LogicFrom Wikipedia, the free encyclopedia

Contents

1 Absence paradox 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Accident (fallacy) 22.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Reference list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Affine logic 33.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Affirmative conclusion from a negative premise 54.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

5 Affirming the consequent 65.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

6 Analytic reasoning 86.1 Kant’s Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86.3 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

7 Animistic fallacy 97.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

8 Antecedent (logic) 108.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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9 Antepredicament 119.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

10 Aporime 1210.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

11 Argumentum a contrario 1311.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

12 Argumentum ad lapidem 1412.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1412.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1412.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

13 Assertoric 1613.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

14 Association for Symbolic Logic 1714.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.2 Meetings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

15 Bar induction 1815.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

16 Baralipton 1916.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

17 Barcan formula 2017.1 The Barcan formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.2 Converse Barcan formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

18 Binary decision 2218.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

19 Calculus of structures 2319.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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19.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

20 Card paradox 2420.1 The paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2420.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2420.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

21 Cayenne (programming language) 2521.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

22 Cirquent calculus 2622.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2622.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

23 Classical modal logic 2823.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2823.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2823.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

24 Cointerpretability 2924.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2924.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

25 Completeness (knowledge bases) 30

26 Comprehension (logic) 3126.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

27 Conditioned disjunction 3227.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

28 Consequent 3328.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3328.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

29 Conservativity theorem 3429.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

30 Consistency (knowledge bases) 35

31 Contradictio in terminis 3631.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

32 Converse accident 3732.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

33 Converse implication 38

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33.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3833.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3833.1.2 Venn diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

33.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3833.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3833.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3833.5 Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3933.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

34 Counterargument 4034.1 Speech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4134.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4134.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4134.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4134.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

35 Counterinduction 4235.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

36 Counting quantification 4336.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4336.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

37 Deep inference 4437.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4437.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

38 Defeasible logic 4538.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4538.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

39 Degree of truth 4639.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4639.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

40 Descriptive fallacy 4740.1 Role of ‘descriptive fallacy’ in Austin’s philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . 4740.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

41 Don't-care term 4841.1 X value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4841.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4841.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

42 Double turnstile 49

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42.1 Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4942.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4942.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

43 Effective method 5043.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5043.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5043.3 Computable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5043.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5143.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

44 Empty domain 5244.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

45 End term 54

46 Enumerative definition 5546.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

47 Existential fallacy 5647.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5647.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5747.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

48 Existential generalization 5848.1 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5848.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5848.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

49 Existential instantiation 5949.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5949.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

50 Explanatory power 6050.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6050.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6050.3 Relation to other criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6150.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

51 Extension (predicate logic) 6251.1 Relationship with characteristic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6251.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

52 Extensionality 6352.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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52.2 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6352.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6452.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

53 Fallacies of illicit transference 6553.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6553.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

54 Fallacy of division 6654.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6654.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6654.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

55 Fallacy of exclusive premises 6855.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6955.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

56 Fallacy of relative privation 7056.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7056.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7056.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

57 Falsism 7157.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7157.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

58 First-order predicate 7258.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7258.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

59 Fluent calculus 7359.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7359.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

60 Fragment (logic) 7460.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

61 Frege’s theorem 7561.1 Frege’s theorem in propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7561.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

62 Guarded logic 7662.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7662.2 Types of Guarded Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7662.3 Definitions of Guarded Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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62.3.1 Guarded Bisimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7762.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

63 Herbrand interpretation 7863.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

64 Hybrid logic 7964.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7964.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7964.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

65 Idempotency of entailment 80

66 Illicit major 8166.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8166.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

67 Illicit minor 8367.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

68 Inclusion (logic) 8468.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

69 Instantiation principle 8569.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

70 Interpretability 8670.1 Informal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8670.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8670.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

71 Interval temporal logic 8771.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8771.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

72 Inverse (logic) 8872.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8872.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

73 Inverse resolution 8973.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

74 Invincible ignorance fallacy 9074.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9074.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9074.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

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74.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

75 Issue trees 9275.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

76 Lambert of Auxerre 9476.1 Works and Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9476.2 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9476.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

77 Lemma (logic) 9577.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

78 Limitation of size 9678.1 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9678.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

79 Lindenbaum’s lemma 9779.1 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9779.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9779.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9779.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9779.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9779.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

80 Literal (mathematical logic) 9880.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9880.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

81 Logic Spectacles 99

82 Logical constant 10082.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10082.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10082.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

83 Logical cube 10183.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10183.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

84 Main contention 10284.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10284.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

85 Material nonimplication 10385.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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85.1.1 Truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10485.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10485.3 Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10485.4 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

85.4.1 Grammatical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10485.4.2 Rhetorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10485.4.3 Colloquial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

85.5 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10485.6 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10485.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10485.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

86 MaxEnt school 105

87 McNamara fallacy 10687.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10687.2 In modern clinical trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10687.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

88 Middle term 10788.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

89 Models And Counter-Examples 10889.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10889.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10889.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

90 Monadic Boolean algebra 10990.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10990.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11090.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

91 Monotonicity of entailment 11191.1 Weakening rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11191.2 Non-monotonic logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11191.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

92 Multimodal logic 11392.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11392.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11392.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

93 Multiple-conclusion logic 11593.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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93.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

94 Neighborhood semantics 11694.1 Correspondence between relational and neighborhood models . . . . . . . . . . . . . . . . . . . . 11694.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

95 Non-wellfounded mereology 11795.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11795.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

96 Nonfirstorderizability 11896.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11896.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

97 Normal form (natural deduction) 120

98 Normal modal logic 12198.1 Common normal modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

99 OBJ3 12299.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12299.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

100Objection (argument) 123100.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

101One-sided argument 124101.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124101.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124101.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

102Otter (theorem prover) 125102.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125102.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125102.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

103Pars destruens/pars construens 126103.1External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

104PhoX 127104.1External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

105Polysyllogism 128105.1Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128105.2Sorites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128105.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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105.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

106Post disputation argument 130

107Predicate logic 131107.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131107.2Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131107.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

108Principle of nonvacuous contrast 133108.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

109Principles of Mathematical Logic 134109.1Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134109.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

110Probabilistic proposition 135

111Problem of multiple generality 136111.1Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

112Proof net 138112.1Correctness criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138112.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138112.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138112.4Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

113Proof-theoretic semantics 139113.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139113.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139113.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

114Propositional variable 140114.1Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140114.2In first order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140114.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140114.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

115Prototype Verification System 141115.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141115.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141115.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

116Provability logic 142116.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142116.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

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116.3Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142116.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142116.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

117Proving a point 144117.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

118Regular modal logic 145118.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

119Robinson’s joint consistency theorem 146119.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

120Rule of replacement 147120.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

121Rules of passage (logic) 148121.1The rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148121.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148121.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148121.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

122Sacrifice of the intellect 150122.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150122.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

123Sanctioned specialisation 152

124Second-order predicate 153124.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

125Second-order propositional logic 154125.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154125.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

126Self-reference puzzle 155126.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155126.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

127Self-verifying theories 156127.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156127.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

128Sentence (logic) 157128.1Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157128.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

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128.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

129Slothful induction 159129.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

130Specialization (logic) 160130.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

131Strict logic 161131.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

132Syllogistic fallacy 162132.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162132.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

133T-schema 163133.1The inductive definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163133.2Natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163133.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164133.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164133.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

134Tacit assumption 165134.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165134.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

135Takeuti’s conjecture 166135.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166135.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166135.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

136Tee (symbol) 167136.1Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167136.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167136.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

137The Game of Logic 168137.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168137.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

138Third-cause fallacy 169138.1Other names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169138.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169138.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

139Transparent Intensional Logic 170

xiv CONTENTS

139.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170139.2Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170139.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

140Triangle of opposition 171140.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

141Truth condition 172141.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

142Two-variable logic 173142.1Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173142.2Counting quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173142.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

143Unique name assumption 174143.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174143.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

144Unsatisfiable core 175144.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

145Vagrant predicate 176145.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

146Valentino Annibale Pastore 177146.1Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177146.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

147Vampire (theorem prover) 179147.1Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179147.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

148Van Gogh fallacy 180148.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

149Vienna Summer of Logic 181149.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182149.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

150Vivid knowledge 184150.1Propositional knowledge base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184150.2First-order knowledge base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184150.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184150.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184150.5Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 186

CONTENTS xv

150.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186150.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193150.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Chapter 1

Absence paradox

The absence paradox, while named a paradox, is more precisely an informal fallacy and humorous misuse of lan-guage which results in the conclusion that “No one is ever present.” The statement of the argument is some formulationof the following:

• No person is ever present because he is either not in Rome or alternatively is not in Beijing.

• Therefore, he must be somewhere else.

• If he is somewhere else he is not here.

The use of this fallacy dates from the 19th century. The fallacy in the argument is that it interprets the relative adverb“else” in an absolute sense.[1]

1.1 References[1] Bothamley, Dictionary of theories

1

Chapter 2

Accident (fallacy)

The informal fallacy of accident (also called destroying the exception or a dicto simpliciter ad dictum secun-dum quid) is a deductively valid but unsound argument occurring in statistical syllogisms (an argument based on ageneralization) when an exception to a rule of thumb[1] is ignored. It is one of the thirteen fallacies originally identifiedby Aristotle in Sophistical Refutations. The fallacy occurs when one attempts to apply a general rule to an irrelevantsituation.For example:

• Cutting people with knives is a crime. →

Surgeons cut people with knives. →Surgeons are criminals.

It is easy to construct fallacious arguments by applying general statements to specific incidents that are obviouslyexceptions.Generalizations that are weak generally have more exceptions (the number of exceptions to the generalization neednot be a minority of cases) and vice versa.This fallacy may occur when we confuse particulars (“some”) for categorical statements (“always and everywhere”). Itmay be encouraged when no qualifying words like “some”, “many”, “rarely” etc. are used to mark the generalization.Related inductive fallacies include: overwhelming exception, hasty generalization. See faulty generalization.The opposing kind of dicto simpliciter fallacy is the converse accident.

2.1 Notes[1] “The Fallacy of Accident”. The Fallacy Files.

2.2 Reference list• S. Morris Engel (1999). With Good Reason: An Introduction to Informal Fallacies. Bedford/St. Martin’s.ISBN 0312157584. Retrieved 2013-02-17.

2

Chapter 3

Affine logic

Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. It can also becharacterized as linear logic with weakening.The name “affine logic” is associated with linear logic, to which it differs by allowing the weakening rule. Jean-YvesGirard introduced the name as part of the geometry of interaction semantics of linear logic, which characterises linearlogic in terms of linear algebra; here he alludes to affine transformations on vector spaces.[1]

The logic predated linear logic. V. N. Grishin used this logic in 1974,[2] after observing that Russell’s paradox cannotbe derived in a set theory without contraction, even with an unbounded comprehension axiom.[3] Likewise, the logicformed the basis of a decidable subtheory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984;Ketonen & Bellin, 1989).Affine logic can be embedded into linear logic by rewriting the affine arrow A→ B as the linear arrow A−◦B⊗⊤ .Whereas full linear logic (i.e. propositional linear logic with multiplicatives, additives and exponentials) is undecid-able, full affine logic is decidable.Affine logic forms the foundation of ludics.

3.1 Notes

[1] Jean-Yves Girard, 1997. 'Affine'. Message to the TYPES mailing list.

[2] Grishin, 1974, and later, Grishin, 1981.

[3] Cf. Frederic Fitch's demonstrably consistent set theory

3.2 References

• V.N. Grishin, 1974. “A nonstandard logic and its application to set theory,” (Russian). Studies in FormalizedLanguages and Nonclassical Logics (Russian), 135-171. Izdat, “Nauka,” Moskow. .

• V.N. Grishin, 1981. “Predicate and set-theoretic calculi based on logic without contraction rules,” (Russian).Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 45(1):47-68. 239. Math. USSR Izv., 18, no.1,Moscow.

• Ketonen and Weyhrauch, 1984, A decidable fragment of predicate calculus. Theoretical Computer Science32:297-307.

• Ketonen and Bellin, 1989. A decision procedure revisited: notes on Direct Logic. In Linear Logic and itsImplementation.

3

4 CHAPTER 3. AFFINE LOGIC

3.3 See also• Strict logic and relevant logic

• Affine type system, a substructural type system

Chapter 4

Affirmative conclusion from a negativepremise

Affirmative conclusion from a negative premise (illicit negative) is a formal fallacy that is committed when acategorical syllogism has a positive conclusion, but one or two negative premises.For example:

No fish are dogs, and no dogs can fly, therefore all fish can fly.

The only thing that can be properly inferred from these premises is that some things that are not fish cannot fly,provided that dogs exist.Or:

We don't read that trash. People who read that trash don't appreciate real literature. Therefore, weappreciate real literature.

This could be illustrated mathematically as

If A ∩B = ∅ and B ∩ C = ∅ then A ⊂ C.

It is a fallacy because any valid forms of categorical syllogism that assert a negative premise must have a negativeconclusion.

4.1 See also• Negative conclusion from affirmative premises, in which a syllogism is invalid because the conclusion is negativeyet the premises are affirmative

• Fallacy of exclusive premises, in which a syllogism is invalid because both premises are negative

4.2 References• The Fallacy Files: Affirmative Conclusion from a Negative Premiss

5

Chapter 5

Affirming the consequent

Affirming the consequent, sometimes called converse error, fallacy of the converse or confusion of necessityand sufficiency, is a formal fallacy of inferring the converse from the original statement. The corresponding argumenthas the general form:

1. If P, then Q.

2. Q.

3. Therefore, P.

An argument of this form is invalid, i.e., the conclusion can be false even when statements 1 and 2 are true. Since Pwas never asserted as the only sufficient condition for Q, other factors could account for Q (while P was false).[1][2]

To put it differently, if P implies Q, the only inference that can be made is non-Q implies non-P. (Non-P and non-Qdesignate the opposite propositions to P and Q.) This is known as logical contraposition. Symbolically:(P → Q) ↔ (¬Q→ ¬P )The name affirming the consequent derives from the premise Q, which affirms the “then” clause of the conditionalpremise.

5.1 Examples

One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but anobviously false conclusion. For example:

If Bill Gates owns Fort Knox, then he is rich.Bill Gates is rich.Therefore, Bill Gates owns Fort Knox.

Owning Fort Knox is not the only way to be rich. Any number of other ways exist to be rich.However, one can affirm with certainty that “if Bill Gates is not rich” (non-Q) then “Bill Gates does not own FortKnox” (non-P). This is the contrapositive of the first statement, and it must be true if the original statement is true.Arguments of the same form can sometimes seem superficially convincing, as in the following example:

If I have the flu, then I have a sore throat.I have a sore throat.Therefore, I have the flu.

But having the flu is not the only cause of a sore throat since many illnesses cause sore throat, such as the commoncold or strep throat.

6

5.2. SEE ALSO 7

5.2 See also• Confusion of the inverse

• Denying the antecedent

• ELIZA effect

• Fallacy of the single cause

• Fallacy of the undistributed middle

• Inference to the best explanation

• Modus ponens

• Modus tollens

• Post hoc ergo propter hoc

• Necessity and sufficiency

5.3 References[1] “Fallacy Files”. http://www.fallacyfiles.org''. Fallacy Files. Retrieved 9 May 2013.

[2] Damer, T. Edward (2001). “Confusion of a Necessary with a Sufficient Condition”. Attacking Faulty Reasoning (4th ed.).Wadsworth. p. 150.

Chapter 6

Analytic reasoning

6.1 Kant’s Usage

In the philosophy of Immanuel Kant, analytic reasoning represents judgments made upon statements that are basedon the virtue of the statement’s own content. No particular experience, beyond an understanding of the meanings ofwords used, is necessary for analytic reasoning.[1]

For example, "John is a bachelor." is a given true statement. Through analytic reasoning, one can make the judgmentthat John is unmarried. One knows this to be true since the state of being unmarried is implied in the word bachelor;no particular experience of John is necessary to make this judgement.To suggest that John is married—given that he is a bachelor—would be self-contradictory.Compare analytic reasoning with synthetic reasoning.

6.2 See also• Analytic-synthetic distinction

6.3 Footnotes[1] See Stephen Palmquist, “Knowledge and Experience - An Examination of the Four Reflective 'Perspectives’ in Kant’s

Critical Philosophy”, Kant-Studien 78:2 (1987), pp.170-200; revised and reprinted as Chapter IV of Kant’s System ofPerspectives (Lanham: University Press of America, 1993).

8

Chapter 7

Animistic fallacy

The animistic fallacy is the informal fallacy of arguing that an event or situation necessarily arose because someoneintentionally acted to cause it.[1] While it could be that someone set out to effect a specific goal, the fallacy appears inan argument that states this must be the case.[1] The name of the fallacy comes from the animistic belief that changesin the physical world are the work of conscious spirits.

7.1 Examples

Thomas Sowell in his book Knowledge and Decisions (1980) presents several arguments as examples of the animisticfallacy:[1]

• that people earn wealth always because of superior choices

• that central planning is necessary to prevent chaos in society

Sowell repeatedly dismisses the necessity that order comes from design, and notes that fallacious animistic argumentstend to provide explanations that require comparatively little time to implement. In this light he contrasts the six-daycreation of the world described in the Bible to the development of life over billions of years described by evolution.

7.2 See also• Anthropomorphism

• Argument from ignorance

• Pathetic fallacy

• Reification (fallacy)

7.3 References[1] Sowell, Thomas (1996). Knowledge and decisions (3rd ed.). Basic Books. pp. 97–100. ISBN 978-0-465-03738-4.

7.4 External links• Google Scholar

• Google Books

9

Chapter 8

Antecedent (logic)

An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause. It isalso known for a person’s principles to a possible or hypothetical issue.Examples:

• If P, then Q.

This is a nonlogical formulation of a hypothetical proposition. In this case, the antecedent is P, and the consequentis Q. In an implication, if ϕ implies ψ then ϕ is called the antecedent and ψ is called the consequent.[1]

• If X is a man, then X is mortal.

“X is a man” is the antecedent for this proposition.

• If men have walked on the moon, then I am the king of France.

Here, “men have walked on the moon” is the antecedent.

8.1 See also• Affirming the consequent (fallacy)

• Denying the antecedent (fallacy)

• Necessity and sufficiency

8.2 References[1] Sets, Functions and Logic - An Introduction to Abstract Mathematics, Keith Devlin, Chapman & Hall/CRC Mathematics,

3rd ed., 2004

10

Chapter 9

Antepredicament

Antepredicaments, in logic, are certain previous matters requisite to a more easy and clear apprehension of thedoctrine of predicaments or categories. Such are definitions of common terms, as equivocals, univocals, etc., withdivisions of things, their differences, etc. They are thus called because Aristotle treated them before the predicaments,hoping that the thread of discourse might not afterwards be interrupted.

9.1 References• This article incorporates text from a publication now in the public domain: Chambers, Ephraim, ed. (1728)."article name needed". Cyclopædia, or an Universal Dictionary of Arts and Sciences (first ed.). James and JohnKnapton, et al.

11

Chapter 10

Aporime

An aporime is a problem difficult to resolve, and which has never been resolved, though it may not be, in itself,impossible.The word is derived from the Greek ἄπορον, which signifies something very difficult and impracticable, being formedfrom the privative α, and πόρος, “passage”. When a question was proposed to any of the ancient Greek philosophers,especially of the sect of Academists, if he could not give a solution, his answer was ἀποροῶ, q.d. “I do not conceiveit; I cannot see through it; I am not able to clear it up.”

10.1 See also• Aporia

10.2 References• This article incorporates text from a publication now in the public domain: Chambers, Ephraim, ed. (1728)."article name needed". Cyclopædia, or an Universal Dictionary of Arts and Sciences (first ed.). James and JohnKnapton, et al.

12

Chapter 11

Argumentum a contrario

In logic, an argumentum a contrario (Latin: “appeal from the contrary” or “argument based on the contrary”) denotesany proposition that is argued to be correct because it is not disproven by a certain case. It is the opposite of theanalogy. Arguments a contrario are often used in the legal system as a way to solve problems not currently coveredby a certain system of laws. Although it might be used as a logical fallacy, arguments a contrario are not by definitionfallacies.

11.1 Examples• "§ 123 of the X-Law says that green cars need to have blue tires. As such, red cars don't have to have bluetires.”

Here the argument is based on the fact that red cars are not green cars and as such § 123 of the X-Law cannot beapplied to them. This requires the law to be interpreted to determine which solution would have been desired if thelawmaker had considered red cars. In this case it’s probably safe to assume that they only wanted to regulate greencars and not regulate cars of other colors.On the other hand, this example:

• "§ 456 of the Y-Law says that it’s irrelevant whether a message is sent by letter or by telegraph. As such,messages cannot be sent by fax machines.”

As with the example above, the argument is based on the fact that the law does not mention fax machines and theymust therefore not be used. Here the interpretation that the lawmaker consciously did not mention fax machines isless valid than the assumption that fax machines did not exist at this time and that, were the law passed today, theywould have been mentioned. Here the argument a contrario is used fallaciously since it places the letter of the lawabove its intent

11.2 See also• Analogy

• A minore ad maius

• A maiore ad minus

11.3 References

13

Chapter 12

Argumentum ad lapidem

Argumentum ad lapidem (Latin: “to the stone”) is a logical fallacy that consists in dismissing a statement as absurdwithout giving proof of its absurdity.[1] The form of argument employed by such dismissals is the argumentum adlapidem, or appeal to the stone.[2][3]

Ad lapidem statements are fallacious because they fail to address the merits of the claim in dispute. Ad hominemarguments, which dispute the merits of a claim’s advocate rather than the merits of the claim itself, are fallacious forthe same reason. The same applies to proof by assertion, where an unproved or disproved claim is asserted as trueon no ground other than that of its truth having been asserted.The name of this fallacy is attributed to Dr. Samuel Johnson, who refuted Bishop Berkeley's immaterialist philosophy(that there are no material objects, only minds and ideas in those minds), by kicking a large stone and asserting, “Irefute it thus.”[3] This action, which fails to prove the existence of the stone outside of the ideas formed by perception,fails to contradict Berkeley’s argument, and has been seen as merely dismissing it.[2]

12.1 Example

• Speaker A: Infectious diseases are caused by microbes.

• Speaker B: What a ridiculous idea!

• Speaker A: How so?

• Speaker B: It’s obviously ridiculous.

Speaker B gives no evidence or reasoning, and when pressed, claims that Speaker A’s statement is inherently absurd,thus applying the fallacy.

12.2 See also

• Proof by assertion

• Ad hominem

• Solvitur ambulando

12.3 References[1] “Definitions of Fallacies”, Dianah Mertz Hsieh, 20 August 1995

[2] Pirie, Madsen (2006). How to win every argument: the use and abuse of logic. Continuum International Publishing Group.p. 101–103. ISBN 978-0-8264-9006-3.

14

12.3. REFERENCES 15

[3] Alexander, Samuel (2000). “Dr. Johnson as a Philosopher”. In Slater, John. Collected works of Samuel Alexander 4.Continuum International Publishing Group. p. 119. ISBN 978-1-85506-853-7.

Chapter 13

Assertoric

An assertoric proposition in Aristotelian logic merely asserts that something is (or is not) the case, in contrast toproblematic propositions which assert the possibility of something being true, or apodeictic propositions which assertthings which are necessarily or self-evidently true or false.[1] For instance, “Chicago is larger than Omaha” is asser-toric. “A corporation could be wealthier than a country” is problematic. “Two plus two equals four” is apodeictic.

13.1 Notes[1] Kant contrasts “apodictic” with “problematic” and “assertoric” in the Critique of Pure Reason, on page A70/B95.

13.2 References• Antony Flew. A Dictionary of Philosophy – Revised Second Edition St. Martin’s Press, NY, 1979

13.3 External links• The dictionary definition of assertoric at Wiktionary

16

Chapter 14

Association for Symbolic Logic

The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic andphilosophical logic—the largest such organization in the world. The ASL was founded in 1936 and its first presidentwas Alonzo Church. The current president of the ASL is Alasdair Urquhart.

14.1 Publications

The ASL publishes books and academic journals. Its three official journals are

• Journal of Symbolic Logic (website) – publishes research in all areas of mathematical logic. Founded in 1936,ISSN 0022-4812.

• Bulletin of Symbolic Logic (website) – publishes primarily expository articles and reviews. Founded in 1995,ISSN 1079-8986.

• Review of Symbolic Logic (website) – publishes research relating to logic, philosophy, science, and their inter-actions. Founded in 2008, ISSN 1755-0203.

In addition, the ASL has a sponsored journal,

• Journal of Logic and Analysis (website) – Publishes research on the interactions between mathematical logicand pure and applied analysis. Founded in 2009 as an open-access successor to the Springer journal Logic andAnalysis. ISSN 1759-9008.

The organization also played an important role in publishing the collected writings of Kurt Gödel.

14.2 Meetings

The ASL holds two main meetings every year, one in the United States and one in Europe (the latter known as theLogic Colloquium). In addition, the ASL regularly holds joint meetings with both the AmericanMathematical Society(“AMS”) and the American Philosophical Association (“APA”), and sponsors meetings in many different countriesevery year.

14.3 External links• ASL website

17

Chapter 15

Bar induction

Bar induction is a reasoning principle used in intuitionistic mathematics, introduced by L.E.J. Brouwer. It is usefulin giving constructive versions of classical results. It is based on an inductive argument.The goal of the principle is to prove properties of infinite streams of natural numbers, called choice sequences inintuitionistic terminology, by inductively reducing them to decidable properties of finite lists.Given two predicates R and S on finite lists of natural numbers, assume the following conditions hold:

• R is decidable;

• Every choice sequence has a finite prefix satisfying R (this is expressed by saying that R is a bar);

• Every list satisfying R also satisfies S;

• If all extensions of a list by one element satisfy S, then that list also satisfies S.

Then we can conclude that S holds for the empty list.In classical reverse mathematics, “bar induction” (BI) denotes the related principle stating that if a relation R is awell-order, then we have the schema of transfinite induction over R for arbitrary formulas.

15.1 References• S.C. Kleene, R.E. Vesley, The foundations of intuitionistic mathematics: especially in relation to recursive func-

tions, North-Holland (1965)

• Michael Dummett, Elements of intuitionism, Clarendon Press (1977)

• A. S. Troelstra, Choice sequences, Clarendon Press (1977)

• Dragalin, A.G. (2001), “Bar induction”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Michael Rathjen, The role of parameters in bar rule and bar induction, Journal of Symbolic Logic 56 (1991),no. 2, pp. 715–730.

18

Chapter 16

Baralipton

In classical logic, baralipton is a mnemonic word used to memorize a syllogism. Specifically, it is when the firsttwo propositions thereof are general, and the third particular; the middle term being the subject of the first, and theattribute of the second. Generally stated, if every M is L, and every S is M, then some L is S. For example,

Every evil ought to be feared.Every violent passion is an evil.Therefore, something that ought to be feared is a violent passion.

16.1 References1. This article incorporates text from a publication now in the public domain: Chambers, Ephraim, ed. (1728).

"article name needed". Cyclopædia, or an Universal Dictionary of Arts and Sciences (first ed.). James and JohnKnapton, et al.

19

Chapter 17

Barcan formula

In quantified modal logic, the Barcan formula and the converse Barcan formula (more accurately, schemata ratherthan formulas) (i) syntactically state principles or interchange between quantifiers and modalities; (ii) semanticallystate a relation between domains of possible worlds. The formulas were introduced as axioms by Ruth BarcanMarcus,in the first extensions of modal propositional logic to include quantification. [1]

Related formulas include the Buridan formula, and the converse Buridan formula.

17.1 The Barcan formula

The Barcan formula is:

∀x□Fx→ □∀xFx

In English, the schema reads: If everything is necessarily F, then it is necessary that everything is F. It is equivalentto

♢∃xFx→ ∃x♢FxThe Barcan formula has generated some controversy because - in terms of possible world semantics - it implies thatall objects which exist in any possible world (accessible to the actual world) exist in the actual world, i.e. that domainscannot grow when one moves to accessible worlds. This thesis is sometimes known as actualism--i.e. that there areno merely possible individuals. There is some debate as to the informal interpretation of the Barcan formula and itsconverse.

17.2 Converse Barcan formula

The converse Barcan formula is:

□∀xFx→ ∀x□FxIf a frame is based on a symmetric accessibility relation, then the Barcan formula will be valid in the frame if, and onlyif, the converse Barcan formula is valid in the frame. It states that domains cannot shrink as one moves to accessibleworlds, i.e. that individuals cannot cease to be possible. The converse Barcan formula is taken to be more plausiblethan the Barcan formula.

17.3 See also

Commutative property

20

17.4. REFERENCES 21

17.4 References[1] Journal of Symbolic Logic (1946),11 and (1947), 12 under Ruth C. Barcan

17.5 External links• Barcan both ways by Melvin Fitting

• Contingent Objects and the Barcan Formula by Hayaki Reina

Chapter 18

Binary decision

A binary decision is a choice between two alternatives, for instance between taking some specific action or not takingit.[1]

Binary decisions are basic to many fields. Examples include:

• Truth values in mathematical logic, and the corresponding Boolean data type in computer science, representinga value which may be chosen to be either true or false.[2]

• Conditional statements (if-then or if-then-else) in computer science, binary decisions about which piece ofcode to execute next.[3]

• Decision trees and binary decision diagrams, representations for sequences of binary decisions.[4]

• Binary choice, a statistical model for the outcome of a binary decision.[5]

18.1 References[1] Snow, Roberta M.; Phillips, Paul H. (2007),Making Critical Decisions: A Practical Guide for Nonprofit Organizations, John

Wiley & Sons, p. 44, ISBN 9780470185032.

[2] Dixit, J. B. (2009), Computer Fundamentals and Programming in C, Firewall Media, p. 61, ISBN 9788170088820.

[3] Yourdon, Edward (March 19, 1975), “Clear thinking vital: Nested IFs not evil plot leading to program bugs”,Computerworld:15.

[4] Clarke, E. M.; Grumberg, Orna; Peled, Doron (1999), Model Checking, MIT Press, p. 51, ISBN 9780262032704.

[5] Ben-Akiva, Moshe E.; Lerman, Steven R. (1985), Discrete Choice Analysis: Theory and Application to Travel Demand,Transportation Studies 9, MIT Press, p. 59, ISBN 9780262022170.

22

Chapter 19

Calculus of structures

The calculus of structures is a proof calculus with deep inference for studying the structural proof theory ofnoncommutative logic. The calculus has since been applied to study linear logic, classical logic, modal logic, andprocess calculi, and many benefits are claimed to follow in these investigations from the way in which deep inferenceis made available in the calculus.

19.1 References• Alessio Guglielmi (2004)., 'A System of Interaction and Structure'. ACM Transactions on ComputationalLogic.

• Kai Brünnler (2004). Deep Inference and Symmetry in Classical Proofs. Logos Verlag.

19.2 External links• Calculus of structures homepage

• CoS in Maude: page documenting implementations of logical systems in the calculus of structures, using theMaude system.

23

Chapter 20

Card paradox

The card paradox is a non-self-referential variant of the liar paradox constructed by Philip Jourdain.[1] It is alsoknown as the postcard paradox, Jourdain paradox or Jourdain’s paradox.

20.1 The paradox

Suppose there is a card with statements printed on both sides:Trying to assign a truth value to either of them leads to a paradox.

1. If the first statement is true, then so is the second. But if the second statement is true, then the first statementis false. It follows that if the first statement is true, then the first statement is false.

2. If the first statement is false, then the second is false, too. But if the second statement is false, then the firststatement is true. It follows that if the first statement is false, then the first statement is true.

The same mechanism applies to the second statement. Neither of the sentences employs self-reference, instead this isa case of circular reference. See Yablo’s paradox for a variation of the liar paradox that does not even rely on circularreference.

20.2 See also• Circular reference

• Liar paradox

• List of paradoxes

• Self-reference

• Yablo’s paradox

20.3 References[1] O'Connor, John J.; Robertson, Edmund F. (February 2005). “Philip Edward Bertrand Jourdain”. The MacTutor History

of Mathematics archive. Retrieved 4 April 2010.

24

Chapter 21

Cayenne (programming language)

Cayenne is a functional programming language with dependent types. The basic types are functions, products, andsums. Functions and products use dependent types to gain additional power.There are very few building blocks in the language, but much syntactic sugar to make it more readable. The syntaxis largely borrowed from Haskell.There is no special module system, because with dependent types records (products) are powerful enough to definemodules.The main aim with Cayenne is not to use the types to express specifications (although this can be done), but ratherto use the type system to give type to more functions. An example of a function that can be given a type in Cayenneis printf.PrintfType :: String -> # PrintfType (Nil) = String PrintfType ('%':('d':cs)) = Int -> PrintfType cs PrintfType('%':('s’:cs)) = String -> PrintfType cs PrintfType ('%':( _ :cs)) = PrintfType cs PrintfType ( _ :cs) = PrintfTypecs aux :: (fmt::String) -> String -> PrintfType fmt aux (Nil) out = out aux ('%':('d':cs)) out = \ (i::Int) -> aux cs (out++ show i) aux ('%':('s’:cs)) out = \ (s::String) -> aux cs (out ++ s) aux ('%':( c :cs)) out = aux cs (out ++ c : Nil) aux(c:cs) out = aux cs (out ++ c : Nil) printf :: (fmt::String) -> PrintfType fmt printf fmt = aux fmt Nil

The Cayenne implementation is written in Haskell, and it also translates to Haskell.

21.1 External links• A description.

25

Chapter 22

Cirquent calculus

Cirquent calculus is a proof calculus which manipulates graph-style constructs termed cirquents, as opposed to thetraditional tree-style objects such as formulas or sequents. Cirquents come in a variety of forms, but they all share onemain characteristic feature, making them different from the more traditional objects of syntactic manipulation. Thisfeature is the ability to explicitly account for possible sharing of subcomponents between different components. Forinstance, it is possible to write an expression where two subexpressions F and E, while neither one is a subexpressionof the other, still have a common occurrence of a subexpression G (as opposed to having two different occurrencesof G, one in F and one in E).The approachwas introduced byG. Japaridze in[1] as an alternative proof theory capable of “taming” various nontrivialfragments his computability logic, which had otherwise resisted all axiomatization attempts within the traditionalproof-theoretic frameworks.[2] [3]

The basic version of cirquent calculus in[4] was accompanied with an "abstract resource semantics" and the claim thatthe latter was an adequate formalization of the resource philosophy traditionally associated with linear logic. Basedon that claim and the fact that the semantics induced a logic properly stronger than (affine) linear logic, Japaridzeargued that linear logic was incomplete as a logic of resources. Furthermore, he argued that not only the deductivepower but also the expressive power of linear logic was weak, for it, unlike cirquent calculus, failed to capture theubiquitous phenomenon of resource sharing.[5]

Among the later-found applications of cirquent calculus was the use of it to define a semantics for purely propositionalindependence-friendly logic.[6] The corresponding logic was axiomatized by W. Xu.[7]

22.1 Literature• M.Bauer, “The computational complexity of propositional cirquent calculus”. Logical Methods is ComputerScience 11 (2015),

Issue 1, Paper 12, pp. 1–16.

• G.Japaridze, “Introduction to cirquent calculus and abstract resource semantics”. Journal of Logic and Com-putation 16 (2006), pp. 489–532.

• G.Japaridze, “Cirquent calculus deepened.” Journal of Logic and Computation 18 (2008), pp. 983–1028.

• G.Japaridze, “From formulas to cirquents in computability logic”. Logical Methods is Computer Science 7(2011), Issue 2 , Paper 1, pp. 1–55.

• G.Japaridze, “The taming of recurrences in computability logic through cirquent calculus, Part I”.Archive forMathematical Logic 52 (2013), pages 173–212.

• G.Japaridze, “The taming of recurrences in computability logic through cirquent calculus, Part II” Archive forMathematical Logic 52 (2013), pages 213–259.

• W.Xu and S.Liu, “Soundness and completeness of the cirquent calculus system CL6 for computability logic”.Logic Journal of the IGPL 20 (2012), pp. 317–330.

26

22.2. REFERENCES 27

• W.Xu and S.Liu, “Cirquent calculus system CL8S versus calculus of structures system SKSg for propositionallogic”. In: Quantitative Logic and Soft Computing. GuojunWang, Bin Zhao andYongming Li, eds. Singapore,World Scientific, 2012, pp. 144–149.

• W.Xu, “A propositional system induced by Japaridze’s approach to IF logic”. Logic Journal of the IGPL 22(2014), pages 982–991.

22.2 References[1] G.Japaridze, “Introduction to cirquent calculus and abstract resource semantics”. Journal of Logic and Computation 16

(2006), pp. 489–532.

[2] G.Japaridze, “The taming of recurrences in computability logic through cirquent calculus, Part I”.Archive forMathematicalLogic 52 (2013), pages 173-212.

[3] G.Japaridze, “The taming of recurrences in computability logic through cirquent calculus, Part II” Archive forMathematicalLogic 52 (2013), pages 213–259.

[4] G.Japaridze, "Introduction to cirquent calculus and abstract resource semantics". Journal of Logic and Computation 16(2006), pp. 489–532.

[5] G.Japaridze, “Cirquent calculus deepened.” Journal of Logic and Computation 18 (2008), pp. 983–1028.

[6] G.Japaridze, “From formulas to cirquents in computability logic”. Logical Methods is Computer Science 7 (2011), Issue2 , Paper 1, pp. 1–55.

[7] W.Xu, “A propositional system induced by Japaridze’s approach to IF logic”. Logic Journal of the IGPL 22 (2014), pages982–991.

Chapter 23

Classical modal logic

In modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem) the duality of themodal operators♢A ≡ ¬□¬Awhich is also closed under the ruleA ≡ B ⊢ □A ≡ □B.Alternatively one can give a dual definition of L by which L is classical iff it contains (as axiom or theorem)□A ≡ ¬♢¬Aand is closed under the ruleA ≡ B ⊢ ♢A ≡ ♢B.The weakest classical system is sometimes referred to as E and is non-normal. Both algebraic and neighborhoodsemantics characterize familiar classical modal systems that are weaker than the weakest normal modal logic K.Every regular modal logic is classical, and every normal modal logic is regular and hence classical.

23.1 References

Chellas, Brian. Modal Logic: An Introduction. Cambridge University Press, 1980.

23.2 Notes

23.3 External links

28

Chapter 24

Cointerpretability

In mathematical logic, cointerpretability is a binary relation on formal theories: a formal theory T is cointerpretablein another such theory S, when the language of S can be translated into the language of T in such a way that S provesevery formula whose translation is a theorem of T. The “translation” here is required to preserve the logical structureof formulas.This concept, in a sense dual to interpretability, was introduced by Dzhaparidze (1993), who also proved that, fortheories of Peano arithmetic and any stronger theories with effective axiomatizations, cointerpretability is equivalentto Σ1 -conservativity.

24.1 See also• Cotolerance

• interpretability logic.

• Tolerance (in logic)

24.2 References• Dzhaparidze, Giorgie (1993), “A generalized notion of weak interpretability and the corresponding modallogic”,Annals of Pure andApplied Logic 61 (1-2): 113–160, doi:10.1016/0168-0072(93)90201-N,MR1218658.

• Japaridze, Giorgi; de Jongh, Dick (1998), “The logic of provability”, in Buss, Samuel R., Handbook of ProofTheory, Studies in Logic and the Foundations of Mathematics 137, Amsterdam: North-Holland, pp. 475–546,doi:10.1016/S0049-237X(98)80022-0, MR 1640331.

29

Chapter 25

Completeness (knowledge bases)

A knowledge base KB is complete if there is no formular α such that KB ⊭ α and KB ⊭ ¬α.Example of knowledge base with incomplete knowledge:KB := { A ∨ B }Then we have KB ⊭ A and KB ⊭ ¬A.In some cases, you can make a consistent knowledge base complete with the closed world assumption - that is, addingall not-entailed literals as negations to the knowledge base. In the above example though, this would not work becauseit would make the knowledge base inconsistent:KB' = { A ∨ B, ¬A, ¬B }In the case you have KB := { P(a), Q(a), Q(b) }, you have KB ⊭ P(b) and KB ⊭ ¬P(b), so with the closed worldassumption you would get KB' = { P(a), ¬P(b), Q(a), Q(b) } where you have KB' ⊨ ¬P(b).See also:

• Vivid knowledge

30

Chapter 26

Comprehension (logic)

In logic, the comprehension of an object is the totality of intensions, that is, attributes, characters, marks, properties,or qualities, that the object possesses, or else the totality of intensions that are pertinent to the context of a givendiscussion. This is the correct technical term for the whole collection of intensions of an object, but it is common inless technical usage to see 'intension' used for both the composite and the primitive ideas.

26.1 See also• Extension

• Extensional definition

• Intension

• Intensional definition

31

Chapter 27

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.

27.1 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.

32

Chapter 28

Consequent

For other uses, see Consequence.

A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it isthe part that follows “then”. In an implication, if ϕ implies ψ then ϕ is called the antecedent and ψ is called theconsequent.[1]

Examples:

• If P, then Q.

Q is the consequent of this hypothetical proposition.

• If X is a mammal, then X is an animal.

Here, “X is an animal” is the consequent.

• If computers can think, then they are alive.

“They are alive” is the consequent.The consequent in a hypothetical proposition is not necessarily a consequence of the antecedent.

• If monkeys are purple, then fish speak Klingon.

“Fish speak Klingon” is the consequent here, but intuitively is not a consequence of (nor does it have anything to dowith) the claim made in the antecedent that “monkeys are purple”.

28.1 See also• Antecedent (logic)

• Necessity and sufficiency

28.2 References[1] Sets, Functions and Logic - An Introduction to Abstract Mathematics, Keith Devlin, Chapman & Hall/CRC Mathematics,

3rd ed., 2004

33

Chapter 29

Conservativity theorem

In mathematical logic, the conservativity theorem states the following: Suppose that a closed formula

∃x1 . . .∃xm φ(x1, . . . , xm)

is a theorem of a first-order theory T . Let T1 be a theory obtained from T by extending its language with newconstants

a1, . . . , am

and adding a new axiom

φ(a1, . . . , am)

Then T1 is a conservative extension of T , which means that the theory T1 has the same set of theorems in the originallanguage (i.e., without constants ai ) as the theory T .In a more general setting, the conservativity theorem is formulated for extensions of a first-order theory by intro-ducing a new functional symbol:

Suppose that a closed formula ∀y⃗ ∃xφ(x, y⃗) is a theorem of a first-order theory T , where we denotey⃗ := (y1, . . . , yn) . Let T1 be a theory obtained from T by extending its language with new functionalsymbol f (of arity n ) and adding a new axiom ∀y⃗ φ(f(y⃗), y⃗) . Then T1 is a conservative extension ofT , i.e. the theories T and T1 prove the same theorems not involving the functional symbol f ).

29.1 References• Elliott Mendelson (1997). Introduction to Mathematical Logic (4th ed.) Chapman & Hall.

• J.R. Shoenfield (1967). Mathematical Logic. Addison-Wesley Publishing Company.

34

Chapter 30

Consistency (knowledge bases)

A knowledge base KB is consistent iff its negation is not a tautology.I.e., a knowledge base KB is inconsistent (not consistent) iff there is no interpretation which entails KB.Example of an inconsistent knowledge base:KB := { a, ¬a }Consistency in terms of knowledge bases is mostly the same as the natural understanding of consistency.

35

Chapter 31

Contradictio in terminis

Contradictio in terminis (Latin for contradiction in terms) refers to a combination of words whose meanings are inconflict with one another. Examples are “liquid ice”, “independent colony”, and “square circle”.If the contradiction is intentional (rhetorical or poetic), then one can speak of an oxymoron.

31.1 See also• Contradiction

• Meinong’s jungle

• Oxymoron

• Paradox

• Principle of contradiction

• Self-refuting idea

36

Chapter 32

Converse accident

The fallacy of converse accident (also called reverse accident, destroying the exception, or a dicto secundumquid ad dictum simpliciter) is an informal fallacy that can occur in a statistical syllogism when an exception to ageneralization is wrongly excluded, and the generalization wrongly called for as applying to all cases.For example:

If we allow people with glaucoma to use medical marijuana, then everyone should be allowed to usemarijuana.

The inductive version of this fallacy is called hasty generalization. See faulty generalization.This fallacy is similar to the slippery slope, where the opposition claims that if a restricted action under debate isallowed, such as allowing people with glaucoma to use medical marijuana, then the action will by stages becomeacceptable in general, such as eventually everyone being allowed to use marijuana. The two arguments imply thereis no difference between the exception and the rule, and in fact fallacious slippery slope arguments often use theconverse accident to the contrary as the basis for the argument. However, a key difference between the two is thepoint and position being argued. The above argument using converse accident is an argument for full legal use ofmarijuana given that glaucoma patients use it. The argument based on the slippery slope argues against medicinaluse of marijuana because it will lead to full use.The opposing kind of dicto simpliciter is accident.

32.1 External links• Stephen’s guide: Converse accident

37

Chapter 33

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

33.1 Definition

33.1.1 Truth table

The truth table of A⊂B

33.1.2 Venn diagram

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

33.2 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.

33.3 Symbol

33.4 Natural language

“Not q without p.”“p if q.”

38

33.5. BOOLEAN ALGEBRA 39

33.5 Boolean Algebra

(A + B')

33.6 See also• Logical connective

• Material implication

Chapter 34

Counterargument

Counterargument on Graham's Hierarchy of Disagreement

In reasoning and argument mapping, a counterargument is an objection to an objection. A counterargument can beused to rebut an objection to a premise, a main contention or a lemma. Synonyms of counterargument may includerebuttal, reply, counterstatement, counterreason, comeback and response. The attempt to rebut an argument mayinvolve generating a counterargument or finding a counterexample.[1]

To speak of counterarguments is not to assume that there are only two sides to a given issue nor that there is onlyone type of counterargument.[2] For a given argument, there are often a large number of counterarguments, some ofwhich are not compatible with each other.[2]

A counterargument might seek to cast doubt on facts of one or more of the first argument’s premises, to show that thefirst argument’s contention does not follow from its premises in a valid manner, or the counterargument might paylittle attention to the premises and common structure of the first argument and simply attempt to demonstrate thatthe truth of a conclusion is incompatible with that of the first argument.

40

34.1. SPEECH 41

34.1 Speech

In a debate or in a speaking context, a counterargument can be handled in a variety of ways.[3]

Responding to a counterargument does not mean utterly obliterating it. You may concede it, min-imize it, dismiss it as irrelevant, or attack the supporting evidence or underlying premise. Even if yougrant the existence of a problem, you can differ from your audience on the best solution.[3]

34.2 See also• Inference objection

34.3 Notes[1] Rahwan, Iyad et al. (2009). Argumentation in Multi-Agent Systems, p. 186., p. 186, at Google Books

[2] Zwiers, Jeff. (2008). Developing Academic Thinking Skills in Grades 6-12, p. 111., p. 111, at Google Books

[3] Sprague, Jo et al. (2008). The Speaker’s Handbook, p. 326., p. 326, at Google Books

34.4 References• Rahwan, Iyad and Pavlos Moraitis. (2009). Argumentation in Multi-Agent Systems. Berlin: Springer. 10-ISBN3642002064/13-ISBN 9783642002069; OCLC 496892202

• Sprague, Jo and Douglas Stuart. (2008). The Speaker’s Handbook. San Diego: Harcourt Brace Jovanovich.10-ISBN 0155831755/13-ISBN 9780155831759; OCLC 10795362

• Zwiers, Jeff. (2004). Developing Academic Thinking Skills in Grades 6-12: a Handbook of Multiple Intel-ligence Activities. Newark, Delaware: International Reading Association. 10-ISBN 0872075575/13-ISBN9780872075573; OCLC 300267606

34.5 External links• Harvey, Gordon: “Counter-Argument”, adapted from The Academic Essay: A Brief Anatomy, The WritingCenter, Harvard College

Chapter 35

Counterinduction

Counterinduction is the rule of inference that one should assume the opposite of what induction suggests. Forexample:

“The Sun has risen every day in the past, therefore I think that it will not rise tomorrow.”

In most references to counterinduction, it is not suggested that counterinduction is valid. It is instead a refutation ofMax Black's proposed inductive justification of induction, since the counterinductive justification of counterinductionis formally identical to the inductive justification of induction.[1]

Paul Feyerabend's anarchist theory popularized the notion of counterinduction.

35.1 References[1] In a nutshell, Hume’s comes up with three basic questions:

42

Chapter 36

Counting quantification

A counting quantifier is a mathematical term for a quantifier of the form “there exists at least k elements that satisfyproperty X". In first-order logic with equality, counting quantifiers can be defined in terms of ordinary quantifiers,so in this context they are a notational shorthand. However, they are interesting in the context of logics such as two-variable logic with counting that restrict the number of variables in formulas. Also, generalized counting quantifiersthat say “there exists infinitely many” are not expressible using a finite number of formulas in first-order logic.

36.1 See also• Uniqueness quantification

36.2 References• Erich Graedel, Martin Otto, and Eric Rosen. “Two-Variable Logic with Counting is Decidable.” In Proceedings

of 12th IEEE Symposium on Logic in Computer Science LICS `97, Warschau. 1997. Postscript file OCLC282402933

43

Chapter 37

Deep inference

Deep inference names a general idea in structural proof theory that breaks with the classical sequent calculus bygeneralising the notion of structure to permit inference to occur in contexts of high structural complexity. The termdeep inference is generally reserved for proof calculi where the structural complexity is unbounded; in this article wewill use non-shallow inference to refer to calculi that have structural complexity greater than the sequent calculus,but not unboundedly so, although this is not at present established terminology.Deep inference is not important in logic outside of structural proof theory, since the phenomena that lead to theproposal of formal systems with deep inference are all related to the cut-elimination theorem. The first calculus ofdeep inference was proposed by Kurt Schütte,[1] but the idea did not generate much interest at the time.Nuel Belnap proposed display logic in an attempt to characterise the essence of structural proof theory. The calculusof structures was proposed in order to give a cut-free characterisation of noncommutative logic.

37.1 Notes[1] Kurt Schütte. Proof Theory. Springer-Verlag, 1977.

37.2 Further reading• Kai Brünnler, “Deep Inference and Symmetry in Classical Proofs” (Ph.D. thesis 2004) , also published in bookform by Logos Verlag (ISBN 978-3-8325-0448-9).

• Deep Inference and the Calculus of Structures Intro and reference web page about ongoing research in deepinference.

44

Chapter 38

Defeasible logic

Defeasible logic is a non-monotonic logic proposed by Donald Nute to formalize defeasible reasoning. In defeasiblelogic, there are three different types of propositions:

strict rules specify that a fact is always a consequence of another;

defeasible rules specify that a fact is typically a consequence of another;

undercutting defeaters specify exceptions to defeasible rules.

A priority ordering over the defeasible rules and the defeaters can be given. During the process of deduction, thestrict rules are always applied, while a defeasible rule can be applied only if no defeater of a higher priority specifiesthat it should not.

38.1 See also• Common sense

• Non-monotonic logic

• Default logic

• Defeasible reasoning

38.2 References• D. Nute (1994). Defeasible logic. InHandbook of logic in artificial intelligence and logic programming, volume3: Nonmonotonic reasoning and uncertain reasoning, pages 353-395. Oxford University Press.

• G. Antoniou, D. Billington, G. Governatori, and M. Maher (2001). Representation results for defeasible logic.ACM Transactions on Computational Logic, 2(2):255-287.

45

Chapter 39

Degree of truth

In standard mathematics, propositions can typically be considered unambiguously true or false. For instance, theproposition zero belongs to the set { 1 } is regarded as simply false; while the proposition one belongs to the set { 1 } isregarded as simply true. However, some mathematicians, computer scientists, and philosophers have been attractedto the idea that a proposition might be more or less true, rather than simply true or simply false. Consider My coffeeis hot.In mathematics, this idea can be developed in terms of fuzzy logic. In computer science, it has found application inartificial intelligence. In philosophy, the idea has proved particularly appealing in the case of vagueness. Degrees oftruth is an important concept in law.

39.1 See also• Artificial intelligence

• Bivalence

• Fuzzy logic

• Fuzzy set

• Half-truth

• Multi-valued logic

• Paradox of the heap

• Truth

• Truth value

• Vagueness

39.2 Bibliography• Zadeh, L.A. (1965). “Fuzzy sets”. Information and Control 8 (3): 338–353. doi:10.1016/S0019-9958(65)90241-X. ISSN 0019-9958.

46

Chapter 40

Descriptive fallacy

The descriptive fallacy refers to reasoning which treats a speech act as a logical proposition, which would bemistakenwhen the meaning of the statement is not based on its truth condition.[1] It was suggested by the British philosopherof language J. L. Austin in 1955 in the lectures now known as How to Do Things With Words. Austin argued thatperformative utterances are not meaningfully evaluated as true or false but rather by other measures, which wouldhold that a statement such as “thank you” is not meant to describe a fact and to interpret it as such would be to committhe descriptive fallacy.

40.1 Role of ‘descriptive fallacy’ in Austin’s philosophy

Austin’s label of ‘descriptive fallacy’ was aimed primarily at logical positivism, and his speech act theory was largelya response to logical positivism’s view that only statements that are logically or empirically verifiable have cognitivemeaning.[2] Logical positivism aimed to approach philosophy on the model of empirical science, seeking to expressphilosophical statements in ways to render them verifiable by empirical means. Statements that cannot be verifiedas either true or false are seen as meaningless. This would exclude many statements about religion, metaphysics,aesthetics, or ethics as meaningless and philosophically uninteresting, making merely emotive or evocative claimsexpressing one’s feelings rather than making verifiable claims about reality.[3]

Austin disagreed with the positivist’s contention that the only philosophically significant use of language is to describereality by stating facts, pointing out that speakers do much more with language than merely describe reality. Forexample, asking questions, making requests or issuing orders, offering invitations, making promises, and many othercommon statements are not descriptive. Rather, they are performative: in making such statements, speakers do thingsrather than describe things.[4]

Based on this distinction of what Austin labeled as constative utterances (statements that describe, which were thefocus of logical positivism) and performative utterances (statements that perform or do something), Austin developedhis speech act theory to investigate how we do things with words.[5]

40.2 References[1] Bunnin, Nicholas; Yu, Jiyuan, eds. (2004). “Descriptive fallacy”. The Blackwell Dictionary of Western Philosophy. ISBN

978-1-4051-0679-5.

[2] Chapman, Siobhan; Routledge, Christopher, eds. (2009). “Speech Act Theory”. Key Ideas in Linguistics and the Philosophyof Language. ISBN 9781849724517.

[3] Honderich, Ted, ed. (2005). “Logical Positivism”. The Oxford Companion to Philosophy. ISBN 9780199264797.

[4] Chapman, Siobhan (2000). Philosophy for Linguists: An Introduction. New York: Routledge. pp. 106–143. ISBN9780415206594.

[5] Hogan, Patrick, ed. (2011). “Performative and Constative”. The Cambridge Encyclopedia of the Language Sciences. ISBN9781139144711.

47

Chapter 41

Don't-care term

In digital logic, a don't-care term for a function is an input-sequence (a series of bits) that is known never to occur.The designer of a logic circuit to implement the function need not care about such inputs, but can choose the circuit’soutput arbitrarily, usually such that the simplest circuit results (minimization). Examples of don't-care terms arethe binary values 1010 through 1111 (10 through 15 in decimal) for a function that takes a binary-coded decimal(BCD) value, because a BCD value never takes on such values (so called pseudo-tetrades); in the pictures, the circuitcomputing the lower left bar of a 7-segment display can be minimized to a b + a c + a d by an appropriate choice ofcircuit outputs for dcba=1010...1111.Don't-care terms are important to consider in minimizing logic circuit design, using Karnaugh maps and the Quine–McCluskey algorithm. Don't care optimization can also be used in the development of highly size-optimized assemblyor machine code taking advantage of side effects.

41.1 X value

“Don't care” may also refer to an unknown value in a multi-valued logic system, in which case it may also be calledan X value. In the Verilog hardware description language such values are denoted by the letter “X”. In the VHDLhardware description language such values are denoted (in the standard logic package) by the letter “X” (forcedunknown) or the letter “W” (weak unknown).[1]

An X value does not exist in hardware. In simulation, an X value can result from two or more sources driving a signalsimultaneously, or the stable output of a flip-flop (electronics) not having been reached. In synthesized hardware,however, the actual value of such a signal will be either 0 or 1, but will not be determinable from the circuit’s inputs.[1]

41.2 See also• Karnaugh map

• Quine–McCluskey algorithm

• Decision table

41.3 References[1] David Naylor and Simon Jones (1997). Vhdl: A Logic Synthesis Approach. Springer. pp. 14–15,219,221. ISBN 0-412-

61650-5.

48

Chapter 42

Double turnstile

Not to be confused with .

In logic, the symbol ⊨, ⊨ or |= is called the double turnstile. It is closely related to the turnstile symbol ⊢ , which hasa single bar across the middle. It is often read as "entails", "models", “is a semantic consequence of” or “is strongerthan”.[1] In TeX, the turnstile symbols ⊨ and |= are obtained from the commands \vDash and \models respectively.In Unicode it is encoded at U+22A8 ⊨ true (HTML ⊨)In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, and iscapable of putting labels below or above it, in the correct places. The article A Tool for Logicians is a tutorial onusing this package.

42.1 Meaning

The double turnstile is a binary relation. It has several different meanings in different contexts:

• To show semantic consequence, with a set of sentences on the left and a single sentence on the right, to denotethat if every sentence on the left is true, the sentence on the right must be true, e.g. Γ ⊨ φ . This usage isclosely related to the single-barred turnstile symbol which denotes syntactic consequence.

• To show satisfaction, with a model (or truth-structure) on the left and a set of sentences on the right, to denotethat the structure is a model for (or satisfies) the set of sentences, e.g. A |= Γ .

• To denote a tautology, ⊨ φ . which is to say that the expression φ is a semantic consequence of the empty set.

42.2 See also• List of logic symbols

• List of mathematical symbols

42.3 References[1] Nederpelt, Rob (2004). “Chapter 7: Strengthening and weakening”. Logical Reasoning: A First Course (3rd revised ed.).

King’s College Publications. p. 62. ISBN 0-9543006-7-X.

49

Chapter 43

Effective method

In logic, mathematics and computer science, especially metalogic and computability theory, an effective method[1]or effective procedure is a procedure for solving a problem from a specific class. An effective method is sometimesalso calledmechanical method or procedure.[2]

43.1 Definition

A method is called effective for a class of problems iff

• it consists of a finite number of exact, finite instructions

• when applied to a problem from its class, it always finishes (terminates) after a finite number of steps

• when applied to a problem from its class, it always produces a correct answer

• in principle, it can be done by a human without any aids, except writing materials

• its instructions need only be followed rigorously to succeed; in particular, it requires no ingenuity to do so.[3]

Optionally, one may require that when an effective method is applied to a problem from outside the class for which itis effective, it may halt without result or diverge, but must not return a result as if it were the answer to the problem.Adding this requirement reduces the set of classes for which there is an effective method.

43.2 Algorithms

An effective method for calculating the values of a function is an algorithm. Functions for which an effective methodexists are sometimes called effectively calculable.

43.3 Computable functions

Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed def-initions (general recursion, Turing machines, λ-calculus) that later were shown to be equivalent. The notion capturedby these definitions is known as recursive or effective computability.The Church–Turing thesis states that the two notions coincide: any number-theoretic function that is effectivelycalculable is recursively computable. As this is not a mathematical statement, it cannot be proven by a mathematicalproof.

50

43.4. SEE ALSO 51

43.4 See also• Decidability (logic)

• Decision problem

• Function problem

• Effective results in number theory

• Recursive set

• Undecidable problem

43.5 References[1] Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California

Press, 1971

[2] Copeland, B.J.; Copeland, Jack; Proudfoot, Diane (June 2000). “The Turing-Church Thesis”. AlanTuring.net. TuringArchive for the History of Computing. Retrieved 23 March 2013.

[3] The Cambridge Dictionary of Philosophy, effective procedure

• S. C. Kleene (1967), Mathematical logic. Reprinted, Dover, 2002, ISBN 0-486-42533-9, pp. 233 ff., esp. p.231.

Chapter 44

Empty domain

In first-order logic the empty domain is the empty set having no members. In traditional and classical logic domainsare restrictedly non-empty in order that certain theorems be valid. Interpretations with an empty domain are shown tobe a trivial case by a convention originating at least in 1927 with Bernays and Schönfinkel (though possibly earlier) butoft-attributed to Quine 1951. The convention is to assign any formula beginning with a universal quantifier the valuetruth while any formula beginning with an existential quantifier is assigned the value falsehood. This follows from theidea that existentially quantified statements have existential import (i.e. they imply the existence of something) whileuniversally quantified statements do not. This interpretation reportedly stems from George Boole in the late 19thcentury but this is debatable. In modern model theory, it follows immediately for the truth conditions for quantifiedsentences:

• A |= ∃xϕ(x) an is there iff a ∈ A that such A |= ϕ[a]

• A |= ∀xϕ(x) every iff a ∈ A that such is A |= ϕ[a]

In other words, an existential quantification of the open formula φ is true in a model iff there is some element in thedomain (of the model) that satisfies the formula; i.e. iff that element has the property denoted by the open formula.A universal quantification of an open formula φ is true in a model iff every element in the domain satisfies thatformula. (Note that in the metalanguage, “everything that is such that X is such that Y” is interpreted as a universalgeneralization of the material conditional “if anything is such that X then it is such that Y”. Also, the quantifiers aregiven their usual objectual readings, so that a positive existential statement has existential import, while a universalone does not.) An analogous case concerns the empty conjunction and the empty disjunction. The semantic clausesfor, respectively, conjunctions and disjunctions are given by

• A |= ϕ1 ∧ · · · ∧ ϕn ⇐⇒ ∀ϕi(1 ≤ i ≤ n), A |= ϕi

• A |= ϕ1 ∨ · · · ∨ ϕn ⇐⇒ ∃ϕi(1 ≤ i ≤ n), A |= ϕi .

It is easy to see that the empty conjunction is trivially true, and the empty disjunction trivially false.Logics whose theorems are valid in every, including the empty, domain were first considered by Jaskowski 1934,Mostowski 1951, Hailperin 1953, Quine 1954, Leonard 1956, and Hintikka 1959. While Quine called such logics“inclusive” logic they are now referred to as free logic.

44.1 See also• Table of logic symbols

52

44.1. SEE ALSO 53

In modern logic only the contradictories in the square of opposition apply, because domains may be empty.(Black areas are empty,red areas are nonempty.)

Chapter 45

End term

The end terms in a categorical syllogism are the major term and the minor term (not the middle term). These twoterms appear together in the conclusion and separately with the middle term in the major premise and minor premise,respectively.Example:

Major premise: All M are P.Minor premise: All S are M.Conclusion: All S are P.

The end terms are in italics. S is the minor term, P is the major term, and M is the middle term.

54

Chapter 46

Enumerative definition

An enumerative definition of a concept or term is a special type of extensional definition that gives an explicit andexhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are onlypossible for finite sets and only practical for relatively small sets.An example of an enumerative definition would be, [types of Car brands] Ford, Chevrolet, Volkswagen, Toyota, etc...

46.1 See also• Definition

• Extension

• Extensional definition

• Set notation

• Enumeration

55

Chapter 47

Existential fallacy

The existential fallacy, or existential instantiation, is a formal fallacy. In the existential fallacy, we presuppose thata class has members when we are not supposed to do so; that is, when we should not assume existential import.One example would be: "Everyone in the room is pretty and smart". It does not imply that there is a pretty, smartperson in the room, because it does not state that there is a person in the room.An existential fallacy is committed in a medieval categorical syllogism because it has two universal premises and aparticular conclusion with no assumption that at least one member of the class exists, which is not established by thepremises.In modern logic, the presupposition that a class has members is seen as unacceptable. In 1905, Bertrand Russellwrote an essay entitled “The Existential Import of Proposition”, in which he called this Boolean approach "Peano'sinterpretation”.The fallacy does not occur in enthymemes, where hidden premises required to make the syllogism valid assume theexistence of at least one member of the class .

One central concern of the Aristotelian tradition in logic is the theory of the categorical syllogism.This is the theory of two-premised arguments in which the premises and conclusion share three termsamong them, with each proposition containing two of them. It is distinctive of this enterprise thateverybody agrees on which syllogisms are valid. The theory of the syllogism partly constrains the inter-pretation of the forms. For example, it determines that the A form has existential import, at least if theI form does. For one of the valid patterns (Darapti) is:

Every C is BEvery C is ASo, some A is B

This is invalid if the A form lacks existential import, and valid if it has existential import. It is held to bevalid, and so we know how the A form is to be interpreted. One then naturally asks about the O form;what do the syllogisms tell us about it? The answer is that they tell us nothing. This is because Aristotledid not discuss weakened forms of syllogisms, in which one concludes a particular proposition when onecould already conclude the corresponding universal. For example, he does not mention the form:

No C is BEvery A is CSo, some A is not B

If people had thoughtfully taken sides for or against the validity of this form, that would clearly be relevantto the understanding of the O form. But the weakened forms were typically ignored.[1]

—Terence Parsons, The Stanford Encyclopedia of Philosophy

47.1 See also• Vacuous truth

56

47.2. REFERENCES 57

47.2 References[1] Parsons, Terence (2012). “The Traditional Square of Opposition”. In Edward N. Zalta. The Stanford Encyclopedia of

Philosophy (Fall 2012 ed.). 3-4.

47.3 External links• Fallacy files: existential fallacy

• FOLDOC: existential fallacy

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008and incorporated under the “relicensing” terms of the GFDL, version 1.3 or later.

Chapter 48

Existential generalization

In predicate logic, existential generalization[1][2] (also known as existential introduction, ∃I) is a valid rule ofinference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, orexistential proposition. In first-order logic, it is often used as a rule for the existential quantifier (∃) in formal proofs.Example: “Rover loves to wag his tail. Therefore, something loves to wag its tail.”In the Fitch-style calculus:

Q(a) → ∃xQ(x)

Where a replaces all free instances of x within Q(x).[3]

48.1 Quine

Universal instantiation andExistential Generalization are two aspects of a single principle, for instead of saying that"∀x x=x" implies “Socrates=Socrates”, we could as well say that the denial “Socrates≠Socrates"' implies "∃x x≠x".The principle embodied in these two operations is the link between quantifications and the singular statements thatare related to them as instances. Yet it is a principle only by courtesy. It holds only in the case where a term namesand, furthermore, occurs referentially.[4]

48.2 See also• Inference rules

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

[2] Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.

[3] pg. 347. Jon Barwise and John Etchemendy, Language proof and logic Second Ed., CSLI Publications, 2008.

[4] Willard van Orman Quine; Roger F. Gibson (2008). “V.24. Reference and Modality”. Quintessence. Cambridge, Mass:Belknap Press of Harvard University Press. Here: p.366.

58

Chapter 49

Existential instantiation

In predicate logic, existential instantiation (also called existential elimination)[1][2][3] is a valid rule of inferencewhich says that, given a formula of the form (∃x)ϕ(x) , one may infer ϕ(c) for a new constant or variable symbolc. The rule has the restriction that the constant or variable c introduced by the rule must be a new term that has notoccurred earlier in the proof.In one formal notation, the rule may be denoted

(∃x)Fx :: Fa,

where a is an arbitrary term that has not been a part of our proof thus far.

49.1 See also• existential fallacy

49.2 References[1] Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.

[2] Copi and Cohen

[3] Moore and Parker

59

Chapter 50

Explanatory power

Explanatory power is the ability of a hypothesis or theory to effectively explain the subject matter it pertains to. Theopposite of explanatory power is explanatory impotence.In the past, various criteria or measures for explanatory power have been proposed. In particular, one hypothesis,theory or explanation can be said to have more explanatory power than another about the same subject matter- if more facts or observations are accounted for;- if it changes more “surprising facts” into “a matter of course” (following Peirce);- if more details of causal relations are provided, leading to a high accuracy and precision of the description;- if it offers greater predictive power, i.e., if it offers more details about what we should expect to see, and what weshould not;- if it depends less on authorities and more on observations;- if it makes fewer assumptions;- if it is more falsifiable, i.e., more testable by observation or experiment (following Popper).Recently, David Deutsch proposed that the correct hypothesis or theory, the one that stands out among all possibleexplanations, is that specific explanation that- is hard to vary.By this expression he intends to state that the correct theory, i.e., the true explanation, provides specific details whichfit together so tightly that it is impossible to change any one detail without affecting the whole theory.

50.1 Introduction

Philosopher and physicist David Deutsch offers a criterion for a good explanation that he says may be just as importantto scientific progress as learning to reject appeals to authority, and adopting formal empiricism and falsifiability. ToDeutsch, these aspects of a good explanation, and more, are contained in any theory that is specific and “hard tovary”. He believes that this criterion helps eliminate “bad explanations” which continuously add justifications, andcan otherwise avoid ever being truly falsified.[1]

50.2 Examples

Deutsch takes examples from Greek mythology. He describes how very specific, and even somewhat falsifiabletheories were provided to explain how the gods’ sadness caused the seasons. Alternatively, Deutsch points out, onecould have just as easily explained the seasons as resulting from the gods’ happiness - making it a bad explanation,because it is so easy to arbitrarily change details.[1] Without Deutsch’s criterion, the 'Greek gods explanation' couldhave just kept adding justifications. This same criterion, of being “hard to vary”, may be what makes the modernexplanation for the seasons a good one: none of the details - about the earth rotating around the sun at a certain anglein a certain orbit - can be easily modified without changing the theory’s coherence.[1]

60

50.3. RELATION TO OTHER CRITERIA 61

Carbon Cycle

Rivers

Storage in GtC

Atmosphere 750

Deep Ocean38.100

Vegetation 610

Sediments 150

Fossil Fuels &Cement Production4,000

Surface Ocean 1,020

Dissolved Organic Carbon<700

Soils1,580

Marine Biota3

Fluxes in GtC/yr

121.3

60

1.6

0.5

9290

50

64

40

6

91.6

0.2

100

5.5

CO2

60

Deutsch says that the truth consists of detailed and “hard to vary assertions about reality”

Special relativity is a description of motion, including extremely rapid motion, that correctly describes all knownobservations in flat space and also made numerous predictions. Special relativity is hard to vary, as it is based on justtwo principles: the principle of relativity and the invariance of the speed of light.General relativity is a description of gravitation that correctly reproduces all known observations and made numerouspredictions. What makes the theory appealing is its internal consistency: it is hard to vary, in the sense that one cannotchange a detail in it without changing all its predictions.String theory is a hypothesis for the unification of general relativity and quantum theory; but since it has not yetprovided an explanation of the standard model of particle physics that is hard to vary, the hypothesis is regularlyquestioned.

50.3 Relation to other criteria

It can be argued that the criterion hard to vary is closely related to Occam’s razor: both imply logical consistency anda minimum of assumptions.

50.4 References[1] David Deutsch, “A new way of explaining explanation”

Chapter 51

Extension (predicate logic)

The extension of a predicate – a truth-valued function – is the set of tuples of values that, used as arguments, satisfythe predicate. Such a set of tuples is a relation.For example the statement "d2 is the weekday following d1" can be seen as a truth function associating to each tuple(d2, d1) the value true or false. The extension of this truth function is, by convention, the set of all such tuplesassociated with the value true, i.e.{(Monday, Sunday), (Tuesday, Monday), (Wednesday, Tuesday), (Thursday, Wednesday), (Friday, Thursday), (Sat-urday, Friday), (Sunday, Saturday)}By examining this extension we can conclude that “Tuesday is the weekday following Saturday” (for example) is false.Using set-builder notation, the extension of the n-ary predicate Φ can be written as

{(x1, ..., xn) | Φ(x1, ..., xn)} .

51.1 Relationship with characteristic function

If the values 0 and 1 in the range of a characteristic function are identified with the values false and true, respectively –making the characteristic function a predicate – , then for all relations R and predicatesΦ the following two statementsare equivalent:

• Φ is the characteristic function of R;

• R is the extension of Φ .

51.2 See also• Extensionality

• Intension

62

Chapter 52

Extensionality

In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have thesame external properties. It stands in contrast to the concept of intensionality, which is concerned with whether theinternal definitions of objects are the same.

52.1 Example

Consider the two functions f and g mapping from and to natural numbers, defined as follows:

• To find f(n), first add 5 to n, then multiply by 2.

• To find g(n), first multiply n by 2, then add 10.

These functions are extensionally equal; given the same input, both functions always produce the same value. But thedefinitions of the functions are not equal, and in that intensional sense the functions are not the same.Similarly, in natural language there are many predicates (relations) that are intensionally different but are extensionallyidentical. For example, suppose that a town has one person named Joe, who is also the oldest person in the town.Then, the two argument predicates “has one person named”, “is the oldest person in” are intensionally distinct, butextensionally equal for “Joe” in that “town” now.

52.2 In mathematics

The extensional definition of function equality, discussed above, is commonly used in mathematics. Sometimesadditional information is attached to a function, such as an explicit codomain, in which case two functions must notonly agree on all values, but must also have the same codomain, in order to be equal.A similar extensional definition is usually employed for relations: two relations are said to be equal if they have thesame extensions.In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements.In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions—withtheir extension as stated above, so that it is impossible for two relations or functions with the same extension to bedistinguished.Other mathematical objects are also constructed in such a way that the intuitive notion of “equality” agrees with set-level extensional equality; thus, equal ordered pairs have equal elements, and elements of a set which are related byan equivalence relation belong to the same equivalence class.Type-theoretical foundations of mathematics are generally not extensional in this sense, and setoids are commonlyused to maintain a difference between intensional equality and a more general equivalence relation (which generallyhas poor constructibility or decidability properties).

63

64 CHAPTER 52. EXTENSIONALITY

52.3 See also• Duck typing

• Structural typing

• Univalence axiom

52.4 References

Chapter 53

Fallacies of illicit transference

A fallacy of illicit transference is an informal fallacy occurring when an argument assumes there is no differencebetween a term in the distributive (referring to every member of a class) and collective (referring to the class itself asa whole) sense.[1]

There are two variations of this fallacy:[1]

• Fallacy of composition - assumes what is true of the parts is true of the whole. This fallacy is also known as“arguing from the specific to the general.”

Since Judy is so diligent in the workplace, this entire company must have an amazing work ethic.

• Fallacy of division - assumes what is true of the whole is true of its parts (or some subset of parts).

Because this company is so corrupt, so must every employee within it be corrupt.

While fallacious, arguments that make these assumptions may be persuasive because of the representativeness heuris-tic.

53.1 References[1] Hurley, Patrick (2014), A Concise Introduction to Logic (12th ed.), Cengage Learning, pp. 161, 172, ISBN 978-1-285-

96556-7

53.2 See also• Existential fallacy

• Ecological fallacy

• Fallacy of the undistributed middle

• Ontogeny recapitulates phylogeny

65

Chapter 54

Fallacy of division

A fallacy of division occurs when one reasons logically that something true for the whole must also be true of all orsome of its parts.An example:

1. A Boeing 747 can fly unaided across the ocean.

2. A Boeing 747 has jet engines.

3. Therefore, one of its jet engines can fly unaided across the ocean.

The converse of this fallacy is called fallacy of composition, which arises when one fallaciously attributes a propertyof some part of a thing to the thing as a whole. Both fallacies were addressed by Aristotle in Sophistical Refutations.In the philosophy of the ancient Greek Anaxagoras, as claimed by the Roman atomist Lucretius,[1] it was assumedthat the atoms constituting a substance must themselves have the salient observed properties of that substance: soatoms of water would be wet, atoms of iron would be hard, atoms of wool would be soft, etc. This doctrine is calledhomoeomeria, and it depends on the fallacy of division.If a system as a whole has some property that none of its constituents has (or perhaps, it has it but not as a result ofsome constituent having that property), this is sometimes called an emergent property of the system.

54.1 Examples

In statistics an ecological fallacy is a logical fallacy in the interpretation of statistical data where inferences about thenature of individuals are deduced from inference for the group to which those individuals belong. The four commonstatistical ecological fallacies are: confusion between ecological correlations and individual correlations, confusionbetween group average and total average, Simpson’s paradox, and other statistical methods.[2]

54.2 See also

• Ecological fallacy

54.3 References[1] Brauneis, Robert (2009). Intellectual Property Protection of Fact-based Works: Copyright and Its Alternatives. Edward

Elgar Publishing. p. 110.

[2] Burnham Terrell, Dailey (1967). Logic: A Modern Introduction to Deductive Reasoning. Holt, Rinehart and Winston. pp.160–163.

66

54.3. REFERENCES 67

• Werner Ebeling; Hans-Michael Voigt. Parallel Problem Solving from Nature - PPSN IV: International Confer-ence on Evolutionary Computation. The 4th International Conference on Parallel Problem Solving from NatureBerlin, Germany, September 22 - 26, 1996. Proceedings, Volume 114. Springer Science & Business Media.pp. 170–173.

• Richard M. Grinnell; Jr., Yvonne A. Unrau. Social Work Research and Evaluation: Foundations of Evidence-Based Practice. Oxford University Press. pp. 393–394.

• “Division”. The Fallacy Files.

Chapter 55

Fallacy of exclusive premises

The fallacy of exclusive premises is a syllogistic fallacy committed in a categorical syllogism that is invalid becauseboth of its premises are negative.Example of an EOO-4 invalid syllogism

E Proposition: No cats are dogs.O Proposition: Some dogs are not pets.O Proposition: Therefore, some pets are not cats.

Explanation of Example 1:

This may seem like a logical conclusion, as it appears to be logically derived that if Some dogs are notpets, then surely some are pets, otherwise, the premise would have stated “No Dogs are pets”, and ifsome pets are dogs, then not all pets can be cats, thus, some pets are not cats. However, this breaks downwhen you apply the same logic to the conclusion: If some pets are not cats then it would seem logical tostate that some pets are cats. But this is not supported by either premise. Cats not being dogs, and thestate of dogs as either pets or not, has nothing to do with whether cats are pets. Two negative premisescannot give a logical foundation for a conclusion, as they will invariably be independent statements thatcannot be directly related, thus the name 'Exclusive Premises’. It is made more clear when the subjectsin the argument are more clearly unrelated such as the following:

Additional Example of an EOO-4 invalid syllogism

E Proposition: No planets are dogs.O Proposition: Some dogs are not pets.O Proposition: Therefore, some pets are not planets.

Explanation of Example 2:

In this example we can more clearly see that the physical difference between a dog and a planet has nocorrelation to the domestication of dogs. The two premises are exclusive and the subsequent conclusionis nonsense, as the transpose would imply that some pets are planets.

Conclusion:

It is important to note that the truthfulness of the final statement is not relevant in this fallacy. The con-clusion of the first example is true, while the final statement in the second is clearly ridiculous; however,both are argued on fallacious logic and would not hold up as valid arguments.

68

55.1. SEE ALSO 69

55.1 See also• affirmative conclusion from a negative premise, in which a syllogism is invalid because the conclusion is affir-mative yet one of the premises is negative

• negative conclusion from affirmative premises, in which a syllogism is invalid because the conclusion is negativeyet the premises are affirmative

55.2 External links• Syllogistic Fallacies: Exclusive Premises

• Stephen Downes Guide to the Logical Fallacies: Exclusive Premises

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008and incorporated under the “relicensing” terms of the GFDL, version 1.3 or later.

Chapter 56

Fallacy of relative privation

For the sociological term, see relative deprivation.

The fallacy of relative privation, or appeal to bigger problems, is an informal fallacy in which it is stated anopponent’s arguments should be dismissed or ignored, on the grounds that more important problems exist, regardlessof whether these problems are relevant to the question at hand or not.A well-known example of this fallacy is the response “but there are children starving in Africa,” with the implicationthat any issue less serious is not worthy of discussion.

56.1 See also• Whataboutism

• False dilemma

• Nirvana fallacy

• Pollyanna

• First World problem

• Think of the children

• Thought-terminating cliché

56.2 References

56.3 External links• Rational Wiki. "Not As Bad As.”

70

Chapter 57

Falsism

A falsism is a claim that is clearly and self-evidently wrong. A falsism is usually used merely as a reminder or as arhetorical or literary device. An example is “pigs can fly.” It is the opposite of truism.[1][2] A falsism is similar to,though not the same as, a fallacy.

57.1 See also• Straw man

57.2 References[1] “Definition: truism”. http://www.websters-online-dictionary.org/: Webster’s Online Dictionary. Retrieved 2010-03-10.

Noun Base (truism)

(a) An obvious truth. Wordnet.(b) An undoubted or self-evident truth; a statement which is pliantly true; a proposition needing no proof or argument;

-- opposed to falsism. Websters.

[2] Basically the opposite of the truth so a lie that is technically not a lie but still a lie

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Chapter 58

First-order predicate

In mathematical logic, a first-order predicate (also called a monad) is a predicate that takes only individual(s)constants or variables as argument(s).[1] Compare second-order predicate and higher-order predicate.

58.1 See also• First-order predicate calculus

• Monadic predicate calculus

58.2 References[1] Flew, Antony (1984), A Dictionary of Philosophy: Revised Second Edition, Macmillan, p. 147, ISBN 9780312209230.

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Chapter 59

Fluent calculus

The fluent calculus is a formalism for expressing dynamical domains in first-order logic. It is a variant of the situationcalculus; the main difference is that situations are considered representations of states. A binary function symbol ◦ isused to concatenate the terms that represent facts that hold in a situation. For example, that the box is on the table inthe situation s is represented by the formula ∃t.s = on(box, table) ◦ t . The frame problem is solved by assertingthat the situation after the execution of an action is identical to the one before but for the conditions changed by theaction. For example, the action of moving the box from the table to the floor is formalized as:

State(Do(move(box, table, floor), s)) ◦ on(box, table) = State(s) ◦ on(box, floor)

This formula states that the state after themove is added the term on(box, floor) and removed the term on(box, table). Axioms specifying that ◦ is commutative and non-idempotent are necessary for such axioms to work.

59.1 See also• Fluent (artificial intelligence)

• Frame problem

• Situation calculus

• Event calculus

59.2 References• M. Thielscher (1998). Introduction to the fluent calculus. Electronic Transactions on Artificial Intelligence,2(3–4):179–192.

• M. Thielscher (2005). Reasoning Robots - The Art and Science of Programming Robotic Agents. Volume 33of Applied Logic Series. Springer, Dordrecht.

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Chapter 60

Fragment (logic)

In mathematical logic, a fragment of a logical language or theory is a subset of this logical language obtained byimposing syntactical restrictions on the language.[1] Hence, the well-formed formulae of the fragment are a subset ofthose in the original logic. However, the semantics of the formulae in the fragment and in the logic coincide, and anyformula of the fragment can be expressed in the original logic.The computational complexity of tasks such as satisfiability or model checking for the logical fragment can be nohigher than the same tasks in the original logic, as there is a reduction from the first problem to the other. Animportant problem in computational logic is to determine fragments of well-known logics such as first-order logicwhich are as expressive as possible yet are decidable or more strongly have low computational complexity.[1] The fieldof descriptive complexity theory aims at establishing a link between logics and computational complexity theory, byidentifying logical fragments that exactly capture certain complexity classes.[2]

60.1 References[1] Bradley, Aaron R.; Manna, Zohar (2007), The Calculus of Computation: Decision Procedures with Applications to Verifi-

cation, Springer, p. 70, ISBN 9783540741138.

[2] Ebbinghaus, Heinz-Dieter; Flum, Jörg (2005), “Chapter 7. Descriptive Complexity Theory”, Finite Model Theory, Per-spectives in mathematical logic, Springer, pp. 119–164, ISBN 9783540287889.

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Chapter 61

Frege’s theorem

In metalogic andmetamathematics, Frege’s theorem is a metatheorem that states that the Peano axioms of arithmeticcan be derived in second-order logic from Hume’s principle. It was first proven, informally, by Gottlob Frege in hisDie Grundlagen der Arithmetik (Foundations of Arithmetic), published in 1884, and proven more formally in hisGrundgesetze der Arithmetik (Basic Laws of Arithmetic), published in two volumes, in 1893 and 1903. The theoremwas re-discovered by Crispin Wright in the early 1980s and has since been the focus of significant work. It is at thecore of the philosophy of mathematics known as neo-logicism.

61.1 Frege’s theorem in propositional logic

In propositional logic, Frege’s theorems refers to this tautology:

(P → (Q→ R)) → ((P → Q) → (P → R))

61.2 References• Zalta, Edward (2013), “Frege’s Theorem and Foundations for Arithmetic”, Stanford Encyclopedia of Philoso-

phy.

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Chapter 62

Guarded logic

Guarded logic is a choice set of dynamic logic involved in choices, where outcomes are limited.A simple example of guarded logic is as follows: if X is true, then Y, else Z can be expressed in dynamic logic as(X?;Y)∪(~X?;Z). This shows a guarded logical choice: if X holds, then X?;Y is equal to Y, and ~X?;Z is blocked, anda ∪block is also equal to Y. Hence, when X is true, the primary performer of the action can only take the Y branch,and when false the Z branch.[1]

A real-world example is the idea of paradox: something cannot be both true and false. A guarded logical choice isone where any change in true affects all decisions made down the line.[2]

62.1 History

Before the use of guarded logic there were two major terms used to interpret modal logic. Mathematical logic anddatabase theory (Artificial Intelligence) were first-order predicate logic. Both terms found sub-classes of first-classlogic and efficiently used in solvable languages which can be used for research. But neither could explain powerfulfixed-point extensions to modal style logics.Later Moshe Y. Vardi[3] made a conjecture that a tree model would work for many modal style logics. The guardedfragment of first-order logic was first introduced by Hajnal Andréka, István Németi and Johan Van Benthem in theirarticle Modal languages and bounded fragments of predicate logic. They successfully transferred key properties ofdescription, modal, and temporal logic to predicate logic. It was found that the robust decidability of guarded logiccould be generalized with a tree model property. The tree model can also be a strong indication that guarded logicextends modal framework which retains the basics of modal logics.Modal logics are generally characterized by invariances under bisimulation. It also so happens that invariance underbisimulation is the root of tree model property which helps towards defining automata theory.

62.2 Types of Guarded Logic

Within Guarded Logic there exists numerous guarded objects. The first being guarded fragment which are first-order logic of modal logic. Guarded fragments generalize modal quantification through finding relative patterns ofquantification. The syntax used to denote guarded fragment isGF. Another object is guarded fixed point logic denotedμGF naturally extends guarded fragment from fixed points of least to greatest. Guarded bisimulations are objectswhich when analyzing guarded logic. All relations in a slightly modified standard relational algebra with guardedbisimulation and first-order definable are known as guarded relational algebra. This is denoted using GRA.Along with first-order guarded logic objects, there are objects of second-order guarded logic. It is known as GuardedSecond-Order Logic and denoted GSO. Similar to second-order logic, guarded second-order logic quantifies whoserange over guarded relations restrict it semantically. This is different from second-order logic which the range isrestricted over arbitrary relations. [4]

76

62.3. DEFINITIONS OF GUARDED LOGIC 77

62.3 Definitions of Guarded Logic

Let B be a relational structure with universe B and vocabulary τ.i) A set X ⊆ B is guarded in B if there exists a ground atom α(b_1, ..., b_k) such that B = α(b_1, ..., b_k)and X ={b_1, ..., b_k}.ii) A τ-structure A, in particular a substructure A ⊆ B, is guarded if its universe is a guarded set in A (in B).iii) A tuple (b_1, ..., b_n) ∈ B^n is guarded in B if {b_1, ..., b_n} ⊆ X for some guarded set X ⊆ B.iv) A tuple (b_1, ..., b_k) ∈ B^k is a guarded list in B if its components are pairwise distinct and {b_1, ..., b_k} is aguarded set. The empty list is taken to be a guarded list.v) A relation X ⊆ B^n is guarded if it only consists of guarded tuples.[5]

62.3.1 Guarded Bisimulation

A guarded bisimulation between two τ-structures A and B is an non-empty set I of finite partial isomorphic f: X →Y from A to B such that the back and forth conditions are satisfied.Back: For every f: X → Y in I and for every guarded set Y` ⊆ B, there exists a partial isomorphic g: X` → Y` in Isuch that f^−1 and g^−1 agree on Y ∩ Y`.Forth For every f: X → Y in I and for every guarded set X` A, there exists a partial isomorphic g: X` → Y` in Isuch that f and g agree on X ∩ X`.

62.4 References[1] “International Conference on Formal Modelling and Analysis of Timed Systems No4”. Paris, France. September 25–27,

2006. |chapter= ignored (help)

[2] Nieuwenhuis, Robert; Andrei Voronkov (2001). Logic for Programming, Artificial Intelligence, and Reasoning. Springer.pp. 88–89. ISBN 3-540-42957-3.

[3] Vardi, Moshe (1998). Reasoning about the Past with Two-Way Automata (PDF).

[4] “Guarded Logics: Algorithms and Bisimulation” (PDF). p. 26 - 48. Retrieved 15 May 2014.

[5] “Guarded Logics: Algorithms and Bisimulation” (PDF). p. 25. Retrieved 15 May 2014.

Chapter 63

Herbrand interpretation

In mathematical logic, a Herbrand interpretation is an interpretation in which all constants and function symbolsare assigned very simple meanings. Specifically, every constant is interpreted as itself, and every function symbol isinterpreted as the function that applies it. The interpretation also defines predicate symbols as denoting a subset ofthe relevant Herbrand base, effectively specifying which ground atoms are true in the interpretation. This allows thesymbols in a set of clauses to be interpreted in a purely syntactic way, separated from any real instantiation.The importance of Herbrand interpretations is that, if any interpretation satisfies a given set of clauses S then thereis a Herbrand interpretation that satisfies them. Moreover, Herbrand’s theorem states that if S is unsatisfiable thenthere is a finite unsatisfiable set of ground instances from the Herbrand universe defined by S. Since this set is finite,its unsatisfiability can be verified in finite time. However there may be an infinite number of such sets to check.It is named after Jacques Herbrand.

