Logic symbols 1Wikipedia

Contents

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

2 Converse implication 22.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Converse nonimplication 43.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1.2 Venn diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Exclusive or 74.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Equivalencies, elimination, and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Relation to modern algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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4.4 Exclusive “or” in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.5 Alternative symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.7 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.7.1 Bitwise operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.8 Encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 False (logic) 145.1 In classical logic and Boolean logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 False, negation and contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Material equivalence 166.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2.3 Origin of iff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.3 Distinction from “if” and “only if” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 Indicative conditional 197.1 Distinctions between the material conditional and the indicative conditional . . . . . . . . . . . . . 197.2 Psychology and indicative conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8 Logical biconditional 218.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.1.2 Venn diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.3 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8.3.1 Biconditional introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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8.3.2 Biconditional elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.4 Colloquial usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9 Logical conjunction 269.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.3 Introduction and elimination rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.5 Applications in computer engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.6 Set-theoretic correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10 Logical connective 3210.1 In language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

10.1.1 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.1.2 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10.2 Common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2.1 List of common logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2.2 History of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2.3 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

10.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.4 Order of precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.5 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11 Logical disjunction 3811.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.2.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.4 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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11.5 Applications in computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.5.1 Bitwise operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.5.2 Logical operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.5.3 Constructive disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

11.6 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.7 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12 Logical equality 4412.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.2 Alternative descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

13 Logical NOR 4713.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

13.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.3 Introduction, elimination, and equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

14 Material conditional 4914.1 Definitions of the material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

14.1.1 As a truth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.1.2 As a formal connective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

14.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.3 Philosophical problems with material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

14.4.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

15 Material equivalence 5415.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

15.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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15.2.3 Origin of iff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.3 Distinction from “if” and “only if” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.4 More general usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

16 Material nonimplication 5716.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

16.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

16.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

16.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5816.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

17 Modal operator 5917.1 Modality interpreted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

17.1.1 Alethic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.1.2 Deontic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.1.3 Axiological . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.1.4 Epistemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.1.5 Doxastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

18 Negation 6018.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6018.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6018.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

18.3.1 Double negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.3.2 Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.3.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.3.4 Self dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

18.4 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.5 Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.6 Kripke semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

vi CONTENTS

18.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

19 Sheffer stroke 6419.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

19.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.4 Introduction, elimination, and equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.5 Formal system based on the Sheffer stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

19.5.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.5.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.5.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.5.4 Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

19.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6819.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

20 Strict conditional 6920.1 Avoiding paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6920.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 72

20.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7220.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7420.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Chapter 1

Conditioned disjunction

In logic, conditioned disjunction (sometimes called conditional disjunction) is a ternary logical connective in-troduced by Church.[1] Given operands p, q, and r, which represent truth-valued propositions, the meaning of theconditioned 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 values of p, q, and r, the value of [p, q, r] is the value of p when qis 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” (?:) operator in many programming languages.In conjunction with truth constants denoting each truth-value, conditioned disjunction is truth-functionally completefor classical logic.[2] Its truth table is the following:There are other truth-functionally complete ternary connectives.

1.1 References[1] Church, Alonzo (1956). Introduction to Mathematical Logic. Princeton University Press.

[2] Wesselkamper, T., “A sole sufficient operator”, Notre Dame 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:

p⊂q, Bpq, or p←q

2.1 Definition

2.1.1 Truth table

The truth table of A⊂B

2.1.2 Venn diagram

The Venn diagram of “If B then A” (the white area shows where the statement is false)

2.2 Properties

truth-preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truthvalue of 'true' as a result of converse implication.

2.3 Symbol

2.4 Natural language

“Not q without p.”“p if q.”

2

2.5. BOOLEAN ALGEBRA 3

2.5 Boolean Algebra

(A + B')

2.6 See also• Logical connective

• Material implication

Chapter 3

Converse nonimplication

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

3.1 Definition

p ̸⊂qwhich is the same as ∼(p⊂q)

3.1.1 Truth table

The truth table of p ̸⊂q .[2]

3.1.2 Venn diagram

The Venn Diagram of “It is not the case that B implies A” (the red area is true)

3.2 Properties

falsehood-preserving: The interpretation under which all variables are assigned a truth value of 'false' produces atruth value of 'false' as a result of converse nonimplication

3.3 Symbol

Alternatives for p ̸⊂q are

• p←̃q : ←̃ combines Converse implication’s left arrow( ← ) with Negation’s tilde( ∼ ).

• Mpq : uses prefixed capital letter.

• p↚q : ↚ combines Converse implication’s left arrow( ← ) denied by means of a stroke( / ).

4

3.4. NATURAL LANGUAGE 5

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 is defined as q↚p=q′p .Example of a 2-element Boolean algebra: the 2 elements {0,1} with 0 as zero and 1 as unity element, operators ∼ ascomplement operator, ∨ as join operator and ∧ as meet operator, build the Boolean algebra of propositional logic.[4] Example of a 4-element Boolean algebra: the 4 divisors {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 algebra.

3.5.1 Properties

Non-associative

r↚(q↚p)=(r↚q)↚p iff rp=0[5] (In a two-element Boolean algebra the latter condition is reduced to r=0or p=0). Hence

in a nontrivial Boolean algebra Converse Nonimplication is nonassociative.

::

(r ↚ q) ↚ p = r′q ↚ p definition) (by= (r′q)′p definition) (by= (r + q′)p laws) Morgan's (De= (r + r′q′)p law) (Absorption= rp+ r′q′p

= rp+ r′(q ↚ p) definition) (by= rp+ r ↚ (q ↚ p) definition) (by

Clearly, it is associative iff rp=0 .

Non-commutative

• q↚p=p↚q iff q=p[6]. Hence Converse Nonimplication is noncommutative.

Neutral and absorbing elements

• 0 is a left neutral element ( 0↚p=p ) and a right absorbing element ( p↚0=0 ).

• 1↚p=0 , p↚1=p′ , and p↚p=0 .

• Implication q→p is the dual of Converse Nonimplication q↚p[7].

[6]

[7]

6 CHAPTER 3. CONVERSE NONIMPLICATION

3.6 Computer science

An example for converse nonimplication in computer science can be found when performing a right outer join on aset of tables from a database, if records not matching the join-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 Programming, 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 operation that outputs true only when both inputs differ (one istrue, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR (/ˌɛksˈɔr/), EOR,EXOR, ⊻, ⊕, ↮, and ≢. The opposite of XOR is logical biconditional, which outputs true only when both inputs arethe same.It gains the name “exclusive or” because the meaning of “or” is ambiguous when both operands are true; the exclusiveor operator excludes that case. This is sometimes thought of as “one or the other but not both”. This could be writtenas “A or B but not A and B”.More generally, XOR is true only when an odd number of inputs is true. A chain of XORs—a XOR b XOR c XORd (and so on)—is true whenever an odd number of the inputs are true and is false whenever an even number of inputsare true.

4.1 Truth table

The truth table of A XOR B shows that it outputs true whenever the inputs differ:0 = FALSE

1 = TRUE

4.2 Equivalencies, elimination, and introduction

Exclusive disjunction essentially means 'either one, but not both'. In other words, if and only if one is true, the othercannot be true. For example, one of the two horses will win the race, but not both of them. The exclusive disjunctionp⊕ q , or Jpq, can be expressed in terms of the 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 in the following way:

p⊕ q = (p ∧ ¬q) ∨ (¬p ∧ q)

This representation of XOR may be found useful when constructing a circuit or network, because it has only one ¬operation and small number of ∧ and ∨ operations. A proof of this identity is given below:

7

8 CHAPTER 4. EXCLUSIVE OR

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

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 p⊕ q in the following way:

p⊕ q = ¬((p ∧ q) ∨ (¬p ∧ ¬q))

This equivalence can be established by applying De Morgan’s laws twice to the fourth line of the above proof.The exclusive or is also equivalent to the negation of a logical biconditional, by the rules of material implication (amaterial conditional is equivalent to the disjunction of the negation of its antecedent and its consequence) and materialequivalence.In summary, we have, in mathematical and in engineering notation:

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 ALGEBRA 9

4.3 Relation to modern algebra

Although the operators ∧ (conjunction) and ∨ (disjunction) are very useful in logic systems, they fail a more gener-alizable structure in the following way:The systems ({T, F},∧) and ({T, F},∨) are monoids. This unfortunately prevents the combination of these twosystems into larger structures, such as a mathematical ring.However, the system using exclusive or ({T, F},⊕) is an abelian group. The combination of operators ∧ and ⊕over elements {T, F} produce the well-known field F2 . This field can represent any logic obtainable with the system(∧,∨) and has the added benefit of the arsenal of algebraic analysis tools for fields.More specifically, if one associates F with 0 and T with 1, one can interpret the logical “AND” operation as multi-plication on F2 and the “XOR” operation as addition on F2 :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 referred to as algebraic normal form.

4.4 Exclusive “or” in English

The Oxford English Dictionary explains “either ... or” as follows:

“The primary function of either, etc., is to emphasize the perfect indifference of the two (or more) thingsor courses ... ; but a secondary function is to emphasize the mutual exclusiveness, = either of the two,but not both.”[1]

The exclusive-or explicitly states “one or the other, but not neither nor both.” However, the mapping correspondencebetween formal Boolean operators and natural language conjunctions is far from simple or one-to-one, and has beenstudied for decades in linguistics and analytic philosophy.Following this kind of common-sense intuition about “or”, it is sometimes argued that in many natural languages,English included, the word “or” has an “exclusive” sense.[2] The exclusive disjunction of a pair of propositions, (p, q),is supposed to mean that p is true or q is true, but not both. For example, it might be argued that the normal intentionof a statement like “You may have coffee, or you may have tea” is to stipulate that exactly one of the conditions can betrue. Certainly under some circumstances a sentence like this example should be taken as forbidding the possibilityof one’s accepting both options. Even so, there is good reason to suppose that this sort of sentence is not disjunctiveat all. If all we know about some disjunction is that it is true overall, we cannot be sure that either of its disjunctsis true. For example, if a woman has been told that her friend is either at the snack bar or on the tennis court, shecannot validly infer that he is on the tennis court. But if her waiter tells her that she may have coffee or she may havetea, she can validly infer that she may have tea. Nothing classically thought of as a disjunction has this property. Thisis so even given that she might reasonably take her waiter as having denied her the possibility of having both coffeeand tea.(Note: If the waiter intends that choosing neither tea nor coffee is an option i.e. ordering nothing, the appropriateoperator is NAND: p NAND q.)In English, the construct “either ... or” is usually used to indicate exclusive or and “or” generally used for inclusive.But in Spanish, the word “o” (or) can be used in the form p o q (exclusive) or the form o p o q (inclusive). Some maycontend that any binary or other n-ary exclusive “or” is true if and only if it has an odd number of true inputs (this isnot, however, the only reasonable definition; for example, digital xor gates with multiple inputs typically do not usethat definition), and that there is no conjunction in English that has this general property. For example, Barrett andStenner contend in the 1971 article “The Myth of the Exclusive 'Or'" (Mind, 80 (317), 116–121) that no author hasproduced an example of an English or-sentence that appears to be false because both of its inputs are true, and brushoff or-sentences such as “The light bulb is either on or off” as reflecting particular facts about the world rather thanthe nature of the word “or”. However, the "barber paradox"—Everybody in town shaves himself or is shaved by thebarber, who shaves the barber? -- would not be paradoxical if “or” could not be exclusive (although a purist couldsay that “either” is required in the statement of the paradox).

10 CHAPTER 4. EXCLUSIVE OR

Whether these examples can be considered “natural language” is another question. Certainly when one sees a menustating “Lunch special: sandwich and soup or salad” (parsed as “sandwich and (soup or salad)" according to commonusage in the restaurant trade), one would not expect to be permitted to order both soup and salad. Nor would oneexpect to order neither soup nor salad, because that belies the nature of the “special”, that ordering the two itemstogether is cheaper than ordering them a la carte. Similarly, a lunch special consisting of one meat, French fries ormashed potatoes and vegetable would consist of three items, only one of which would be a form of potato. If onewanted to have meat and both kinds of potatoes, one would ask if it were possible to substitute a second order ofpotatoes for the vegetable. And, one would not expect to be permitted to have both types of potato and vegetable,because the result would be a vegetable plate rather than a meat plate.

4.5 Alternative symbols

The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on theproperties being emphasized in a given context of discussion. In addition to the abbreviation “XOR”, any of thefollowing symbols may also be seen:

• Aplus sign (+). This makes sensemathematically because exclusive disjunction corresponds to additionmodulo2, which has the following addition table, clearly isomorphic to the one above:

• The use of the plus sign has the added advantage that all of the ordinary algebraic properties of mathematicalrings and fields can be used without further ado. However, the plus sign is also used for Inclusive disjunctionin some notation systems.

• A plus sign that is modified in some way, such as being encircled (⊕ ). This usage faces the objection that thissame symbol is already used in mathematics for the direct sum of algebraic structures.

• A prefixed J, as in Jpq.

• An inclusive disjunction symbol ( ∨ ) that is modified in some way, such as being underlined ( ∨ ) or with dotabove ( ∨̇ ).

• In several programming languages, such as C, C++, C#, Java, Perl, Ruby and Python, a caret (^) is used todenote the bitwise XOR operator. This is not used outside of programming contexts because it is too easilyconfused with other uses of the caret.

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

• In IEC symbology, an exclusive or is marked "=1”.

4.6 Properties

Commutativity: yesAssociativity: yesDistributivity: The exclusive or doesn't distribute over any binary function (not even itself),but logical conjunction (see there) distributes over exclusive or.(Conjunction and exclusive or form the multiplication and addition operations of a field GF(2), and as in any fieldthey obey the distributive law.)Idempotency: no

Monotonicity: noTruth-preserving: noWhen all inputs are true, the output is not true.Falsehood-preserving: yesWhen all inputs are false, the output is false.

4.7. COMPUTER SCIENCE 11

Walsh spectrum: (2,0,0,−2)Non-linearity: 0 (the function is linear)If using binary values for true (1) and false (0), then exclusive or works exactly like addition modulo 2.

4.7 Computer science

AB

Q

Traditional symbolic representation of an XOR logic gate

4.7.1 Bitwise operation

Main article: Bitwise operationExclusive disjunction is often used for bitwise operations. Examples:

• 1 xor 1 = 0

• 1 xor 0 = 1

• 0 xor 1 = 1

• 0 xor 0 = 0

• 1110 xor 1001 = 0111 (this is equivalent to addition without carry)

As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of twon-bit strings is identical to the standard vector of addition in the vector space (Z/2Z)n .In computer science, exclusive disjunction has several uses:

• It tells whether two bits are unequal.

• It is an optional bit-flipper (the deciding input chooses whether to invert the data input).

• It tells whether there is an odd number of 1 bits (A⊕B⊕C⊕D⊕E is true iff an odd number of the variablesare true).

In logical circuits, a simple adder can be made with an XOR gate to add the numbers, and a series of AND, OR andNOT gates to create the carry output.On some computer architectures, it is more efficient to store a zero in a register by xor-ing the register with itself (bitsxor-ed with themselves are always zero) instead of loading and storing the value zero.

12 CHAPTER 4. EXCLUSIVE OR

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

In simple threshold activated neural networks, modeling the 'xor' function requires a second layer because 'xor' is nota linearly separable function.Exclusive-or is sometimes used as a simple mixing function in cryptography, for example, with one-time pad or Feistelnetwork systems.Similarly, XOR can be used in generating entropy pools for hardware random number generators. The XOR operationpreserves randomness, meaning that a random bit XORed with a non-random bit will result in a random bit. Multiplesources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteedto be at least as good as the best individual source.[3]

4.8. ENCODINGS 13

XOR is used in RAID 3–6 for creating parity information. For example, RAID can “back up” bytes 10011100 and01101100 from two (or more) hard drives by XORing the just mentioned bytes, resulting in (11110000) and writingit to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-createdby XORing bytes from the remaining drives. For instance, if the drive containing 01101100 is lost, 10011100 and11110000 can be XORed to recover the lost byte.XOR is also used to detect an overflow in the result of a signed binary arithmetic operation. If the leftmost retainedbit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORingthose two bits will give a “1” if there is an overflow.XOR can be used to swap two numeric variables in computers, using the XOR swap algorithm; however this isregarded as more of a curiosity and not encouraged in practice.XOR linked lists leverage XOR properties in order to save space to represent doubly linked list data structures.In computer graphics, XOR-based drawing methods are often used to manage such items as bounding boxes andcursors on systems without alpha channels or overlay planes.

4.8 Encodings

Apart from the ASCII codes, the operator is encoded at U+22BB ⊻ xor (HTML &#8891;) and U+2295 ⊕ circledplus (HTML &#8853; · &oplus;), both in block Mathematical 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 First Myth of'Or'":Jennings, R. E. (1994). The Genealogy of Disjunction. New York: Oxford University Press.

[3] Davies, Robert B (28 February 2002). “Exclusive OR (XOR) and hardware random number generators” (PDF). Retrieved28 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 nullary logical connective. In a truth-functional system of propositionallogic it is one of two postulated truth values, along with its negation, truth.[1] Usual notations of the false are 0(especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ⊥.[2]

Another approach is used for several formal theories (for example, 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 Boolean logic

Boolean logic defines the false in both senses mentioned above: “0” is a propositional constant, whose value bydefinition is 0. In a classical propositional calculus, depending on the chosen set of fundamental connectives, the falsemay or may not have a dedicated symbol. Such formulas as p ∧ ¬p and ¬(p → p) may be used instead.In both systems the negation of the truth gives false. The negation of false is equivalent to the truth not only in classicallogic and Boolean logic, but also in most other logical systems, as explained below.

5.2 False, negation and contradiction

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

¬p ⇔ (p → ⊥)

This is the definition of negation in some systems,[6] such as intuitionistic logic, and can be proven in propositionalcalculi where negation is a fundamental connective. Because p → p is usually a theorem or axiom, a consequence isthat the negation of false (¬ ⊥) is true.The contradiction is a statement which entails the false, i.e. φ ⊢ ⊥. Using the equivalence above, the fact that φ is acontradiction may be derived, for example, from ⊢ ¬φ. Contradiction and the false are sometimes not distinguished,especially due to Latin term falsum denoting both. Contradiction means a statement is proven to be false, but thefalse itself is a proposition which is defined to be opposite to the truth.Logical systems may or may not contain the principle of explosion (in Latin, ex falso quodlibet), ⊥ ⊢ φ.

5.3 Consistency

Main article: Consistency

14

5.4. SEE ALSO 15

A formal theory using "⊥" connective is defined to be consistent if and only if the false is not its theorem. Inthe absence of propositional constants, some substitutes such as mentioned above may be used instead to defineconsistency.

5.4 See also• Contradiction

• Logical truth

• Tautology (logic) (for symbolism of logical truth)

5.5 References[1] Jennifer Fisher, On the Philosophy of Logic, Thomson Wadsworth, 2007, ISBN 0-495-00888-5, p. 17.

[2] Willard Van Orman Quine, Methods of Logic, 4th ed, Harvard 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 Companion to Philosophical Logic, Continuum International PublishingGroup, 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, ISBN0-521-85433-4, p. 105.

[6] Dov M. Gabbay and Franz Guenthner (eds), Handbook of Philosophical Logic, Volume 6, 2nd ed, Springer, 2002, ISBN1-4020-0583-0, p. 12.

Chapter 6

Material equivalence

“Iff” redirects here. For other uses, see IFF (disambiguation)."↔" redirects here. It is not to be confused with Bidirectional traffic.

↔⇔≡Logical symbols representing iff

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logicalconnective between statements.In that it is biconditional, the connective can be likened to the 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 both statements are true, or both are false). It is controversialwhether the connective thus defined is properly rendered by the English “if and only if”, with its pre-existing meaning.There is nothing to stop one from stipulating that we may read this connective as “only if and if”, although this maylead to confusion.In writing, phrases commonly used, with debatable propriety, as alternatives to P “if and only if” Q include Q isnecessary and sufficient for P, P is equivalent (or materially equivalent) to Q (comparematerial implication), P preciselyif Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. Many authors regard “iff” as unsuitablein formal writing;[1] others use it freely.[2]

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

6.1 Definition

The truth table of p ↔ q is as follows:[3]

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

6.2 Usage

6.2.1 Notation

The corresponding logical symbols are "↔", "⇔" and "≡", and sometimes “iff”. These are usually treated as equiv-alent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositionallogic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is usedin reasoning about those logic formulas (e.g., in metalogic). In Łukasiewicz's notation, it is the prefix symbol 'E'.Another term for this logical connective is exclusive nor.

16

6.3. DISTINCTION FROM “IF” AND “ONLY IF” 17

6.2.2 Proofs

In most logical systems, one proves a statement of the form “P iff Q” by proving “if P, then Q” and “if Q, then P”.Proving this pair of statements sometimes leads to a more natural proof, since there are not obvious conditions inwhich one would infer a biconditional directly. An alternative is to prove the disjunction "(P and Q) or (not-P andnot-Q)", which itself can be inferred directly from either of its disjuncts—that is, because “iff” is truth-functional, “Piff Q” follows if P and Q have both been shown true, or both false.

6.2.3 Origin of iff

Usage of the abbreviation “iff” first appeared in print in John L. Kelley's 1955 bookGeneral Topology.[4] Its inventionis often credited to Paul Halmos, who wrote “I invented 'iff,' for 'if and only if'—but I could never believe I was reallyits first inventor.”[5]

6.3 Distinction from “if” and “only if”1. “Madison will eat the fruit if it is an apple.” (equivalent to “Only if Madison will eat the fruit, is it anapple;" or “Madison will eat the fruit← fruit is an apple”)

This states simply that Madison will eat fruits that are apples. It does not, however, exclude thepossibility that Madison might also eat bananas or other types of fruit. All that is known for certainis that she will eat any and all apples that she happens upon. That the fruit is an apple is a sufficientcondition for Madison to eat the fruit.

2. “Madison will eat the fruit only if it is an apple.” (equivalent to “If Madison will eat the fruit, then it isan apple” or “Madison will eat the fruit→ fruit is an apple”)

This states that the only fruit Madison will eat is an apple. It does not, however, exclude the possi-bility that Madison will refuse an apple if it is made available, in contrast with (1), which requiresMadison to eat any available apple. In this case, that a given fruit is an apple is a necessary conditionfor Madison to be eating it. It is not a sufficient condition since Madison might not eat all the applesshe is given.

3. “Madison will eat the fruit if and only if it is an apple” (equivalent to “Madison will eat the fruit↔ fruitis an apple”)

This statement makes it clear that Madison will eat all and only those fruits that are apples. Shewill not leave any apple uneaten, and she will not eat any other type of fruit. That a given fruit is anapple is both a necessary and a sufficient condition for Madison to eat the fruit.

Sufficiency is the inverse of necessity. That is to say, given P→Q (i.e. if P then Q), P would be a sufficient conditionfor Q, and Q would be a necessary condition for P. Also, given P→Q, it is true that ¬Q→¬P (where ¬ is the negationoperator, i.e. “not”). This means that the relationship between P and Q, established by P→Q, can be expressed in thefollowing, all equivalent, ways:

P is sufficient for QQ is necessary for P¬Q is sufficient for ¬P

¬P is necessary for ¬Q

As an example, take (1), above, which states P→Q, where P is “the fruit in question is an apple” and Q is “Madisonwill eat the fruit in question”. The following are four equivalent ways of expressing this very relationship:

If the fruit in question is an apple, then Madison 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, will Madison not eat it.

18 CHAPTER 6. MATERIAL EQUIVALENCE

So we see that (2), above, can be restated in the form of if...then as “If Madison will eat the fruit in question, then itis an apple"; taking this in conjunction with (1), we find that (3) can be stated as “If the fruit in question is an apple,then Madison will eat it; AND if Madison will eat the fruit, then it is an apple”.

6.4 More general usage

Iff is used outside the field of logic, wherever logic is applied, especially in mathematical discussions. It has the samemeaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficientfor the other. This is an example of mathematical jargon. (However, as noted above, if, rather than iff, is more oftenused in statements of definition.)The elements of X are all and only the elements of Y is used to mean: “for any z in the domain of discourse, z is inX if and only if z is in Y.”

6.5 See also• Covariance

• Logical biconditional

• Logical equality

• Necessary and sufficient condition

• Polysyllogism

6.6 Footnotes[1] E.g. Daepp, Ulrich; Gorkin, Pamela (2011), Reading, Writing, and Proving: A Closer Look at Mathematics, Undergraduate

Texts in Mathematics, Springer, p. 52, ISBN 9781441994790, While it can be a real time-saver, we don't recommend itin formal writing.

[2] Rothwell, Edward J.; Cloud, Michael J. (2014), Engineering Writing by Design: Creating Formal Documents of LastingValue, CRC Press, p. 98, ISBN 9781482234312, It is common in mathematical writing.

[3] p <=> q. Wolfram|Alpha

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

[5] Nicholas J. Higham (1998). Handbook of writing for the mathematical sciences (2nd ed.). SIAM. p. 24. ISBN 978-0-89871-420-3.

6.7 External links• Language Log: “Just in Case”

• Southern California Philosophy for philosophy graduate students: “Just in Case”

Chapter 7

Indicative conditional

In natural languages, an indicative conditional[1][2] is the logical operation given by statements of the form “If A thenB”. Unlike the material conditional, an indicative conditional does not have a stipulated definition. The philosophicalliterature on this operation is broad, and no clear consensus has been reached.

7.1 Distinctions between the material conditional and the indicative con-ditional

The material conditional does not always function in accordance with everyday if-then reasoning. Therefore thereare drawbacks with using the material conditional to represent if-then statements.One problem is that the material conditional allows implications to be true even when the antecedent is irrelevant tothe consequent. For example, it’s commonly accepted that the sun is made of gas, on one hand, and that 3 is a primenumber, on the other. The standard definition of implication allows us to conclude that, if the sun is made of gas,then 3 is a prime number. This is arguably synonymous to the following: the sun’s being made of gas makes 3 bea prime number. Many people intuitively think that this is false, because the sun and the number three simply havenothing to do with one another. Logicians have tried to 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 designed to deal with counterfactuals and other cases that peopleoften find in if-then reasoning. This has inspired people to develop modal logic.A further problem is that the material conditional is such that P AND ¬P→Q, regardless of what Q is taken to mean.That is, a contradiction implies that absolutely everything is true. Logicians concerned with this have tried to developparaconsistent logics.

7.2 Psychology and indicative conditionals

Most behavioral experiments on conditionals in the psychology of reasoning have been carried out with indicativeconditionals, causal conditionals, and counterfactual conditionals. People readily make the modus ponens inference,that is, given if A then B, and given A, they conclude B, but only about half of participants in experiments make themodus tollens inference, that is, given if A then B, and given not-B, only about half of participants conclude not-A,the remainder say that nothing follows (Evans et al., 1993). When participants are given counterfactual conditionals,they make both the modus ponens and the modus tollens inferences (Byrne, 2005).

7.3 See also

• Material conditional

19

20 CHAPTER 7. INDICATIVE 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 to Reality.Cambridge, MA: MIT Press.

• Edgington, Dorothy. (2006). “Conditionals”. The Stanford Encyclopedia of Philosophy, Edward Zalta (ed.).http://plato.stanford.edu/entries/conditionals/.

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

Chapter 8

Logical biconditional

In logic and mathematics, the logical biconditional (sometimes known as thematerial biconditional) is the logicalconnective of two statements asserting "p if and only if q", where q is an antecedent and p is a consequent.[1] This isoften abbreviated p iff q. The operator is denoted using a doubleheaded arrow (↔), a prefixed E (Epq), an equalitysign (=), an equivalence sign (≡), or EQV. It is logically equivalent to (p → q) ∧ (q → p), or the XNOR (exclusivenor) boolean operator. It is equivalent to "(not p or q) and (not q or p)". It is also logically equivalent to "(p and q) or(not p and not q)", meaning “both or neither”.The only difference from material conditional is the case when the hypothesis is false but the conclusion is true. Inthat case, in the conditional, the result is true, yet in the biconditional the result is false.In the conceptual interpretation, a = b means “All a 's are b 's and all b 's are a 's"; in other words, the sets a and bcoincide: they are identical. This does not mean that the concepts have the same meaning. Examples: “triangle” and“trilateral”, “equiangular trilateral” and “equilateral triangle”. The antecedent is the subject and the consequent is thepredicate of a universal affirmative proposition.In the propositional interpretation, a⇔ bmeans that a implies b and b implies a; in other words, that the propositionsare equivalent, that is to say, either true or false at the same time. This does not mean that they have the samemeaning.Example: “The triangle ABC has two equal sides”, and “The triangle ABC has two equal angles”. The antecedent isthe premise or the cause and the consequent is the consequence. When an implication is translated by a hypothetical(or conditional) judgment the antecedent is called the hypothesis (or the condition) and the consequent is called thethesis.A commonway of demonstrating a biconditional is to use its equivalence to the conjunction of two converse conditionals,demonstrating these separately.When both members of the biconditional are propositions, it can be separated into two conditionals, of which oneis called a theorem and the other its reciprocal. Thus whenever a theorem and its reciprocal are true we have abiconditional. A simple theorem gives rise to an implication whose antecedent is the hypothesis and whose consequentis the thesis of the theorem.It is often said that the hypothesis is the sufficient condition of the thesis, and the thesis the necessary condition ofthe hypothesis; that is to say, it is sufficient that the hypothesis be true for the thesis to be true; while it is necessarythat the thesis be true for the hypothesis to be true also. When a theorem and its reciprocal are true we say that itshypothesis is the necessary and sufficient condition of the thesis; that is to say, that it is at the same time both causeand consequence.

8.1 Definition

Logical equality (also known as biconditional) is an operation on two logical values, typically the values of twopropositions, that produces a value of true if and only if both operands are false or both operands are true.

21

22 CHAPTER 8. LOGICAL BICONDITIONAL

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 ambiguous:x1 ↔ x2 ↔ x3 ↔ ...↔ xn may be meant as (((x1 ↔ x2) ↔ x3) ↔ ...) ↔ xn ,or may be used to say that all xi are together true or together false: ( x1 ∧ ... ∧ xn ) ∨ (¬x1 ∧ ... ∧ ¬xn)Only for zero or two arguments this is the same.The following truth tables show the same bit pattern only in the line with no argument and in the lines with twoarguments:

x1 ↔ ... ↔ xn

meant as equivalent to¬ (¬x1 ⊕ ...⊕ ¬xn)The central Venn diagram below,and line (ABC ) in this matrixrepresent the same operation.

The left Venn diagram below, and the lines (AB ) in these matrices represent the same operation.

8.1.2 Venn diagrams

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

8.2. PROPERTIES 23

x1 ↔ ... ↔ xn

meant as shorthand for( x1 ∧ ... ∧ xn )∨ (¬x1 ∧ ... ∧ ¬xn)The Venn diagram directly below,and line (ABC ) in this matrixrepresent the same operation.

8.2 Properties

commutativity: yesassociativity: yesdistributivity: Biconditional doesn't distribute over any binary function (not even itself),but logical disjunction (see there) distributes over biconditional.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)

24 CHAPTER 8. LOGICAL BICONDITIONAL

8.3 Rules of inference

Main article: Rules of inference

Like all connectives in first-order logic, the biconditional has 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 B follows from A, and A follows from B, then A if and only ifB.For example, from the statements “if I'm breathing, then I'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 a conditional from a biconditional: if ( A ↔ B ) is true, then one mayinfer one direction of the biconditional, ( A → B ) and ( B → A ).For example, if it’s true that I'm breathing if and only if I'm alive, then it’s true that if I'm breathing, I'm alive; likewise,it’s true that if I'm alive, I'm breathing.Formally:( A ↔ B ) ∴ ( A → B )also( A ↔ B ) ∴ ( B → A )

8.4 Colloquial usage

One unambiguous way of stating a biconditional in plain English is of the form "b if a and a if b". Another is "aif and only if b". Slightly more formally, one could say "b implies a and a implies b". The plain English “if'" maysometimes be used as a biconditional. One must weigh context heavily.For example, “I'll buy you a new wallet if you need one” may be meant as a biconditional, since the speaker doesn'tintend a valid outcome to be buying the wallet whether or not 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 25

8.6 Notes[1] Handbook of Logic, page 81

8.7 References• Brennan, Joseph G. Handbook of Logic, 2nd Edition. Harper & Row. 1961

This article incorporates material from Biconditional on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

Chapter 9

Logical conjunction

"∧" redirects here. For the logic gate, see AND gate. For exterior product, see Exterior algebra.In logic and mathematics, and is the truth-functional operator of logical conjunction; the and of a set of operands is

Venn diagram of A∧B

true if and only if all of its operands are true. The logical connective that represents this operator is typically writtenas ∧ 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 fields are:

• In natural language, the coordinating conjunction “and”.

• In programming languages, the short-circuit and control structure.

26

9.1. NOTATION 27

Venn diagram of A∧B∧C

• In set theory, intersection.

• In predicate logic, universal quantification.

9.1 Notation

And is usually expressed with an infix operator: in mathematics and logic, ∧; in electronics, · ; and in programminglanguages,& or and. In Jan Łukasiewicz's prefix notation for logic, the operator is K, for Polish koniunkcja.[1]

9.2 Definition

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces avalue of true if and only if both of its operands are 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 defined as an operator or function of

28 CHAPTER 9. LOGICAL CONJUNCTION

arbitrary arity, the empty conjunction (AND-ing over an empty set of operands) is often defined as having the result1.

9.2.1 Truth table

Conjunctions of the arguments on the left — The true bits form a Sierpinski triangle.

The truth table of A ∧B :

9.3 Introduction and elimination rules

As a rule of inference, conjunction introduction is a classically valid, simple argument form. The argument form hastwo premises, A and B. Intuitively, it permits the inference of their conjunction.

A,B.Therefore, A and B.

or in logical operator notation:

A,

B

9.4. PROPERTIES 29

⊢ A ∧B

Here is an example of an argument that fits the form conjunction introduction:

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

Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference fromany conjunction of either element of that conjunction.

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 Properties

commutativity: yesassociativity: yesdistributivity: with various operations, especially with or

idempotency: 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), then logical conjunctionworks exactly like normal arithmeticmultiplication.

30 CHAPTER 9. LOGICAL CONJUNCTION

AND logic gate

9.5 Applications in computer engineering

In high-level computer programming and digital electronics, logical conjunction is commonly represented by an infixoperator, usually as a keyword such as “AND”, an algebraic multiplication, or the ampersand symbol "&". Manylanguages also provide short-circuit control structures 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 words viewed as bitstrings of equal length, by taking the bitwiseAND of each pair of bits at corresponding positions. For example:

• 11000110 AND 10100011 = 10000010.

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

9.6 Set-theoretic correspondence

The membership of an element of an intersection set in set theory is defined in terms of a logical conjunction: x ∈ A∩ B if and only if (x ∈ A) ∧ (x ∈ B). Through this correspondence, set-theoretic intersection shares several propertieswith logical conjunction, such as associativity, commutativity, and idempotence.

9.7 Natural language

As 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 conjunction. For example, “and” sometimes implies order. Forexample, “They got married and had a child” in common discourse means that the marriage came before the child.

9.8. SEE ALSO 31

The word “and” can also imply a partition of a thing into parts, as “The American flag is red, white, and blue.” Hereit is not meant that the flag is at once red, white, and blue, 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 Morgan’s laws

• First-order logic

• Fréchet inequalities

• Grammatical conjunction

• Logical disjunction

• Logical negation

• Logical graph

• Logical value

• Operation

• Peano-Russell notation

• Propositional calculus

9.9 References[1] Józef Maria Bocheński (1959), A Précis of Mathematical Logic, translated by Otto Bird from the French and German

editions, Dordrecht, North Holland: D. Reidel, passim.

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 discourse connective.For other logical symbols, see List of logic symbols.

In logic, a logical connective (also called a logical operator) is a symbol or word used to connect two or moresentences (of either a formal or a natural language) in a grammatically valid way, such that the sense of the compoundsentence produced depends only on the original sentences.The most common logical connectives are binary connectives (also called dyadic connectives) which join twosentences which can be thought of as the function’s operands. Also commonly, negation is considered to be a unaryconnective.Logical connectives along with quantifiers are the two main types of logical constants used in formal systems suchas propositional logic and predicate logic. Semantics of a logical connective is often, but not always, presented as atruth function.A logical connective is similar to but not equivalent to a conditional operator. [1]

10.1 In language

10.1.1 Natural language

In the grammar of natural languages two sentences may be joined by a grammatical conjunction to form a gram-matically compound sentence. Some but not all such grammatical conjunctions are truth functions. For example,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 the hill.D: Jack went up the hill so Jill went up the hill.

Thewords and and so are grammatical conjunctions joining the sentences (A) and (B) to form the compound sentences(C) and (D). The and in (C) is a logical connective, since the truth of (C) is completely determined by (A) and (B):it would make no sense to affirm (A) and (B) but deny (C). However, so in (D) is not a logical connective, since itwould be quite reasonable to affirm (A) and (B) but deny (D): perhaps, after all, Jill went up the hill to fetch a pail ofwater, not because Jack had gone up the hill at all.Various English words and word pairs express logical connectives, and some of them are synonymous. Examples(with the name of the relationship in parentheses) are:

• “and” (conjunction)

• “and then” (conjunction)

32

10.2. COMMON LOGICAL CONNECTIVES 33

• “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) alsoexpress a logical connective – even though they are applied to a single statement, and do not connect two statements.

10.1.2 Formal languages

In formal languages, truth functions are represented by unambiguous symbols. These symbols are called “logicalconnectives”, “logical operators”, “propositional operators”, or, in classical logic, "truth-functional connectives”. Seewell-formed formula for the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives.Logical connectives can be used to link more than two statements, so one can speak about "n-ary logical connective”.

10.2 Common logical connectives

10.2.1 List of common logical connectives

Commonly used logical connectives include

• Negation (not): ¬ , N (prefix), ~

• Conjunction (and): ∧ , K (prefix), & , ∙

• Disjunction (or): ∨ , A (prefix)

• Material implication (if...then): → , C (prefix),⇒ , ⊃

• Biconditional (if and only if): ↔ , E (prefix), ≡ , =

Alternative names for biconditional are “iff”, “xnor” and “bi-implication”.For example, the meaning of the statements it is raining and I am indoors is transformed when the two are combinedwith 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)

34 CHAPTER 10. LOGICAL CONNECTIVE

• 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 and the always false formula to be connective:

• True formula (⊤, 1, V [prefix], or T)• False formula (⊥, 0, O [prefix], or F)

10.2.2 History of notations

• Negation: the symbol ¬ appeared inHeyting in 1929.[2][3] (compare to Frege's symbol A in his Begriffsschrift);the symbol ~ appeared in Russell in 1908;[4] an alternative notation is to add an horizontal line on top of theformula, as in P ; another alternative notation is to use a prime symbol as in P'.

• Conjunction: the symbol ∧ appeared in Heyting in 1929[2] (compare to Peano's use of the set-theoretic notationof intersection ∩[5]); & appeared at least in Schönfinkel in 1924;[6] . comes from Boole's interpretation of logicas an elementary algebra.

• Disjunction: the symbol ∨ appeared in Russell in 1908[4] (compare to Peano's use of the set-theoretic notationof union ∪); the symbol + is also used, in spite of the ambiguity coming from the fact that the + of ordinaryelementary algebra is an exclusive or when interpreted logically in a two-element ring; punctually in the historya + together with a dot in the lower right corner has been used by Peirce,[7]

• Implication: the symbol → can be seen in Hilbert in 1917;[8] ⊃ was used by Russell in 1908[4] (compare toPeano’s inverted C notation);⇒ was used in Vax.[9]

• Biconditional: the symbol ≡ was used at least by Russell in 1908;[4]↔was used at least by Tarski in 1940;[10]⇔was used in Vax; other symbols appeared punctually in the history such as ⊃⊂ in Gentzen,[11] ~ in Schönfinkel[6]or ⊂⊃ in Chazal.[12]

• True: the symbol 1 comes from Boole's interpretation of logic as an elementary algebra over the two-elementBoolean algebra; other notations include

∧to be found in Peano.

• False: the symbol 0 comes also from Boole’s interpretation of logic as a ring; other notations include∨to be

found in Peano.

Some authors used letters for connectives at some time of the history: u. for conjunction (German’s “und” for“and”) and o. for disjunction (German’s “oder” for “or”) in earlier works by Hilbert (1904); Np for negation, Kpqfor conjunction, Dpq for alternative denial, Apq for disjunction, Xpq for joint denial, Cpq for implication, Epq forbiconditional in Łukasiewicz (1929);[13] cf. Polish notation.

10.2.3 Redundancy

Such logical connective as converse implication ← is actually the same as material conditional with swapped ar-guments, so the symbol for converse implication is redundant. In some logical calculi (notably, in classical logic)certain essentially different compound statements are logically equivalent. A less trivial example of a redundancy isthe classical equivalence between ¬P ∨ Q and P → Q. Therefore, a classical-based logical system does not need theconditional operator "→" if "¬" (not) and "∨" (or) are already in use, or may use the "→" only as a syntactic sugarfor a compound having one negation and one disjunction.There are sixteen Boolean functions associating the input truth values P and Q with four-digit binary outputs. Thesecorrespond to possible choices of binary logical connectives for classical logic. Different implementation of classicallogic can choose different functionally complete subsets of connectives.One approach is to choose a minimal set, and define other connectives by some logical form, like in the example withmaterial conditional above. The following are the minimal functionally complete sets of operators in classical logicwhose arities do not exceed 2:

10.3. PROPERTIES 35

One element {↑}, {↓}.Two elements { ∨ , ¬}, { ∧ , ¬}, {→, ¬}, {←, ¬}, {→, ⊥ }, {←, ⊥ }, {→, ↮ }, {←, ↮ }, {→, ↛ }, {→, ↚ },

{←,↛ }, {←,↚ }, {↛ , ¬}, {↚ , ¬}, {↛ , ⊤ }, {↚ , ⊤ }, {↛ ,↔ }, {↚ ,↔ }.Three elements { ∨ ,↔ , ⊥ }, { ∨ ,↔ ,↮ }, { ∨ ,↮ , ⊤ }, { ∧ ,↔ , ⊥ }, { ∧ ,↔ ,↮ }, { ∧ ,↮ , ⊤ }.

See more details about functional completeness in classical logic at Functional completeness in truth function.Another approach is to use on equal rights connectives of a certain convenient and functionally complete, but notminimal set. This approach requires more propositional axioms and each equivalence between logical forms must beeither an axiom or provable as a theorem.But intuitionistic logic has the situation more complicated. Of its five connectives {∧, ∨, →, ¬, ⊥} only negation ¬has to be reduced to other connectives (see details). Neither of conjunction, disjunction and material conditional hasan equivalent form constructed of other four logical connectives.

10.3 Properties

Some logical connectives possess properties whichmay be expressed in the theorems containing the connective. Someof those properties that a logical connective may have are:

• Associativity: Within an expression containing two or more of the same associative connectives in a row, theorder of the operations does not matter as long as the sequence of the operands is not changed.

• Commutativity: The operands of the connective may be swapped preserving logical equivalence to the originalexpression.

• Distributivity: A connective denoted by · distributes 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 operation are the same, the compound is logically equivalent tothe operand.

• Absorption: A pair of connectives ∧ , ∨ satisfies the absorption law if a ∧ (a ∨ b) = a for all operands a, b.• Monotonicity: If f(a1, ..., an) ≤ f(b1, ..., bn) for all a1, ..., an, b1, ..., bn ∈ {0,1} such that a1 ≤ b1, a2 ≤ b2,..., an ≤ bn. E.g., ∨ , ∧ , ⊤ , ⊥ .

• Affinity: Each variable always makes a difference in the truth-value of the operation or it never makes adifference. E.g., ¬ ,↔ ,↮ , ⊤ , ⊥ .

• Duality: To read the truth-value assignments for the operation from top to bottom on its truth table is the sameas taking the complement of reading the table of the same or another connective from bottom to top. Withoutresorting to truth tables it may be formulated as g̃ (¬a1, ..., ¬an) = ¬g(a1, ..., an). E.g., ¬ .

• Truth-preserving: The compound all those argument are tautologies is a tautology itself. E.g., ∨ , ∧ , ⊤ ,→,↔ , ⊂. (see validity)

• Falsehood-preserving: The compound all those argument 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 "…←…" forlogical compounds can be both proved as theorems, and the "≤" symbol means that "…→…" for logical compoundsis a consequence of corresponding "…→…" connectives for propositional variables. Some many-valued logics mayhave incompatible definitions of equivalence and order (entailment).Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties ofmany-valued logic and intuitionistic logic. The same is true about distributivity of conjunction over disjunction anddisjunction over conjunction, as well as for the absorption law.In classical logic and some varieties of many-valued logic, conjunction and disjunction are dual, and negation isself-dual, the latter is also self-dual in intuitionistic logic.

36 CHAPTER 10. LOGICAL CONNECTIVE

10.4 Order of precedence

As a way of reducing the number of necessary parentheses, 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 precedence of logical operators.[14]

However not all authors use the same order; for instance, an ordering in which disjunction is lower precedence thanimplication or bi-implication has also been used.[15] Sometimes precedence between conjunction and disjunction isunspecified requiring to provide it explicitly in given formula with parentheses. The order of precedence determineswhich connective is the “main connective” when interpreting a non-atomic formula.

10.5 Computer science

A truth-functional approach to logical operators is implemented as logic gates in digital circuits. Practically all digitalcircuits (the major exception is DRAM) are built up from NAND, NOR, NOT, and transmission gates; see moredetails in Truth function in computer science. Logical operators over bit vectors (corresponding to finite Booleanalgebras) are bitwise operations.But not every usage of a logical connective in computer programming has a Boolean semantic. For example, lazyevaluation is sometimes implemented for P ∧ Q and P ∨ Q, so these connectives are not commutative if some ofexpressions P, Q has side effects. Also, a conditional, which in some sense corresponds to the material conditionalconnective, is essentially non-Boolean because for if (P) then Q; the consequent Q is not executed if the antecedentP is false (although a compound as a whole is successful ≈ “true” in such case). This is closer to intuitionist andconstructivist views on the material conditional, rather than to classical logic’s ones.

10.6 See also

10.7 Notes[1] Cogwheel. “What is the difference between logical and conditional /operator/". Stack Overflow. Retrieved 9 April 2015.

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

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

[4] Russell (1908) Mathematical logic as based on the theory of types (American Journal of Mathematics 30, p222–262, alsoin From Frege to Gödel edited by van Heijenoort).

[5] Peano (1889) Arithmetices principia, nova methodo exposita.

[6] Schönfinkel (1924) Über die Bausteine der mathematischen Logik, translated as On the building blocks of mathematicallogic in From Frege to Gödel edited by van Heijenoort.

[7] Peirce (1867) On an improvement in Boole’s calculus of logic.

[8] Hilbert (1917/1918) Prinzipien der Mathematik (Bernays’ course notes).

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

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

[11] Gentzen (1934) Untersuchungen über das logische Schließen.

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

[13] See Roegel

10.8. REFERENCES 37

[14] O'Donnell, John; Hall, Cordelia; Page, Rex (2007), Discrete Mathematics Using a Computer, Springer, p. 120, ISBN9781846285981.

[15] Jackson, Daniel (2012), Software Abstractions: Logic, Language, and Analysis, MIT Press, p. 263, ISBN 9780262017152.

10.8 References• Bocheński, Józef Maria (1959), A Précis of Mathematical Logic, translated from the French and German edi-tions by Otto Bird, D. Reidel, Dordrecht, South Holland.

• Enderton, Herbert (2001), A Mathematical Introduction to Logic (2nd ed.), Boston, MA: Academic Press,ISBN 978-0-12-238452-3

• Gamut, L.T.F (1991), “Chapter 2”, Logic, Language and Meaning 1, University of Chicago Press, pp. 54–64,OCLC 21372380

• Rautenberg, W. (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: Springer Sci-ence+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6.

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

10.10 External links• Hazewinkel, Michiel, ed. (2001), “Propositional connective”, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Lloyd Humberstone (2010), "Sentence Connectives in Formal Logic", Stanford Encyclopedia of Philosophy(An abstract algebraic logic approach to connectives.)

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

Chapter 11

Logical disjunction

“Disjunction” redirects here. For the logic gate, see OR gate. For separation of chromosomes, see Meiosis. Fordisjunctions in distribution, see Disjunct distribution.In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation;

Venn diagram of A∨B

the or of a set of operands is true if and only if one or more of its operands is true. The logical connective thatrepresents this operator is typically written as ∨ or + ."A or B" is true if A is true, or if B is true, or if both A and B are true.In logic, or by itself means the inclusive or, distinguished from an exclusive or, which is false when both of itsarguments are true, while an “or” is true in that case.An operand of a disjunction is called a disjunct.Related concepts in other fields are:

38

11.1. NOTATION 39

Venn diagram of A∨B∨C

• In natural language, the coordinating conjunction “or”.

• In programming languages, the short-circuit or control structure.

• In set theory, union.

• In predicate logic, existential quantification.

11.1 Notation

Or is usually expressed with an infix operator: in mathematics and logic, ∨; in electronics, +; and in programminglanguages, | or or. In Jan Łukasiewicz's prefix notation for logic, the operator is A, for Polish alternatywa.[1]

11.2 Definition

Logical disjunction is an operation on two logical values, typically the values of two propositions, that has a valueof false if and only if both of its operands are false. More generally, a disjunction is a logical formula that can have

40 CHAPTER 11. LOGICAL DISJUNCTION

one or more literals separated only by ORs. A single literal is often considered to be a degenerate disjunction.The disjunctive identity is false, which is to say that the or of an expression with false has the same value as the originalexpression. In keeping with the concept of vacuous truth, when disjunction is defined as an operator or function ofarbitrary arity, the empty disjunction (OR-ing over an empty set of operands) is generally defined as false.

11.2.1 Truth table

Disjunctions of the arguments on the left — The false bits form a Sierpinski triangle.

The truth table of A ∨B :

11.3 Properties

• Commutativity

• Associativity

• Distributivity with various operations, especially with and

• Idempotency

• Monotonicity

• Truth-preserving validity

11.4. SYMBOL 41

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 logical disjunction works almost like binary addition. The onlydifference is that 1 ∨ 1 = 1 , while 1 + 1 = 10 .

11.4 Symbol

The mathematical symbol for logical disjunction varies in the literature. In addition to the word “or”, and the formula“Apq", the symbol " ∨ ", deriving from the Latin word vel (“either”, “or”) is commonly used for disjunction. Forexample: "A ∨ B " is read as "A or B ". Such a disjunction is false if both A and B are false. In all other cases it istrue.All of the following are disjunctions:

A ∨B

¬A ∨B

A ∨ ¬B ∨ ¬C ∨D ∨ ¬E.

The corresponding operation in set theory is the set-theoretic union.

11.5 Applications in computer science

AB

out

OR logic gate

Operators corresponding to logical disjunction exist in most programming languages.

11.5.1 Bitwise operation

Disjunction is often used for bitwise operations. Examples:

42 CHAPTER 11. LOGICAL DISJUNCTION

• 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 field to 1, by or-ing the field with a constant field with the relevant bitsset to 1. For example, x = x | 0b00000001 will force the final bit to 1 while leaving other bits unchanged.

11.5.2 Logical operation

Many languages distinguish between bitwise and logical disjunction by providing two distinct operators; in languagesfollowing C, bitwise disjunction is performed with the single pipe (|) and logical disjunction with the double pipe (||)operators.Logical disjunction is usually short-circuited; that is, if the first (left) operand evaluates to true then the second (right)operand is not evaluated. The logical disjunction operator thus usually constitutes a sequence point.In a parallel (concurrent) language, it is possible to short-circuit both sides: they are evaluated in parallel, and if oneterminates with value true, the other is interrupted. This operator is thus called the parallel or.Although in most languages the type of a logical disjunction expression is boolean and thus can only have the valuetrue or false, in some (such as Python and JavaScript) the logical disjunction operator returns one of its operands: thefirst operand if it evaluates to a true value, and the second operand otherwise.

11.5.3 Constructive disjunction

The Curry–Howard correspondence relates a constructivist form of disjunction to tagged union types.

11.6 Union

The membership of an element of an union set in set theory is defined in terms of a logical disjunction: x ∈ A ∪ B ifand only if (x ∈ A) ∨ (x ∈ B). Because of this, logical disjunction satisfies many of the same identities as set-theoreticunion, such as associativity, commutativity, distributivity, and de Morgan’s laws.

11.7 Natural language

Aswith other notions formalized in mathematical logic, the meaning of the natural-language coordinating conjunctionor is closely related to, but different from the logical or. For example, “Please ring me or send an email” likely means“do one or the other, but not both”. On the other hand, “Her grades are so good that she’s either very bright or studieshard” does not exclude the possibility of both. In other words, in ordinary language “or” can mean the inclusive orexclusive or.

11.8 See also

11.9 Notes• George Boole, closely following analogy with ordinary mathematics, premised, as a necessary condition tothe definition of “x + y”, that x and y were mutually exclusive. Jevons, and practically all mathematical logi-cians after him, advocated, on various grounds, the definition of “logical addition” in a form which does notnecessitate mutual exclusiveness.

11.10. EXTERNAL LINKS 43

11.10 External links• Hazewinkel, Michiel, ed. (2001), “Disjunction”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Stanford Encyclopedia of Philosophy entry

• Eric W. Weisstein. “Disjunction.” From MathWorld--A Wolfram Web Resource

11.11 References[1] Józef Maria Bocheński (1959), A Précis of Mathematical Logic, translated by Otto Bird from the French and German

editions, Dordrecht, North Holland: D. Reidel, passim.

Chapter 12

Logical equality

For the corresponding concept in combinational logic, see XNOR gate.Logical equality is a logical operator that corresponds to equality in Boolean algebra and to the logical biconditional

AB

Q

XNOR Logic Gate Symbol

in propositional calculus. It gives the functional value true if both functional arguments have the same logical value,and false if they are different.It is customary practice in various applications, if not always technically precise, to indicate the operation of logicalequality on the logical operands x and y by any of the following forms:x↔ y x⇔ y Exyx EQ y x = y

Some logicians, however, draw a firm distinction between a functional form, like those in the lefthand column, whichthey interpret as an application of a function to a pair of arguments — and thus a mere indication that the value ofthe compound expression depends on the values of the component expressions — and an equational form, like thosein the righthand column, which they interpret as an assertion that the arguments have equal values, in other words,that the functional value of the compound expression is true.In mathematics, the plus sign "+" almost invariably indicates an operation that satisfies the axioms assigned to additionin the type of algebraic structure that is known as a field. For boolean algebra, this means that the logical operationsignified by "+" is not the same as the inclusive disjunction signified by "∨" but is actually equivalent to the logicalinequality operator signified by "≠", or what amounts to the same thing, the exclusive disjunction signified by “XOR”.Naturally, these variations in usage have caused some failures to communicate between mathematicians and switchingengineers over the years. At any rate, one has the following array of corresponding forms for the symbols associatedwith logical inequality:

44

12.1. DEFINITION 45

x+ y x ̸≡ y Jxyx XOR y x ̸= y

This explains why “EQ” is often called "XNOR" in the combinational logic of circuit engineers, since it is the Nega-tion of the XOR operation; NXOR is a less commonly used alternative.[1] Another rationalization of the admittedlycircuitous name “XNOR” is that one begins with the “both false” operator NOR and then adds the eXception, “orboth true”.

12.1 Definition

Logical equality is an operation on two logical values, typically the values of two propositions, that produces a valueof true if and only if both operands are false or both operands are true.The truth table of p EQ q (also written as p = q, p ↔ q, or p ≡ q, or p == q) is as follows:

The Venn diagram of A EQ B (red part is true)

12.2 Alternative descriptions

The form (x = y) is equivalent to the form (x ∧ y) ∨ (¬x ∧ ¬y).(x = y) = ¬(x⊕ y) = ¬x⊕ y = x⊕ ¬y = (x ∧ y) ∨ (¬x ∧ ¬y) = (¬x ∨ y) ∧ (x ∨ ¬y)For the operands x and y, the truth table of the logical equality operator is as follows:

12.3 See also• Boolean function

46 CHAPTER 12. LOGICAL EQUALITY

• If and only if

• Logical equivalence

• Logical biconditional

• Propositional calculus

12.4 References[1] Keeton, Brian; Cavaness, Chuck; Friesen, Geoff (2001), Using Java 2, Que Publishing, p. 112, ISBN 9780789724687.

12.5 External links• Mathworld, XNOR

Chapter 13

Logical NOR

This article is about NOR in the logical sense. For the electronic gate, see NOR gate. For other uses, see Nor.“Peirce arrow” redirects here. It is not to be confused with Pierce-Arrow, an automobile manufacturer.In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation

Venn diagram of A ↓ B(nor part in red)

of logical or. That is, a sentence of the form (p NOR q) is true precisely when neither p nor q is true—i.e. when bothof p and q are false. In grammar, nor is a coordinating conjunction.The NOR operator is also known as Peirce’s arrow—Charles Sanders Peirce introduced the symbol ↓ for it,[1] anddemonstrated that the logical NOR is completely expressible: by combining uses of the logical NOR it is possibleto express any logical operation on two variables. Thus, as with its dual, the NAND operator (a.k.a. the Shefferstroke — symbolized as either | or /), NOR can be used by itself, without any other logical operator, to constitute alogical formal system (making NOR functionally complete). It is also known as Quine's dagger (his symbol was †),the ampheck (from Greek αμφηκης, cutting both ways; compare amphi-) by Peirce,[2] or “neither-nor”.

47

48 CHAPTER 13. LOGICAL NOR

One way of expressing p NOR q is p ∨ q , where the symbol ∨ signifies OR and the bar signifies the negation of theexpression under it: in essence, simply ¬(p ∨ q) . Other ways of expressing p NOR q are Xpq, and p+ q .The computer used in the spacecraft that first carried humans to the moon, the Apollo Guidance Computer, wasconstructed entirely using NOR gates with three inputs.[3]

13.1 Definition

TheNORoperation is a logical operation on two logical values, typically the values of two propositions, that producesa value of true if and only if both operands are false. In other words, it produces a value of false if and only if at leastone operand is true.

13.1.1 Truth table

The truth table of A NOR B (also written as A ↓ B) is as follows:

13.2 Properties

Logical NOR does not possess any of the five qualities (truth-preserving, false-preserving, linear, monotonic, self-dual) required to be absent from at least one member of a set of functionally complete operators. Thus, the setcontaining only NOR suffices as a complete set.

13.3 Introduction, elimination, and equivalencies

NOR has the interesting feature that all other logical operators can be expressed by interlaced NOR operations. Thelogical NAND operator also has this ability.The logical NOR ↓ is the negation of the disjunction:Expressed in terms of NOR ↓ , the usual operators of propositional logic are:

13.4 See also

13.5 References[1] Hans Kleine Büning; Theodor Lettmann (1999). Propositional logic: deduction and algorithms. Cambridge University

Press. p. 2. ISBN 978-0-521-63017-7.

[2] C.S. Peirce, CP 4.264

[3] Hall, Eldon C. (1996), Journey to the Moon: The History of the Apollo Guidance Computer, Reston, Virginia, USA: AIAA,p. 196, ISBN 1-56347-185-X

Chapter 14

Material conditional

“Logical conditional” redirects here. For other related meanings, see Conditional statement.Not to be confused with material inference.Thematerial conditional (also known as "material implication", "material consequence", or simply "implication",

Venn diagram of A → B .If a member of the set described by this diagram (the red areas) is a member of A , it is in the intersection of A and B , and ittherefore is also in B .

"implies" or "conditional") is a logical connective (or a binary operator) that is often symbolized by a forward ar-row "→". The material conditional is used to form statements of the form "p→q" (termed a conditional statement)which is read as “if p then q” or “p only if q” and conventionally compared to the English construction “If...then...”.But unlike the English construction, the material conditional statement "p→q" does not specify a causal relationshipbetween p and q and is to be understood to mean “if p is true, then q is also true” such that the statement "p→q"is false only when p is true and q is false.[1] Intuitively, consider that a given p being true and q being false wouldprove an “if p is true, q is always also true” statement false, even when the “if p then q” does not represent a causal

49

50 CHAPTER 14. MATERIAL CONDITIONAL

relationship between p and q. Instead, the statement describes p and q as each only being true when the other istrue, and makes no claims that p causes q. However, note that such a general and informal way of thinking aboutthe material conditional is not always acceptable, as will be discussed. As such, the material conditional is also to bedistinguished from logical consequence.The material conditional is also symbolized using:

1. p ⊃ q (Although this symbol may be used for the superset symbol in set theory.);2. p ⇒ q (Although this symbol is often used for logical consequence (i.e. logical implication) rather than for

material conditional.)

With respect to the material conditionals above, p is termed the antecedent, and q the consequent of the conditional.Conditional statements may be nested such that either or both of the antecedent or the consequent may themselvesbe conditional statements. In the example "(p→q) → (r→s)" both the antecedent and the consequent are conditionalstatements.In classical logic p→ q is logically equivalent to ¬(p ∧ ¬q) and by De Morgan’s Law logically equivalent to ¬p ∨ q.[2] Whereas, in minimal logic (and therefore also intuitionistic logic) p→ q only logically entails ¬(p∧¬q) ; and inintuitionistic logic (but not minimal logic) ¬p ∨ q entails p→ q .

14.1 Definitions of the material conditional

Logicians have many different views on the nature of material implication and approaches to explain its sense.[3]

14.1.1 As a truth function

In classical logic, the compound p→q is logically equivalent to the negative compound: not both p and not q. Thusthe compound p→q is false if and only if both p is true and q is false. By the same stroke, p→q is true if and only ifeither p is false or q is true (or both). Thus → is a function from pairs of truth values of the components p, q to truthvalues of the compound p→q, whose truth value is entirely a function of the truth values of the components. Hence,this interpretation is called truth-functional. The compound p→q is logically equivalent also to ¬p∨q (either not p, orq (or both)), and to ¬q→¬p (if not q then not p). But it is not equivalent to ¬p→¬q, which is equivalent to q→p.

Truth table

The truth table associated with the material conditional p→q is identical to that of ¬p∨q and is also denoted by Cpq.It is as follows:It may also be useful to note that in Boolean algebra, true and false can be denoted as 1 and 0 respectively with anequivalent table.

14.1.2 As a formal connective

The material conditional can be considered as a symbol of a formal theory, taken as a set of sentences, satisfying allthe classical inferences involving →, in particular the following characteristic rules:

1. Modus ponens;2. Conditional proof;3. Classical contraposition;4. Classical reductio ad absurdum.

Unlike the truth-functional one, this approach to logical connectives permits the examination of structurally identi-cal propositional forms in various logical systems, where somewhat different properties may be demonstrated. Forexample, in intuitionistic logic which rejects proofs by contraposition as valid rules of inference, (p → q) ⇒ ¬p ∨ qis not a propositional theorem, but the material conditional is used to define negation.

14.2. FORMAL PROPERTIES 51

14.2 Formal properties

When studying logic formally, the material conditional is distinguished from the semantic consequence relation |= .We say A |= B if every interpretation that makes A true also makes B true. However, there is a close relationshipbetween the two in most logics, including classical logic. For example, the following principles hold:

• If Γ |= ψ then ∅ |= (φ1 ∧ · · · ∧ φn → ψ) for some φ1, . . . , φn ∈ Γ . (This is a particular form of thededuction theorem. In words, it says that if Γ models ψ this means that ψ can be deduced just from somesubset of the theorems in Γ.)

• The converse of the above

• Both→ and |= are monotonic; i.e., if Γ |= ψ then ∆ ∪ Γ |= ψ , and if φ → ψ then (φ ∧ α) → ψ for any α,Δ. (In terms of structural rules, this is often referred to as weakening or thinning.)

These principles do not hold in all logics, however. Obviously they do not hold in non-monotonic logics, nor do theyhold in relevance logics.Other properties of implication (the following expressions are always true, for any logical values of variables):

• distributivity: (s→ (p→ q)) → ((s→ p) → (s→ q))

• transitivity: (a→ b) → ((b→ c) → (a→ c))

• reflexivity: a→ a

• totality: (a→ b) ∨ (b→ a)

• truth preserving: The interpretation under which all variables are assigned a truth value of 'true' produces atruth value of 'true' as a result of material implication.

• commutativity of antecedents: (a→ (b→ c)) ≡ (b→ (a→ c))

Note that a → (b → c) is logically equivalent to (a ∧ b) → c ; this property is sometimes called un/currying.Because of these properties, it is convenient to adopt a right-associative notation for → where a → b → c denotesa→ (b→ c) .Comparison of Boolean truth tables shows that a→ b is equivalent to ¬a ∨ b , and one is an equivalent replacementfor the other in classical logic. See material implication (rule of inference).

14.3 Philosophical problems with material conditional

Outside of mathematics, it is a matter of some controversy as to whether the truth function for material implica-tion provides an adequate treatment of conditional statements in English (a sentence in the indicative mood with aconditional clause attached, i.e., an indicative conditional, or false-to-fact sentences in the subjunctive mood, i.e., acounterfactual conditional).[4] That is to say, critics argue that in some non-mathematical cases, the truth value ofa compound statement, “if p then q", is not adequately determined by the truth values of p and q.[4] Examples ofnon-truth-functional statements include: "q because p", "p before q" and “it is possible that p".[4] “[Of] the sixteenpossible truth-functions of A and B, material implication is the only serious candidate. First, it is uncontroversial thatwhen A is true and B is false, “If A, B" is false. A basic rule of inference is modus ponens: from “If A, B" and A, wecan infer B. If it were possible to have A true, B false and “If A, B" true, this inference would be invalid. Second, it isuncontroversial that “If A, B" is sometimes true when A and B are respectively (true, true), or (false, true), or (false,false)… Non-truth-functional accounts agree that “If A, B" is false when A is true and B is false; and they agree thatthe conditional is sometimes true for the other three combinations of truth-values for the components; but they denythat the conditional is always true in each of these three cases. Some agree with the truth-functionalist that when Aand B are both true, “If A, B" must be true. Some do not, demanding a further relation between the facts that A andthat B.”[4]

52 CHAPTER 14. MATERIAL CONDITIONAL

The truth-functional theory of the conditional was integral to Frege's new logic (1879). It was takenup enthusiastically by Russell (who called it “material implication”), Wittgenstein in the Tractatus, andthe logical positivists, and it is now found in every logic text. It is the first theory of conditionals whichstudents encounter. Typically, it does not strike students as obviously correct. It is logic’s first surprise.Yet, as the textbooks testify, it does a creditable job in many circumstances. And it has many defenders.It is a strikingly simple theory: “If A, B" is false when A is true and B is false. In all other cases, “If A,B" is true. It is thus equivalent to "~(A&~B)" and to "~A or B". "A ⊃ B" has, by stipulation, these truthconditions.

— Dorothy Edgington, The Stanford Encyclopedia of Philosophy, “Conditionals”[4]

The meaning of the material conditional can sometimes be used in the natural language English “if condition thenconsequence" construction (a kind of conditional sentence), where condition and consequence are to be filled withEnglish sentences. However, this construction also implies a “reasonable” connection between the condition (protasis)and consequence (apodosis) (see Connexive logic).The material conditional can yield some unexpected truths when expressed in natural language. For example, anymaterial conditional statement with a false antecedent is true (see vacuous truth). So the statement “if 2 is odd then 2is even” is true. Similarly, any material conditional with a true consequent is true. So the statement “if I have a pennyin my pocket then Paris is in France” is always true, regardless of whether or not there is a penny in my pocket. Theseproblems are known as the paradoxes of material implication, though they are not really paradoxes in the strict sense;that is, they do not elicit logical contradictions. These unexpected truths arise because speakers of English (and othernatural languages) are tempted to equivocate between the material conditional and the indicative conditional, or otherconditional statements, like the counterfactual conditional and the material biconditional. It is not surprising that arigorously defined truth-functional operator does not correspond exactly to all notions of implication or otherwiseexpressed by 'if...then...' sentences in English (or their equivalents in other natural languages). For an overview ofsome the various analyses, formal and informal, of conditionals, see the “References” section below.

14.4 See also

14.4.1 Conditionals

• Counterfactual conditional

• Indicative conditional

• Corresponding conditional

• Strict conditional

14.5 References[1] Magnus, P.D (January 6, 2012). “forallx: An Introduction to Formal Logic” (PDF). Creative Commons. p. 25. Retrieved

28 May 2013.

[2] Teller, Paul (January 10, 1989). “A Modern Formal Logic Primer: Sentence Logic Volume 1” (PDF). Prentice Hall. p.54. Retrieved 28 May 2013.

[3] Clarke, Matthew C. (March 1996). “A Comparison of Techniques for Introducing Material Implication”. Cornell Univer-sity. Retrieved March 4, 2012.

[4] Edgington, Dorothy (2008). Edward N. Zalta, ed. “Conditionals”. The Stanford Encyclopedia of Philosophy (Winter 2008ed.).

14.6 Further reading• Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, KluwerAcademic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

14.7. EXTERNAL LINKS 53

• Edgington, Dorothy (2001), “Conditionals”, in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic,Blackwell.

• Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, HarvardUniversity Press, Cambridge, MA.

• Stalnaker, Robert, “Indicative Conditionals”, Philosophia, 5 (1975): 269–286.

14.7 External links• Conditionals entry by Edgington, Dorothy in the Stanford Encyclopedia of Philosophy

Chapter 15

Material equivalence

“Iff” redirects here. For other uses, see IFF (disambiguation)."↔" redirects here. It is not to be confused with Bidirectional traffic.

↔⇔≡Logical symbols representing iff

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logicalconnective between statements.In that it is biconditional, the connective can be likened to the 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 both statements are true, or both are false). It is controversialwhether the connective thus defined is properly rendered by the English “if and only if”, with its pre-existing meaning.There is nothing to stop one from stipulating that we may read this connective as “only if and if”, although this maylead to confusion.In writing, phrases commonly used, with debatable propriety, as alternatives to P “if and only if” Q include Q isnecessary and sufficient for P, P is equivalent (or materially equivalent) to Q (comparematerial implication), P preciselyif Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. Many authors regard “iff” as unsuitablein formal writing;[1] others use it freely.[2]

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

15.1 Definition

The truth table of p ↔ q is as follows:[3]

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

15.2 Usage

15.2.1 Notation

The corresponding logical symbols are "↔", "⇔" and "≡", and sometimes “iff”. These are usually treated as equiv-alent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositionallogic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is usedin reasoning about those logic formulas (e.g., in metalogic). In Łukasiewicz's notation, it is the prefix symbol 'E'.Another term for this logical connective is exclusive nor.

54

15.3. DISTINCTION FROM “IF” AND “ONLY IF” 55

15.2.2 Proofs

In most logical systems, one proves a statement of the form “P iff Q” by proving “if P, then Q” and “if Q, then P”.Proving this pair of statements sometimes leads to a more natural proof, since there are not obvious conditions inwhich one would infer a biconditional directly. An alternative is to prove the disjunction "(P and Q) or (not-P andnot-Q)", which itself can be inferred directly from either of its disjuncts—that is, because “iff” is truth-functional, “Piff Q” follows if P and Q have both been shown true, or both false.

15.2.3 Origin of iff

Usage of the abbreviation “iff” first appeared in print in John L. Kelley's 1955 bookGeneral Topology.[4] Its inventionis often credited to Paul Halmos, who wrote “I invented 'iff,' for 'if and only if'—but I could never believe I was reallyits first inventor.”[5]

15.3 Distinction from “if” and “only if”1. “Madison will eat the fruit if it is an apple.” (equivalent to “Only if Madison will eat the fruit, is it anapple;" or “Madison will eat the fruit← fruit is an apple”)

This states simply that Madison will eat fruits that are apples. It does not, however, exclude thepossibility that Madison might also eat bananas or other types of fruit. All that is known for certainis that she will eat any and all apples that she happens upon. That the fruit is an apple is a sufficientcondition for Madison to eat the fruit.

2. “Madison will eat the fruit only if it is an apple.” (equivalent to “If Madison will eat the fruit, then it isan apple” or “Madison will eat the fruit→ fruit is an apple”)

This states that the only fruit Madison will eat is an apple. It does not, however, exclude the possi-bility that Madison will refuse an apple if it is made available, in contrast with (1), which requiresMadison to eat any available apple. In this case, that a given fruit is an apple is a necessary conditionfor Madison to be eating it. It is not a sufficient condition since Madison might not eat all the applesshe is given.

3. “Madison will eat the fruit if and only if it is an apple” (equivalent to “Madison will eat the fruit↔ fruitis an apple”)

This statement makes it clear that Madison will eat all and only those fruits that are apples. Shewill not leave any apple uneaten, and she will not eat any other type of fruit. That a given fruit is anapple is both a necessary and a sufficient condition for Madison to eat the fruit.

Sufficiency is the inverse of necessity. That is to say, given P→Q (i.e. if P then Q), P would be a sufficient conditionfor Q, and Q would be a necessary condition for P. Also, given P→Q, it is true that ¬Q→¬P (where ¬ is the negationoperator, i.e. “not”). This means that the relationship between P and Q, established by P→Q, can be expressed in thefollowing, all equivalent, ways:

P is sufficient for QQ is necessary for P¬Q is sufficient for ¬P

¬P is necessary for ¬Q

As an example, take (1), above, which states P→Q, where P is “the fruit in question is an apple” and Q is “Madisonwill eat the fruit in question”. The following are four equivalent ways of expressing this very relationship:

If the fruit in question is an apple, then Madison 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, will Madison not eat it.

56 CHAPTER 15. MATERIAL EQUIVALENCE

So we see that (2), above, can be restated in the form of if...then as “If Madison will eat the fruit in question, then itis an apple"; taking this in conjunction with (1), we find that (3) can be stated as “If the fruit in question is an apple,then Madison will eat it; AND if Madison will eat the fruit, then it is an apple”.

15.4 More general usage

Iff is used outside the field of logic, wherever logic is applied, especially in mathematical discussions. It has the samemeaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficientfor the other. This is an example of mathematical jargon. (However, as noted above, if, rather than iff, is more oftenused in statements of definition.)The elements of X are all and only the elements of Y is used to mean: “for any z in the domain of discourse, z is inX if and only if z is in Y.”

15.5 See also• Covariance

• Logical biconditional

• Logical equality

• Necessary and sufficient condition

• Polysyllogism

15.6 Footnotes[1] E.g. Daepp, Ulrich; Gorkin, Pamela (2011), Reading, Writing, and Proving: A Closer Look at Mathematics, Undergraduate

Texts in Mathematics, Springer, p. 52, ISBN 9781441994790, While it can be a real time-saver, we don't recommend itin formal writing.

[2] Rothwell, Edward J.; Cloud, Michael J. (2014), Engineering Writing by Design: Creating Formal Documents of LastingValue, CRC Press, p. 98, ISBN 9781482234312, It is common in mathematical writing.

[3] p <=> q. Wolfram|Alpha

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

[5] Nicholas J. Higham (1998). Handbook of writing for the mathematical sciences (2nd ed.). SIAM. p. 24. ISBN 978-0-89871-420-3.

15.7 External links• Language Log: “Just in Case”

• Southern California Philosophy for philosophy graduate students: “Just in Case”

Chapter 16

Material nonimplication

Venn diagram of A ↛ B∧⇔⇔ ¬

Material nonimplication or abjunction (latin ab = “from”, junctio =–"joining”) is the negation of material impli-cation. That is to say that for any two propositions P and Q, the material nonimplication from P to Q is true if andonly if P does not imply Q.It may be written using logical notation as:

p⊅qLpqp↛q

57

58 CHAPTER 16. MATERIAL NONIMPLICATION

16.1 Definition

16.1.1 Truth table

16.2 Properties

falsehood-preserving: The interpretation under which all variables are assigned a truth value of “false” produces atruth value of “false” as a result of material nonimplication.

16.3 Symbol

The symbol for material nonimplication is simply a crossed-out material implication symbol. Its Unicode symbol is8603 (decimal).

16.4 Natural language

16.4.1 Grammatical

16.4.2 Rhetorical

“It’s not the case that p implies q.”“p but not q.”

16.4.3 Colloquial

“Just because p, doesn't mean q.”

16.5 Boolean algebra

Further information: Boolean algebra

(A'+B)'

16.6 Computer science

Bitwise operation: A&(~B)Logical operation: A&&(!B)

16.7 See also• Implication

16.8 References

Chapter 17

Modal operator

A modal connective (or modal operator) is a logical connective for modal logic. It is an operator which formspropositions from propositions. In general, a modal operator has the “formal” property of being non-truth-functional,and is “intuitively” characterized by expressing a modal attitude (such as necessity, possibility, belief, or knowledge)about the proposition to which the operator is applied.

17.1 Modality interpreted

There are several ways to interpret modal operators in modal logic, including: alethic, deontic, axiological, epistemic,and doxastic.

17.1.1 Alethic

Alethic modal operators (M-operators) determine the fundamental conditions of possible worlds, especially causality,time-space parameters, and the action capacity of persons. They indicate the possibility, impossibility and necessityof actions, states of affairs, events, people, and qualities in the possible worlds.

17.1.2 Deontic

Deontic modal operators (P-operators) influence the construction of possible worlds as proscriptive or prescriptivenorms, i.e. they indicate what is prohibited, obligatory, or permitted.

17.1.3 Axiological

Axiological modal operators (G-operators) transform the world’s entities into values and disvalues as seen by a socialgroup, a culture, or a historical period. Axiological modalities are highly subjective categories: what is good for oneperson may be considered as bad by another one.

17.1.4 Epistemic

Epistemic modal operators (K-operators) reflect the level of knowledge, ignorance and belief in the possible world.

17.1.5 Doxastic

Doxastic modal operators express belief in statements.

59

Chapter 18

Negation

For other uses, see Negation (disambiguation) and NOT gate.

In logic, negation, also called logical complement, is an operation that takes a proposition p to another proposition“not p", written ¬p, which is interpreted intuitively as being true when p is false and false when p is true. Negationis thus a unary (single-argument) logical connective. It may be applied as an operation on propositions, truth values,or semantic values more generally. In classical logic, negation is normally identified with the truth function that takestruth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation,the negation of a proposition p is the proposition whose proofs are the refutations of p.

18.1 Definition

No agreement exists as to the possibility of defining negation, as to its logical status, function, and meaning, as to itsfield of applicability..., and as to the interpretation of the negative judgment, (F.H. Heinemann 1944).[1]

Classical negation is an operation on one logical value, typically the value of a proposition, that produces a valueof true when its operand is false and a value of false when its operand is true. So, if statement A is true, then ¬A(pronounced “not A”) would therefore be false; and conversely, if ¬A is true, then A would be false.The truth table of ¬p is as follows:Classical negation can be defined in terms of other logical operations. For example, ¬p can be defined as p→ F, where"→" is logical consequence and F is absolute falsehood. Conversely, one can define F as p & ¬p for any propositionp, where "&" is logical conjunction. The idea here is that any contradiction is false. While these ideas work in bothclassical and intuitionistic logic, they do not work in Brazilian logic, where contradictions are not necessarily false.But in classical logic, we get a further identity: p→ q can be defined as ¬p ∨ q, where "∨" is logical disjunction: “notp, or q".Algebraically, classical negation corresponds to complementation in a Boolean algebra, and intuitionistic negation topseudocomplementation in a Heyting algebra. These algebras provide a semantics for classical and intuitionistic logicrespectively.

18.2 Notation

The negation of a proposition p is notated in different ways in various contexts of discussion and fields of application.Among these variants are the following:In set theory \ is also used to indicate 'not member of': U \ A is the set of all members of U that are not members ofA.No matter how it is notated or symbolized, the negation ¬p / −p can be read as “it is not the case that p", “not that p",or usually more simply (though not grammatically) as “not p".

60

18.3. PROPERTIES 61

18.3 Properties

18.3.1 Double negation

Within a system of classical logic, double negation, that is, the negation of the negation of a proposition p, is logicallyequivalent to p. Expressed in symbolic terms, ¬¬p⇔ p. In intuitionistic logic, a proposition implies its double negationbut not conversely. This marks one important difference between classical and intuitionistic negation. Algebraically,classical negation is called an involution of period two.However, in intuitionistic logic we do have the equivalence of ¬¬¬p and ¬p. Moreover, in the propositional case, asentence is classically provable if its double negation is intuitionistically provable. This result is known as Glivenko’stheorem.

18.3.2 Distributivity

De Morgan’s laws provide a way of distributing negation over disjunction and conjunction :

¬(a ∨ b) ≡ (¬a ∧ ¬b) , and¬(a ∧ b) ≡ (¬a ∨ ¬b) .

18.3.3 Linearity

In Boolean algebra, a linear function is one such that:If there exists a0, a1, ..., a ∈ {0,1} such that f(b1, ..., b ) = a0 ⊕ (a1 ∧ b1) ⊕ ... ⊕ (a ∧ b ), for all b1, ..., b ∈ {0,1}.Another way to express this is that each variable always makes a difference in the truth-value of the operation or itnever makes a difference. Negation is a linear logical operator.

18.3.4 Self dual

In Boolean algebra a self dual function is one such that:f(a1, ..., a ) = ~f(~a1, ..., ~a ) for all a1, ..., a ∈ {0,1}. Negation is a self dual logical operator.

18.4 Rules of inference

There are a number of equivalent ways to formulate rules for negation. One usual way to formulate classical negationin a natural deduction setting is to take as primitive rules of inference negation introduction (from a derivation of p toboth q and ¬q, infer ¬p; this rule also being called reductio ad absurdum), negation elimination (from p and ¬p inferq; this rule also being called ex falso quodlibet), and double negation elimination (from ¬¬p infer p). One obtains therules for intuitionistic negation the same way but by excluding double negation elimination.Negation introduction states that if an absurdity can be drawn as conclusion from p then p must not be the case (i.e.p is false (classically) or refutable (intuitionistically) or etc.). Negation elimination states that anything follows froman absurdity. Sometimes negation elimination is formulated using a primitive absurdity sign ⊥. In this case the rulesays that from p and ¬p follows an absurdity. Together with double negation elimination one may infer our originallyformulated rule, namely that anything follows from an absurdity.Typically the intuitionistic negation ¬p of p is defined as p→⊥. Then negation introduction and elimination are justspecial cases of implication introduction (conditional proof) and elimination (modus ponens). In this case one mustalso add as a primitive rule ex falso quodlibet.

18.5 Programming

As in mathematics, negation is used in computer science to construct logical statements.

62 CHAPTER 18. NEGATION

if (!(r == t)) { /*...statements executed when r does NOT equal t...*/ }

The "!" signifies logical NOT in B, C, and languages with a C-inspired syntax such as C++, Java, JavaScript, Perl,and PHP. “NOT” is the operator used in ALGOL 60, BASIC, and languages with an ALGOL- or BASIC-inspiredsyntax such as Pascal, Ada, Eiffel and Seed7. Some languages (C++, Perl, etc.) provide more than one operator fornegation. A few languages like PL/I and Ratfor use ¬ for negation. Some modern computers and operating systemswill display ¬ as ! on files encoded in ASCII. Most modern languages allow the above statement to be shortened fromif (!(r == t)) to if (r != t), which allows sometimes, when the compiler/interpreter is not able to optimize it, fasterprograms.In computer science there is also bitwise negation. This takes the value given and switches all the binary 1s to 0sand 0s to 1s. See bitwise operation. This is often used to create ones’ complement or "~" in C or C++ and two’scomplement (just simplified to "-" or the negative sign since this is equivalent to taking the arithmetic negative valueof the number) as it basically creates the opposite (negative value equivalent) or mathematical complement of thevalue (where both values are added together they create a whole).To get the absolute (positive equivalent) value of a given integer the following would work as the "-" changes it fromnegative to positive (it is negative because “x < 0” yields true)unsigned int abs(int x) { if (x < 0) return -x; else return x; }

To demonstrate logical negation:unsigned int abs(int x) { if (!(x < 0)) return x; else return -x; }

Inverting the condition and reversing the outcomes produces code that is logically equivalent to the original code, i.e.will have identical results for any input (note that depending on the compiler used, the actual instructions performedby the computer may differ).This convention occasionally surfaces in written speech, as computer-related slang for not. The phrase !voting, forexample, means “not voting”.

18.6 Kripke semantics

In Kripke semantics where the semantic values of formulae are sets of possible worlds, negation can be taken to meanset-theoretic complementation. (See also possible world semantics.)

18.7 See also

• Logical conjunction

• Logical disjunction

• NOT gate

• Bitwise NOT

• Ampheck

• Apophasis

• Cyclic negation

• Double negative elimination

• Grammatical polarity

• Negation (linguistics)

• Negation as failure

18.8. REFERENCES 63

• Square of opposition

• Binary opposition

18.8 References[1] Horn, Laurence R (2001). “Chapter 1”. A NATURAL HISTORY OF NEGATION (PDF). Stanford University: CLSI Pub-

lications. p. 1. ISBN 1-57586-336-7. Retrieved 29 Dec 2013.

18.9 Further reading• Gabbay, Dov, and Wansing, Heinrich, eds., 1999. What is Negation?, Kluwer.

• Horn, L., 2001. A Natural History of Negation, University of Chicago Press.

• G. H. von Wright, 1953–59, “On the Logic of Negation”, Commentationes Physico-Mathematicae 22.

• Wansing, Heinrich, 2001, “Negation”, inGoble, Lou, ed., The Blackwell Guide to Philosophical Logic, Blackwell.

• Marco Tettamanti, Rosa Manenti, Pasquale A. Della Rosa, Andrea Falini, Daniela Perani, Stefano F. Cappaand Andrea Moro (2008). “Negation in the brain: Modulating action representation”, NeuroImage Volume43, Issue 2, 1 November 2008, pages 358–367, http://dx.doi.org/10.1016/j.neuroimage.2008.08.004/

18.10 External links• Negation entry by Laurence R. Horn & Heinrich Wansing in the Stanford Encyclopedia of Philosophy

• Hazewinkel, Michiel, ed. (2001), “Negation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• NOT, on MathWorld

Chapter 19

Sheffer stroke

Venn diagram of A ↑ B

In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "|" (seevertical bar, not to be confused with "||" which is often used to represent disjunction), “Dpq", or "↑" (an upwardsarrow), denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinarylanguage as “not both”. It is also called nand (“not and”) or the alternative denial, since it says in effect that at leastone of its operands is false. In Boolean algebra and digital electronics it is known as the NAND operation.Like its dual, the NOR operator (also known as the Peirce arrow or Quine dagger), NAND can be used by itself,without any other logical operator, to constitute a logical formal system (making NAND functionally complete). Thisproperty makes the NAND gate crucial to modern digital electronics, including its use in NAND flash memory andcomputer processor design.

64

19.1. DEFINITION 65

19.1 Definition

The NAND operation is a logical operation on two logical values. It produces a value of true, if — and only if — atleast one of the propositions is false.

19.1.1 Truth table

The truth table of A NAND B (also written as A | B, Dpq, or A ↑ B) is as follows:

19.2 History

The stroke is named after Henry M. Sheffer, who in 1913 published a paper in the Transactions of the AmericanMathematical Society (Sheffer 1913) providing an axiomatization of Boolean algebras using the stroke, and provedits equivalence to a standard formulation thereof by Huntington employing the familiar operators of propositionallogic (and, or, not). Because of self-duality of Boolean algebras, Sheffer’s axioms are equally valid for either of theNAND or NOR operations in place of the stroke. Sheffer interpreted the stroke as a sign for non-disjunction (NOR)in his paper, mentioning non-conjunction only in a footnote and without a special sign for it. It was Jean Nicod whofirst used the stroke as a sign for non-conjunction (NAND) in a paper of 1917 and which has since become currentpractice.[1] Russell and Whitehead used the Sheffer stroke in the 1927 second edition of Principia Mathematica andsuggested it as a replacement for the “or” and “not” operations of the first edition.Charles Sanders Peirce (1880) had discovered the functional completeness of NAND or NOR more than 30 yearsearlier, using the term ampheck (for 'cutting both ways’), but he never published his finding.

19.3 Properties

NAND does not possess any of the following five properties, each of which is required to be absent from, and theabsence of all of which is sufficient for, at least one member of a set of functionally complete operators: truth-preservation, falsity-preservation, linearity, monotonicity, self-duality. (An operator is truth- (falsity-) preservingif its value is truth (falsity) whenever all of its arguments are truth (falsity).) Therefore {NAND} is a functionallycomplete set.This can also be realized as follows: All three elements of the functionally complete set {AND, OR, NOT} can beconstructed using only NAND. Thus the set {NAND} must be functionally complete as well.

19.4 Introduction, elimination, and equivalencies

The Sheffer stroke ↑ is the negation of the conjunction:Expressed in terms of NAND ↑ , the usual operators of propositional logic are:

19.5 Formal system based on the Sheffer stroke

The following is an example of a formal system based entirely on the Sheffer stroke, yet having the functional ex-pressiveness of the propositional logic:

19.5.1 Symbols

pn for natural numbers n( | )The Sheffer stroke commutes but does not associate (e.g., (T|T)|F = T, but T|(T|F) = F). Hence any formal systemincluding the Sheffer stroke must also include a means of indicating grouping. We shall employ '(' and ')' to this effect.

66 CHAPTER 19. SHEFFER STROKE

We also write p, q, r, … instead of p0, p1, p2.

19.5.2 Syntax

Construction Rule I: For each natural number n, the symbol pn is a well-formed formula (wff), called an atom.Construction Rule II: If X and Y are wffs, then (X|Y) is a wff.Closure Rule: Any formulae which cannot be constructed by means of the first two Construction Rules are not wffs.The letters U, V,W, X, and Y are metavariables standing for wffs.A decision procedure for determining whether a formula is well-formed goes as follows: “deconstruct” the formulaby applying the Construction Rules backwards, thereby breaking the formula into smaller subformulae. Then repeatthis recursive deconstruction process to each of the subformulae. Eventually the formula should be reduced to itsatoms, but if some subformula cannot be so reduced, then the formula is not a wff.

19.5.3 Calculus

All wffs of the form

((U |(V |W))|((Y |(Y |Y))|((X|V)|((U |X)|(U |X)))))

are axioms. Instances of

(U |(V |W)), U ⊢ W

are inference rules.

19.5.4 Simplification

Since the only connective of this logic is |, the symbol | could be discarded altogether, leaving only the parentheses togroup the letters. A pair of parentheses must always enclose a pair of wffs. Examples of theorems in this simplifiednotation are

(p(p(q(q((pq)(pq)))))),

(p(p((qq)(pp)))).

The notation can be simplified further, by letting

(U) := (UU)((U)) ≡ U

for any U. This simplification causes the need to change some rules:

1. More than two letters are allowed within parentheses.

2. Letters or wffs within parentheses are allowed to commute.

3. Repeated letters or wffs within a same set of parentheses can be eliminated.

The result is a parenthetical version of the Peirce existential graphs.Another way to simplify the notation is to eliminate parenthesis by using Polish Notation. For example, the earlierexamples with only parenthesis could be rewritten using only strokes as follows

19.6. SEE ALSO 67

(p(p(q(q((pq)(pq)))))) becomes|p|p|q|q||pq|pq, and

(p(p((qq)(pp)))) becomes,

: |p|p||qq|pp.This follows the same rules as the parenthesis version, with opening parenthesis replaced with a Sheffer stroke andthe (redundant) closing parenthesis removed.

19.6 See also

• List of logic symbols

• AND gate

• Boolean domain

• CMOS

• Gate equivalent (GE)

• Laws of Form

• Logic gate

• Logical graph

• NAND Flash Memory

• NAND logic

• NAND gate

• NOR gate

• NOT gate

• OR gate

• Peirce’s law

• Peirce arrow = NOR

• Propositional logic

• Sole sufficient operator

• XOR gate

• Peirce arrow

19.7 Notes

[1] Church (1956:134)

68 CHAPTER 19. SHEFFER STROKE

19.8 References• Bocheński, Józef Maria (1960), Précis of Mathematical Logic, translated from the French and German editionsby Otto Bird, Dordrecht, South Holland: D. Reidel.

• Church, Alonzo, (1956) Introduction to mathematical logic, Vol. 1, Princeton: Princeton University Press.

• Nicod, Jean G. P., (1917) “A Reduction in the Number of Primitive Propositions of Logic”, Proceedings ofthe Cambridge Philosophical Society, Vol. 19, pp. 32–41.

• Charles Sanders Peirce, 1880, “A Boolian[sic] Algebra with One Constant”, in Hartshorne, C. and Weiss, P.,eds., (1931–35) Collected Papers of Charles Sanders Peirce, Vol. 4: 12–20, Cambridge: Harvard UniversityPress.

• Sheffer, H. M. (1913), “A set of five independent postulates for Boolean algebras, with application to logicalconstants”, Transactions of the American Mathematical Society 14: 481–488, JSTOR 1988701

19.9 External links• http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nand.html

• implementations of 2 and 4-input NAND gates

• Proofs of some axioms by Stroke function by Yasuo Setô @ Project Euclid

Chapter 20

Strict conditional

In logic, a strict conditional is a modal operator, that is, a logical connective of modal logic. It is logically equivalentto the material conditional of classical logic, combined with the necessity operator from modal logic. For any twopropositions p and q, the formula p→ q says that pmaterially implies qwhile□(p→ q) says that p strictly implies q.[1]Strict conditionals are the result of Clarence Irving Lewis's attempt to find a conditional for logic that can adequatelyexpress indicative conditionals in natural language.[2] They have also been used in studying Molinist theology.[3]

20.1 Avoiding paradoxes

The strict conditionals may avoid paradoxes of material implication. The following statement, for example, is notcorrectly formalized by material implication:

If Bill Gates had graduated in Medicine, then Elvis never died.

This condition should clearly be false: the degree of Bill Gates has nothing to do with whether Elvis is still alive.However, the direct encoding of this formula in classical logic using material implication leads to:

Bill Gates graduated in Medicine → Elvis never died.

This formula is true because a formula A → B is true whenever the antecedent A is false. Hence, this formula is notan adequate translation of the original sentence. An encoding using the strict conditional is:

□

In modal logic, this formula means (roughly) that, in every possible world in which Bill Gates graduated in Medicine,Elvis never died. Since one can easily imagine a world where Bill Gates is a Medicine graduate and Elvis is dead, thisformula is false. Hence, this formula seems a correct translation of the original sentence.

20.2 Problems

Although the strict conditional is much closer to being able to express natural language conditionals than the materialconditional, it has its own problems with consequents that are necessarily true (such as 2 + 2 = 4) or antecedents thatare necessarily false.[4] The following sentence, for example, is not correctly formalized by a strict conditional:

If Bill Gates graduated in Medicine, then 2 + 2 = 4.

Using strict conditionals, this sentence is expressed as:

69

70 CHAPTER 20. STRICT CONDITIONAL

□

In modal logic, this formula means that, in every possible world where Bill Gates graduated in medicine, it holds that2 + 2 = 4. Since 2 + 2 is equal to 4 in all possible worlds, this formula is true, although it does not seem that theoriginal sentence should be. A similar situation arises with 2 + 2 = 5, which is necessarily false:

If 2 + 2 = 5, then Bill Gates graduated in Medicine.

Some logicians view this situation as indicating that the strict conditional is still unsatisfactory. Others have notedthat the strict conditional cannot adequately express counterfactual conditionals,[5] and that it does not satisfy certainlogical properties.[6] In particular, the strict conditional is transitive, while the counterfactual conditional is not.[7]

Some logicians, such as Paul Grice, have used conversational implicature to argue that, despite apparent difficulties,the material conditional is just fine as a translation for the natural language 'if...then...'. Others still have turned torelevance logic to supply a connection between the antecedent and consequent of provable conditionals.

20.3 See also• Counterfactual conditional

• Indicative conditional

• Material conditional

• Logical consequence

• Corresponding conditional

20.4 References[1] Graham Priest, An Introduction to Non-Classical Logic: From if to is, 2nd ed, Cambridge University Press, 2008, ISBN

0-521-85433-4, p. 72.

[2] Nicholas Bunnin and Jiyuan Yu (eds), The Blackwell Dictionary ofWestern Philosophy, Wiley, 2004, ISBN 1-4051-0679-4,“strict implication,” p. 660.

[3] Jonathan L. Kvanvig, “Creation, Deliberation, and Molinism,” in Destiny and Deliberation: Essays in Philosophical Theol-ogy, Oxford University Press, 2011, ISBN 0-19-969657-8, p. 127–136.

[4] Roy A. Sorensen, A Brief History of the Paradox: Philosophy and the labyrinths of the mind, Oxford University Press, 2003,ISBN 0-19-515903-9, p. 105.

[5] Jens S. Allwood, Lars-Gunnar Andersson, and Östen Dahl, Logic in Linguistics, Cambridge University Press, 1977, ISBN0-521-29174-7, p. 120.

[6] Hans Rott and Vítezslav Horák, Possibility and Reality: Metaphysics and Logic, ontos verlag, 2003, ISBN 3-937202-24-2,p. 271.

[7] John Bigelow and Robert Pargetter, Science and Necessity, Cambridge University Press, 1990, ISBN 0-521-39027-3, p.116.

20.5 Bibliography• Edgington, Dorothy, 2001, “Conditionals,” in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic.Blackwell.

For an introduction to non-classical logic as an attempt to find a better translation of the conditional, see:

20.5. BIBLIOGRAPHY 71

• Priest, Graham, 2001. An Introduction to Non-Classical Logic. Cambridge Univ. Press.

For an extended philosophical discussion of the issues mentioned in this article, see:

• Mark Sainsbury, 2001. Logical Forms. Blackwell Publishers.

• Jonathan Bennett, 2003. A Philosophical Guide to Conditionals. Oxford Univ. Press.

72 CHAPTER 20. STRICT CONDITIONAL

20.6 Text and image sources, contributors, and licenses

20.6.1 Text• Conditioned disjunction Source: http://en.wikipedia.org/wiki/Conditioned_disjunction?oldid=577876317Contributors: DavidWBrooks,

Nortexoid, GregorB, BD2412, SmackBot, Elonka, Lambiam, CBM, Gregbard, Addbot, Yobot, HRoestBot, ZéroBot and Matthew Kastor• Converse implication Source: http://en.wikipedia.org/wiki/Converse_implication?oldid=602814694 Contributors: Urhixidur, GregorB,

BD2412, Kbdank71, DouglasCalvert, CBM, Gregbard, Cydebot, David Eppstein, Francvs, Watchduck, Hans Adler, Addbot, Meisam,Yobot, Dante Cardoso Pinto de Almeida and Anonymous: 5

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• Logical equality Source: http://en.wikipedia.org/wiki/Logical_equality?oldid=641602683 Contributors: Toby Bartels, Patrick, Ixfd64,Lethe, Peruvianllama, Paul August, AzaToth, Oleg Alexandrov, Mindmatrix, BD2412, Canderson7, FlaBot, YurikBot, Daverocks, Dijx-tra, Trovatore, Arthur Rubin, SmackBot, Melchoir, Bluebot, Jerome Charles Potts, Jjbeard~enwiki, Radagast83, Jon Awbrey, Gregbard,Cydebot, Julian Mendez, Letranova, Thijs!bot, Escarbot, David Eppstein, Infovarius, MartinBot, Santiago Saint James, R'n'B, SieBot,BotMultichill, Aeoza, Sitush, Francvs, Rumping, Hans Adler, Addbot, AnomieBOT, 2ndjpeg, Gamewizard71, Kuzmaka, Wcherowi,Masssly, MerlIwBot, Faus, Trinitresque, MikeShafe and Anonymous: 22

• Logical NOR Source: http://en.wikipedia.org/wiki/Logical_NOR?oldid=634224903 Contributors: Dreamyshade, Bryan Derksen, Stev-ertigo, Edward, Ixfd64, Hermeneus, Jallan, Colin Marquardt, Furrykef, Grendelkhan, Cameronc, SirPeebles, Jerzy, Gubbubu, Urhixidur,MementoVivere, ArnoldReinhold, SocratesJedi, Paul August, Sietse Snel, EmilJ, Nortexoid, Iolar~enwiki, Emvee~enwiki, Bookandcof-fee, Mindmatrix, BD2412, Qwertyus, Kbdank71, Maxim Razin, Chobot, YurikBot, RussBot, Dijxtra, Trovatore, JMRyan, Krymzon, Ty-omitch, Melchoir, Stimpy, Ohnoitsjamie, Mhss, Chris the speller, Bluebot, Nbarth, Cícero, UU, Jon Awbrey, Bjankuloski06en~enwiki,Loadmaster, Pukkie, CBM, Gregbard, Cydebot, Thijs!bot, Hut 8.5, Vujke, Santiago Saint James, Policron, The Tetrast, IronMaiden-Rocks, BotKung, Jean-Frédéric, Dogah, WarrenPlatts, Prestonmag, Francvs, Adrianwn, Watchduck, Hans Adler, Addbot, Nesarose,MrOllie, A:-)Brunuś, Meisam, GorgeUbuasha, Nallimbot, Queen of the Dishpan, AnomieBOT, RokerHRO, RibotBOT, SassoBot,Branzillo, Lantern Leatherhead, Dega180, Gamewizard71, Igotta Lemma, Igor Yalovecky, Webber Jocky, Metadjinn~enwiki, PangurBan My Cat, Iserdo, Tijfo098, Masssly, Helpful Pixie Bot, GKFX, YiFeiBot and Anonymous: 46

• Material conditional Source: http://en.wikipedia.org/wiki/Material_conditional?oldid=665659334 Contributors: William Avery, Dcljr,AugPi, Charles Matthews, Dcoetzee, Doradus, Cholling, Giftlite, Jason Quinn, Nayuki, TedPavlic, Elwikipedista~enwiki, Nortexoid,Vesal, Eric Kvaalen, BD2412, Kbdank71, Martin von Gagern, Joel D. Reid, Fresheneesz, Vonkje, NevilleDNZ, RussBot, KSchutte,NawlinWiki, Trovatore, Avraham, Closedmouth, Arthur Rubin, SyntaxPC, Fctk~enwiki, SmackBot, Amcbride, Incnis Mrsi, Pokipsy76,BiT, Mhss, Jaymay, Tisthammerw, Sholto Maud, Robma, Cybercobra, Jon Awbrey, Oceanofperceptions, Byelf2007, Grumpyyoung-man01, Clark Mobarry, Beefyt, Rory O'Kane, Dreftymac, Eassin, JRSpriggs, Gregbard, FilipeS, Cydebot, Julian Mendez, Thijs!bot,Egriffin, Jojan, Escarbot, Applemeister, WinBot, Salgueiro~enwiki, JAnDbot, Olaf, Alastair Haines, Arno Matthias, JaGa, SantiagoSaint James, Pharaoh of the Wizards, Pyrospirit, SFinside, Anonymous Dissident, The Tetrast, Cnilep, Radagast3, Newbyguesses, Light-breather, Paradoctor, Iamthedeus, Soler97, Francvs, Classicalecon, Josang, Ruy thompson, Watchduck, Hans Adler, Djk3, Marc van

74 CHAPTER 20. STRICT CONDITIONAL

Leeuwen, Tbsdy lives, Addbot, Melab-1, Fyrael, Morriswa, SpellingBot, CarsracBot, Chzz, Jarble, Meisam, Luckas-bot, AnomieBOT,Sonia, Pnq, Bearnfæder, FrescoBot, Greyfriars, Machine Elf 1735, RedBot, MoreNet, Beyond My Ken, John of Reading, Hgetnet, Hi-bou57, ClueBot NG, Movses-bot, Jiri 1984, Masssly, Dooooot, Noobnubcakes, Hanlon1755, Leif Czerny, CarrieVS, Jochen Burghardt,Lukekfreeman, ArchReader, NickDragonRyder, Indomitavis, Rathkirani, AnotherPseudonym, Xerula, Matthew Kastor, MathematicalTruth and Anonymous: 73

• Material equivalence Source: http://en.wikipedia.org/wiki/If_and_only_if?oldid=666023709 Contributors: Damian Yerrick, AxelBoldt,Matthew Woodcraft, Vicki Rosenzweig, Zundark, Tarquin, Larry_Sanger, Toby Bartels, Ark~enwiki, Camembert, Stevertigo, Patrick,Chas zzz brown, Michael Hardy, Wshun, DopefishJustin, Dante Alighieri, Dominus, SGBailey, Wwwwolf, Delirium, Geoffrey~enwiki,Stevenj, Kingturtle, UserGoogol, Andres, Evercat, Jacquerie27, Adam Conover, Revolver, Wikiborg, Dysprosia, Itai, Ledge, McKay,Robbot, Psychonaut, Henrygb, Ruakh, Diberri, Tobias Bergemann, Adam78, Enochlau, Giftlite, DavidCary, Ævar Arnfjörð Bjarmason,Mellum, Chinasaur, Jason Quinn, Taak, Superfrank~enwiki, Wmahan, LiDaobing, Mvc, Neutrality, Urhixidur, Ropers, Jewbacca, KarlDickman, PhotoBox, Brianjd, Paul August, Sunborn, Elwikipedista~enwiki, Chalst, Edward Z. Yang, Pearle, Ekevu, IgorekSF, Msh210,Interiot, ABCD, Stillnotelf, Suruena, Voltagedrop, Rhialto, Forderud, Oleg Alexandrov, Joriki, Velho, Woohookitty, Mindmatrix, RuudKoot, Ryan Reich, Adjam, Pope on a Rope, Eyu100, R.e.b., FlaBot, Mathbot, Rbonvall, Glenn L, BMF81, Bgwhite, RussBot, Post-glock, Voidxor, El Pollo Diablo, Gadget850, Jkelly, Danielpi, Lt-wiki-bot, TheMadBaron, Nzzl, Arthur Rubin, Netrapt, SmackBot, Ttzz,Gloin~enwiki, Incnis Mrsi, InverseHypercube, Melchoir, GoOdCoNtEnT, Thumperward, Jonatan Swift, Peterwhy, Acdx, Shirifan, Evil-dictaitor, Abolen, Rainwarrior, Dicklyon, Mets501, Yuide, Shoeofdeath, CRGreathouse, CBM, Joshwa, Picaroon, Gregbard, Eu.stefan,Letranova, Thijs!bot, Egriffin, Schneau, Jojan, Davkal, AgentPeppermint, Urdutext, Holyknight33, Escarbot, WinBot, Serpent’s Choice,Timlevin, Singularity, David Eppstein, Msknathan, MartinBot, AstroHurricane001, Vanished user 47736712, Kenneth M Burke, Dorgan-Bot, Bsroiaadn, TXiKiBoT, Anonymous Dissident, Abyaly, Ichtaca, Mouse is back, Rjgodoy, TrippingTroubadour, KjellG, AlleborgoBot,Lillingen, SieBot, Iamthedeus, This, that and the other, Smaug123, Skippydo, Warman06~enwiki, Tiny plastic Grey Knight, Francvs,Minehava, ClueBot, Surfeited, BANZ111, Master11218, WestwoodMatt, Excirial, He7d3r, Pfhorrest, Kmddmk, Addbot, Ronhjones,Wikimichael22, Lightbot, Jarble, Yobot, Bryan.burgers, AnomieBOT, Nejatarn, Ciphers, Quintus314, Лев Дубовой, WissensDürster,Ex13, Rapsar, Mikespedia, Lotje, Igor Yalovecky, EmausBot, Ruxkor, Chricho, Sugarfoot1001, Tijfo098, ClueBot NG,Wcherowi, Widr,MerlIwBot, Helpful Pixie Bot, Chmarkine, CarrieVS, Ekips39, Epicgenius, Dr Lindsay B Yeates, Matthew Kastor and Anonymous: 143

• Material nonimplication Source: http://en.wikipedia.org/wiki/Material_nonimplication?oldid=662913831Contributors: Kaldari, BD2412,Kbdank71, Chris Capoccia, MacMog, SmackBot, Cybercobra, Bjankuloski06en~enwiki, Gregbard, Cydebot, David Eppstein, MauriceCarbonaro, Anzurio, Francvs, Classicalecon, BANZ111, Alex836, Watchduck, Addbot, Meisam, Luckas-bot, Yobot, FrescoBot, JesseV., EmausBot, Jontturi, Matthew Kastor and Anonymous: 5

• Modal operator Source: http://en.wikipedia.org/wiki/Modal_operator?oldid=630168849 Contributors: Markhurd, Hyacinth, Alten-mann, Filemon, Velho, BD2412, SmackBot, BenetD, JRSpriggs, CBM, Gregbard, Nick Number, Gwern, R'n'B, Erhasalz, Tanfil13,Mild Bill Hiccup, TheOldJacobite, Addbot, EmbraceParadox, Jim1138, PhnomPencil, Sydactive and Anonymous: 2

• Negation Source: http://en.wikipedia.org/wiki/Negation?oldid=652849228 Contributors: AxelBoldt, Zundark, Arvindn, Toby Bartels,William Avery, Ryguasu, Youandme, Stevertigo, Edward, Patrick, Ihcoyc, Andres, Hyacinth, David Shay, Omegatron, Francs2000,Robbot, Lowellian, Hadal, Wikibot, Benc, Tobias Bergemann, Adam78, Giftlite, DocWatson42, Nayuki, Chameleon, Neilc, JonathanGrynspan, Poccil, Paul August, Chalst, EmilJ, Spoon!, John Vandenberg, Nortexoid, Rajah, Deryck Chan, Daf, Pazouzou, ObradovicGoran, Pearle, Knucmo2, Musiphil, Cesarschirmer~enwiki, RainbowOfLight, Forderud, Eric Qel-Droma, Oleg Alexandrov, Mindma-trix, Troels.jensen~enwiki, Bluemoose, BrydoF1989, TAKASUGI Shinji, BD2412, Kbdank71, FlaBot, Gparker, Chobot, DTOx, Vi-sor, Roboto de Ajvol, YurikBot, RussBot, Xihr, RJC, Gaius Cornelius, Cookman, Trovatore, Vanished user 1029384756, Dhollm,Mditto, SmackBot, KnowledgeOfSelf, Eskimbot, Bluebot, Iain.dalton, Thumperward, Ewjw, Furby100, Rrburke, Mr.Z-man, UU, Cy-bercobra, Revengeful Lobster, Decltype, Jon Awbrey, Quatloo, Byelf2007, Lambiam, Christoffel K~enwiki, Loadmaster, Hans Bauer,Adambiswanger1,Mudd1, TheTito, Andkore, Simeon, Gregbard, FilipeS, Cydebot, Reywas92, Thijs!bot, Zron, Escarbot, Djihed, Slacka123,Catgut, WhatamIdoing, Gwern, Santiago Saint James, R'n'B, Kavadi carrier, Policron, Enix150, VolkovBot, Semmelweiss, Pasixxxx,Cs-Reaperman, PGSONIC, TXiKiBoT, Gwib, Anonymous Dissident, Ontoraul, HeirloomGardener, AlleborgoBot, SieBot, Ivan Štam-buk, Soler97, Francvs, ClueBot, PixelBot, Alejandrocaro35, Holothurion, AHRtbA==, HumphreyW, DumZiBoT, Mifter, Addbot, Con-CompS,AndersBot, Gail, Jarble, Meisam, Qwertymith, Legobot, Luckas-bot, AnomieBOT,Nasnema, Oursipan, Zhentmdfan, Pinethicket,Half price, MastiBot, DixonDBot, Mayoife, Xnn, Shadex9999, EmausBot, ClueBot NG,Wcherowi, Strcat,Wdchk,Masssly, Hadi Payami,Victor Yus, ChrisGualtieri, YFdyh-bot, Dexbot, Arfæst Ealdwrítere, GinAndChronically, Solid Frog, Reybansingh and Anonymous: 93

• Sheffer stroke Source: http://en.wikipedia.org/wiki/Sheffer_stroke?oldid=659255469 Contributors: AxelBoldt, Fubar Obfusco, Davidspector, Vik-Thor, Michael Hardy, AugPi, Jouster, Dcoetzee, Dysprosia, Markhurd, Hyacinth, Cameronc, Johnleemk, Robbot, Saaska,Rorro, Paul Murray, Snobot, Giftlite, DocWatson42, Brouhaha, Zigger, Gubbubu, Halo, Sam, Urhixidur, Ratiocinate, Rich Farmbrough,Leibniz, Pie4all88, TheJames, SocratesJedi, Paul August, Chalst, EmilJ, Nortexoid, Redfarmer, Emvee~enwiki, Dominic, Bookandcoffee,Drakferion, Woohookitty, Mindmatrix, Steven Luo, Ruud Koot, Wayward, BD2412, Qwertyus, Kbdank71, R.e.b., Ademkader, FlaBot,Mathbot, George Leung, Algebraist, RobotE, Sceptre, Imagist, Archelon, Ksyrie, NormalAsylum, Dijxtra, Trovatore, Nad, Yahya Abdal-Aziz, Prolineserver, JMRyan, Rohanmittal, Luethi, JoanneB, SmackBot, Melchoir, Mhss, Chris the speller, Bluebot, Thumperward, UU,Cybercobra, Jon Awbrey, Lambiam, Loadmaster, Yoderj, CBM, Ezrakilty, Gregbard, Nilfanion, Rotiro, Cydebot, Julian Mendez, Ase-nine, SpK, Royas, MER-C, Magioladitis, VoABot II, Vujke, Seba5618, Santiago Saint James, Kloisiie, Olmsfam, Somejan, Josephholsten,The Tetrast, Philogo, Manusharma, Jamelan, Inductiveload, Dogah, CultureDrone, Francvs, ClueBot, Plastikspork, Achlaug, Watchduck,Dspark76, Hans Adler, Addbot, Meisam, Bunnyhop11, TaBOT-zerem, Erud, M&M987, Dante Cardoso Pinto de Almeida, LittleWink,Dhanyavaada, Omerta-ve, Dega180, Gamewizard71, RjwilmsiBot, Arielkoiman, Set theorist, Zahnradzacken, Hpvpp, SporkBot, ClueBotNG, Masssly, Jones11235813, MerlIwBot, SOFooBah, Yamaha5, Brianlen, SarahRMadden, Sofia Koutsouveli and Anonymous: 65

• Strict conditional Source: http://en.wikipedia.org/wiki/Strict_conditional?oldid=630178932Contributors: Michael Hardy, CharlesMatthews,Hyacinth, J heisenberg, Guanaco, Mu, Aquafleur, Citizensunshine, Chalst, Cdegough, Alansohn, Kbdank71, Tizio, KSchutte, Arthur Ru-bin, Incnis Mrsi, Mhss, Melbournian, Ocanter, Beefyt, Gregbard, Cydebot, Magioladitis, Philogo, Graymornings, Addbot, Tide rolls,Luckas-bot, AnomieBOT, Gemtpm, Machine Elf 1735, Hobbes Goodyear, D.Lazard, ClueBot NG, Helpful Pixie Bot, Hanlon1755 andAnonymous: 16

20.6.2 Images• File:AND_Gate_diagram.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/41/AND_Gate_diagram.svg License: Pub-

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