-dimensional categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.7 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Composite number 395.1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 CornishFisher expansion 446.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Counterexample 467.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.1.1 Rectangle example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.1.2 Other mathematical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.2 In philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

8 Domain of discourse 498.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.3 Universe of discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.4 Booles 1854 denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

9 Element (mathematics) 51

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9.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.2 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.3 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

10 Existential quantication 5410.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5410.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

10.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.2.2 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.2.3 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10.3 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

11 Expression (mathematics) 5811.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.2 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.3 Syntax versus semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.3.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.3.3 Formal languages and lambda calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

11.4 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5911.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6011.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

12 False (logic) 6112.1 In classical logic and Boolean logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.2 False, negation and contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

13 First-order logic 6313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6313.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

13.2.1 Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.2.2 Formation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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13.2.3 Free and bound variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6713.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

13.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6813.3.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.3.2 Evaluation of truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.3.3 Validity, satisability, and logical consequence . . . . . . . . . . . . . . . . . . . . . . . . 7013.3.4 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7013.3.5 First-order theories, models, and elementary classes . . . . . . . . . . . . . . . . . . . . . 7113.3.6 Empty domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13.4 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.4.1 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.4.2 Hilbert-style systems and natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.4.3 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.4.4 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.4.6 Provable identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

13.5 Equality and its axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.5.1 First-order logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.5.2 Dening equality within a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

13.6 Metalogical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.6.1 Completeness and undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.6.2 The LwenheimSkolem theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.6.3 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.6.4 Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

13.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.7.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.7.2 Formalizing natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

13.8 Restrictions, extensions, and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.8.1 Restricted languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.8.2 Many-sorted logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.8.3 Additional quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7813.8.4 Innitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7813.8.5 Non-classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7813.8.6 Fixpoint logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7913.8.7 Higher-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

13.9 Automated theorem proving and formal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 7913.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8013.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8013.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8113.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

14 Functor 84

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14.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8414.1.1 Covariance and contravariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8414.1.2 Opposite functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8514.1.3 Bifunctors and multifunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

14.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8514.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8614.4 Relation to other categorical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.5 Computer implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8714.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8814.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

15 Ground expression 8915.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8915.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

15.2.1 Ground terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8915.2.2 Ground atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.2.3 Ground formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

15.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

16 Image (mathematics) 9116.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

16.1.1 Image of an element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9216.1.2 Image of a subset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9216.1.3 Image of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

16.2 Inverse image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9216.3 Notation for image and inverse image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

16.3.1 Arrow notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9216.3.2 Star notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9216.3.3 Other terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

16.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.5 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9316.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9416.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9416.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

17 Interpretation (logic) 9517.1 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

17.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9517.1.2 Logical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

17.2 General properties of truth-functional interpretations . . . . . . . . . . . . . . . . . . . . . . . . . 96

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17.2.1 Logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9617.3 Interpretation of a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.4 Interpretations for propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.5 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

17.5.1 Formal languages for rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9717.5.2 Interpretations of a rst-order language . . . . . . . . . . . . . . . . . . . . . . . . . . . 9817.5.3 Example of a rst-order interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9817.5.4 Non-empty domain requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9917.5.5 Interpreting equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9917.5.6 Many-sorted rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

17.6 Higher-order predicate logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10017.7 Non-classical interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10017.8 Intended interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

17.8.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10117.9 Other concepts of interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10117.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10117.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10217.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

18 List of logic symbols 10318.1 Basic logic symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10318.2 Advanced and rarely used logical symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

18.2.1 Poland and Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10418.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10518.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10518.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10518.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

19 Logic 10619.1 The study of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

19.1.1 Logical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10619.1.2 Deductive and inductive reasoning, and abductive inference . . . . . . . . . . . . . . . . . 10719.1.3 Consistency, validity, soundness, and completeness . . . . . . . . . . . . . . . . . . . . . . 10819.1.4 Rival conceptions of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