63.1 See also• Herbrand structure

• Interpretation (logic)

• Interpretation (model theory)

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Chapter 64

Hybrid logic

Hybrid logic refers to a number of extensions to propositional modal logic with more expressive power, though stillless than first-order logic. In formal logic, there is a trade-off between expressiveness and computational tractability(how easy it is to compute/reason with logical languages). The history of hybrid logic began with Arthur Prior’s workin tense logic.[1]

Unlike ordinary modal logic, hybrid logic makes it possible to refer to states (possible worlds) in formulas. This isachieved by a class of formulas called nominals, which are true in exactly one state, and by the use of the @ operator,which is defined as follows:

@i p is true if and only if p is true in the unique state named by the nominal i (i.e., the state where i istrue).

Hybrid logics with extra or other operators exist, but @ is more-or-less “standard.”Hybrid logics have many features in common with temporal logics (which use nominal-like constructs to denotespecific points in time), and they are a rich source of ideas for researchers in modern modal logic. They also haveapplications in the areas of feature logic, model theory, proof theory, and the logical analysis of natural language.It is also deeply connected to description logic because the use of nominals allows one to perform assertional ABoxreasoning, as well as the more standard terminological TBox reasoning.

64.1 References[1] Torben Braüner (2008). “Hybrid Logic”. Stanford Encyclopedia of Philosophy. Retrieved 1 February 2011.

64.2 Further reading• P. Blackburn. 2000. Representation, reasoning and relational structures: a hybrid logic manifesto. Logic

Journal of the IGPL, 8(3):339-365.

64.3 External links• Hybrid Logics’ Home Page

• Stanford Encyclopedia of Philosophy entry on Hybrid Logic

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Chapter 65

Idempotency of entailment

Idempotency of entailment is a property of logical systems that states that one may derive the same consequencesfrom many instances of a hypothesis as from just one. This property can be captured by a structural rule calledcontraction and in such systems one may say that entailment is idempotent if and only if contraction is an admissiblerule.Rule of Contraction: fromA,C,C -> Bis derivedA,C -> B.Or in sequent calculus notation,

Γ, C, C ⊢ BΓ, C ⊢ B

80

Chapter 66

Illicit major

Illicit major is a formal fallacy committed in a categorical syllogism that is invalid because its major term isundistributed in the major premise but distributed in the conclusion.This fallacy has the following argument form:

1. All A are B

2. No C are A

3. Therefore, no C are B

Example:

1. All dogs are mammals

2. No cats are dogs

3. Therefore, no cats are mammals

In this argument, the major term is “mammals”. This is distributed in the conclusion (the last statement) because weare making a claim about a property of allmammals: that they are not cats. However, it is not distributed in the majorpremise (the first statement) where we are only talking about a property of somemammals: Only some mammals aredogs.The error is in assuming that the converse of the first statement (that all mammals are dogs) is also true.However, an argument in the following form differs from the above, and is valid (Camestres):

1. All A are B

2. No B are C

3. Therefore, no C are A

66.1 See also

• Illicit minor

• Syllogistic fallacy

81

82 CHAPTER 66. ILLICIT MAJOR

66.2 External links• Illicit Major The Fallacy Files

This article was originally based on material from the Free On-line Dictionary of Philosophy, which islicensed under the GFDL.

Chapter 67

Illicit minor

Illicit minor is a formal fallacy committed in a categorical syllogism that is invalid because its minor term isundistributed in the minor premise but distributed in the conclusion.This fallacy has the following argument form:

All A are B.All A are C.Therefore, all C are B.

Example:

All cats are felines.All cats are mammals.Therefore, all mammals are felines.

The minor term here is mammal, which is not distributed in the minor premise “All cats are mammals,” because thispremise is only defining a property of possibly some mammals (i.e., that they're cats.) However, in the conclusion“All mammals are felines,” mammal is distributed (it is talking about all mammals being felines). It is shown to befalse by any mammal that is not a feline; for example, a dog.Example:

Pie is good.Pie is unhealthy.Thus, all good things are unhealthy.

67.1 See also• Illicit major

• Syllogistic fallacy

This article was originally based on material from the Free On-line Dictionary of Philosophy, which islicensed under the GFDL.

83

Chapter 68

Inclusion (logic)

In logic and mathematics, inclusion is the concept that all the contents of one object are also contained within asecond object.[1]

The modern symbol for inclusion first appears in Gergonne (1816), who defines it as one idea 'containing' or being'contained' by another, using the backward letter 'C' to express this. Peirce articulated this clearly in 1870, arguingalso that inclusion was a wider concept than equality, and hence a logically simpler one.[2] Schröder (also Frege) callsthe same concept 'subordination'.[3]

68.1 References[1] Quine, W. V. (December 1937). “Logic based on inclusion and abstraction”. The Journal of Symbolic Logic 2 (4). pp.

145–152. doi:10.2307/2268279.

[2] “Descr. of a notation”, CP III 28.

[3] Vorlesungen I., 127.

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Chapter 69

Instantiation principle

The principle of instantiation or principle of exemplification is the concept in metaphysics and logic that therecan be no uninstantiated or unexemplified properties (or universals). In other words, it is impossible for a propertyto exist which is not had by some object. Aristotle is well known for endorsing the principle and Plato for denying it.Consider a chair. Presumably chairs did not exist 150,000 years ago. Thus, according to the Principle of Instantiation,the property of being a chair did not exist 150,000 years ago either. Similarly (and assuming objects are colored), ifall red objects were to suddenly go out of existence, then the property of being red would likewise go out of existence.To make the Principle of Instantiation more plausible in the light of these examples, the existence of properties oruniversals is not tied to their actual existence now, but to their existence in space-time considered as a whole.[1] Thus,any property which is instantiated, has been instantiated, or will be instantiated exists. The property of being redwould exist even if all red things were to be destroyed, because it has been instantiated. This broadens the range ofproperties which exist if the principle is true.Those who endorse the principle of instantiation are known as in re realists or “immanent realists”.[2]

69.1 References[1] Armstrong, David (1989). Universals: An Opinionated Introduction (PAPERBACK) (book). Colorado: Westview Press.

[2] Loux, Michael (2006). “Aristotle’s Constituent Ontology”. In Zimmerman, Dean W. Oxford Studies in Metaphysics (PA-PERBACK) (book). Oxford University Press. ISBN 978-0-19-929058-1. Retrieved 2012-06-25.

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Chapter 70

Interpretability

In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of inter-preting or translating one into the other.

70.1 Informal definition

Assume T and S are formal theories. Slightly simplified, T is said to be interpretable in S if and only if the languageof T can be translated into the language of S in such a way that S proves the translation of every theorem of T. Ofcourse, there are some natural conditions on admissible translations here, such as the necessity for a translation topreserve the logical structure of formulas.This concept, together with weak interpretability, was introduced by Alfred Tarski in 1953. Three other relatedconcepts are cointerpretability, logical tolerance, and cotolerance, introduced by Giorgi Japaridze in 1992-1993.

70.2 References• Japaridze, G., and De Jongh, D. (1998) “The logic of provability” in Buss, S., ed., Handbook of Proof Theory.North-Holland: 476-546.

• Alfred Tarski, Andrzej Mostowski, and Raphael Robinson (1953) Undecidable Theories. North-Holland.

70.3 See also

86

Chapter 71

Interval temporal logic

Interval temporal logic (also interval logic) is a temporal logic for representing both propositional and first-orderlogical reasoning about periods of time that is capable of handling both sequential and parallel composition. Insteadof dealing with infinite sequences of state, interval temporal logics deal with finite sequences.Interval temporal logics find application in computer science, artificial intelligence and linguistics. First-order intervaltemporal logic was initially developed in 1980s for the specification and verification of hardware protocols. IntervalTemporal Logic (ITL) is a specific form of temporal logic, originally developed by Ben Moszkowski for his thesisat Stanford University. It is useful in the formal description of hardware and software for computer-based systems.Tools are available to aid in this process. Tempura provides an executable ITL framework. Compositionality is asignificant issue and consideration in the design of ITL.Notable derivatives of interval temporal logic are graphical interval logic, signed interval logic and future intervallogic.

71.1 See also• Duration Calculus

• Formal methods

• Temporal Logic of Actions

71.2 External links• ITL: Interval Temporal Logic

87

Chapter 72

Inverse (logic)

In logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditionalsentence. Any conditional sentence has an inverse: the contrapositive of the converse. The inverse of P → Q is thus¬P → ¬Q .For example, substituting propositions in natural language for logical variables, the inverse of the conditional propo-sition, “If it’s raining, then Sam will meet Jack at the movies” is “If it’s not raining, then Sam will not meet Jack atthe movies.”The inverse of the inverse, that is, the inverse of ¬P → ¬Q , is ¬¬P → ¬¬Q . Since a double negation hasno logical effect, the inverse of the inverse is logically equivalent to the original conditional P → Q . Thus it ispermissible to say that ¬P → ¬Q and P → Q are inverses of each other. Likewise, P → ¬Q and ¬P → Q areinverses of each other.The inverse and the converse of a conditional are logically equivalent to each other, just as the conditional andits contrapositive are logically equivalent to each other. But the inverse of a conditional is not inferable from theconditional. For example, “If it’s not raining, then Sam will not meet Jack at the movies” cannot be inferred from “Ifit’s raining, then Sam will meet Jack at the movies.” It could easily be the case that Sam and Jack are attending themovies no matter the weather.In traditional logic, where there are four named types of categorical propositions, only forms A and E have an inverse.To find the inverse of these categorical propositions one must: replace the subject and the predicate of the invertendby their respective contradictories and change the quantity from universal to particular.[1]

• All S are P (A form) becomes Some non-S are non-P

• All S are not P (E form) becomes Some non-S are not non-P

72.1 See also• Contraposition

• Converse (logic)

• Obversion

• Transposition (logic)

72.2 Notes[1] Toohey, John Joseph. An Elementary Handbook of Logic. Schwartz, Kirwin and Fauss, 1918

88

Chapter 73

Inverse resolution

Inverse resolution is an inductive reasoning technique that involves inverting the resolution operator.

73.1 References

Inverse resolution

89

Chapter 74

Invincible ignorance fallacy

Not to be confused with Vincible ignorance.

The invincible ignorance fallacy[1] is a deductive fallacy of circularity where the person in question simply refusesto believe the argument, ignoring any evidence given. It is not so much a fallacious tactic in argument as it is arefusal to argue in the proper sense of the word, the method instead being to make assertions with no considerationof objections.

74.1 History

The term “invincible ignorance” has its roots in Catholic theology, where — as the opposite of the term vincibleignorance — it is used to refer to the state of persons (such as pagans and infants) who are ignorant of the Christianmessage because they have not yet had an opportunity to hear it. The first Pope to use the term officially seems tohave been Pope Pius IX in the allocution Singulari Quadam (9 December 1854) and the encyclicals Singulari Quidem(17 March 1856) and Quanto Conficiamur Moerore (10 August 1863). The term, however, is far older than that.Aquinas, for instance, uses it in his Summa Theologica (written 1265–1274),[2] and discussion of the concept canbe found as far back as Origen (3rd century). When and how the term was taken by logicians to refer to the verydifferent state of persons who deliberately refuse to attend to evidence remains unclear, but one of its first uses wasin the book Fallacy: The Counterfeit of Argument by W. Ward Fearnside and William B. Holther.[3]

74.2 See also

• A posteriori

• A priori

• Argumentum ad lapidem

• Future probation

• Vincible ignorance

74.3 References

[1] “Invincible Ignorance” by Bruce Thompson, Department of Humanities (Philosophy), Cuyamaca College

[2] Aquinas, Summa Theologica Ia IIae q.76 a.2

[3] Fearnside, W. Ward and William B. Holther, Fallacy: The Counterfeit of Argument, 1959. ISBN 978-0-13-301770-0.

90

74.4. EXTERNAL LINKS 91

74.4 External links• Listing of the fallacy on philosophicalsociety.com

• Discovery.org article: “Naturalism’s Argument from Invincible Ignorance”

• Pius IX, Quanto Conficiamur Moerore (On Promotion Of False Doctrines), 10 August 1863

Chapter 75

Issue trees

An issue tree, also called a logic tree, is a graphical breakdown of a question that dissects it into its different com-ponents vertically and that progresses into details as it reads to the right.Issue trees are useful in problem solving to identify the root causes of a problem as well as to identify its potentialsolutions. They also provide a reference point to see how each piece fits into the whole picture of a problem.[1]

There are two types of issue trees: diagnostic ones and solution ones.Diagnostic trees breakdown a “why” key question, identifying all the possible root causes for the problem. Solutiontree breakdown a “how” key question, identifying all the possible alternatives to fix the problem.To be effective, an issue tree needs to obey four basic rules:[2]

1. Consistently answer a “why” or a “how” question

2. Progress from the key question to the analysis as it moves to the right

3. Have branches that are mutually exclusive and collectively exhaustive (MECE)

4. Use an insightful breakdown

The requirement for issue trees to be collectively exhaustive implies that divergent thinking is a critical skill.A profitability tree is an example of an issue tree. It looks at different ways in which a company can increase itsprofitability. Starting from the key question on the right, it breaks it down between revenues and costs, and breakthese down into further details.

75.1 References[1] http://webarchive.nationalarchives.gov.uk/20060213205515/http://strategy.gov.uk/downloads/survivalguide/downloads/ssg_

v2.1.pdf

[2] http://powerful-problem-solving.com/build-logic-trees

92

75.1. REFERENCES 93

An issue tree showing how a company can increase profitability

Chapter 76

Lambert of Auxerre

Lambert of Auxerre was a medieval 13th century logician best known for writing the book "Summa Lamberti" orsimply "Logica" [1] in the mid 1250s which became an authoritative textbook on logic in the Western tradition.[2] Hewas a Dominican in the Dominican house at Auxerre. His contemporaries were Peter of Spain, William of Sherwood,and Roger Bacon.[3]

76.1 Works and Translations• Logica (Summa Lamberti), First edition of the Latin text by Franco Alessio, Firenze, La Nuova Italia, 1971.

• Logica, or Summa Lamberti, translated with notes and introduction by Thomas S. Maloney, Notre Dame Uni-versity Press, 2015.

• Properties of Terms, in Norman Kretzmann, Eleonore Stump, trans., in Cambridge Translations of MedievalPhilosophical Texts, Vol. 1: Logic and the Philosophy of Language, Cambridge: Cambridge University Press,1988, pp. 102-162.

• Alain de Libera, Le traité De appellatione de Lambert de Lagny (Lambert d’Auxerre), Archives d’histoire doc-trinale et littéraire du Moyen Age, 48, pp. 227–285, 1982.

76.2 Further Readings• L. M. de Rijk, On the genuine text of Peter of Spain’s Summule logicales, IV: The Lectura tractatuum by Guillel-

mus Arnaldi, master of arts at Toulouse (1235–1244). With a note on the date of Lambert of Auxerre’s Summule.Vivarium 7, 1969, pp. 120–162.

• A. de Libera, De la logique a` la grammaire: remarques sur la théorie de la détermination chez Roger Bacon etLambert d’Auxerre (Lambert de Lagny), In: Geoffrey L. Bursill-Hall, Sten Ebbesen and E.F.K. Koerner (eds.),De Ortu Grammaticae. Studies in Medieval Grammar and Linguistic Theory in Memory of Jan Pinborg, JohnBenjamin, Amsterdam/Philadelphia, 1990, pp. 209–226.

76.3 Notes[1] The Summa Lamberti is now attributed to Lambert of Lagny (Lambertus de Latiniaco) (fl. 1250): see A. de Libera (1982).

[2] http://books.google.com/books?id=c2nmqagEz6QC&pg=PA531&lpg=PA531&dq=Lambert+of+Auxerre&source=bl&ots=qVv4hocSYp&sig=9314KYEdxmWQLLWuBZEokJ_N-9k&hl=en&ei=mbVXSu_uOJPQtgPnnJnaBw&sa=X&oi=book_result&ct=result&resnum=4

[3] http://books.google.com/books?id=MOqtNYn8qzkC&pg=PA102&lpg=PA102&dq=Lambert+of+Auxerre&source=bl&ots=79ROEuKSnP&sig=dl7RXamiexlD7DrInyV8cUZk_6o&hl=en&ei=mbVXSu_uOJPQtgPnnJnaBw&sa=X&oi=book_result&ct=result&resnum=7

94

Chapter 77

Lemma (logic)

For other uses, see Lemma (disambiguation).

In informal logic and argument mapping, a lemma is simultaneously a contention for premises below it and a premisefor a contention above it. Transitivity: If one has proof that B follows from A and proof of A, then one has proof ofB.

77.1 See also• Co-premise

• Inference objection

• Objection

95

Chapter 78

Limitation of size

In the philosophy of mathematics, specifically the philosophical foundations of set theory, limitation of size is aconcept developed by Philip Jourdain and/or Georg Cantor to avoid Cantor’s paradox. It identifies certain “inconsis-tent multiplicities”, in Cantor’s terminology, that cannot be sets because they are “too large”. In modern terminologythese are called proper classes.

78.1 Use

The axiom of limitation of size is an axiom in some versions of von Neumann–Bernays–Gödel set theory or Morse–Kelley set theory. This axiom says that any class which is not “too large” is a set, and a set cannot be “too large”.“Too large” is defined as being large enough that the class of all sets can be mapped one-to-one into it.

78.2 References• Hallett, Michael (1986). Cantorian Set Theory and Limitation of Size. Oxford University Press. ISBN 0-19-853283-0.

96

Chapter 79

Lindenbaum’s lemma

In mathematical logic, Lindenbaum’s lemma states that any consistent theory of predicate logic can be extended toa complete consistent theory. The lemma is a special case of the ultrafilter lemma for Boolean algebras, applied tothe Lindenbaum algebra of a theory.

79.1 Uses

It is used in the proof of Gödel’s completeness theorem, among other places.

79.2 Extensions

The effective version of the lemma’s statement, “every consistent computably enumerable theory can be extendedto a complete consistent computably enumerable theory,” fails (provided Peano Arithmetic is consistent) by Gödel’sincompleteness theorem.

79.3 History

The lemma was not published by Adolf Lindenbaum; it is originally attributed to him by Alfred Tarski.[1]

79.4 Notes[1] Tarski, A. On Fundamental Concepts of Metamathematics, 1930.

79.5 References• Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). What is mathematical logic?.London-Oxford-New York: Oxford University Press. p. 16. ISBN 0-19-888087-1. Zbl 0251.02001.

79.6 External links• University of Texas, A Causal Theory of Modal Knowledge (Including Logical and Mathematical Knowledge

97

Chapter 80

Literal (mathematical logic)

In mathematical logic, a literal is an atomic formula (atom) or its negation. The definition mostly appears in prooftheory (of classical logic), e.g. in conjunctive normal form and the method of resolution.Literals can be divided into two types:

• A positive literal is just an atom.

• A negative literal is the negation of an atom.

For a literal l , the complementary literal is a literal corresponding to the negation of l , we can write l̄ to denotethe complementary literal of l . More precisely, if l ≡ x then l̄ is ¬x and if l ≡ ¬x then l̄ is x .In the context of a formula in the conjunctive normal form, a literal is pure if the literal’s complement does not appearin the formula.

80.1 Examples

In propositional calculus a literal is simply a propositional variable or its negation.In predicate calculus a literal is an atomic formula or its negation, where an atomic formula is a predicate symbolapplied to some terms, P (t1, . . . , tn) with the terms recursively defined starting from constant symbols, variablesymbols, and function symbols. For example, ¬Q(f(g(x), y, 2), x) is a negative literal with the constant symbol 2,the variable symbols x, y, the function symbols f, g, and the predicate symbol Q.

80.2 References• Samuel R. Buss (1998). “An introduction to proof theory”. In Samuel R. Buss. Handbook of proof theory.Elsevier. pp. 1–78. ISBN 0-444-89840-9.

98

Chapter 81

Logic Spectacles

Logic Spectacles, Thomas Carlyle's name for eyes that can discern only the external relations of things, but not theinner nature of them.This article incorporates text from a publication now in the public domain: Wood, James, ed. (1907). "article name needed".The Nuttall Encyclopædia. London and New York: Frederick Warne.

99

Chapter 82

Logical constant

In logic, a logical constant of a languageL is a symbol that has the same semantic value under every interpretation ofL . Two important types of logical constants are logical connectives and quantifiers. The equality predicate (usuallywritten '=') is also treated as a logical constant in many systems of logic. One of the fundamental questions in thephilosophy of logic is “What is a logical constant?"; that is, what special feature of certain constants makes themlogical in nature?[1]

Some symbols that are commonly treated as logical constants are:Many of these logical constants are sometimes denoted by alternate symbols (e.g., the use of the symbol "&" ratherthan "∧" to denote the logical and).

82.1 See also• Non-logical symbol

• Logical value

• Logical connective

82.2 References[1] Carnap

82.3 External links• Stanford Encyclopedia of Philosophy entry on logical constants

100

Chapter 83

Logical cube

Not to be confused with Lambda cube.

In the system of Aristotelian logic, the logical cube is a diagram representing the different ways in which each of theeight propositions of the system is logically related ('opposed') to each of the others. The system is also useful in theanalysis of syllogistic logic, serving to identify the allowed logical conversions from one type to another.

83.1 See also• Square of opposition

• Logical hexagon

83.2 References

101

Chapter 84

Main contention

In both formal and informal logic, a main contention or conclusion is a thought which is capable of being eithertrue or false and is usually the most controversial proposition being argued for. In reasoning, a main contention isrepresented by the top of an argument map, with all supporting and objecting premises which bear upon it placedunderneath.In the context of argumentative text, it is the point that the author wants to convince you to believe - the culminationof all their reasoning. The main contention provides an answer to the following types of questions:[1]

• “Why is the author bothering to tell me these things?"

• “What is the main point the author is trying to convince me of?"

• “What is the most important thing the author is arguing for or against?"

84.1 See also• Argumentation theory

• Co-premise

• Logical consequence

• Inference objection

• Premise

84.2 References[1] http://learn.austhink.com/topics/IdentifyingContentions.pdf

102

Chapter 85

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

103

104 CHAPTER 85. MATERIAL NONIMPLICATION

85.1 Definition

85.1.1 Truth table

85.2 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.

85.3 Symbol

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

85.4 Natural language

85.4.1 Grammatical

85.4.2 Rhetorical

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

85.4.3 Colloquial

“Just because p, doesn't mean q.”

85.5 Boolean algebra

Further information: Boolean algebra

(A'+B)'

85.6 Computer science

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

85.7 See also• Implication

85.8 References

Chapter 86

MaxEnt school

The MaxEnt school is a school of logic associated with the theories of American physicist Edwin Jaynes and hiscollaborators.

105

Chapter 87

McNamara fallacy

The McNamara fallacy (also known as quantitative fallacy[1]), named for Robert McNamara, the United StatesSecretary of Defense from 1961 to 1968, involves making a decision based solely on quantitative observations andignoring all others. The reason given is often that these other observations cannot be proven. (See the example below.)It refers to McNamara’s belief as to what led the United States to defeat in the Vietnam War—specifically, hisquantification of success in the war (e.g. in terms of enemy body count), ignoring other variables.

The first step is to measure whatever can be easily measured. This is OK as far as it goes. The secondstep is to disregard that which can't be easily measured or to give it an arbitrary quantitative value. Thisis artificial and misleading. The third step is to presume that what can't be measured easily really isn'timportant. This is blindness. The fourth step is to say that what can't be easily measured really doesn'texist. This is suicide.—Daniel Yankelovich “Corporate Priorities: A continuing study of the new demands on business.”(1972)

87.1 Examples

Ted has a lot of money. Lots of money makes a person happy. Ted says that he is depressed. What Ted says doesnot necessarily indicate how he feels. Depression cannot be proven. Therefore, Ted is happy.

87.2 In modern clinical trials

There has been increasing discussion of theMcNamara fallacy in medical literature.[2][3] In particular, theMcNamarafallacy is invoked to describe the futility of using progression-free survival (PFS) as a primary endpoint in clinicaltrials for agents treating metastatic solid tumors simply because PFS is an endpoint which is merely measurable, whilefailing to capture outcomes which are more meaningful such as overall quality of life or overall survival.

87.3 References[1] Fischer, D. H. (June 1970). Historians’ fallacies: toward a logic of historical thought. Harper torchbooks (first ed.). New

York: HarperCollins. p. 90. ISBN 978-0-06-131545-9. OCLC 185446787.

[2] Basler, Michael H. (2009). “Utility of the McNamara fallacy”. BMJ 339: b3141. doi:10.1136/bmj.b3141.

[3] Booth, ChristopherM.; Eisenhauer, Elizabeth A. (2012). “Progression-Free Survival: Meaningful or SimplyMeasurable?".Journal of Clinical Oncology 30 (10): 1030–1033. doi:10.1200/JCO.2011.38.7571.

106

Chapter 88

Middle term

In logic, a middle term is a term that appears (as a subject or predicate of a categorical proposition)[1] in bothpremises but not in the conclusion of a categorical syllogism.[2] Themiddle term (in bold below) must be distributedin at least one premise but not in the conclusion. The major term and the minor terms, also called the end terms, doappear in the conclusion.Example:

Major premise: All men are mortal.Minor premise: Socrates is a man.Conclusion: Socrates is mortal.

The middle term is bolded above.

88.1 References[1] Distribution of terms#Quantity

[2] http://www.philosophypages.com/dy/m7.htm#midt

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008and incorporated under the “relicensing” terms of the GFDL, version 1.3 or later.

107

Chapter 89

Models And Counter-Examples

Mace stands for “Models And Counter-Examples”, and is an automated theorem prover based on model generation[1]It is GNU GPL licensed.[2]

89.1 References[1] William McCune home site

[2] See COPYING file in the tarball.

89.2 See also• Otter (theorem prover)

89.3 External links• System download

108

Chapter 90

Monadic Boolean algebra

In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature

⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩,

where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the receivedprefix notation for ∃):

• ∃0 = 0

• ∃x ≥ x

• ∃(x + y) = ∃x + ∃y

• ∃x∃y = ∃(x∃y).

∃x is the existential closure of x. Dual to ∃ is the unary operator ∀, the universal quantifier, defined as ∀x := (∃x' )'.A monadic Boolean algebra has a dual definition and notation that take ∀ as primitive and ∃ as defined, so that ∃x:= (∀x ' )' . (Compare this with the definition of the dual Boolean algebra.) Hence, with this notation, an algebra Ahas signature ⟨·, +, ', 0, 1, ∀⟩, with ⟨A, ·, +, ', 0, 1⟩ a Boolean algebra, as before. Moreover, ∀ satisfies the followingdualized version of the above identities:

1. ∀1 = 1

2. ∀x ≤ x

3. ∀(xy) = ∀x∀y

4. ∀x + ∀y = ∀(x + ∀y).

∀x is the universal closure of x.

90.1 Discussion

Monadic Boolean algebras have an important connection to topology. If ∀ is interpreted as the interior operatorof topology, (1)-(3) above plus the axiom ∀(∀x) = ∀x make up the axioms for an interior algebra. But ∀(∀x) =∀x can be proved from (1)-(4). Moreover, an alternative axiomatization of monadic Boolean algebras consists ofthe (reinterpreted) axioms for an interior algebra, plus ∀(∀x)' = (∀x)' (Halmos 1962: 22). Hence monadic Booleanalgebras are the semisimple interior/closure algebras such that:

• The universal (dually, existential) quantifier interprets the interior (closure) operator;

109

110 CHAPTER 90. MONADIC BOOLEAN ALGEBRA

• All open (or closed) elements are also clopen.

A more concise axiomatization of monadic Boolean algebra is (1) and (2) above, plus ∀(x∨∀y) = ∀x∨∀y (Halmos1962: 21). This axiomatization obscures the connection to topology.Monadic Boolean algebras form a variety. They are tomonadic predicate logic what Boolean algebras are to propositionallogic, and what polyadic algebras are to first-order logic. Paul Halmos discovered monadic Boolean algebras whileworking on polyadic algebras; Halmos (1962) reprints the relevant papers. Halmos and Givant (1998) includes anundergraduate treatment of monadic Boolean algebra.Monadic Boolean algebras also have an important connection to modal logic. The modal logic S5, viewed as atheory in S4, is a model of monadic Boolean algebras in the same way that S4 is a model of interior algebra. Like-wise, monadic Boolean algebras supply the algebraic semantics for S5. Hence S5-algebra is a synonym for monadicBoolean algebra.

90.2 See also• clopen set

• interior algebra

• Kuratowski closure axioms

• Łukasiewicz–Moisil algebra

• modal logic

• monadic logic

90.3 References• Paul Halmos, 1962. Algebraic Logic. New York: Chelsea.

• ------ and Steven Givant, 1998. Logic as Algebra. Mathematical Association of America.

Chapter 91

Monotonicity of entailment

Monotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived factmay be freely extended with additional assumptions. In sequent calculi this property can be captured by an inferencerule called weakening, or sometimes thinning, and in such systems one may say that entailment is monotone if andonly if the rule is admissible. Logical systems with this property are occasionally called monotonic logics in order todifferentiate them from non-monotonic logics.

91.1 Weakening rule

To illustrate, starting from the natural deduction sequent:

Γ ⊢ C

weakening allows one to conclude:

Γ, A ⊢ C

91.2 Non-monotonic logics

Main article: Non-monotonic logic

In most logics, weakening is either an inference rule or a metatheorem if the logic doesn't have an explicit rule.Notable exceptions are:

• Strict logic or relevant logic, where every hypothesis must be necessary for the conclusion.

• Linear logic which disallows arbitrary contraction in addition to arbitrary weakening.

• Bunched implications where weakening is restricted to additive composition.

• Various types of default reasoning.

• Abductive reasoning, the process of deriving the most likely explanations of the known facts.

• Reasoning about knowledge, where statements specifying that something is not known need to be retractedwhen that thing is learned.

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112 CHAPTER 91. MONOTONICITY OF ENTAILMENT

91.3 See also• Contraction

• Exchange rule

• Substructural logic

Chapter 92

Multimodal logic

A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial appli-cations in theoretical computer science.A modal logic with n primitive unary modal operators □i, i ∈ {1, . . . , n} is called an n-modal logic. Given theseoperators and negation, one can always add ♢i modal operators defined as ♢iP if and only if ¬□i¬P .The first substantive example of a 2-modal logic is perhaps Arthur Prior's tense logic, with two modalities, F and P,corresponding to “sometime in the future” and “sometime in the past”. A logic[1] with infinitely many modalities is(propositional) dynamic logic, introduced in 1976 and having a separate modal operator for every regular expression.A version of temporal logic introduced in 1977 and intended for program verification has two modalities, corre-sponding to dynamic logic’s [A] and [A*] modalities for a single program A, understood as the whole universe takingone step forwards in time. The term “multimodal logic” itself was not introduced until 1980. Another example ofa multimodal logic is the Hennessy-Milner logic, itself a fragment of the more expressive modal μ-calculus, whichadditionally is also a fixed-point logic.Multimodal logic can be used also to formalize a kind of knowledge representation: the motivation of epistemic logicis allowing several agents (they are regarded as subjects capable of forming beliefs, knowledge); and managing thebelief or knowledge of each agent, so that epistemic assertions can be formed about them. The modal operator □must be capable of bookkeeping the cognition of each agent, thus □i must be indexed on the set of the agents. Themotivation is that □iα should assert “The subject i has knowledge about α being true”. But it can be used also forformalizing “the subject i believes α ". For formalization of meaning based on the possible world semantics approach,a multimodal generalization of Kripke semantics can be used: instead of a single “common” accessibility relation,there is a series of them indexed on the set of agents.[2]

92.1 Notes

[1] Sergio Tessaris; Enrico Franconi; Thomas Eiter (2009). Reasoning Web. Semantic Technologies for Information Systems:5th International Summer School 2009, Brixen-Bressanone, Italy, August 30 - September 4, 2009, Tutorial Lectures. Springer.p. 112. ISBN 978-3-642-03753-5.

[2] Ferenczi 2002: 257

92.2 References

• Ferenczi, Miklós (2002). Matematikai logika (in Hungarian). Budapest: Műszaki könyvkiadó. ISBN 963-16-2870-1.

• Dov M. Gabbay, A. Kurucz, F. Wolter, M. Zakharyaschev (2003). Many-dimensional modal logics: theoryand applications. Elsevier. ISBN 978-0-444-50826-3.

• Walter Carnielli; Claudio Pizzi (2008). Modalities and Multimodalities. Springer. ISBN 978-1-4020-8589-5.

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114 CHAPTER 92. MULTIMODAL LOGIC

92.3 External links• Stanford Encyclopedia of Philosophy: "Modal logic" – by James Garson.

Chapter 93

Multiple-conclusion logic

Amultiple-conclusion logic is one in which logical consequence is a relation, ⊢ , between two sets of sentences (orpropositions). Γ ⊢ ∆ is typically interpreted as meaning that whenever each element of Γ is true, some element of∆ is true; and whenever each element of∆ is false, some element of Γ is false.This form of logic was developed in the 1970s by D. J. Shoesmith and Timothy Smiley[1] but has not been widelyadopted.Some logicians favor a multiple-conclusion consequence relation over the more traditional single-conclusion relationon the grounds that the latter is asymmetric (in the informal, non-mathematical sense) and favors truth over falsity(or assertion over denial).

93.1 See also• Sequent calculus

93.2 References[1] D. J. Shoesmith and T. J. Smiley, Multiple Conclusion Logic, Cambridge University Press, 1978

115

Chapter 94

Neighborhood semantics

Neighborhood semantics, also known as Scott-Montague semantics, is a formal semantics for modal logics. It is ageneralization, developed independently by Dana Scott and Richard Montague, of the more widely known relationalsemantics for modal logic. Whereas a relational frame ⟨W,R⟩ consists of a set W of worlds (or states) and anaccessibility relation R intended to indicate which worlds are alternatives to (or, accessible from) others, a neighbor-hood frame ⟨W,N⟩ still has a setW of worlds, but has instead of an accessibility relation a neighborhood function

N :W → 22W

that assigns to each element ofW a set of subsets ofW. Intuitively, each family of subsets assigned to a world are thepropositions necessary at that world, where 'proposition' is defined as a subset of W (i.e. the set of worlds at whichthe proposition is true). Specifically, if M is a model on the frame, then

M,w |= □A⇐⇒ (A)M ∈ N(w),

where

(A)M = {u ∈W |M,u |= A}

is the truth set of A.Neighborhood semantics is used for the classical modal logics that are strictly weaker than the normal modal logicK.

94.1 Correspondence between relational and neighborhood models

To every relational model M = (W,R,V) there corresponds an equivalent (in the sense of having point-wise equivalentmodal theories) neighborhood model M' = (W,N,V) defined by

N(w) = {(A)M :M,w |= □A}.The fact that the converse fails gives a precise sense to the remark that neighborhood models are a generalization ofrelational ones. Another (perhapsmore natural) generalization of relational structures are general relational structures.

94.2 References• Scott, D. “Advice in modal logic”, in Philosophical Problems in Logic, ed. Karel Lambert. Reidel, 1970.• Montague, R. “Universal Grammar”, Theoria 36, 373-98, 1970.• Chellas, B.F. Modal Logic. Cambridge University Press, 1980.

116

Chapter 95

Non-wellfounded mereology

In philosophy, specifically metaphysics, mereology is the study of parthood relationships. In mathematics and formallogic, wellfoundedness prohibits · · · < x < · · · < x < · · · for any x.Thus non-wellfoundedmereology treats topologically circular, cyclical, repetitive, or other eventual self-containment.More formally, non-wellfounded partial orders may exhibit · · · < x < · · · < x < · · · for some x whereas well-founded orders prohibit that.

95.1 See also• Aczel’s anti-foundation axiom

• Peter Aczel

• John Barwise

• Steve Awodey

• Dana Scott

95.2 External links• http://plato.stanford.edu/entries/nonwellfounded-set-theory/

117

Chapter 96

Nonfirstorderizability

In formal logic, nonfirstorderizability is the inability of an expression to be adequately captured in particular theoriesin first-order logic. Nonfirstorderizable sentences are sometimes presented as evidence that first-order logic is notadequate to capture the nuances of meaning in natural language.The term was coined by George Boolos in his well-known paper “To Be is to Be a Value of a Variable (or to Be SomeValues of Some Variables).” Boolos argued that such sentences call for second-order symbolization, which can beinterpreted as plural quantification over the same domain as first-order quantifiers use, without postulation of distinct“second-order objects” (properties, sets, etc.).A standard example, known as the Geach–Kaplan sentence, is:

Some critics admire only one another.

If Axy is understood to mean "x admires y,” and the universe of discourse is the set of all critics, then a reasonabletranslation of the sentence into second order logic is:

∃X(∃x, y(Xx ∧Xy ∧Axy) ∧ ∃x¬Xx ∧ ∀x∀y(Xx ∧Axy → Xy))

That this formula has no first-order equivalent can be seen as follows. Substitute the formula (y = x + 1 v x = y + 1)for Axy. The result,

∃X(∃x, y(Xx ∧Xy ∧ (y = x+ 1 ∨ x = y + 1)) ∧ ∃x¬Xx ∧ ∀x ∀y(Xx ∧ (y = x+ 1 ∨ x = y + 1) → Xy))

states that there is a nonempty set which is closed under the predecessor and successor operations and yet does notcontain all numbers. Thus, it is true in all nonstandard models of arithmetic but false in the standard model. Sinceno first-order sentence has this property, the result follows.

96.1 See also• Plural quantification

• Reification (linguistics)

• Branching quantifier

• Generalized quantifier

96.2 References• George Boolos (1984). “To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)".

Journal of Philosophy (The Journal of Philosophy, Vol. 81, No. 8) 81 (8): 430–49. doi:10.2307/2026308.

118

96.2. REFERENCES 119

JSTOR 2026308. Reprinted in Boolos, George (1998). Logic, Logic, and Logic. Cambridge, MA: HarvardUniversity Press. ISBN 0-674-53767-X.

Chapter 97

Normal form (natural deduction)

An inference of natural deduction is a normal form, according to Dag Prawitz, if no formula occurrence is both theprincipal premise of an elimination rule and the conclusion of an introduction rule.

120

Chapter 98

Normal modal logic

In logic, a normal modal logic is a set L of modal formulas such that L contains:

• All propositional tautologies;

• All instances of the Kripke schema: □(A→ B) → (□A→ □B)

and it is closed under:

• Detachment rule (Modus Ponens): A→ B,A ⊢ B ;

• Necessitation rule: ⊢ A implies ⊢ □A .

The smallest logic satisfying the above conditions is calledK. Most modal logics commonly used nowadays (in termsof having philosophical motivations), e.g. C. I. Lewis's S4 and S5, are extensions ofK. However a number of deonticand epistemic logics, for example, are non-normal, often because they give up the Kripke schema.

98.1 Common normal modal logics

The following table lists several common normal modal systems. The notation refers to the table at Kripke semantics§ Commonmodal axiom schemata. Frame conditions for some of the systems were simplified: the logics are completewith respect to the frame classes given in the table, but they may correspond to a larger class of frames.

121

Chapter 99

OBJ3

OBJ3 is a version of OBJ based on order-sorted rewriting. OBJ3 is agent-oriented and runs on Kyoto Common LispAKCL. It is now of (important) historical interest since newer versions of the OBJ family are available.

99.1 References• Introducing OBJ3, Joseph Goguen et al., SRI-CSL-88-9, SRI International, USA, 1988.

99.2 External links• Information and OBJ3 manual, PostScript format

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008and incorporated under the “relicensing” terms of the GFDL, version 1.3 or later.

122

Chapter 100

Objection (argument)

“Refute” redirects here. For the Transformer, see Refute (Transformers).In informal logic an objection (also called expostulation or refutation), is a reason arguing against a premise,

Refutation on Graham's Hierarchy of Disagreement

lemma, or main contention. An objection to an objection is known as a rebuttal.

100.1 See also• Argument map• Argumentation theory• Inference objection

123

Chapter 101

One-sided argument

A one-sided argument (also known as card stacking, stacking the deck, ignoring the counterevidence, slanting,and suppressed evidence)[1] is an informal fallacy that occurs when only the reasons supporting a proposition aresupplied, while all reasons opposing it are omitted.Peter Suber has written: “The one-sidedness fallacy does not make an argument invalid. It may not even make theargument unsound. The fallacy consists in persuading readers, and perhaps ourselves, that we have said enough to tiltthe scale of evidence and therefore enough to justify a judgment. If we have been one-sided, though, then we haven'tyet said enough to justify a judgment. The arguments on the other side may be stronger than our own. We won'tknow until we examine them. So the one-sidedness fallacy doesn't mean that your premises are false or irrelevant,only that they are incomplete.”[2]

“With rational messages, you need to decide if you want to use a one-sided argument or a two-sided argument. A one-sided argument only presents the pro side of the argument, while a two-sided argument presents both sides. Whichone you use will depend on which one meets your needs and the type of audience. Generally, one-sided argumentsare better with audiences already favorable to your message. Two-sided arguments are best with audiences who areopposed to your argument, are better educated or have already been exposed to counter arguments.”[2]

101.1 See also• Pars destruens/pars construens

• Special pleading

• Confirmation bias

• Cherry picking

101.2 References[1] “One-Sidedness - The Fallacy Files”. Retrieved October 2014.

[2] Peter Suber. “The One-Sidedness Fallacy”. Retrieved 25 September 2012.

101.3 External links• The One-Sidedness Fallacy - Peter Suber, Philosophy Department, Earlham College

• Developing a Promotional Strategy - Michigan State University, Extension Bulletin E-1939

• Acharacterization of the one-sidedness fallacywithin the framework of the cognitive distortions, Paul Franceschi,2009

124

Chapter 102

Otter (theorem prover)

Otter is an automated theorem prover developed by William McCune at Argonne National Laboratory in Illinois.Otter was the first widely distributed, high-performance theorem prover for first-order logic, and it pioneered a numberof important implementation techniques. Otter is an acronym for Organized Techniques for Theorem-proving andEffective Research.Otter has been very stable for a number of years but is no longer actively developed. As of November 2008, the lastchangelog entry was dated 14 September 2004. A successor to Otter is Prover9.The software is in the public domain. The University of Chicago has declined to assert its copyrights in this software,and it may be used, modified, and redistributed (with or without modifications) by the public. However, “NEITHERTHE UNITED STATES GOVERNMENT NOR ANY AGENCY THEREOF [...] REPRESENTS THAT ITS USEWOULD NOT INFRINGE PRIVATELY OWNED RIGHTS.”[1]

According to Wos and Pieper, OTTER is written in approximately 28,000 lines of C programming language.

102.1 See also• MACE

102.2 References[1] File name Legal in the tarball

• McCune, William; Larry Wos (1997). “Otter: The CADE-13 Competition Incarnations”. Journal of Auto-mated Reasoning 18 (2): 211–220. doi:10.1023/A:1005843632307.

• Kalman, John Arnold. Automated Reasoning with OTTER. ISBN 1-58949-004-5

102.3 External links• Otter home page

• Prover9 home page

125

Chapter 103

Pars destruens/pars construens

Pars destruens/pars construens (lat.) is in common parlance about different parts of an argumentation. Thenegative part of criticizing views is the pars destruens. And the positive part of stating one’s own position andarguments is the pars construens.The distinction goes back to Francis Bacon and his work Novum Organum, 1620. There he puts forth his inductivemethod that has two parts. A negative part, pars destruens, that removes all prejudices and errors. And the positivepart, pars construens, that is about gaining knowledge and truth.

103.1 External links• A site about Francis Bacon

126

Chapter 104

PhoX

In automated theorem proving, PhoX is a proof assistant based on higher-order logic which is eXtensible. The usergives PhoX an initial goal and guides it through subgoals and evidence to prove that goal; internally, it constructsnatural deduction trees. Each previously proven formula can become a rule for later proofs.PhoX was originally designed and implemented by Christophe Raffalli in the OCaml programming language. He hascontinued to lead the current development team, a joint effort of Savoy University and University Paris VII.The primary aim of the PhoX project creating a user friendly proof checker using the type system developed by Jean-Louis Krivine at University Paris VII. It is meant to be more intuitive than other systems while remaining extensible,efficient, and expressive. Compared to other systems, the proof-building syntax is simplified and closer to naturallanguage. Other features includeGUI-driven proof construction, rendering formatted output, and proof of correctnessof programs in the ML programming language.PhoX is currently used to teach logic at Savoy University. It is in an experimental but usable state. It is released underCeCILL 2.0.

104.1 External links• PhoX website

127

Chapter 105

Polysyllogism

Not to be confused with Polylogism.

A polysyllogism (also called multi-premise syllogism, sorites, climax, or gradatio) is a string of any number ofpropositions forming together a sequence of syllogisms such that the conclusion of each syllogism, together with thenext proposition, is a premise for the next, and so on. Each constituent syllogism is called a prosyllogism except thevery last, because the conclusion of the last syllogism is not a premise for another syllogism.

105.1 Example

An example for a polysyllogism is:

It is raining.If we go out while it is raining we will get wet.If we get wet, we will get cold.Therefore, if we go out we will get cold.

Examination of the structure of the argument reveals the following sequence of constituent (pro)syllogisms:

It is raining.If we go out while it is raining we will get wet.Therefore, if we go out we will get wet.

If we go out we will get wet.If we get wet, we will get cold.Therefore, if we go out we will get cold.

105.2 Sorites

A sorites is a specific kind of polysyllogism in which the predicate of each proposition is the subject of the nextpremise. Example:

All lions are big cats.All big cats are predators.All predators are carnivores.Therefore, all lions are carnivores.

128

105.3. SEE ALSO 129

The word “sorites” /sɒˈraɪtiːz/ comes from Ancient Greek: σωρίτης, “heaped up”, from σωρός “heap” or “pile”. Inother words, a sorites is a heap of propositions chained together. A sorites polysyllogism should not be confused withthe sorites paradox, a.k.a. fallacy of the heap.Lewis Carroll uses sorites in his book Symbolic Logic (1896). Here is an example:[1]

No experienced person is incompetent;Jenkins is always blundering;No competent person is always blundering.∴ Jenkins is inexperienced.

Carroll’s example may be translated thus

All experienced persons are competent persons.No competent persons are blunderers.Jenkins is a blunderer.∴ Jenkins is not an experienced person.

105.3 See also• Anadiplosis - the rhetorical grounds of polysyllogism.

• Transitive relation

105.4 References• B. P. Bairan. An Introduction to Syllogistic Logic. Goodwill Trading. p. 342. ISBN 971-574-094-4.

[1] p.113

Chapter 106

Post disputation argument

A post disputation argument is an argument in which one party attempts to alter their view on the disputed factsafter the answer has already been discovered by an outside medium. It is an example of a fallacy.Example:

Party one claims A is true.Party two claims B is true and A is false.Party one claims B is false.Party one and party two look in reliable source for the truth.According to the source party one is correct because A is true.Party two claims that they are correct and then restates their claim into argument B'.

Party two has changed their view after the truth was discovered and by changing their argument appears to have beenright since the beginning. This is a post disputation argument. A post disputation argument is consideredmanipulativein the sense that often it is not used to clarify a party’s initial stance but to change it so they appear correct after thetruth has been discovered.

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Chapter 107

Predicate logic

For the specific term, see First-order logic.

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic, or infinitary logic. This formal system is distinguished from other systems in that itsformulae contain variables which can be quantified. Two common quantifiers are the existential ∃ (“there exists”)and universal ∀ (“for all”) quantifiers. The variables could be elements in the universe under discussion, or perhapsrelations or functions over that universe. For instance, an existential quantifier over a function symbol would beinterpreted as modifier “there is a function”. The foundations of predicate logic were developed independently byGottlob Frege and Charles Sanders Peirce.[1]

In informal usage, the term “predicate logic” occasionally refers to first-order logic. Some authors consider thepredicate calculus to be an axiomatized form of predicate logic, and the predicate logic to be derived from aninformal, more intuitive development.[2]

Predicate logics also include logics mixing modal operators and quantifiers. See Modal logic, Saul Kripke, BarcanMarcus formulae, A. N. Prior, and Nicholas Rescher.

107.1 See also

• First-order logic

• Propositional logic

• Existential graph

107.2 Footnotes[1] Eric M. Hammer: Semantics for Existential Graphs, Journal of Philosophical Logic, Volume 27, Issue 5 (October 1998),

page 489: “Development of first-order logic independently of Frege, anticipating prenex and Skolem normal forms”

[2] Among these authors is Stolyar, p. 166. Hamilton considers both to be calculi but divides them into an informal calculusand a formal calculus.

107.3 References

• A. G. Hamilton 1978, Logic for Mathematicians, Cambridge University Press, Cambridge UK ISBN 0-521-21838-1

• Abram Aronovic Stolyar 1970, Introduction to Elementary Mathematical Logic, Dover Publications, Inc. NY.ISBN 0-486-645614

131

132 CHAPTER 107. PREDICATE LOGIC

• George F Luger, Artificial Intelligence, Pearson Education, ISBN 978-81-317-2327-2

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

Chapter 108

Principle of nonvacuous contrast

The principle of nonvacuous contrast is a logical ormethodological principle which requires that a genuine predicatenever refer to everything, or to nothing, within its universe of discourse.

108.1 References• William Dray (1964). Philosophy of History. Englewood Cliffs, NJ: Prentice-Hall, Inc. pp. 29 pp.

133

Chapter 109

Principles of Mathematical Logic

Principles ofMathematical Logic is the 1950[1] American translation of the 1938 second edition[2] of David Hilbert'sand Wilhelm Ackermann's classic text Grundzüge der theoretischen Logik,[3] on elementary mathematical logic. The1928 first edition thereof is considered the first elementary text clearly grounded in the formalism now known asfirst-order logic (FOL). Hilbert and Ackermann also formalized FOL in a way that subsequently achieved canonicalstatus. FOL is now a core formalism of mathematical logic, and is presupposed by contemporary treatments of Peanoarithmetic and nearly all treatments of axiomatic set theory.The 1928 edition included a clear statement of the Entscheidungsproblem (decision problem) for FOL, and also askedwhether that logic was complete (i.e., whether all semantic truths of FOL were theorems derivable from the FOLaxioms and rules). The first problem was answered in the negative by Alonzo Church in 1936. The second wasanswered affirmatively by Kurt Gödel in 1929.The text also touched on set theory and relational algebra as ways of going beyond FOL. Contemporary notation forlogic owes more to this text than it does to the notation of Principia Mathematica, long popular in the English speakingworld.

109.1 Notes[1] Curry, Haskell B. (1953). “Review: Grundzüge der theoretischen Logik (3rd edition)" (PDF). Bull. Amer. Math. Soc. 59

(3): 263–267. The translation of the 1938 2nd German edition into English was published in 1950, while the 3rd Germanedition was published in 1949.

[2] Rosser, Barkley (1938). “Review: Grundzüge der theoretischen Logik (2nd edition)" (PDF). Bull. Amer. Math. Soc. 44(7): 474–475.

[3] Langford, C. H (1930). “Review of Grundzüge der theoretischen Logik by D. Hilbert and W. Ackermann” (PDF). Bull.Amer. Math. Soc. 36 (1): 22–25. doi:10.1090/s0002-9904-1930-04859-4.

109.2 References• David Hilbert andWilhelmAckermann (1928). Grundzüge der theoretischen Logik (Principles of Mathematical

Logic). Springer-Verlag, ISBN 0-8218-2024-9. This text went into four subsequent German editions, the lastin 1972.

• Hendricks, Neuhaus, Petersen, Scheffler and Wansing (eds.) (2004). First-order logic revisited. Logos Verlag,ISBN 3-8325-0475-3. Proceedings of a workshop, FOL-75, commemorating the 75th anniversary of thepublication of Hilbert and Ackermann (1928).

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Chapter 110

Probabilistic proposition

A probabilistic proposition is a proposition with a measured probability of being true for an arbitrary person at anarbitrary time.These are some examples of probabilistic propositions collected by the Mindpixel project:

• You are not human 0.17• Color and colour are the same word spelled differently 0.95• You do not think the sun will rise tomorrow 0.15• You have never seen the sky 0.13• You are a rock 0.01• Is westlife a pop group? 0.50• I exist 0.98• Bread is raw toast 0.89• Do you know how to talk? 0.89

135

Chapter 111

Problem of multiple generality

The problem ofmultiple generality names a failure in traditional logic to describe certain intuitively valid inferences.For example, it is intuitively clear that if:

Some cat is feared by every mouse

then it follows logically that:

All mice are afraid of at least one cat

The syntax of traditional logic (TL) permits exactly four sentence types: “All As are Bs”, “No As are Bs”, “SomeAs are Bs” and “Some As are not Bs”. Each type is a quantified sentence containing exactly one quantifier. Sincethe sentences above each contain two quantifiers ('some' and 'every' in the first sentence and 'all' and 'at least one' inthe second sentence), they cannot be adequately represented in TL. The best TL can do is to incorporate the secondquantifier from each sentence into the second term, thus rendering the artificial-sounding terms 'feared-by-every-mouse' and 'afraid-of-at-least-one-cat'. This in effect “buries” these quantifiers, which are essential to the inference’svalidity, within the hyphenated terms. Hence the sentence “Some cat is feared by every mouse” is allotted the samelogical form as the sentence “Some cat is hungry”. And so the logical form in TL is:

Some As are Bs

All Cs are Ds

which is clearly invalid.The first logical calculus capable of dealing with such inferences was Gottlob Frege's Begriffsschrift (1879), the an-cestor of modern predicate logic, which dealt with quantifiers by means of variable bindings. Modestly, Frege didnot argue that his logic was more expressive than extant logical calculi, but commentators on Frege’s logic regard thisas one of his key achievements.Using modern predicate calculus, we quickly discover that the statement is ambiguous.

Some cat is feared by every mouse

could mean (Some cat is feared) by every mouse, i.e.

For every mouse m, there exists a cat c, such that c is feared by m,

∀m. (Mouse(m) → ∃c. (Cat(c) ∧ Fears(m, c)) )

in which case the conclusion is trivial.But it could also mean Some cat is (feared by every mouse), i.e.

There exists one cat c, such that for every mouse m, c is feared by m.

∃c. (Cat(c) ∧ ∀m. (Mouse(m) → Fears(m, c)) )

This example illustrates the importance of specifying the scope of quantifiers as for all and there exists.

136

111.1. FURTHER READING 137

111.1 Further reading• Patrick Suppes, Introduction to Logic, D. Van Nostrand, 1957, ISBN 978-0-442-08072-3.

• A. G. Hamilton, Logic for Mathematicians, Cambridge University Press, 1978, ISBN 0-521-29291-3.

• Paul Halmos and Steven Givant, Logic as Algebra, MAA, 1998, ISBN 0-88385-327-2.

Chapter 112

Proof net

In proof theory, proof nets are a geometrical method of representing proofs that eliminates two forms of bureaucracythat differentiates proofs: (A) irrelevant syntactical features of regular proof calculi such as the natural deductioncalculus and the sequent calculus, and (B) the order of rules applied in a derivation. In this way, the formal propertiesof proof identity correspond more closely to the intuitively desirable properties. Proof nets were introduced by Jean-Yves Girard.For instance, these two linear logic proofs are “morally” identical:And their corresponding nets will be the same.

112.1 Correctness criteria

Several correctness criteria are known to check if a sequential proof structure (i.e. something which seems to be aproof net) is actually a concrete proof structure (i.e. something which encodes a valid derivation in linear logic). Thefirst such criterion is the long-trip criterion[1] which was described by Jean-Yves Girard.

112.2 See also• Linear logic• Ludics• Geometry of interaction• Coherent space• Deep inference• Interaction nets

112.3 References[1] Girard, Jean-Yves. Linear logic, Theoretical Computer Science, Vol 50, no 1, pp. 1–102, 1987

112.4 Sources• Proofs and Types. Girard J-Y, Lafont Y, and Taylor P. Cambridge Press, 1989.• Roberto Di Cosmo and Vincent Danos, The Linear Logic Primer• Sean A. Fulop, A survey of proof nets and matrices for substructural logics

138

Chapter 113

Proof-theoretic semantics

Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate themeaning of propositionsand logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that theproposition or logical connective plays within the system of inference.Gerhard Gentzen is the founder of proof-theoretic semantics, providing the formal basis for it in his account ofcut-elimination for the sequent calculus, and some provocative philosophical remarks about locating the meaning oflogical connectives in their introduction rules within natural deduction. The history of proof-theoretic semantics sincethen has been devoted to exploring the consequences of these ideas.Dag Prawitz extended Gentzen’s notion of analytic proof to natural deduction, and suggested that the value of aproof in natural deduction may be understood as its normal form. This idea lies at the basis of the Curry–Howardisomorphism, and of intuitionistic type theory. His inversion principle lies at the heart of most modern accounts ofproof-theoretic semantics.Michael Dummett introduced the very fundamental idea of logical harmony, building on a suggestion of Nuel Belnap.In brief, a language, which is understood to be associated with certain patterns of inference, has logical harmony if itis always possible to recover analytic proofs from arbitrary demonstrations, as can be shown for the sequent calculusby means of cut-elimination theorems and for natural deduction by means of normalisation theorems. A language thatlacks logical harmony will suffer from the existence of incoherent forms of inference: it will likely be inconsistent.

113.1 References• Proof-Theoretic Semantics, at the Stanford Encyclopedia of Philosophy

• Logical Consequence, Deductive-Theoretic Conceptions, at the Internet Encyclopedia of Philosophy.

• Nissim Francez, “On a Distinction of Two Facets of Meaning and its Role in Proof-theoretic Semantics”,Logica Universalis 9, 2015. doi:10.1007/s11787-015-0118-8

113.2 See also• Inferential role semantics

• Truth-conditional semantics

113.3 External links• Arché Bibliography on Proof-Theoretic Semantics.

139

Chapter 114

Propositional variable

In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is a variablewhich can either be true or false. Propositional variables are the basic building-blocks of propositional formulas,used in propositional logic and higher logics.

114.1 Uses

Formulas in logic are typically built up recursively from some propositional variables, some number of logical con-nectives, and some logical quantifiers. Propositional variables are the atomic formulas of propositional logic.

Example

In a given propositional logic, we might define a formula as follows:

• Every propositional variable is a formula.

• Given a formula X the negation ¬X is a formula.

• Given two formulas X and Y, and a binary connective b (such as the logical conjunction ∧), then (X b Y) is aformula. (Note the parentheses.)

In this way, all of the formulas of propositional logic are built up from propositional variables as a basic unit. Propo-sitional variables should not be confused with the metavariables which appear in the typical axioms of propositionalcalculus; the latter effectively range over well-formed formulae.

114.2 In first order logic

Propositional variables are represented as nullary predicates in first order logic.

114.3 See also

114.4 References• Smullyan, RaymondM. First-Order Logic. 1968. Dover edition, 1995. Chapter 1.1: Formulas of PropositionalLogic.

140

Chapter 115

Prototype Verification System

ThePrototype Verification System (PVS) is a specification language integrated with support tools and an automatedtheorem prover, developed at the Computer Science Laboratory of SRI International in Menlo Park, California.PVS is based on a kernel consisting of an extension of Church's theory of types with dependent types, and is funda-mentally a classical typed higher-order logic. The base types include uninterpreted types that may be introduced bythe user, and built-in types such as the booleans, integers, reals, and the ordinals. Type-constructors include func-tions, sets, tuples, records, enumerations, and abstract data types. Predicate subtypes and dependent types can beused to introduce constraints; these constrained types may incur proof obligations (called type-correctness conditionsor TCCs) during typechecking. PVS specifications are organized into parameterized theories.The system is implemented in Common Lisp, and is released under the GNU General Public License (GPL).

115.1 See also• Formal methods

115.2 References• Owre, Shankar, and Rushby, 1992. PVS: A Prototype Verification System. Published in theCADE 11 conferenceproceedings.

115.3 External links• PVS website at SRI International's Computer Science Laboratory

• Summary of PVS by John Rushby at the Mechanized Reasoning database of Michael Kohlhase and CarolynTalcott

141

Chapter 116

Provability logic

Provability logic is a modal logic, in which the box (or “necessity”) operator is interpreted as 'it is provable that'.The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic.

116.1 Examples

There are a number of provability logics, some of which are covered in the literature mentioned in the Referencessection. The basic system is generally referred to as GL (for Gödel-Löb) or L or K4W. It can be obtained by addingthe modal version of Löb’s theorem to the logic K (or K4).Namely, the axioms of GL are all tautologies of classical propositional logic plus all formulas of one of the followingforms:

• Distribution Axiom: □(p→ q) → (□p→□q);

• Löb’s Axiom: □(□p→ p) → □p.

And the rules of inference are:

• Modus Ponens: From p→ q and p conclude q;

• Necessitation: From p conclude □p.

116.2 History

The GL model was pioneered by Robert M. Solovay in 1976. Since then until his death in 1996 the prime inspirerof the field was George Boolos. Significant contributions to the field have been made by Sergei N. Artemov, LevBeklemishev, Giorgi Japaridze, Dick de Jongh, Franco Montagna, Vladimir Shavrukov, Albert Visser and others.

116.3 Generalizations

Interpretability logics and Japaridze’s Polymodal Logic present natural extensions of provability logic.

116.4 See also• Interpretability logic

• Kripke semantics

• Japaridze’s Polymodal Logic

142

116.5. REFERENCES 143

116.5 References• George Boolos, The Logic of Provability. Cambridge University Press, 1993.

• Giorgi Japaridze and Dick de Jongh, The logic of provability. In: Handbook of Proof Theory, S. Buss, ed.Elsevier, 1998, pp. 475-546.

• Sergei N. Artemov and Lev Beklemishev, Provability logic. In: Handbook of Philosophical Logic, D. Gabbayand F. Guenthner, eds., vol. 13, 2nd ed., pp. 189-360. Springer, 2005.

• Per Lindström, Provability logic - a short introduction. Theoria 62 (1996), pp. 19-61.

• Craig Smoryński, Self-reference and modal logic. Springer, Berlin, 1985.

• Robert M. Solovay, ``Provability Interpretations of Modal Logic``, Israel Journal of Mathematics, Vol. 25(1976): 287-304.

• Provability logic, from the Stanford Encyclopedia of Philosophy.

Chapter 117

Proving a point

Proving a point is an element of debate or argument in which the logical truth of a position is established.[1]

In mathematics, when a proof is complete and the point is made, Q.E.D. or a tombstone (∎) may be used as aconcluding flourish.

117.1 References[1] John D. Mullen, Hard thinking: the reintroduction of logic into everyday life

and vagebrus

144

Chapter 118

Regular modal logic

In modal logic, a regular modal logic L is a modal logic closed under the duality of the modal operators:♢A ≡ ¬□¬Aand the rule(A ∧B) → C ⊢ (□A ∧□B) → □C.Every regular modal logic is classical, and every normal modal logic is regular and hence classical.

118.1 References

Chellas, Brian. Modal Logic: An Introduction. Cambridge University Press, 1980.

145

Chapter 119

Robinson’s joint consistency theorem

Robinson’s joint consistency theorem is an important theorem of mathematical logic. It is related to Craig inter-polation and Beth definability.The classical formulation of Robinson’s joint consistency theorem is as follows:Let T1 and T2 be first-order theories. If T1 and T2 are consistent and the intersection T1 ∩ T2 is complete (in thecommon language of T1 and T2 ), then the union T1 ∪ T2 is consistent. Note that a theory is complete if it decidesevery formula, i.e. either T ⊢ φ or T ⊢ ¬φ .Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem:Let T1 and T2 be first-order theories. If T1 and T2 are consistent and if there is no formula φ in the common languageof T1 and T2 such that T1 ⊢ φ and T2 ⊢ ¬φ , then the union T1 ∪ T2 is consistent.

119.1 References• Boolos, George S.; Burgess, John P.; Jeffrey, Richard C. (2002). Computability and Logic. Cambridge Uni-versity Press. p. 264. ISBN 0-521-00758-5.

146

Chapter 120

Rule of replacement

In logic, a rule of replacement[1][2][3] is a transformation rule that may be applied to only a particular segmentof an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both astransformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a wholelogical expression, a rule of replacement may be applied to only a particular segment. Within the context of a logicalproof, logically equivalent expressions may replace each other. Rules of replacement are used in propositional logicto manipulate propositions.Common rules of replacement include de Morgan’s laws, commutation, association, distribution, double negation,[4]transposition, material implication, material equivalence, exportation, and tautology.

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

[2] Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.

[3] Moore and Parker

[4] not admitted in intuitionistic logic

147

Chapter 121

Rules of passage (logic)

In mathematical logic, the rules of passage govern how quantifiers distribute over the basic logical connectives offirst-order logic. The rules of passage govern the “passage” (translation) from any formula of first-order logic to theequivalent formula in prenex normal form, and vice versa.

121.1 The rules

See Quine (1982: 119, chpt. 23). Let Q and Q 'denote ∀ and ∃ or vice versa. β denotes a closed formula inwhich x does not appear. The rules of passage then include the following sentences, whose main connective is thebiconditional:

• Qx[¬α(x)] ↔ ¬Q′x[α(x)].

The following conditional sentences can also be taken as rules of passage:

• ∃x[α(x) ∧ γ(x)] → (∃xα(x) ∧ ∃xγ(x)).

• (∀xα(x) ∨ ∀x γ(x)) → ∀x [α(x) ∨ γ(x)].

• (∃xα(x) ∧ ∀x γ(x)) → ∃x [α(x) ∧ γ(x)].

“Rules of passage” first appeared in French, in the writings of Jacques Herbrand. Quine employed the Englishtranslation of the phrase in each edition of his Methods of Logic, starting in 1950.

121.2 See also

• First-order logic

• Prenex normal form

• Quantifier

121.3 References

• Willard Quine, 1982. Methods of Logic, 4th ed. Harvard Univ. Press.

• Jean Van Heijenoort, 1967. From Frege to Gödel: A Source Book on Mathematical Logic. Harvard Univ. Press.

148

121.4. EXTERNAL LINKS 149

121.4 External links• Stanford Encyclopedia of Philosophy: "Classical Logic -- by Stewart Shapiro.

Chapter 122

Sacrifice of the intellect

The sacrifice of the intellect (sacrificium intellectus, sometimes rendered in Italian, sacrifizio dell'intelletto) is a con-cept associated with Christian devotion, particularly with the Jesuit order. It was the “third sacrifice” demanded bythe founder of the Jesuits, St. Ignatius Loyola, who required

besides entire outward submission to command, also the complete identification of the inferior’s willwith that of the superior . [Loyola] lays down that the superior is to be obeyed simply as such and asstanding in the place of God, without reference to his personal wisdom, piety or discretion; that anyobedience which falls short of making the superior’s will one’s own, in inward affection as well as inoutward effect, is lax and imperfect; that going beyond the letter of command, even in things abstractlygood and praise-worthy, is disobedience, and that the “sacrifice of the intellect” is the third and highestgrade of obedience, well pleasing to God, when the inferior not only wills what the superior wills, butthinks what he thinks, submitting his judgment, so far as it is possible for the will to influence and leadthe judgment.[1]

The concept was taken up in a more individualistic sense by the Jansenist thinker Blaise Pascal, and particularly by theexistentialist thinker Søren Kierkegaard, who thought that the act of faith requires a leap into the void, which amountsto a sacrifice of the intellect and reason.[2] This was quintessentially expressed in the traditional dictum, credo quiaabsurdum, “I believe because it is absurd.” This view of faith is rejected by the Catholic church, which regards reasonas a path towards direct knowledge of God.[3]

The phrase is often used in a pejorative sense in writings on the psychology and sociology of religion - e.g.:

• Max Weber states: “There is absolutely no 'unbroken' religion working as a vital force, which is not compelledat some point to demand the credo non quod, sed credo quia absurdum - the “sacrifice of the intellect.""[4]

• According to Paul Pruyser, “Sacrifice of the intellect, demanded by a good many religious movements andblithely if not joyously made by a good many religious persons, is surely one of the ominous features of neuroticreligion.”[5]

122.1 References[1] Encyclopædia Britannica, 11th edition, “JESUIT”. Available online.

[2] S. Kierkegaard, Fear and Trembling. (Copenhagen: 1843)

[3] See St. Thomas Aquinas, Summa Theologica; Pope John Paul II, encyclical Fides et ratio.

[4] MaxWeber, “World Rejection and Theodicy”, ch.8. See H.H. Gerth and C.W. Mills, FromMaxWeber: essays in sociology(1948, 2002), p.352

[5] Paul Pruyser, “The Seamy Side of Current Religious Beliefs,” in H. Newton Maloney & Bernard Spilka, eds. Religion inPsychodynamic Perspective: The Contributions of Paul Pruyser (Oxford 1991): 51.

150

122.2. SEE ALSO 151

122.2 See also

Theological veto

Chapter 123

Sanctioned specialisation

Sanctioned specialisation is a mechanism for extending the content of a controlled terminology system by combiningexisting components of the system using specific mechanisms also defined in the system.An example: pneumonia can be described as a special type of inflammation:Inflammation which has_location Lungand this can be further specialised into pneumococcal pneumonia (caused by pneumococcus)Inflammation which has_location Lung is_caused_by PneumococcusSanctioned specialisation is used in description logic systems such as SNOMED CT. The concept is defined in abroader context and more examples given in ISO 17115, an ISO standard.

152

Chapter 124

Second-order predicate

In mathematical logic, a second-order predicate is a predicate that takes a first-order predicate as an argument.[1]Compare higher-order predicate.The idea of second order predication was introduced by the German mathematician and philosopher Frege. It isbased on his idea that a predicate such as “is a philosopher” designates a concept, rather than an object.[2] Sometimesa concept can itself be the subject of a proposition, such as in “There are no Albanian philosophers”. In this case, weare not saying anything of any Albanian philosophers, but of the concept “is an Albanian philosopher” that it is notsatisfied. Thus the predicate “is not satisfied” attributes something to the concept “is an Albanian philosopher”, andis thus a second-level predicate.This idea is the basis of Frege’s theory of number.[3]

124.1 References[1] Yaqub, Aladdin M. (2013), An Introduction to Logical Theory, Broadview Press, p. 288, ISBN 9781551119939.

[2] Oppy, Graham (2007),Ontological Arguments and Belief in God, CambridgeUniversity Press, p. 145, ISBN9780521039000.

[3] Kremer, Michael (1985), “Frege’s theory of number and the distinction between function and object”, Philosophical Studies47 (3): 313–323, doi:10.1007/BF00355206, MR 788101.

153

Chapter 125

Second-order propositional logic

A second-order propositional logic is a propositional logic extended with quantification over propositions. A specialcase are the logics that allow second-order Boolean propositions, where quantifiers may range either just over theBoolean truth values, or over the Boolean-valued truth functions.The most widely known formalism is the intuitionistic logic with impredicative quantification, system F. Parigot(1997) showed how this calculus can be extended to admit classical logic.

125.1 See also• Boolean satisfiability problem

• Second-order arithmetic

• Second-order logic

• Type theory

125.2 References

Parigot, Michel (1997). Proofs of strong normalisation for second order classical natural deduction. Journal ofSymbolic Logic 62(4):1461–1479.

154

Chapter 126

Self-reference puzzle

A self-reference puzzle is a type of logical puzzle where the question in the puzzle refers to the attributes of thepuzzle itself.[1] A common example is that a “fill in the blanks” style sentence is given, but what is filled in the blankscan contribute to the sentence itself. An example is “There are _____ e’s in this sentence.”, for which a solution is“eight” (since including the “eight”, there are 8 e’s in the sentence).

126.1 Examples•

126.2 References[1] Base 3 self-reference puzzles

155

Chapter 127

Self-verifying theories

Self-verifying theories are consistent first-order systems of arithmetic much weaker than Peano arithmetic thatare capable of proving their own consistency. Dan Willard was the first to investigate their properties, and he hasdescribed a family of such systems. According to Gödel’s incompleteness theorem, these systems cannot contain thetheory of Peano arithmetic, and in fact, not even the weak fragment of Robinson arithmetic; nonetheless, they cancontain strong theorems.In outline, the key to Willard’s construction of his system is to formalise enough of the Gödel machinery to talkabout provability internally without being able to formalise diagonalisation. Diagonalisation depends upon being ableto prove that multiplication is a total function (and in the earlier versions of the result, addition also). Addition andmultiplication are not function symbols of Willard’s language; instead, subtraction and division are, with the additionand multiplication predicates being defined in terms of these. Here, one cannot prove the Π0

2 sentence expressingtotality of multiplication:

(∀x, y) (∃z) multiply(x, y, z).

where multiply is the three-place predicate which stands for z/y = x . When the operations are expressed in this way,provability of a given sentence can be encoded as an arithmetic sentence describing termination of an analytic tableau.Provability of consistency can then simply be added as an axiom. The resulting system can be proven consistent bymeans of a relative consistency argument with respect to ordinary arithmetic.We can add any true Π0

1 sentence of arithmetic to the theory and still remain consistent.

127.1 References• Solovay, R., 1989. “Injecting Inconsistencies into Models of PA”. Annals of Pure and Applied Logic 44(1-2):101—132.

• Willard, D., 2001. “Self Verifying Axiom Systems, the Incompleteness Theorem and the Tangibility ReflectionPrinciple”. Journal of Symbolic Logic 66:536—596.

• Willard, D., 2002. “How to Extend the Semantic Tableaux and Cut-Free Versions of the Second Incomplete-ness Theorem to Robinson’s Arithmetic Q” . Journal of Symbolic Logic 67:465—496.

127.2 External links• Dan Willard’s home page.

156

Chapter 128

Sentence (logic)

This article is a technical mathematical article in the area of predicate logic. For the ordinary Englishlanguage meaning see Sentence (linguistics), for a less technical introductory article see Statement (logic).

In mathematical logic, a sentence of a predicate logic is a boolean-valued well-formed formula with no free variables.A sentence can be viewed as expressing a proposition, something that may be true or false. The restriction of havingno free variables is needed to make sure that sentences can have concrete, fixed truth values: As the free variables ofa (general) formula can range over several values, the truth value of such a formula may vary.Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomicformula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers.A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate thetruth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories,interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truthvalue. A theory is satisfiable when all of its sentences are true. The study of algorithms to automatically discoverinterpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem.

128.1 Example

The following example is in first-order logic.

∀y∃x(x2 = y)

is a sentence. This sentence is true in the positive real numbers ℝ+, false in the real numbers ℝ, and true in the complexnumbers ℂ. (In plain English, this sentence is interpreted to mean that every member of the structure concerned isthe square of a member of that particular structure.) On the other hand, the formula

∃x(x2 = y)

is not a sentence, because of the presence of the free variable y. In the structure of the real numbers, this formula istrue if we substitute (arbitrarily) y = 2, but is false if y = –2.

128.2 See also• Ground expression

• Open sentence

• Statement (logic)

• Proposition

157

158 CHAPTER 128. SENTENCE (LOGIC)

128.3 References• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.

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

Chapter 129

Slothful induction

Slothful induction, also called appeal to coincidence, is a fallacy in which an inductive argument is denied its properconclusion, despite strong evidence for inference. An example of slothful induction might be that of a careless manwho has had twelve accidents in the last six months and it is strongly evident that it was due to his negligence orrashness, yet keeps insisting that it is just a coincidence and not his fault.[1] Its logical form is: evidence suggests Xresults in Y, yet the person in question insists Y was caused by something else.[2]

129.1 References[1] Barker, Stephen F. (24 July 2002). The Elements of Logic (6th ed.). McGraw-Hill. ISBN 0-07-283235-5.

[2] Bennett, Bo. Logically Fallacious: The Ultimate Collection of Over 300 Logical Fallacies ...

159

Chapter 130

Specialization (logic)

Specialisation, (or specialization) is an important way to generate propositional knowledge, by applying generalknowledge, such as the theory of gravity, to specific instances, such as “when I release this apple, it will fall to thefloor”. Specialisation is the opposite of generalisation.Concept B is a specialisation of concept A if and only if:

• every instance of concept B is also an instance of concept A; and

• there are instances of concept A which are not instances of concept B.

130.1 See also• Generalisation

160

Chapter 131

Strict logic

Strict logic is essentially synonymous with relevant logic, though it can be characterized proof-theoretically as

• ordinary logic without Disjunction introduction, or

• linear logic with contraction.

131.1 See also• Substructural logic

161

Chapter 132

Syllogistic fallacy

Syllogistic fallacies are formal fallacies that occur in syllogisms. They include:Any syllogism type (other than polysyllogism and disjunctive):

• fallacy of four terms

Occurring in categorical syllogisms:

• related to affirmative or negative premises:• affirmative conclusion from a negative premise• fallacy of exclusive premises• negative conclusion from affirmative premises

• existential fallacy• fallacy of the undistributed middle• illicit major• illicit minor• fallacy of necessity

Occurring in disjunctive syllogisms:

• affirming a disjunct

Occurring in statistical syllogisms (dicto simpliciter fallacies):

• accident• converse accident

132.1 See also• Argumentation theory

132.2 External links• Fallacy files: Syllogistic fallacy• Online Syllogistic Machine An interactive syllogistic machine for exploring all the fallacies, figures, terms, andmodes of syllogisms.

162

Chapter 133

T-schema

The T-schema or truth schema (not to be confused with 'Convention T') is used to give an inductive definition oftruth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth. Some authors refer to it asthe “Equivalence Schema”, a synonym introduced by Michael Dummett.[1]

The T-schema is often expressed in natural language, but it can be formalized in many-sorted predicate logic or modallogic; such a formalisation is called a T-theory. T-theories form the basis of much fundamental work in philosophicallogic, where they are applied in several important controversies in analytic philosophy.As expressed in semi-natural language (where 'S' is the name of the sentence abbreviated to S): 'S' is true if and onlyif SExample: 'snow is white' is true if and only if snow is white.

133.1 The inductive definition

By using the schema one can give an inductive definition for the truth of compound sentences. Atomic sentencesare assigned truth values disquotationally. For example, the sentence "'Snow is white' is true” becomes materiallyequivalent with the sentence “snow is white”, i.e. 'snow is white' is true if and only if snow is white. The truth ofmore complex sentences is defined in terms of the components of the sentence:

• A sentence of the form “A and B” is true if and only if A is true and B is true

• A sentence of the form “A or B” is true if and only if A is true or B is true

• A sentence of the form “if A then B” is true if and only if A is false or B is true; see material implication.

• A sentence of the form “not A” is true if and only if A is false

• A sentence of the form “for all x, A(x)" is true if and only if, for every possible value of x, A(x) is true.

• A sentence of the form “for some x, A(x)" is true if and only if, for some possible value of x, A(x) is true.

133.2 Natural languages

Joseph Heath points out[2] that “The analysis of the truth predicate provided by Tarski’s Schema T is not capable ofhandling all occurrences of the truth predicate in natural language. In particular, Schema T treats only “freestanding”uses of the predicate—cases when it is applied to complete sentences.” He gives as “obvious problem” the sentence:

• Everything that Bill believes is true.

Heath argues that analyzing this sentence using T-schema generates the sentence fragment—“everything that Billbelieves”—on the righthand side of the Logical biconditional.

163

164 CHAPTER 133. T-SCHEMA

133.3 See also• Principle of bivalence

• Law of excluded middle

133.4 References[1] Wolfgang Künne (2003). Conceptions of truth. Clarendon Press. p. 18. ISBN 978-0-19-928019-3.

[2] Joseph Heath (2001). Communicative action and rational choice. MIT Press. p. 186. ISBN 978-0-262-08291-4.

133.5 External links• Tarski’s Truth Definitions entry in the Stanford Encyclopedia of Philosophy

• Consequences of the Semantic Paradoxes entry in the Stanford Encyclopedia of Philosophy

Chapter 134

Tacit assumption

A tacit assumption or implicit assumption is an assumption that includes the underlying agreements or statementsmade in the development of a logical argument, course of action, decision, or judgment that are not explicitly voicednor necessarily understood by the decision maker or judge. Often, these assumptions are made based on personallife experiences, and are not consciously apparent in the decision making environment. These assumptions can bethe source of apparent paradoxes, misunderstandings and resistance to change in human organizational behavior.

134.1 See also• Implied consent

134.2 References• Edgar H. Schein, Organizational Culture and Leadership, Jossey-Bass, 2004, ISBN 0-7879-7597-4

165

Chapter 135

Takeuti’s conjecture

In mathematics, Takeuti’s conjecture is the conjecture of Gaisi Takeuti that a sequent formalisation of second-orderlogic has cut-elimination (Takeuti 1953). It was settled positively:

• By Tait, using a semantic technique for proving cut-elimination, based on work by Schütte (Tait 1966);

• Independently by Takahashi by a similar technique (Takahashi 1967);

• It is a corollary of Jean-Yves Girard's syntactic proof of strong normalization for System F.

Takeuti’s conjecture is equivalent1 to the consistency of second-order arithmetic and to the strong normalization ofthe Girard/Reynold’s System F.

135.1 See also• Hilbert’s second problem

135.2 Notes• ^ Equivalent in the sense that each of the statements can be derived from each other in the weak system PRAof arithmetic; consistency refers here to the truth of the Gödel sentence for second-order arithmetic. Seeconsistency proof for more discussion.

135.3 References• William W. Tait, 1966. A nonconstructive proof of Gentzen's Hauptsatz for second order predicate logic. In

Bulletin of the American Mathematical Society, 72:980–983.

• Gaisi Takeuti, 1953. On a generalized logic calculus. In Japanese Journal of Mathematics, 23:39–96. Anerrata to this article was published in the same journal, 24:149–156, 1954.

• Moto-o Takahashi, 1967. A proof of cut-elimination in simple type theory. In Japanese Mathematical Society,10:44–45.

166

Chapter 136

Tee (symbol)

The tee (⊤), also called down tack (as opposed to the up tack) or verum is a symbol used to represent:

• The top element in lattice theory.

• A logical constant denoting a tautology in logic.

• The top type in type theory.

136.1 Encoding

In Unicode, the tee character is encoded as U+22A4 ⊤ down tack (HTML &#8868;).[1] The symbol is encoded inLaTeX as \top.

136.2 See also• Turnstile (⊢)

• Up tack (⊥)

• Falsum

• List of logic symbols

• List of mathematical symbols

136.3 Notes[1] “Mathematical Operators – Unicode” (PDF). Retrieved 2013-07-20.

167

Chapter 137

The Game of Logic

The Game of Logic is a book written by Lewis Carroll, published in 1886.[1] [2] [3] [4] [5]

137.1 References[1] “Scanned copy of the book at archive.org”.

[2] “The Game Of Logic”. goodreads.com. Retrieved 23 December 2013.

[3] “The game of logic”. amazon.com. Retrieved 23 December 2013.

[4] “The Game of Logic”. barnesandnoble.com. Retrieved 23 December 2013.

[5] “The Game of Logic: By Lewis Carroll”. books.google.com. Retrieved 23 December 2013.

137.2 External links• Scanned copy at archive.org

• Entry at gutenberg.org

168

Chapter 138

Third-cause fallacy

The third cause fallacy is a logical fallacy where a spurious relationship is confused for causation. It asserts that Xcauses Y when, in reality, X and Y are both caused by Z. It is a variation on the post hoc ergo propter hoc fallacy anda member of the questionable cause group of fallacies.When third causes are ignored, it becomes possible to corral shocking statistical evidence in support of a nonexistentcausality. For example:

“It seems that every time empty beer cans are piled up in a car, an accident occurs. It seems that theexcess weight and shape of the cans must cause other cars to want to crash into the victim’s car.”

The fallacy in this situation would be the fact that the arguer focused on the first (beer cans) and second (car crashes)facts without looking for possible causes of both phenomena, such as drunk driving.

138.1 Other names• Ignoring a common cause[1]

• Questionable cause[1]

138.2 See also• Spurious relationship

• Questionable cause

138.3 References[1] Labossiere, M.C., Dr. LaBossiere’s Philosophy Pages

169

Chapter 139

Transparent Intensional Logic

Transparent Intensional Logic (frequently abbreviated as TIL) is a logical system created by Pavel Tichý. Dueto its rich procedural semantics TIL is in particular apt for the logical analysis of natural language. From the formalpoint of view, TIL is a hyperintensional, partial, typed lambda-calculus. TIL applications cover a wide range of topicsfrom formal semantics, philosophy of language, epistemic logic, philosophical, and formal logic. TIL provides anoverarching semantic framework for all sorts of discourse, whether colloquial, scientific, mathematical or logical.The semantic theory is a procedural one, according to which sense is an abstract, pre-linguistic procedure detailingwhat operations to apply to what procedural constituents to arrive at the product (if any) of the procedure. TILprocedures, known as constructions, are hyperintensionally individuated. Construction is the single most importantnotion of Transparent Intensional Logic, being a philosophically well-motivated and formally worked-out conceptionof Frege’s notion of mode of presentation. Constructions, and the entities they construct, are organized into a ramifiedtype theory incorporating a simple type theory. The semantics is tailored to the hardest case, as constituted byhyperintensional contexts, and generalized from there to intensional and extensional contexts. The underlying logicis a Frege-style function/argument one, treating functions, rather than relations or sets, as primitive, together with aChurch-style logic, centred on the operations of functional abstraction and application.Key constraints informing TIL approach to semantic analysis are compositionality and anti-contextualism. The assign-ment of constructions to expressions as their meanings is context-invariant. Depending on the sort of logical contextin which a construction occurs, what is context-dependent is the logical manipulation of the respective meaning itselfrather than the meaning assignment.

139.1 See also• Intensional logic

139.2 Bibliography• P. Tichý (1988): The Foundations of Frege’s Logic. De Gruyter, Berlin and New York 1988, 333 pp.

• M. Duží, B. Jespersen and P. Materna: Procedural Semantics for Hyperintensional Logic. Foundations andApplications of TIL. Springer, 2010.

139.3 External links• TIL home page

170

Chapter 140

Triangle of opposition

In the system of Aristotelian logic, the triangle of opposition is a diagram representing the different ways in whicheach of the three propositions of the system is logically related ('opposed') to each of the others. The system is alsouseful in the analysis of syllogistic logic, serving to identify the allowed logical conversions from one type to another.

140.1 See also• Square of opposition

• Logical hexagon

171

Chapter 141

Truth condition

In semantics, truth conditions are which obtain precisely when a sentence is true. For example, “It is snowing inNebraska” is true precisely when it is snowing in Nebraska.More formally, we can think of a truth condition as what makes for the truth of a sentence in an inductive definition oftruth (for details, see the semantic theory of truth). Understood this way, truth conditions are theoretical entities. Toillustrate with an example: suppose that, in a particular truth theory, the word “Nixon” refers to Richard M. Nixon,and “is alive” is associated with the set of currently living things. Then one way of representing the truth condition of“Nixon is alive” is as the ordered pair <Nixon, {x: x is alive}>. And we say that “Nixon is alive” is true if and only ifthe referent (or referent of) “Nixon” belongs to the set associated with “is alive”, that is, if and only if Nixon is alive.In semantics, the truth condition of a sentence is almost universally considered to be distinct from its meaning. Themeaning of a sentence is conveyed if the truth conditions for the sentence are understood. Additionally, there aremany sentences that are understood although their truth condition is uncertain. One popular argument for this viewis that some sentences are necessarily true —that is, they are true whatever happens to obtain. All such sentenceshave the same truth conditions, but arguably do not thereby have the same meaning. Likewise, the sets {x: x is alive}and {x: x is alive and x is not a rock} are identical—they have precisely the same members—but presumably thesentences “Nixon is alive” and “Nixon is alive and is not a rock” have different meanings.

141.1 See also• Slingshot argument

172

Chapter 142

Two-variable logic

In mathematical logic and computer science, two-variable logic is the fragment of first-order logic where formulaecan be written using only two different variables. This fragment is usually studied without function symbols.

142.1 Decidability

One of the main points is that some important problems about two-variable logic, such as satisfiability and finitesatisfiability, are decidable. This result generalizes results about the decidability of fragments of two-variable logic,such as certain description logics; however, some fragments of two-variable logic enjoy a much lower computationalcomplexity for their satisfiability problems.By contrast, for the three-variable fragment of first-order logic without function symbols, satisfiability is undecidable.[1]

142.2 Counting quantifiers

The two-variable fragment of first-order logic with no function symbols is known to be decidable even with theaddition of counting quantifiers, and thus of uniqueness quantification. This is a more powerful result, as countingquantifiers for high numerical values are not expressible in that logic.

142.3 References[1] A. S. Kahr, Edward F. Moore and Hao Wang. Entscheidungsproblem Reduced to the ∀ ∃ ∀ Case, 1962, noting that their ∀

∃ ∀ formulas use only three variables.

173

Chapter 143

Unique name assumption

The unique name assumption is a simplifying assumption made in some ontology languages and description logics.In logics with the unique name assumption, different names always refer to different entities in the world.[1]

The standard ontology language OWL does not make this assumption, but provides explicit constructs to expresswhether two names denote the same or distinct entities.[2][3]

• owl:sameAs is the OWL property that asserts that two given names or identifiers (e.g., URIs) refer to the sameindividual or entity.

• owl:differentFrom is the OWL property that asserts that two given names or identifiers (e.g., URIs) refer todifferent individuals or entities.

143.1 See also• Closed-world assumption

• Coreference

143.2 References[1] