Logical connectivesWikipedia

Contents

1 Conditioned disjunction 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Converse implication 22.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Venn diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Converse nonimplication 33.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.2 Venn diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.5.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Exclusive or 54.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Equivalencies, elimination, and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.3 Relation to modern algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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4.4 Exclusive or in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.5 Alternative symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.7 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.7.1 Bitwise operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.8 Encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 False (logic) 105.1 In classical logic and Boolean logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 False, negation and contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 Material equivalence 116.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Indicative conditional 147.1 Distinctions between the material conditional and the indicative conditional . . . . . . . . . . . . . 147.2 Psychology and indicative conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

8 Logical biconditional 168.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.1.2 Venn diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8.3.1 Biconditional introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3.2 Biconditional elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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8.4 Colloquial usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9 Logical conjunction 199.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.3 Introduction and elimination rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.5 Applications in computer engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.6 Set-theoretic correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 Logical connective 2210.1 In language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

10.1.1 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.1.2 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

10.2 Common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2.1 List of common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2.2 History of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2.3 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

10.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.4 Order of precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.5 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11 Logical disjunction 2711.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.4 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.5 Applications in computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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11.5.1 Bitwise operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.5.2 Logical operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.5.3 Constructive disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11.6 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12 Logical equality 3012.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.2 Alternative descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

13 Logical NOR 3213.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

13.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213.3 Introduction, elimination, and equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

14 Material conditional 3414.1 Denitions of the material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

14.1.1 As a truth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.1.2 As a formal connective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

14.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.3 Philosophical problems with material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

14.4.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

15 Material equivalence 3815.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

15.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.2.3 Origin of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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15.3 Distinction from if and only if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

16 Material nonimplication 4116.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

16.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

16.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

16.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

17 Modal operator 4217.1 Modality interpreted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

17.1.1 Alethic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.1.2 Deontic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.1.3 Axiological . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.1.4 Epistemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.1.5 Doxastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

18 Negation 4318.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4318.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4318.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

18.3.1 Double negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4318.3.2 Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4318.3.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.3.4 Self dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

18.4 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.5 Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.6 Kripke semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

vi CONTENTS

18.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

19 Sheer stroke 4619.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

19.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4619.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4619.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4619.4 Introduction, elimination, and equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4719.5 Formal system based on the Sheer stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

19.5.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4719.5.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4719.5.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4719.5.4 Simplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

19.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4819.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4819.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4819.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

20 Statement (logic) 4920.1 Statement as an abstract entity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4920.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4920.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4920.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

21 Strict conditional 5121.1 Avoiding paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5221.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5221.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

22 Tautology (logic) 5322.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5322.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422.3 Denition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422.4 Verifying tautologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422.5 Tautological implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5522.6 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5522.7 Ecient verication and the Boolean satisability problem . . . . . . . . . . . . . . . . . . . . . . 5522.8 Tautologies versus validities in rst-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5522.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

22.9.1 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5622.9.2 Related logical topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

CONTENTS vii

22.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5622.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5622.12Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 57

22.12.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5722.12.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6022.12.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter 1

Conditioned disjunction

In logic, conditioned disjunction (sometimes calledconditional disjunction) is a ternary logical connectiveintroduced by Church.[1] Given operands p, q, and r,which represent truth-valued propositions, the meaningof the conditioned disjunction [p, q, r] is given by:

[p; q; r] $ (q ! p) ^ (:q ! r)

In words, [p, q, r] is equivalent to: if q then p, else r", or"p or r, according as q or not q". This may also be statedas "q implies p and, not q implies r". So, for any valuesof p, q, and r, the value of [p, q, r] is the value of p whenq is true, and is the value of r otherwise.The conditioned disjunction is also equivalent to:

(q ^ p) _ (:q ^ r)

and has the same truth table as the ternary (?:) operatorin many programming languages.In conjunction with truth constants denoting each truth-value, conditioned disjunction is truth-functionally com-plete for classical logic.[2] Its truth table is the following:There are other truth-functionally complete ternary con-nectives.

1.1 References[1] Church, Alonzo (1956). Introduction to Mathematical

Logic. Princeton University Press.

[2] Wesselkamper, T., A sole sucient operator, NotreDame Journal of Formal Logic, Vol. XVI, No. 1 (1975),pp. 86-88.

1

Chapter 2

Converse implication

Converse implication is the converse of implication.That is to say; that for any two propositions P and Q, if Qimplies P, then P is the converse implication of Q.It may take the following forms:

pq, Bpq, or pq

2.1 Denition

2.1.1 Truth tableThe truth table of AB

2.1.2 Venn diagramThe Venn diagram of If B then A (the white area showswhere the statement is false)

2.2 Propertiestruth-preserving: The interpretation under which allvariables are assigned a truth value of 'true' produces atruth value of 'true' as a result of converse implication.

2.3 Symbol

2.4 Natural languageNot q without p.p if q.

2.5 Boolean Algebra(A + B')

2.6 See also Logical connective Material implication

2

Chapter 3

Converse nonimplication

In logic, converse nonimplication[1] is a logical connec-tive which is the negation of the converse of implication.

3.1 Denition

p 6qwhich is the same as (pq)

3.1.1 Truth tableThe truth table of p 6q .[2]

3.1.2 Venn diagramThe Venn Diagram of It is not the case that B impliesA (the red area is true)

3.2 Propertiesfalsehood-preserving: The interpretation under whichall variables are assigned a truth value of 'false' produces atruth value of 'false' as a result of converse nonimplication

3.3 SymbolAlternatives for p 6q are

p ~ q : ~ combines Converse implications left arrow( ) with Negations tilde( ).

Mpq : uses prexed capital letter. p8q : 8combines Converse implications left arrow( ) denied by means of a stroke( / ).

3.4 Natural language

3.4.1 Grammatical

3.4.2 Rhetorical

not A but B

3.4.3 Colloquial

3.5 Boolean algebra

Converse Nonimplication in a general Boolean algebra isdened as q8p=q0p .Example of a 2-element Boolean algebra: the 2 elements{0,1} with 0 as zero and 1 as unity element, operators as complement operator, _ as join operator and ^ as meetoperator, build the Boolean algebra of propositional logic.[4] Example of a 4-element Boolean algebra: the 4 divi-sors {1,2,3,6} of 6 with 1 as zero and 6 as unity element,operators c (codivisor of 6) as complement operator, _(least common multiple) as join operator and ^ (greatestcommon divisor) as meet operator, build a Boolean alge-bra.

3.5.1 Properties

Non-associative

r8(q8p)=(r8q)8p i rp=0 [5] (In a two-element Booleanalgebra the latter condition is reduced to r=0 or p=0 ).Hence in a nontrivial Boolean algebra Converse Nonim-plication is nonassociative.

3

4 CHAPTER 3. CONVERSE NONIMPLICATION

::

(r 8 q)8 p = r0q 8 p denition) (by= (r0q)0p denition) (by= (r + q0)p laws) Morgan's (De= (r + r0q0)p law) (Absorption= rp+ r0q0p= rp+ r0(q 8 p) denition) (by= rp+ r 8 (q 8 p) denition) (by

Clearly, it is associative i rp=0 .

Non-commutative

q8p=p8q i q=p [6]. Hence Converse Nonimplica-tion is noncommutative.

Neutral and absorbing elements

0 is a left neutral element ( 08p=p ) and a rightabsorbing element ( p80=0 ).

18p=0 , p81=p0 , and p8p=0 . Implication q!p is the dual of Converse Nonimpli-cation q8p [7].

[6]

[7]

3.6 Computer scienceAn example for converse nonimplication in computer sci-ence can be found when performing a right outer join on aset of tables from a database, if records not matching thejoin-condition from the left table are being excluded.[3]

3.7 Notes[1] Lehtonen, Eero, and Poikonen, J.H.

[2] Knuth 2011, p. 49

[3] http://www.codinghorror.com/blog/2007/10/a-visual-explanation-of-sql-joins.html

3.8 References Knuth, Donald E. (2011). The Art of Computer Pro-gramming, Volume 4A: Combinatorial Algorithms,Part 1 (1st ed.). Addison-Wesley Professional.ISBN 0-201-03804-8.

Chapter 4

Exclusive or

XOR redirects here. For the logic gate, see XOR gate.For other uses, see XOR (disambiguation).

Exclusive disjunction or exclusive or is a logical oper-ation that outputs true only when both inputs dier (oneis true, the other is false). It is symbolized by the pre-x operator J and by the inx operators XOR (/ksr/),EOR, EXOR, , , , and . The opposite of XOR islogical biconditional, which outputs true only when bothinputs are the same.It gains the name exclusive or because the meaning ofor is ambiguous when both operands are true; exclusiveor excludes that case. This is sometimes thought of asone or the other but not both. This could be written asA or B but not A and B.More generally, XOR is true only when an odd numberof inputs is true. A chain of XORsa XOR b XOR cXOR d (and so on)is true whenever an odd number ofthe inputs are true and is false whenever an even numberof inputs are true.

4.1 Truth table

Arguments on the left combined by XORThis is a binary Walsh matrix(compare: Hadamard code)

The truth table of A XOR B shows that it outputs truewhenever the inputs dier:0 = FALSE1 = TRUE

4.2 Equivalencies, elimination, andintroduction

Exclusive disjunction essentially means 'either one, butnot both'. In other words, if and only if one is true, theother cannot be true. For example, one of the two horseswill win the race, but not both of them. The exclusivedisjunction p q , or Jpq, can be expressed in terms ofthe logical conjunction (logical and, ^ ), the disjunction(logical or, _ ), and the negation ( : ) as follows:

p q = (p _ q) ^ :(p ^ q)The exclusive disjunction p q can also be expressed inthe following way:

p q = (p ^ :q) _ (:p ^ q)This representation of XOR may be found useful whenconstructing a circuit or network, because it has only one: operation and small number of ^ and _ operations. Aproof of this identity is given below:

p q = (p ^ :q) _ (:p ^ q)= ((p ^ :q) _ :p) ^ ((p ^ :q) _ q)= ((p _ :p) ^ (:q _ :p)) ^ ((p _ q) ^ (:q _ q))= (:p _ :q) ^ (p _ q)= :(p ^ q) ^ (p _ q)

It is sometimes useful to write pq in the following way:

p q = :((p ^ q) _ (:p ^ :q))This equivalence can be established by applying De Mor-gans laws twice to the fourth line of the above proof.

5

6 CHAPTER 4. EXCLUSIVE OR

The exclusive or is also equivalent to the negation of alogical biconditional, by the rules of material implication(a material conditional is equivalent to the disjunction ofthe negation of its antecedent and its consequence) andmaterial equivalence.In summary, we have, in mathematical and in engineeringnotation:

p q = (p ^ :q) _ (:p ^ q) = pq + pq

= (p _ q) ^ (:p _ :q) = (p+ q)(p+ q)

= (p _ q) ^ :(p ^ q) = (p+ q)(pq)

4.3 Relation to modern algebraAlthough the operators ^ (conjunction) and _(disjunction) are very useful in logic systems, theyfail a more generalizable structure in the following way:The systems (fT; Fg;^) and (fT; Fg;_) are monoids.This unfortunately prevents the combination of these twosystems into larger structures, such as a mathematicalring.However, the system using exclusive or (fT; Fg;) isan abelian group. The combination of operators ^ andover elements fT; Fg produce the well-known eld F2. This eld can represent any logic obtainable with thesystem (^;_) and has the added benet of the arsenal ofalgebraic analysis tools for elds.More specically, if one associates F with 0 and T with1, one can interpret the logical AND operation as mul-tiplication on F2 and the XOR operation as addition onF2 :r = p ^ q , r = p q (mod 2)

r = p q , r = p+ q (mod 2)Using this basis to describe a boolean system is referredto as algebraic normal form.

4.4 Exclusive or in EnglishThe Oxford English Dictionary explains either ... or asfollows:

The primary function of either, etc., is to em-phasize the perfect indierence of the two (ormore) things or courses ... ; but a secondaryfunction is to emphasize the mutual exclusive-ness, = either of the two, but not both.[1]

The exclusive-or explicitly states one or the other, butnot neither nor both. However, the mapping correspon-

dence between formal Boolean operators and natural lan-guage conjunctions is far from simple or one-to-one, andhas been studied for decades in linguistics and analyticphilosophy.Following this kind of common-sense intuition aboutor, it is sometimes argued that in many natural lan-guages, English included, the word or has an exclu-sive sense.[2] The exclusive disjunction of a pair ofpropositions, (p, q), is supposed to mean that p is trueor q is true, but not both. For example, it might be ar-gued that the normal intention of a statement like Youmay have coee, or you may have tea is to stipulate thatexactly one of the conditions can be true. Certainly undersome circumstances a sentence like this example shouldbe taken as forbidding the possibility of ones acceptingboth options. Even so, there is good reason to supposethat this sort of sentence is not disjunctive at all. If all weknow about some disjunction is that it is true overall, wecannot be sure that either of its disjuncts is true. For ex-ample, if a woman has been told that her friend is eitherat the snack bar or on the tennis court, she cannot validlyinfer that he is on the tennis court. But if her waiter tellsher that she may have coee or she may have tea, she canvalidly infer that she may have tea. Nothing classicallythought of as a disjunction has this property. This is soeven given that she might reasonably take her waiter ashaving denied her the possibility of having both coeeand tea.(Note: If the waiter intends that choosing neither tea norcoee is an option i.e. ordering nothing, the appropriateoperator is NAND: p NAND q.)In English, the construct either ... or is usually used toindicate exclusive or and or generally used for inclu-sive. But in Spanish, the word o (or) can be used inthe form p o q (exclusive) or the form o p o q (inclusive).Some may contend that any binary or other n-ary exclu-sive or is true if and only if it has an odd number oftrue inputs (this is not, however, the only reasonable def-inition; for example, digital xor gates with multiple inputstypically do not use that denition), and that there is noconjunction in English that has this general property. Forexample, Barrett and Stenner contend in the 1971 arti-cle The Myth of the Exclusive 'Or'" (Mind, 80 (317),116121) that no author has produced an example of anEnglish or-sentence that appears to be false because bothof its inputs are true, and brush o or-sentences such asThe light bulb is either on or o as reecting partic-ular facts about the world rather than the nature of theword or. However, the "barber paradox"Everybodyin town shaves himself or is shaved by the barber, whoshaves the barber? -- would not be paradoxical if orcould not be exclusive (although a purist could say thateither is required in the statement of the paradox).Whether these examples can be considered natural lan-guage is another question. Certainly when one sees amenu stating Lunch special: sandwich and soup or salad

4.6. PROPERTIES 7

(parsed as sandwich and (soup or salad)" according tocommon usage in the restaurant trade), one would not ex-pect to be permitted to order both soup and salad. Norwould one expect to order neither soup nor salad, be-cause that belies the nature of the special, that orderingthe two items together is cheaper than ordering them a lacarte. Similarly, a lunch special consisting of one meat,French fries or mashed potatoes and vegetable would con-sist of three items, only one of which would be a form ofpotato. If one wanted to have meat and both kinds ofpotatoes, one would ask if it were possible to substitutea second order of potatoes for the vegetable. And, onewould not expect to be permitted to have both types ofpotato and vegetable, because the result would be a veg-etable plate rather than a meat plate.

4.5 Alternative symbolsThe symbol used for exclusive disjunction varies fromone eld of application to the next, and even depends onthe properties being emphasized in a given context of dis-cussion. In addition to the abbreviation XOR, any ofthe following symbols may also be seen:

A plus sign (+). This makes sense mathemati-cally because exclusive disjunction corresponds toaddition modulo 2, which has the following additiontable, clearly isomorphic to the one above:

The use of the plus sign has the added advantage thatall of the ordinary algebraic properties of mathemat-ical rings and elds can be used without further ado.However, the plus sign is also used for Inclusive dis-junction in some notation systems.

A plus sign that is modied in some way, such asbeing encircled ( ). This usage faces the objectionthat this same symbol is already used inmathematicsfor the direct sum of algebraic structures.

A prexed J, as in Jpq.

An inclusive disjunction symbol (_ ) that is modiedin some way, such as being underlined ( _ ) or withdot above ( __ ).

In several programming languages, such as C, C++,C#, Java, Perl, Ruby and Python, a caret (^) is usedto denote the bitwise XOR operator. This is not usedoutside of programming contexts because it is tooeasily confused with other uses of the caret.

The symbol , sometimes written as >< or as >-

8 CHAPTER 4. EXCLUSIVE OR

Nimber addition is the exclusive or of nonnegative integersin binary representation. This is also the vector addition in(Z/2Z)4 .

As noted above, since exclusive disjunction is identicalto addition modulo 2, the bitwise exclusive disjunctionof two n-bit strings is identical to the standard vector ofaddition in the vector space (Z/2Z)n .In computer science, exclusive disjunction has severaluses:

It tells whether two bits are unequal.

It is an optional bit-ipper (the deciding inputchooses whether to invert the data input).

It tells whether there is an odd number of 1 bits (A B C D E is true i an odd number ofthe variables are true).

In logical circuits, a simple adder can be made with anXOR gate to add the numbers, and a series of AND, ORand NOT gates to create the carry output.On some computer architectures, it is more ecient tostore a zero in a register by xor-ing the register with itself(bits xor-ed with themselves are always zero) instead ofloading and storing the value zero.In simple threshold activated neural networks, modelingthe 'xor' function requires a second layer because 'xor' isnot a linearly separable function.Exclusive-or is sometimes used as a simple mixing func-tion in cryptography, for example, with one-time pad orFeistel network systems.

Similarly, XOR can be used in generating entropy poolsfor hardware random number generators. The XOR op-eration preserves randomness, meaning that a random bitXORed with a non-random bit will result in a random bit.Multiple sources of potentially random data can be com-bined using XOR, and the unpredictability of the outputis guaranteed to be at least as good as the best individualsource.[3]

XOR is used in RAID 36 for creating parity informa-tion. For example, RAID can back up bytes 10011100and 01101100 from two (or more) hard drives by XOR-ing the just mentioned bytes, resulting in (11110000) andwriting it to another drive. Under this method, if anyone of the three hard drives are lost, the lost byte can bere-created by XORing bytes from the remaining drives.For instance, if the drive containing 01101100 is lost,10011100 and 11110000 can be XORed to recover thelost byte.XOR is also used to detect an overow in the result ofa signed binary arithmetic operation. If the leftmost re-tained bit of the result is not the same as the innite num-ber of digits to the left, then that means overow oc-curred. XORing those two bits will give a 1 if thereis an overow.XOR can be used to swap two numeric variables in com-puters, using the XOR swap algorithm; however this isregarded as more of a curiosity and not encouraged inpractice.XOR linked lists leverage XOR properties in order to savespace to represent doubly linked list data structures.In computer graphics, XOR-based drawing methods areoften used to manage such items as bounding boxes andcursors on systems without alpha channels or overlayplanes.

4.8 EncodingsApart from the ASCII codes, the operator is encoded atU+22BB xor (HTML ) and U+2295 cir-cled plus (HTML ), both in blockMathematical Operators.

4.9 See also

4.10 Notes[1] or, conj.2 (adv.3) 2a Oxford English Dictionary, second

edition (1989). OED Online.[2] Jennings quotes numerous authors saying that the word

or has an exclusive sense. See Chapter 3, The FirstMyth of 'Or'":Jennings, R. E. (1994). The Genealogy of Disjunction.New York: Oxford University Press.

4.11. EXTERNAL LINKS 9

[3] Davies, Robert B (28 February 2002). Exclusive OR(XOR) and hardware random number generators (PDF).Retrieved 28 August 2013.

4.11 External links An example of XOR being used in cryptography

Chapter 5

False (logic)

See also: Falsity

In logic, false or untrue is a truth value or a nullarylogical connective. In a truth-functional system of propo-sitional logic it is one of two postulated truth values, alongwith its negation, truth.[1] Usual notations of the false are0 (especially in Boolean logic and computer science), O(in prex notation, Opq), and the up tack symbol .[2]

Another approach is used for several formal theories (forexample, intuitionistic propositional calculus) where thefalse is a propositional constant (i.e. a nullary connective), the truth value of this constant being always false in thesense above.[3][4][5]

5.1 In classical logic and Booleanlogic

Boolean logic denes the false in both senses mentionedabove: 0 is a propositional constant, whose value bydenition is 0. In a classical propositional calculus, de-pending on the chosen set of fundamental connectives,the false may or may not have a dedicated symbol. Suchformulas as p p and (p p) may be used instead.In both systems the negation of the truth gives false. Thenegation of false is equivalent to the truth not only in clas-sical logic and Boolean logic, but also in most other logi-cal systems, as explained below.

5.2 False, negation and contradic-tion

In most logical systems, negation, material conditionaland false are related as:

p (p )

This is the denition of negation in some systems,[6] suchas intuitionistic logic, and can be proven in propositionalcalculi where negation is a fundamental connective. Be-

cause p p is usually a theorem or axiom, a consequenceis that the negation of false ( ) is true.The contradiction is a statement which entails the false,i.e. . Using the equivalence above, the fact that is a contradiction may be derived, for example, from . Contradiction and the false are sometimes not dis-tinguished, especially due to Latin term falsum denotingboth. Contradiction means a statement is proven to befalse, but the false itself is a proposition which is denedto be opposite to the truth.Logical systems may or may not contain the principle ofexplosion (in Latin, ex falso quodlibet), .

5.3 ConsistencyMain article: Consistency

A formal theory using "" connective is dened to beconsistent if and only if the false is not its theorem. Inthe absence of propositional constants, some substitutessuch as mentioned above may be used instead to deneconsistency.

5.4 References[1] Jennifer Fisher, On the Philosophy of Logic, Thomson

Wadsworth, 2007, ISBN 0-495-00888-5, p. 17.[2] Willard VanOrmanQuine,Methods of Logic, 4th ed, Har-

vard University Press, 1982, ISBN 0-674-57176-2, p. 34.[3] George Edward Hughes and D.E. Londey, The Elements

of Formal Logic, Methuen, 1965, p. 151.[4] Leon Horsten and Richard Pettigrew, Continuum Com-

panion to Philosophical Logic, Continuum InternationalPublishing Group, 2011, ISBN 1-4411-5423-X, p. 199.

[5] Graham Priest, An Introduction to Non-Classical Logic:From If to Is, 2nd ed, Cambridge University Press, 2008,ISBN 0-521-85433-4, p. 105.

[6] Dov M. Gabbay and Franz Guenthner (eds), Handbookof Philosophical Logic, Volume 6, 2nd ed, Springer, 2002,ISBN 1-4020-0583-0, p. 12.

10

Chapter 6

Material equivalence

I redirects here. For other uses, see IFF (disambigua-tion)."" redirects here. It is not to be confused withBidirectional trac.

Logical symbols representing iIn logic and related elds such as mathematics andphilosophy, if and only if (shortened i) is abiconditional logical connective between statements.In that it is biconditional, the connective can be likened tothe standard material conditional (only if, equal to if ...then) combined with its reverse (if); hence the name.The result is that the truth of either one of the connectedstatements requires the truth of the other (i.e. either bothstatements are true, or both are false). It is controversialwhether the connective thus dened is properly renderedby the English if and only if, with its pre-existing mean-ing. There is nothing to stop one from stipulating that wemay read this connective as only if and if, although thismay lead to confusion.In writing, phrases commonly used, with debatable pro-priety, as alternatives to P if and only if Q include Q isnecessary and sucient for P, P is equivalent (or mate-rially equivalent) to Q (compare material implication), Pprecisely if Q, P precisely (or exactly) when Q, P exactly incase Q, and P just in case Q. Many authors regard i asunsuitable in formal writing;[1] others use it freely.[2]

In logic formulae, logical symbols are used instead ofthese phrases; see the discussion of notation.

6.1 DenitionThe truth table of p q is as follows:[3]

Note that it is equivalent to that produced by the XNORgate, and opposite to that produced by the XOR gate.

6.2 Usage

6.2.1 NotationThe corresponding logical symbols are "", "" and"", and sometimes i. These are usually treatedas equivalent. However, some texts of mathematicallogic (particularly those on rst-order logic, rather thanpropositional logic) make a distinction between these, inwhich the rst, , is used as a symbol in logic formulas,while is used in reasoning about those logic formulas(e.g., in metalogic). In ukasiewicz's notation, it is theprex symbol 'E'.Another term for this logical connective is exclusive nor.

6.2.2 ProofsIn most logical systems, one proves a statement of theform P i Q by proving if P, then Q and if Q, thenP. Proving this pair of statements sometimes leads toa more natural proof, since there are not obvious condi-tions in which one would infer a biconditional directly.An alternative is to prove the disjunction "(P and Q) or(not-P and not-Q)", which itself can be inferred directlyfrom either of its disjunctsthat is, because i is truth-functional, P i Q follows if P and Q have both beenshown true, or both false.

6.2.3 Origin of iUsage of the abbreviation i rst appeared in print inJohn L. Kelley's 1955 book General Topology.[4] Its in-vention is often credited to Paul Halmos, who wrote Iinvented 'i,' for 'if and only if'but I could never be-lieve I was really its rst inventor.[5]

6.3 Distinction from if and onlyif

1. Madison will eat the fruit if it is an apple.(equivalent to Only if Madison will eat the fruit,is it an apple;" or Madison will eat the fruitfruit is an apple)

11

12 CHAPTER 6. MATERIAL EQUIVALENCE

This states simply that Madison will eatfruits that are apples. It does not, how-ever, exclude the possibility that Madi-son might also eat bananas or other typesof fruit. All that is known for certain isthat she will eat any and all apples thatshe happens upon. That the fruit is anapple is a sucient condition for Madi-son to eat the fruit.

2. Madison will eat the fruit only if it is an apple.(equivalent to If Madison will eat the fruit, thenit is an apple or Madison will eat the fruit fruit is an apple)

This states that the only fruit Madisonwill eat is an apple. It does not, however,exclude the possibility that Madison willrefuse an apple if it is made available, incontrast with (1), which requires Madi-son to eat any available apple. In thiscase, that a given fruit is an apple is a nec-essary condition for Madison to be eat-ing it. It is not a sucient condition sinceMadison might not eat all the apples sheis given.

3. Madison will eat the fruit if and only if it is anapple (equivalent to Madison will eat the fruit fruit is an apple)

This statement makes it clear that Madi-son will eat all and only those fruits thatare apples. She will not leave any appleuneaten, and she will not eat any othertype of fruit. That a given fruit is an ap-ple is both a necessary and a sucientcondition for Madison to eat the fruit.

Suciency is the inverse of necessity. That is to say,given PQ (i.e. if P then Q), P would be a sucientcondition for Q, and Q would be a necessary conditionfor P. Also, given PQ, it is true that QP (where is the negation operator, i.e. not). This means that therelationship between P and Q, established by PQ, canbe expressed in the following, all equivalent, ways:

P is sucient for QQ is necessary for PQ is sucient for PP is necessary for Q

As an example, take (1), above, which states PQ, whereP is the fruit in question is an apple and Q is Madi-son will eat the fruit in question. The following are fourequivalent ways of expressing this very relationship:

If the fruit in question is an apple, then Madi-son will eat it.Only if Madison will eat the fruit in question,is it an apple.If Madison will not eat the fruit in question,then it is not an apple.Only if the fruit in question is not an apple, willMadison not eat it.

So we see that (2), above, can be restated in the formof if...then as If Madison will eat the fruit in question,then it is an apple"; taking this in conjunction with (1),we nd that (3) can be stated as If the fruit in question isan apple, then Madison will eat it; AND if Madison willeat the fruit, then it is an apple.

6.4 More general usageI is used outside the eld of logic, wherever logic isapplied, especially in mathematical discussions. It hasthe same meaning as above: it is an abbreviation forif and only if, indicating that one statement is bothnecessary and sucient for the other. This is an exampleof mathematical jargon. (However, as noted above, if,rather than i, is more often used in statements of de-nition.)The elements of X are all and only the elements of Y isused to mean: for any z in the domain of discourse, z isin X if and only if z is in Y.

6.5 See also Covariance Logical biconditional Logical equality Necessary and sucient condition Polysyllogism

6.6 Footnotes[1] E.g. Daepp, Ulrich; Gorkin, Pamela (2011), Reading,

Writing, and Proving: A Closer Look at Mathematics, Un-dergraduate Texts in Mathematics, Springer, p. 52, ISBN9781441994790, While it can be a real time-saver, wedon't recommend it in formal writing.

[2] Rothwell, Edward J.; Cloud, Michael J. (2014),Engineering Writing by Design: Creating Formal Doc-uments of Lasting Value, CRC Press, p. 98, ISBN9781482234312, It is common in mathematical writing.

6.7. EXTERNAL LINKS 13

[3] p q. Wolfram|Alpha

[4] General Topology, reissue ISBN 978-0-387-90125-1

[5] Nicholas J. Higham (1998). Handbook of writing for themathematical sciences (2nd ed.). SIAM. p. 24. ISBN978-0-89871-420-3.

6.7 External links Language Log: Just in Case Southern California Philosophy for philosophygraduate students: Just in Case

Chapter 7

Indicative conditional

In natural languages, an indicative conditional[1][2] is thelogical operation given by statements of the form If Athen B. Unlike the material conditional, an indicativeconditional does not have a stipulated denition. Thephilosophical literature on this operation is broad, and noclear consensus has been reached.

7.1 Distinctions between the mate-rial conditional and the indica-tive conditional

The material conditional does not always function in ac-cordance with everyday if-then reasoning. Thereforethere are drawbacks with using the material conditionalto represent if-then statements.One problem is that the material conditional allows impli-cations to be true even when the antecedent is irrelevantto the consequent. For example, its commonly acceptedthat the sun is made of gas, on one hand, and that 3 is aprime number, on the other. The standard denition ofimplication allows us to conclude that, if the sun is madeof gas, then 3 is a prime number. This is arguably synony-mous to the following: the suns being made of gas makes3 be a prime number. Many people intuitively think thatthis is false, because the sun and the number three simplyhave nothing to do with one another. Logicians have triedto address this concern by developing alternative logics,e.g., relevant logic.For a related problem, see vacuous truth.Another issue is that the material conditional is not de-signed to deal with counterfactuals and other cases thatpeople often nd in if-then reasoning. This has inspiredpeople to develop modal logic.A further problem is that the material conditional is suchthat P AND P Q, regardless of what Q is taken tomean. That is, a contradiction implies that absolutely ev-erything is true. Logicians concerned with this have triedto develop paraconsistent logics.

7.2 Psychology and indicative con-ditionals

Most behavioral experiments on conditionals in the psy-chology of reasoning have been carried out with indica-tive conditionals, causal conditionals, and counterfactualconditionals. People readily make the modus ponens in-ference, that is, given if A then B, and given A, they con-clude B, but only about half of participants in experi-ments make the modus tollens inference, that is, given ifA then B, and given not-B, only about half of participantsconclude not-A, the remainder say that nothing follows(Evans et al., 1993). When participants are given coun-terfactual conditionals, they make both the modus ponensand the modus tollens inferences (Byrne, 2005).

7.3 See also Material conditional Counterfactual conditional Logical consequence Strict conditional

7.4 References[1] Stalnaker, R, Philosophia (1975)

[2] Ellis, B, Australasian Journal of Philosophy (1984)

7.5 Further reading Byrne, R.M.J. (2005). The Rational Imagination:How People Create Counterfactual Alternatives toReality. Cambridge, MA: MIT Press.

Edgington, Dorothy. (2006). Conditionals.The Stanford Encyclopedia of Philosophy, Ed-ward Zalta (ed.). http://plato.stanford.edu/entries/conditionals/.

14

7.5. FURTHER READING 15

Evans, J. St. B. T., Newstead, S. and Byrne, R. M. J.(1993). Human Reasoning: The Psychology of De-duction. Hove, Psychology Press.

Chapter 8

Logical biconditional

In logic and mathematics, the logical biconditional(sometimes known as the material biconditional) isthe logical connective of two statements asserting "p ifand only if q", where q is an antecedent and p is aconsequent.[1] This is often abbreviated p i q. The op-erator is denoted using a doubleheaded arrow (), a pre-xed E (Epq), an equality sign (=), an equivalence sign(), or EQV. It is logically equivalent to (p q) (q p), or the XNOR (exclusive nor) boolean operator. It isequivalent to "(not p or q) and (not q or p)". It is alsologically equivalent to "(p and q) or (not p and not q)",meaning both or neither.The only dierence from material conditional is the casewhen the hypothesis is false but the conclusion is true. Inthat case, in the conditional, the result is true, yet in thebiconditional the result is false.In the conceptual interpretation, a = bmeans All a 's areb 's and all b 's are a 's"; in other words, the sets a and bcoincide: they are identical. This does not mean that theconcepts have the same meaning. Examples: triangleand trilateral, equiangular trilateral and equilateraltriangle. The antecedent is the subject and the conse-quent is the predicate of a universal armative proposi-tion.In the propositional interpretation, a b means that aimplies b and b implies a; in other words, that the propo-sitions are equivalent, that is to say, either true or falseat the same time. This does not mean that they have thesame meaning. Example: The triangle ABC has twoequal sides, and The triangle ABC has two equal an-gles. The antecedent is the premise or the cause and theconsequent is the consequence. When an implication istranslated by a hypothetical (or conditional) judgment theantecedent is called the hypothesis (or the condition) andthe consequent is called the thesis.A common way of demonstrating a biconditional is touse its equivalence to the conjunction of two converseconditionals, demonstrating these separately.When both members of the biconditional are proposi-tions, it can be separated into two conditionals, of whichone is called a theorem and the other its reciprocal. Thuswhenever a theorem and its reciprocal are true we havea biconditional. A simple theorem gives rise to an im-

plication whose antecedent is the hypothesis and whoseconsequent is the thesis of the theorem.It is often said that the hypothesis is the sucient condi-tion of the thesis, and the thesis the necessary conditionof the hypothesis; that is to say, it is sucient that thehypothesis be true for the thesis to be true; while it isnecessary that the thesis be true for the hypothesis to betrue also. When a theorem and its reciprocal are true wesay that its hypothesis is the necessary and sucient con-dition of the thesis; that is to say, that it is at the sametime both cause and consequence.

8.1 Denition

Logical equality (also known as biconditional) is anoperation on two logical values, typically the values oftwo propositions, that produces a value of true if and onlyif both operands are false or both operands are true.

8.1.1 Truth table

The truth table for A$ B (also written as A B, A =B, or A EQ B) is as follows:More than two statements combined by $ are am-biguous:x1 $ x2 $ x3 $ :::$ xn may be meant as (((x1 $x2)$ x3)$ :::)$ xn ,or may be used to say that all xi are together true ortogether false: ( x1 ^ ::: ^ xn ) _ (:x1 ^ ::: ^ :xn)Only for zero or two arguments this is the same.The following truth tables show the same bit pattern onlyin the line with no argument and in the lines with twoarguments:The left Venn diagram below, and the lines (AB ) in thesematrices represent the same operation.

16

8.3. RULES OF INFERENCE 17

x1 $ :::$ xnmeant as equivalent to: (:x1 ::: :xn)The central Venn diagram below,and line (ABC ) in this matrixrepresent the same operation.

x1 $ :::$ xnmeant as shorthand for( x1 ^ ::: ^ xn )_ (:x1 ^ ::: ^ :xn)The Venn diagram directly below,and line (ABC ) in this matrixrepresent the same operation.

8.1.2 Venn diagrams

Red areas stand for true (as in for and).

8.2 Propertiescommutativity: yesassociativity: yesdistributivity: Biconditional doesn't distribute over anybinary function (not even itself),

but logical disjunction (see there) distributes over bicon-ditional.idempotency: no

monotonicity: notruth-preserving: yesWhen all inputs are true, the output is true.falsehood-preserving: noWhen all inputs are false, the output is not false.Walsh spectrum: (2,0,0,2)Nonlinearity: 0 (the function is linear)

8.3 Rules of inference

Main article: Rules of inference

Like all connectives in rst-order logic, the biconditionalhas rules of inference that govern its use in formal proofs.

8.3.1 Biconditional introduction

Main article: Biconditional introduction

Biconditional introduction allows you to infer that, if Bfollows from A, and A follows from B, then A if and onlyif B.For example, from the statements if I'm breathing, thenI'm alive and if I'm alive, then I'm breathing, it can beinferred that I'm breathing if and only if I'm alive or,equally inferrable, I'm alive if and only if I'm breathing.B A A B A B B A A B B A

8.3.2 Biconditional elimination

Biconditional elimination allows one to infer aconditional from a biconditional: if ( A B ) is true,then one may infer one direction of the biconditional, ( A B ) and ( B A ).For example, if its true that I'm breathing if and only ifI'm alive, then its true that if I'm breathing, I'm alive;likewise, its true that if I'm alive, I'm breathing.Formally:( A B ) ( A B )also( A B ) ( B A )

18 CHAPTER 8. LOGICAL BICONDITIONAL

8.4 Colloquial usageOne unambiguous way of stating a biconditional in plainEnglish is of the form "b if a and a if b". Another is "a ifand only if b". Slightly more formally, one could say "bimplies a and a implies b". The plain English if'" maysometimes be used as a biconditional. One must weighcontext heavily.For example, I'll buy you a new wallet if you need onemay bemeant as a biconditional, since the speaker doesn'tintend a valid outcome to be buying the wallet whether ornot the wallet is needed (as in a conditional). However, itis cloudy if it is raining is not meant as a biconditional,since it can be cloudy while not raining.

8.5 See also If and only if Logical equivalence Logical equality XNOR gate Biconditional elimination Biconditional introduction

8.6 Notes[1] Handbook of Logic, page 81

8.7 References Brennan, Joseph G. Handbook of Logic, 2nd Edi-tion. Harper & Row. 1961

This article incorporates material from Biconditional onPlanetMath, which is licensed under the Creative Com-mons Attribution/Share-Alike License.

Chapter 9

Logical conjunction

"" redirects here. For the logic gate, see AND gate. Forexterior product, see Exterior algebra.In logic and mathematics, and is the truth-functional

Venn diagram of A^B

Venn diagram of A^B^C

operator of logical conjunction; the and of a set ofoperands is true if and only if all of its operands are true.The logical connective that represents this operator is typ-ically written as ^ or .

"A and B" is true only if A is true and B is true.An operand of a conjunction is a conjunct.Related concepts in other elds are:

In natural language, the coordinating conjunctionand.

In programming languages, the short-circuit andcontrol structure.

In set theory, intersection.

In predicate logic, universal quantication.

9.1 NotationAnd is usually expressed with an inx operator: in math-ematics and logic, ; in electronics, ; and in program-ming languages, & or and. In Jan ukasiewicz's prexnotation for logic, the operator is K, for Polish koni-unkcja.[1]

9.2 DenitionLogical conjunction is an operation on two logical val-ues, typically the values of two propositions, that pro-duces a value of true if and only if both of its operandsare true.The conjunctive identity is 1, which is to say that AND-ing an expression with 1 will never change the value of theexpression. In keeping with the concept of vacuous truth,when conjunction is dened as an operator or functionof arbitrary arity, the empty conjunction (AND-ing overan empty set of operands) is often dened as having theresult 1.

9.2.1 Truth table

The truth table of A ^B :

19

20 CHAPTER 9. LOGICAL CONJUNCTION

Conjunctions of the arguments on the left The true bits form aSierpinski triangle.

9.3 Introduction and eliminationrules

As a rule of inference, conjunction introduction is a clas-sically valid, simple argument form. The argument formhas two premises, A and B. Intuitively, it permits the in-ference of their conjunction.

A,B.Therefore, A and B.

or in logical operator notation:

A;

B

` A ^BHere is an example of an argument that ts the formconjunction introduction:

Bob likes apples.Bob likes oranges.Therefore, Bob likes apples and oranges.

Conjunction elimination is another classically valid, sim-ple argument form. Intuitively, it permits the inferencefrom any conjunction of either element of that conjunc-tion.

A and B.Therefore, A.

...or alternately,

A and B.Therefore, B.

In logical operator notation:

A ^B

` A...or alternately,

A ^B

` B

9.4 Propertiescommutativity: yesassociativity: yesdistributivity: with various operations, especially withoridempotency: yes

monotonicity: yestruth-preserving: yesWhen all inputs are true, the output is true.falsehood-preserving: yesWhen all inputs are false, the output is false.Walsh spectrum: (1,1,1,1)Nonlinearity: 1 (the function is bent)If using binary values for true (1) and false (0), thenlogical conjunction works exactly like normal arithmeticmultiplication.

9.5 Applications in computer engi-neering

AND logic gate

In high-level computer programming and digital electron-ics, logical conjunction is commonly represented by aninx operator, usually as a keyword such as AND, an

9.8. SEE ALSO 21

algebraic multiplication, or the ampersand symbol "&".Many languages also provide short-circuit control struc-tures corresponding to logical conjunction.Logical conjunction is often used for bitwise operations,where 0 corresponds to false and 1 to true:

0 AND 0 = 0, 0 AND 1 = 0, 1 AND 0 = 0, 1 AND 1 = 1.

The operation can also be applied to two binary wordsviewed as bitstrings of equal length, by taking the bitwiseAND of each pair of bits at corresponding positions. Forexample:

11000110 AND 10100011 = 10000010.

This can be used to select part of a bitstring using abit mask. For example, 10011101 AND 00001000 =00001000 extracts the fth bit of an 8-bit bitstring.In computer networking, bit masks are used to derive thenetwork address of a subnet within an existing networkfrom a given IP address, by ANDing the IP address andthe subnet mask.Logical conjunction AND is also used in SQL opera-tions to form database queries.The Curry-Howard correspondence relates logical con-junction to product types.

9.6 Set-theoretic correspondenceThe membership of an element of an intersection set inset theory is dened in terms of a logical conjunction: x A B if and only if (x A) (x B). Through thiscorrespondence, set-theoretic intersection shares severalproperties with logical conjunction, such as associativity,commutativity, and idempotence.

9.7 Natural languageAs with other notions formalized in mathematical logic,the logical conjunction and is related to, but not the sameas, the grammatical conjunction and in natural languages.English and has properties not captured by logical con-junction. For example, and sometimes implies order.For example, They got married and had a child in com-mon discourse means that the marriage came before thechild. The word and can also imply a partition of a thinginto parts, as The American ag is red, white, and blue.Here it is not meant that the ag is at once red, white, andblue, but rather that it has a part of each color.

9.8 See also And-inverter graph AND gate Binary and Bitwise AND Boolean algebra (logic) Boolean algebra topics Boolean conjunctive query Boolean domain Boolean function Boolean-valued function Conjunction introduction Conjunction elimination De Morgans laws First-order logic Frchet inequalities Grammatical conjunction Logical disjunction Logical negation Logical graph Logical value Operation Peano-Russell notation Propositional calculus

9.9 References[1] Jzef Maria Bocheski (1959), A Prcis of Mathematical

Logic, translated by Otto Bird from the French and Ger-man editions, Dordrecht, North Holland: D. Reidel, pas-sim.

9.10 External links Hazewinkel, Michiel, ed. (2001), Conjunction,Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Wolfram MathWorld: Conjunction

Chapter 10

Logical connective

This article is about connectives in logical systems.For connectors in natural languages, see discourseconnective. For other logical symbols, see List of logicsymbols.

In logic, a logical connective (also called a logical op-erator) is a symbol or word used to connect two or moresentences (of either a formal or a natural language) in agrammatically valid way, such that the sense of the com-pound sentence produced depends only on the originalsentences.The most common logical connectives are binary con-nectives (also called dyadic connectives) which jointwo sentences which can be thought of as the functionsoperands. Also commonly, negation is considered to bea unary connective.Logical connectives along with quantiers are the twomain types of logical constants used in formal systemssuch as propositional logic and predicate logic. Semanticsof a logical connective is often, but not always, presentedas a truth function.A logical connective is similar to but not equivalent to aconditional operator. [1]

10.1 In language

10.1.1 Natural language

In the grammar of natural languages two sentences maybe joined by a grammatical conjunction to form a gram-matically compound sentence. Some but not all suchgrammatical conjunctions are truth functions. For exam-ple, consider the following sentences:

A: Jack went up the hill.B: Jill went up the hill.C: Jack went up the hill and Jill went up thehill.D: Jack went up the hill so Jill went up the hill.

The words and and so are grammatical conjunctions join-ing the sentences (A) and (B) to form the compound sen-tences (C) and (D). The and in (C) is a logical connec-tive, since the truth of (C) is completely determined by(A) and (B): it would make no sense to arm (A) and(B) but deny (C). However, so in (D) is not a logical con-nective, since it would be quite reasonable to arm (A)and (B) but deny (D): perhaps, after all, Jill went up thehill to fetch a pail of water, not because Jack had gone upthe hill at all.Various English words and word pairs express logicalconnectives, and some of them are synonymous. Exam-ples (with the name of the relationship in parentheses)are:

and (conjunction) and then (conjunction) and then within (conjunction) or (disjunction) either...or (exclusive disjunction) implies (implication) if...then (implication) if and only if (equivalence) only if (implication) just in case (biconditional) but (conjunction) however (conjunction) not both (alternative denial) neither...nor (joint denial)

The word not (negation) and the phrases it is false that(negation) and it is not the case that (negation) also ex-press a logical connective even though they are appliedto a single statement, and do not connect two statements.

22

10.2. COMMON LOGICAL CONNECTIVES 23

10.1.2 Formal languages

In formal languages, truth functions are represented byunambiguous symbols. These symbols are called logicalconnectives, logical operators, propositional opera-tors, or, in classical logic, "truth-functional connectives.See well-formed formula for the rules which allow newwell-formed formulas to be constructed by joining otherwell-formed formulas using truth-functional connectives.Logical connectives can be used to link more than twostatements, so one can speak about "n-ary logical con-nective.

10.2 Common logical connectives

10.2.1 List of common logical connectives

Commonly used logical connectives include

Negation (not): , N (prex), ~

Conjunction (and): ^ , K (prex), & ,

Disjunction (or): _ , A (prex)

Material implication (if...then): ! , C (prex),) ,

Biconditional (if and only if): $ , E (prex), , =

Alternative names for biconditional are i, xnor andbi-implication.For example, the meaning of the statements it is rainingand I am indoors is transformed when the two are com-bined with logical connectives:

It is not raining (P)

It is raining and I am indoors (P ^ Q)

It is raining or I am indoors (P _ Q)

If it is raining, then I am indoors (P! Q)

If I am indoors, then it is raining (Q! P)

I am indoors if and only if it is raining (P$ Q)

For statement P = It is raining and Q = I am indoors.It is also common to consider the always true formula andthe always false formula to be connective:

True formula (, 1, V [prex], or T)

False formula (, 0, O [prex], or F)

10.2.2 History of notations Negation: the symbol appeared in Heyting in1929.[2][3] (compare to Frege's symbol A inhis Begrisschrift); the symbol ~ appeared in Rus-sell in 1908;[4] an alternative notation is to add anhorizontal line on top of the formula, as in P ; an-other alternative notation is to use a prime symbolas in P'.

Conjunction: the symbol appeared in Heyting in1929[2] (compare to Peano's use of the set-theoreticnotation of intersection [5]); & appeared at least inSchnnkel in 1924;[6] . comes from Boole's inter-pretation of logic as an elementary algebra.

Disjunction: the symbol appeared in Russell in1908[4] (compare to Peano's use of the set-theoreticnotation of union ); the symbol + is also used, inspite of the ambiguity coming from the fact thatthe + of ordinary elementary algebra is an exclusiveor when interpreted logically in a two-element ring;punctually in the history a + together with a dot inthe lower right corner has been used by Peirce,[7]

Implication: the symbol can be seen in Hilbertin 1917;[8] was used by Russell in 1908[4] (com-pare to Peanos inverted C notation);) was used inVax.[9]

Biconditional: the symbol was used at least byRussell in 1908;[4] was used at least by Tarskiin 1940;[10] was used in Vax; other symbolsappeared punctually in the history such as inGentzen,[11] ~ in Schnnkel[6] or in Chazal.[12]

True: the symbol 1 comes from Boole's interpreta-tion of logic as an elementary algebra over the two-element Boolean algebra; other notations include Vto be found in Peano.

False: the symbol 0 comes also from Booles inter-pretation of logic as a ring; other notations includeW to be found in Peano.

Some authors used letters for connectives at some timeof the history: u. for conjunction (Germans und forand) and o. for disjunction (Germans oder for or)in earlier works by Hilbert (1904); Np for negation, Kpqfor conjunction, Apq for disjunction, Cpq for implica-tion, Epq for biconditional in ukasiewicz (1929);[13] cf.Polish notation.

10.2.3 RedundancySuch logical connective as converse implication is ac-tually the same as material conditional with swapped ar-guments, so the symbol for converse implication is re-dundant. In some logical calculi (notably, in classical

24 CHAPTER 10. LOGICAL CONNECTIVE

logic) certain essentially dierent compound statementsare logically equivalent. A less trivial example of a redun-dancy is the classical equivalence between P Q andP Q. Therefore, a classical-based logical system doesnot need the conditional operator "" if "" (not) and"" (or) are already in use, or may use the "" only as asyntactic sugar for a compound having one negation andone disjunction.There are sixteen Boolean functions associating the in-put truth values P and Q with four-digit binary outputs.These correspond to possible choices of binary logicalconnectives for classical logic. Dierent implementationof classical logic can choose dierent functionally com-plete subsets of connectives.One approach is to choose aminimal set, and dene otherconnectives by some logical form, like in the examplewith material conditional above. The following are theminimal functionally complete sets of operators in clas-sical logic whose arities do not exceed 2:

One element {}, {}.

Two elements { _ , }, { ^ , }, {, }, {, }, {,? }, {, ? }, {,= }, {,= }, {,9 }, {,8 }, {,9 }, {,8 }, {9 , }, {8 , }, {9, > }, {8 , > }, {9 ,$ }, {8 ,$ }.

Three elements { _ ,$ , ? }, { _ ,$ ,= }, { _ ,=, > }, { ^ ,$ , ? }, { ^ ,$ ,= }, { ^ ,= , > }.

See more details about functional completeness in classi-cal logic at Functional completeness in truth function.Another approach is to use on equal rights connectives ofa certain convenient and functionally complete, but notminimal set. This approach requires more propositionalaxioms and each equivalence between logical forms mustbe either an axiom or provable as a theorem.But intuitionistic logic has the situation more compli-cated. Of its ve connectives {, , , , } only nega-tion has to be reduced to other connectives (see details).Neither of conjunction, disjunction and material condi-tional has an equivalent form constructed of other fourlogical connectives.

10.3 PropertiesSome logical connectives possess properties which maybe expressed in the theorems containing the connective.Some of those properties that a logical connective mayhave are:

Associativity: Within an expression containing twoormore of the same associative connectives in a row,the order of the operations does not matter as longas the sequence of the operands is not changed.

Commutativity: The operands of the connectivemay be swapped preserving logical equivalence tothe original expression.

Distributivity: A connective denoted by dis-tributes over another connective denoted by +, if a (b + c) = (a b) + (a c) for all operands a, b, c.

Idempotence: Whenever the operands of the oper-ation are the same, the compound is logically equiv-alent to the operand.

Absorption: A pair of connectives^ ,_ satises theabsorption law if a^ (a_ b) = a for all operands a,b.

Monotonicity: If f(a1, ..., an) f(b1, ..., bn) forall a1, ..., an, b1, ..., bn {0,1} such that a1 b1,a2 b2, ..., an bn. E.g., _ , ^ , > , ? .

Anity: Each variable always makes a dierence inthe truth-value of the operation or it never makes adierence. E.g., : ,$ ,= , > , ? .

Duality: To read the truth-value assignments for theoperation from top to bottom on its truth table is thesame as taking the complement of reading the tableof the same or another connective from bottom totop. Without resorting to truth tables it may be for-mulated as g (a1, ..., an) = g(a1, ..., an). E.g., :.

Truth-preserving: The compound all those argu-ment are tautologies is a tautology itself. E.g., _ , ^, > ,! ,$ , . (see validity)

Falsehood-preserving: The compound all those ar-gument are contradictions is a contradiction itself.E.g., _ , ^ ,= , ? , , . (see validity)

Involutivity (for unary connectives): f(f(a)) = a.E.g. negation in classical logic.

For classical and intuitionistic logic, the "=" symbolmeans that corresponding implications "" and"" for logical compounds can be both proved astheorems, and the "" symbol means that "" forlogical compounds is a consequence of corresponding"" connectives for propositional variables. Somemany-valued logics may have incompatible denitions ofequivalence and order (entailment).Both conjunction and disjunction are associative, com-mutative and idempotent in classical logic, most varietiesof many-valued logic and intuitionistic logic. The sameis true about distributivity of conjunction over disjunc-tion and disjunction over conjunction, as well as for theabsorption law.In classical logic and some varieties of many-valued logic,conjunction and disjunction are dual, and negation is self-dual, the latter is also self-dual in intuitionistic logic.

10.8. REFERENCES 25

10.4 Order of precedenceAs a way of reducing the number of necessary parenthe-ses, one may introduce precedence rules: : has higherprecedence than ^ , ^ higher than _ , and _ higher than! . So for example, P _ Q ^ :R ! S is short for(P _ (Q ^ (:R)))! S .Here is a table that shows a commonly used precedenceof logical operators.[14]

However not all authors use the same order; for instance,an ordering in which disjunction is lower precedencethan implication or bi-implication has also been used.[15]Sometimes precedence between conjunction and disjunc-tion is unspecied requiring to provide it explicitly ingiven formula with parentheses. The order of precedencedetermines which connective is the main connectivewhen interpreting a non-atomic formula.

10.5 Computer scienceA truth-functional approach to logical operators is imple-mented as logic gates in digital circuits. Practically alldigital circuits (the major exception is DRAM) are builtup from NAND, NOR, NOT, and transmission gates; seemore details in Truth function in computer science. Log-ical operators over bit vectors (corresponding to niteBoolean algebras) are bitwise operations.But not every usage of a logical connective in computerprogramming has a Boolean semantic. For example, lazyevaluation is sometimes implemented for P Q and P Q, so these connectives are not commutative if someof expressions P, Q has side eects. Also, a conditional,which in some sense corresponds to the material condi-tional connective, is essentially non-Boolean because forif (P) then Q; the consequent Q is not executed if theantecedent P is false (although a compound as a wholeis successful true in such case). This is closer to in-tuitionist and constructivist views on the material condi-tional, rather than to classical logics ones.

10.6 See also

10.7 Notes[1] Cogwheel. What is the dierence between logical and

conditional /operator/". Stack Overow. Retrieved 9April2015.

[2] Heyting (1929) Die formalen Regeln der intuitionistischenLogik.

[3] Denis Roegel (2002), Petit panorama des notationslogiques du 20e sicle (see chart on page 2).

[4] Russell (1908) Mathematical logic as based on the theoryof types (American Journal of Mathematics 30, p222262, also in From Frege to Gdel edited by van Hei-jenoort).

[5] Peano (1889) Arithmetices principia, nova methodo ex-posita.

[6] Schnnkel (1924) ber die Bausteine der mathematis-chen Logik, translated as On the building blocks of math-ematical logic in From Frege to Gdel edited by van Hei-jenoort.

[7] Peirce (1867) On an improvement in Booles calculus oflogic.

[8] Hilbert (1917/1918) Prinzipien der Mathematik (Bernayscourse notes).

[9] Vax (1982) Lexique logique, Presses Universitaires deFrance.

[10] Tarski (1940) Introduction to logic and to the methodologyof deductive sciences.

[11] Gentzen (1934) Untersuchungen ber das logischeSchlieen.

[12] Chazal (1996) : lments de logique formelle.

[13] See Roegel

[14] O'Donnell, John; Hall, Cordelia; Page, Rex (2007),Discrete Mathematics Using a Computer, Springer, p. 120,ISBN 9781846285981.

[15] Jackson, Daniel (2012), Software Abstractions: Logic,Language, and Analysis, MIT Press, p. 263, ISBN9780262017152.

10.8 References Bocheski, Jzef Maria (1959), A Prcis of Mathe-matical Logic, translated from the French and Ger-man editions by Otto Bird, D. Reidel, Dordrecht,South Holland.

Enderton, Herbert (2001), A Mathematical Intro-duction to Logic (2nd ed.), Boston, MA: AcademicPress, ISBN 978-0-12-238452-3

Gamut, L.T.F (1991), Chapter 2, Logic, Languageand Meaning 1, University of Chicago Press, pp.5464, OCLC 21372380

Rautenberg, W. (2010), A Concise Introduction toMathematical Logic (3rd ed.), New York: SpringerScience+BusinessMedia, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6.

26 CHAPTER 10. LOGICAL CONNECTIVE

10.9 Further reading Lloyd Humberstone (2011). The Connectives. MITPress. ISBN 978-0-262-01654-4.

10.10 External links Hazewinkel, Michiel, ed. (2001), Propositionalconnective, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4

Lloyd Humberstone (2010), "Sentence Connectivesin Formal Logic", Stanford Encyclopedia of Philos-ophy (An abstract algebraic logic approach to con-nectives.)

John MacFarlane (2005), "Logical constants",Stanford Encyclopedia of Philosophy.

Chapter 11

Logical disjunction

Disjunction redirects here. For the logic gate, see ORgate. For separation of chromosomes, see Meiosis. Fordisjunctions in distribution, see Disjunct distribution.In logic and mathematics, or is the truth-functional op-

Venn diagram of A_B

Venn diagram of A_B_C

erator of (inclusive) disjunction, also known as alter-nation; the or of a set of operands is true if and only ifone or more of its operands is true. The logical connec-

tive that represents this operator is typically written as _or + ."A or B" is true if A is true, or if B is true, or if both Aand B are true.In logic, or by itself means the inclusive or, distinguishedfrom an exclusive or, which is false when both of its ar-guments are true, while an or is true in that case.An operand of a disjunction is called a disjunct.Related concepts in other elds are:

In natural language, the coordinating conjunctionor.

In programming languages, the short-circuit orcontrol structure.

In set theory, union. In predicate logic, existential quantication.

11.1 NotationOr is usually expressed with an inx operator: in math-ematics and logic, ; in electronics, +; and in program-ming languages, | or or. In Jan ukasiewicz's prex nota-tion for logic, the operator is A, for Polish alternatywa.[1]

11.2 DenitionLogical disjunction is an operation on two logical val-ues, typically the values of two propositions, that has avalue of false if and only if both of its operands are false.More generally, a disjunction is a logical formula that canhave one or more literals separated only by ORs. A singleliteral is often considered to be a degenerate disjunction.The disjunctive identity is false, which is to say that the orof an expression with false has the same value as the orig-inal expression. In keeping with the concept of vacuoustruth, when disjunction is dened as an operator or func-tion of arbitrary arity, the empty disjunction (OR-ing

27

28 CHAPTER 11. LOGICAL DISJUNCTION

over an empty set of operands) is generally dened asfalse.

11.2.1 Truth table

Disjunctions of the arguments on the left The false bits forma Sierpinski triangle.

The truth table of A _B :

11.3 Properties Commutativity

Associativity

Distributivity with various operations, especiallywith and

Idempotency

Monotonicity

Truth-preserving validity

When all inputs are true, the output is true.

False-preserving validity

When all inputs are false, the output is false.

Walsh spectrum: (3,1,1,1)

Nonlinearity: 1 (the function is bent)

If using binary values for true (1) and false (0), then log-ical disjunction works almost like binary addition. Theonly dierence is that 1 _ 1 = 1 , while 1 + 1 = 10 .

11.4 SymbolThe mathematical symbol for logical disjunction varies inthe literature. In addition to the word or, and the for-mula Apq", the symbol " _ ", deriving from the Latinword vel (either, or) is commonly used for disjunc-tion. For example: "A _ B " is read as "A or B ". Such adisjunction is false if both A and B are false. In all othercases it is true.All of the following are disjunctions:

A _B

:A _BA _ :B _ :C _D _ :E:The corresponding operation in set theory is the set-theoretic union.

11.5 Applications in computer sci-ence

AB out

OR logic gate

Operators corresponding to logical disjunction exist inmost programming languages.

11.5.1 Bitwise operationDisjunction is often used for bitwise operations. Exam-ples:

0 or 0 = 0 0 or 1 = 1 1 or 0 = 1 1 or 1 = 1 1010 or 1100 = 1110

The or operator can be used to set bits in a bit eld to 1,by or-ing the eld with a constant eld with the relevantbits set to 1. For example, x = x | 0b00000001 will forcethe nal bit to 1 while leaving other bits unchanged.

11.8. SEE ALSO 29

11.5.2 Logical operation

Many languages distinguish between bitwise and logicaldisjunction by providing two distinct operators; in lan-guages following C, bitwise disjunction is performed withthe single pipe (|) and logical disjunction with the doublepipe (||) operators.Logical disjunction is usually short-circuited; that is, ifthe rst (left) operand evaluates to true then the second(right) operand is not evaluated.