19.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10819.3 Types of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

19.3.1 Syllogistic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11019.3.2 Propositional logic (sentential logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11019.3.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11019.3.4 Modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11119.3.5 Informal reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11119.3.6 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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19.3.7 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11219.3.8 Computational logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11219.3.9 Bivalence and the law of the excluded middle; non-classical logics . . . . . . . . . . . . . 11319.3.10 Is logic empirical?" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11419.3.11 Implication: strict or material? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11419.3.12 Tolerating the impossible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11419.3.13 Rejection of logical truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

19.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11519.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11619.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11819.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

20 Logical conjunction 12020.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12120.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

20.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12220.3 Introduction and elimination rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12220.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12320.5 Applications in computer engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12420.6 Set-theoretic correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12420.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12420.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12520.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12520.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

21 Logical connective 12621.1 In language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

21.1.1 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12621.1.2 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

21.2 Common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12721.2.1 List of common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12721.2.2 History of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12821.2.3 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

21.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12921.4 Order of precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13021.5 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13021.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13021.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13021.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13121.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13121.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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22 Logical constant 13222.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13222.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13222.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

23 Logical equivalence 13323.1 Logical equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13323.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13323.3 Relation to material equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13423.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13423.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

24 Material conditional 13524.1 Denitions of the material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

24.1.1 As a truth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13624.1.2 As a formal connective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

24.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13724.3 Philosophical problems with material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . 13724.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

24.4.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13824.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13824.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13824.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

25 Natural number 14025.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

25.1.1 Modern denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14225.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14225.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

25.3.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14225.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14225.3.3 Relationship between addition and multiplication . . . . . . . . . . . . . . . . . . . . . . . 14325.3.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14325.3.5 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14325.3.6 Algebraic properties satised by the natural numbers . . . . . . . . . . . . . . . . . . . . . 143

25.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14425.5 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

25.5.1 Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14425.5.2 Constructions based on set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

25.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14625.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14625.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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25.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

26 Open sentence 15126.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

27 Polish notation 15327.1 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15327.2 Computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15427.3 Order of operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15427.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15527.5 Polish notation for logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15527.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15527.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15627.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

28 Power set 15728.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15728.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15828.3 Representing subsets as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15828.4 Relation to binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15928.5 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15928.6 Subsets of limited cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15928.7 Power object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16028.8 Functors and quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16028.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16028.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16028.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16128.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

29 Predicate (mathematical logic) 16229.1 Simplied overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16229.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16229.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16329.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16329.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

30 Predicate logic 16430.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16430.2 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16430.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

31 Presheaf (category theory) 16631.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16631.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

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31.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16631.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

32 Projection (linear algebra) 16832.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

32.1.1 Orthogonal projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16932.1.2 Oblique projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

32.2 Properties and classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17032.2.1 Orthogonal projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17132.2.2 Oblique projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

32.3 Canonical forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17332.4 Projections on normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17432.5 Applications and further considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17432.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17432.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17532.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17532.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17532.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

33 Property (philosophy) 17633.1 Essential and accidental properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17633.2 Determinate and determinable properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17633.3 Lovely and suspect qualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17633.4 Property dualism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17633.5 Properties in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17733.6 Properties and predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17733.7 Intrinsic and extrinsic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17833.8 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17833.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17833.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17833.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

34 Proposition 17934.1 Historical usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

34.1.1 By Aristotle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17934.1.2 By the logical positivists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17934.1.3 By Russell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

34.2 Relation to the mind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18034.3 Treatment in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18034.4 Objections to propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18034.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18134.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

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34.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

35 Propositional formula 18235.1 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

35.1.1 Relationship between propositional and predicate formulas . . . . . . . . . . . . . . . . . 18335.1.2 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

35.2 An algebra of propositions, the propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . 18335.2.1 Usefulness of propositional formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18435.2.2 Propositional variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18435.2.3 Truth-value assignments, formula evaluations . . . . . . . . . . . . . . . . . . . . . . . . 184

35.3 Propositional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18535.3.1 Connectives of rhetoric, philosophy and mathematics . . . . . . . . . . . . . . . . . . . . 18535.3.2 Engineering connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18535.3.3 CASE connective: IF ... THEN ... ELSE ... . . . . . . . . . . . . . . . . . . . . . . . . . 18535.3.4 IDENTITY and evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

35.4 More complex formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18735.4.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18735.4.2 Axiom and denition schemas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18835.4.3 Substitution versus replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

35.5 Inductive denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18835.6 Parsing formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

35.6.1 Connective seniority (symbol rank) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18935.6.2 Commutative and associative laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19035.6.3 Distributive laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19035.6.4 De Morgans laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19035.6.5 Laws of absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19135.6.6 Laws of evaluation: Identity, nullity, and complement . . . . . . . . . . . . . . . . . . . . 19135.6.7 Double negative (Involution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

35.7 Well-formed formulas (ws) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19135.7.1 Ws versus valid formulas in inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

35.8 Reduced sets of connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19235.8.1 The stroke (NAND) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19235.8.2 IF ... THEN ... ELSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

35.9 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19435.9.1 Reduction to normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19435.9.2 Reduction by use of the map method (Veitch, Karnaugh) . . . . . . . . . . . . . . . . . . 195

35.10Impredicative propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19635.11Propositional formula with feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

35.11.1 Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19735.11.2 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

35.12Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19835.13Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

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35.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

36 Propositional function 20836.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

37 Quantier (logic) 20937.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20937.2 Algebraic approaches to quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20937.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21037.4 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21137.5 Equivalent expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21137.6 Range of quantication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21237.7 Formal semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21237.8 Paucal, multal and other degree quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21437.9 Other quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21437.10History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21537.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21537.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21537.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

38 Rule of inference 21738.1 The standard form of rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21738.2 Axiom schemas and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21838.3 Example: Hilbert systems for two propositional logics . . . . . . . . . . . . . . . . . . . . . . . . 21838.4 Admissibility and derivability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21938.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21938.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

39 Rule of replacement 22139.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

40 Satisability 22240.1 Reduction of validity to satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22240.2 Propositional satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22240.3 Satisability in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22340.4 Satisability in model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22340.5 Finite satisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22340.6 Numerical constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22440.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22440.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22440.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22440.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

41 Sequent 225

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41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22541.1.1 The form and semantics of sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22541.1.2 Syntax details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22641.1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22641.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22741.1.5 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

41.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22741.2.1 History of the meaning of sequent assertions . . . . . . . . . . . . . . . . . . . . . . . . . 22741.2.2 Intuitive meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

41.3 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22841.4 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22841.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22941.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22941.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23041.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

42 Set (mathematics) 23142.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23242.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23242.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

42.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23442.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

42.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23542.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23542.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

42.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23642.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23742.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23742.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

42.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24042.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24042.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24142.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24142.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24242.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24242.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24242.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

43 T-schema 24343.1 The inductive denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24343.2 Natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24343.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

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43.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24443.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

44 Tautology (logic) 24544.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24544.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24644.3 Denition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24644.4 Verifying tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24744.5 Tautological implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24744.6 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24744.7 Ecient verication and the Boolean satisability problem . . . . . . . . . . . . . . . . . . . . . . 24844.8 Tautologies versus validities in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24844.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

44.9.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24944.9.2 Related logical topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

44.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24944.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

45 Theorem 25045.1 Informal account of theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25045.2 Provability and theoremhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25145.3 Relation with scientic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25145.4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25145.5 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25245.6 Lore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25345.7 Theorems in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

45.7.1 Syntax and semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25445.7.2 Derivation of a theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25445.7.3 Interpretation of a formal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25545.7.4 Theorems and theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

45.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25545.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25545.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25645.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

46 Theory (mathematical logic) 26146.1 Theories expressed in formal language generally . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

46.1.1 Subtheories and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26146.1.2 Deductive theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26146.1.3 Consistency and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26146.1.4 Interpretation of a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26246.1.5 Theories associated with a structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

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46.2 First-order theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26246.2.1 Derivation in a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26246.2.2 Syntactic consequence in a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . 26346.2.3 Interpretation of a rst order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26346.2.4 First order theories with identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26346.2.5 Topics related to rst order theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

46.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26346.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26446.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26446.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

47 Topos 26547.1 Grothendieck topoi (topoi in geometry) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

47.1.1 Equivalent denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26547.1.2 Geometric morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26647.1.3 Ringed topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26647.1.4 Homotopy theory of topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

47.2 Elementary topoi (topoi in logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26747.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26747.2.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26747.2.3 Logical functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26847.2.4 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26847.2.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

47.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27047.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27047.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

48 Universal generalization 27248.1 Generalization with hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27248.2 Example of a proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27248.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27248.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

49 Universal instantiation 27449.1 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27449.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27449.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

50 Universal quantication 27650.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

50.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27750.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

50.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

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50.2.2 Other connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27850.2.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27950.2.4 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

50.3 Universal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27950.4 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28050.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28050.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28050.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

51 Vacuous truth 28151.1 Scope of the concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28151.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28151.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28251.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28251.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28251.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

52 Well-formed formula 28352.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28452.2 Propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28452.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28552.4 Atomic and open formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28552.5 Closed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28652.6 Properties applicable to formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28652.7 Usage of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28652.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28652.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28652.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28752.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28752.12Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 288

52.12.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28852.12.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29652.12.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Chapter 1

Adjoint functors

For the construction in eld theory, see Adjunction (eld theory). For the construction in topology, see Adjunctionspace.

In mathematics, specically category theory, adjunction is a possible relationship between two functors.Adjunction is ubiquitous in mathematics, as it species intuitive notions of optimization and eciency.In the most concise symmetric denition, an adjunction between categories C and D is a pair of functors,

F : D ! C and G : C ! D

and a family of bijections

homC(FY;X) = homD(Y;GX)

which is natural in the variables X and Y. The functor F is called a left adjoint functor, while G is called a rightadjoint functor. The relationship F is left adjoint to G (or equivalently, G is right adjoint to F) is sometimeswritten

F a G:

This denition and others are made precise below.

1.1 IntroductionThe slogan is Adjoint functors arise everywhere. (Saunders Mac Lane, Categories for the working mathematician)The long list of examples in this article is only a partial indication of how often an interesting mathematical construc-tion is an adjoint functor. As a result, general theorems about left/right adjoint functors, such as the equivalence oftheir various denitions or the fact that they respectively preserve colimits/limits (which are also found in every areaof mathematics), can encode the details of many useful and otherwise non-trivial results.

1.1.1 Spelling (or morphology)One can observe (e.g. in this article), two dierent roots are used: adjunct and adjoint. From Oxford shorterEnglish dictionary, adjunct is from Latin, adjoint is from French.In Mac Lane, Categories for the working mathematician, chap. 4, Adjoints, one can verify the following usage.' : homC(FY;X) = homD(Y;GX)

1

2 CHAPTER 1. ADJOINT FUNCTORS

The hom-set bijection ' is an adjunction.If f an arrow in homC(FY;X) , 'f is the right adjunct of f (p. 81).The functor F is left adjoint for G .

1.2 Motivation

1.2.1 Solutions to optimization problems

It can be said that an adjoint functor is a way of giving the most ecient solution to some problem via a method whichis formulaic. For example, an elementary problem in ring theory is how to turn a rng (which is like a ring that mightnot have a multiplicative identity) into a ring. The most ecient way is to adjoin an element '1' to the rng, adjoinall (and only) the elements which are necessary for satisfying the ring axioms (e.g. r+1 for each r in the ring), andimpose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is formulaicin the sense that it works in essentially the same way for any rng.This is rather vague, though suggestive, and can be made precise in the language of category theory: a constructionis most ecient if it satises a universal property, and is formulaic if it denes a functor. Universal properties comein two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessaryto discuss one of them.The idea of using an initial property is to set up the problem in terms of some auxiliary category E, and then identifythat what we want is to nd an initial object of E. This has an advantage that the optimization the sense that weare nding the most ecient solution means something rigorous and is recognisable, rather like the attainment ofa supremum. The category E is also formulaic in this construction, since it is always the category of elements of thefunctor to which one is constructing an adjoint. In fact, this latter category is precisely the comma category over thefunctor in question.As an example, take the given rng R, and make a category E whose objects are rng homomorphisms R S, with S aring having a multiplicative identity. The morphisms in E between R S1 and R S2 are commutative triangles ofthe form (R S1,R S2, S1 S2) where S1 S2 is a ring map (which preserves the identity). Note that this isprecisely the denition of the comma category of R over the inclusion of unitary rings into rng. The existence of amorphism between R S1 and R S2 implies that S1 is at least as ecient a solution as S2 to our problem: S2 canhave more adjoined elements and/or more relations not imposed by axioms than S1. Therefore, the assertion that anobject R R* is initial in E, that is, that there is a morphism from it to any other element of E, means that the ringR* is a most ecient solution to our problem.The two facts that this method of turning rngs into rings ismost ecient and formulaic can be expressed simultaneouslyby saying that it denes an adjoint functor.

1.2.2 Symmetry of optimization problems

Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is therea problem to which F is the most ecient solution?The notion that F is the most ecient solution to the problem posed by G is, in a certain rigorous sense, equivalent tothe notion that G poses the most dicult problem that F solves.This has the intuitive meaning that adjoint functors should occur in pairs, and in fact they do, but this is not trivial fromthe universal morphism denitions. The equivalent symmetric denitions involving adjunctions and the symmetriclanguage of adjoint functors (we can say either F is left adjoint to G or G is right adjoint to F) have the advantage ofmaking this fact explicit.

1.3 Formal denitionsThere are various denitions for adjoint functors. Their equivalence is elementary but not at all trivial and in facthighly useful. This article provides several such denitions:

1.3. FORMAL DEFINITIONS 3

The denitions via universal morphisms are easy to state, and require minimal verications when constructingan adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuitioninvolving optimizations.

The denition via counit-unit adjunction is convenient for proofs about functors which are known to be adjoint,because they provide formulas that can be directly manipulated.

The denition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint.

Adjoint functors arise everywhere, in all areas of mathematics. Their full usefulness lies in that the structure in anyof these denitions gives rise to the structures in the others via a long but trivial series of deductions. Thus, switchingbetween them makes implicit use of a great deal of tedious details that would otherwise have to be repeated separatelyin every subject area. For example, naturality and terminality of the counit can be used to prove that any right adjointfunctor preserves limits.

1.3.1 ConventionsThe theory of adjoints has the terms left and right at its foundation, and there are many components which live in oneof two categories C and D which are under consideration. It can therefore be extremely helpful to choose letters inalphabetical order according to whether they live in the lefthand category C or the righthand category D, and alsoto write them down in this order whenever possible.In this article for example, the letters X, F, f, will consistently denote things which live in the category C, the lettersY, G, g, will consistently denote things which live in the category D, and whenever possible such things will bereferred to in order from left to right (a functor F:CD can be thought of as living where its outputs are, in C).

1.3.2 Universal morphismsA functor F : C D is a left adjoint functor if for each object X in C, there exists a terminal morphism from F toX. If, for each object X in C, we choose an object G0X of D for which there is a terminal morphism X : F(G0X) X from F to X, then there is a unique functor G : C D such that GX = G0X and X FG(f) = f X for f : X X a morphism in C; F is then called a left adjoint to G.A functor G : C D is a right adjoint functor if for each object Y in D, there exists an initial morphism from Y toG. If, for each object Y in D, we choose an object F0Y of C and an initial morphism Y : Y G(F0Y) from Y to G,then there is a unique functor F : C D such that FY = F0Y and GF(g) Y = Y g for g : Y Y a morphismin D; G is then called a right adjoint to F.Remarks:It is true, as the terminology implies, that F is left adjoint to G if and only ifG is right adjoint to F. This is apparent fromthe symmetric denitions given below. The denitions via universal morphisms are often useful for establishing thata given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitivelymeaningful in that nding a universal morphism is like solving an optimization problem.

1.3.3 Counit-unit adjunctionA counit-unit adjunction between two categories C and D consists of two functors F : C D and G : C D andtwo natural transformations

" : FG! 1C : 1D ! GFrespectively called the counit and the unit of the adjunction (terminology from universal algebra), such that thecompositions

FF!FGF "F!F

4 CHAPTER 1. ADJOINT FUNCTORS

GG!GFG G"!G

are the identity transformations 1F and 1G on F and G respectively.In this situation we say that F is left adjoint to G and G is right adjoint to F , and may indicate this relationshipby writing ("; ) : F a G , or simply F a G .In equation form, the above conditions on (,) are the counit-unit equations

1F = "F F1G = G" Gwhich mean that for each X in C and each Y in D,

1FY = "FY F (Y )1GX = G("X) GXNote that here 1 denotes identity functors, while above the same symbol was used for identity natural transformations.These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimescalled the zig-zag equations because of the appearance of the corresponding string diagrams. A way to rememberthem is to rst write down the nonsensical equation 1 = " and then ll in either F or G in one of the two simpleways which make the compositions dened.Note: The use of the prex co in counit here is not consistent with the terminology of limits and colimits, becausea colimit satises an initial property whereas the counit morphisms will satisfy terminal properties, and dually. Theterm unit here is borrowed from the theory of monads where it looks like the insertion of the identity 1 into a monoid.

1.3.4 Hom-set adjunctionA hom-set adjunction between two categories C and D consists of two functors F : C D and G : C D and anatural isomorphism

: homC(F;)! homD(; G)This species a family of bijections

Y;X : homC(FY;X)! homD(Y;GX)for all objects X in C and Y in D.In this situation we say that F is left adjoint to G and G is right adjoint to F , and may indicate this relationshipby writing : F a G , or simply F a G .This denition is a logical compromise in that it is somewhat more dicult to satisfy than the universal morphismdenitions, and has fewer immediate implications than the counit-unit denition. It is useful because of its obvioussymmetry, and as a stepping-stone between the other denitions.In order to interpret as a natural isomorphism, one must recognize homC(F, ) and homD(, G) as functors. Infact, they are both bifunctors from Dop C to Set (the category of sets). For details, see the article on hom functors.Explicitly, the naturality of means that for all morphisms f : X X in C and all morphisms g : Y Y in D thefollowing diagram commutes:The vertical arrows in this diagram are those induced by composition with f and g. Formally, Hom(Fg, f) : HomC(FY,X) HomC(FY, X ) is given by h f o h o Fg for each h in HomC(FY, X). Hom(g, Gf) is similar.

1.4 Adjunctions in fullThere are hence numerous functors and natural transformations associated with every adjunction, and only a smallportion is sucient to determine the rest.

1.4. ADJUNCTIONS IN FULL 5

Naturality of

An adjunction between categories C and D consists of

A functor F : C D called the left adjoint A functor G : C D called the right adjoint A natural isomorphism : homC(F,) homD(,G) A natural transformation : FG 1C called the counit A natural transformation : 1D GF called the unit

An equivalent formulation, where X denotes any object of C and Y denotes any object of D:For every C-morphism f : FY X, there is a unique D-morphism Y, X(f) = g : Y GX such that the diagramsbelow commute, and for every D-morphism g : Y GX, there is a unique C-morphism 1Y, X(g) = f : FY X inC such that the diagrams below commute:

From this assertion, one can recover that:

6 CHAPTER 1. ADJOINT FUNCTORS

The transformations , , and are related by the equations

f = 1Y;X(g) = "X F (g) 2 homC(F (Y ); X)g = Y;X(f) = G(f) Y 2 homD(Y;G(X))1GX;X(1GX) = "X 2 homC(FG(X); X)Y;FY (1FY ) = Y 2 homD(Y;GF (Y ))

The transformations , satisfy the counit-unit equations

1FY = "FY F (Y )1GX = G("X) GX

Each pair (GX, X) is a terminal morphism from F to X in C Each pair (FY, Y) is an initial morphism from Y to G in D

In particular, the equations above allow one to dene , , and in terms of any one of the three. However, theadjoint functors F and G alone are in general not sucient to determine the adjunction. We will demonstrate theequivalence of these situations below.

1.4.1 Universal morphisms induce hom-set adjunctionGiven a right adjoint functor G : C D; in the sense of initial morphisms, one may construct the induced hom-setadjunction by doing the following steps.

Construct a functor F : C D and a natural transformation . For each object Y in D, choose an initial morphism (F(Y), Y) from Y to G, so we have Y : Y G(F(Y)). We have the map of F on objects and the family of morphisms .

For each f : Y0 Y1, as (F(Y0), Y0) is an initial morphism, then factorize Y1 o f with Y0 and getF(f) : F(Y0) F(Y1). This is the map of F on morphisms.

The commuting diagram of that factorization implies the commuting diagram of natural transformations,so : 1D G o F is a natural transformation.

Uniqueness of that factorization and that G is a functor implies that the map of F on morphisms preservescompositions and identities.

Construct a natural isomorphism : homC(F-,-) homD(-,G-). For each object X in C, each object Y in D, as (F(Y), Y) is an initial morphism, then Y, X is a bijection,

where Y, X(f : F(Y) X) = G(f) o Y . is a natural transformation, G is a functor, then for any objects X0, X1 in C, any objects Y0, Y1 in D,

any x : X0 X1, any y : Y1 Y0, we have Y1, X1(x o f o F(y)) = G(x) o G(f) o G(F(y)) o Y1 =G(x) o G(f) o Y0 o y = G(x) o Y0, X0(f) o y, and then is natural in both arguments.

A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor.(The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairsis a trivially dened inclusion or forgetful functor.)

1.4.2 Counit-unit adjunction induces hom-set adjunctionGiven functors F : C D, G : C D, and a counit-unit adjunction (, ) : F a G, we can construct a hom-setadjunction by nding the natural transformation : homC(F-,-) homD(-,G-) in the following steps:

For each f : FY X and each g : Y GX, dene

1.4. ADJUNCTIONS IN FULL 7

Y;X(f) = G(f) YY;X(g) = "X F (g)The transformations and are natural because and are natural.

Using, in order, that F is a functor, that is natural, and the counit-unit equation 1FY = FY o F(Y), we obtain

f = "X FG(f) F (Y )= f "FY F (Y )= f 1FY = f

hence is the identity transformation.

Dually, using that G is a functor, that is natural, and the counit-unit equation 1GX = G(X) o GX, we obtain

g = G("X) GF (g) Y= G("X) GX g= 1GX g = g

hence is the identity transformation. Thus is a natural isomorphism with inverse 1 = .

1.4.3 Hom-set adjunction induces all of the above

Given functors F : C D, G : C D, and a hom-set adjunction : homC(F-,-) homD(-,G-), we can construct acounit-unit adjunction

("; ) : F a G ,

which denes families of initial and terminal morphisms, in the following steps:

Let "X = 1GX;X(1GX) 2 homC(FGX;X) for each X in C, where 1GX 2 homD(GX;GX) is the identitymorphism.

Let Y = Y;FY (1FY ) 2 homD(Y;GFY ) for each Y in D, where 1FY 2 homC(FY; FY ) is the identitymorphism.

The bijectivity and naturality of imply that each (GX, X) is a terminal morphism from F to X in C, andeach (FY, Y) is an initial morphism from Y to G in D.

The naturality of implies the naturality of and , and the two formulas

Y;X(f) = G(f) Y1Y;X(g) = "X F (g)

for each f: FY X and g: Y GX (which completely determine ).

Substituting FY for X and Y = Y, FY(1FY) for g in the second formula gives the rst counit-unit equation

1FY = "FY F (Y ) ,and substitutingGX for Y and X = 1GX, X(1GX) for f in the rst formula gives the second counit-unitequation1GX = G("X) GX .

8 CHAPTER 1. ADJOINT FUNCTORS

1.5 History

1.5.1 Ubiquity

The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory,it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those facedwith giving tidy, systematic presentations of the subject would have noticed relations such as

hom(F(X), Y) = hom(X, G(Y))

in the category of abelian groups, where F was the functor A (i.e. take the tensor product with A), and G wasthe functor hom(A,). The use of the equals sign is an abuse of notation; those two groups are not really identicalbut there is a way of identifying them that is natural. It can be seen to be natural on the basis, rstly, that these aretwo alternative descriptions of the bilinear mappings from X A to Y. That is, however, something particular to thecase of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a naturalisomorphism.The terminology comes from the Hilbert space idea of adjoint operators T, U with hTx; yi = hx; Uyi , which isformally similar to the above relation between hom-sets. We say that F is left adjoint to G, and G is right adjoint toF. Note that G may have itself a right adjoint that is quite dierent from F (see below for an example). The analogyto adjoint maps of Hilbert spaces can be made precise in certain contexts.[1]

If one starts looking for these adjoint pairs of functors, they turn out to be very common in abstract algebra, andelsewhere as well. The example section below provides evidence of this; furthermore, universal constructions, whichmay be more familiar to some, give rise to numerous adjoint pairs of functors.In accordance with the thinking of Saunders Mac Lane, any idea such as adjoint functors that occurs widely enoughin mathematics should be studied for its own sake.

1.5.2 Problems formulations

Mathematicians do not generally need the full adjoint functor concept. Concepts can be judged according to theiruse in solving problems, as well as for their use in building theories. The tension between these two motivations wasespecially great during the 1950s when category theory was initially developed. Enter Alexander Grothendieck, whoused category theory to take compass bearings in other work in functional analysis, homological algebra and nallyalgebraic geometry.It is probably wrong to say that he promoted the adjoint functor concept in isolation: but recognition of the role ofadjunction was inherent in Grothendiecks approach. For example, one of his major achievements was the formulationof Serre duality in relative form loosely, in a continuous family of algebraic varieties. The entire proof turned onthe existence of a right adjoint to a certain functor. This is something undeniably abstract, and non-constructive, butalso powerful in its own way.

1.5.3 Posets

Every partially ordered set can be viewed as a category (with a single morphism between x and y if and only if x y).A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant,an antitone Galois connection). See that article for a number of examples: the case of Galois theory of course is aleading one. Any Galois connection gives rise to closure operators and to inverse order-preserving bijections betweenthe corresponding closed elements.As is the case for Galois groups, the real interest lies often in rening a correspondence to a duality (i.e. antitoneorder isomorphism). A treatment of Galois theory along these lines by Kaplansky was inuential in the recognitionof the general structure here.The partial order case collapses the adjunction denitions quite noticeably, but can provide several themes:

adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status

1.6. EXAMPLES 9

closure operators may indicate the presence of adjunctions, as corresponding monads (cf. the Kuratowskiclosure axioms)

a very general comment of William Lawvere[2] is that syntax and semantics are adjoint: take C to be the setof all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For atheory T in C, let F(T) be the set of all structures that satisfy the axioms T ; for a set of mathematical structuresS, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if Tlogically implies G(S): th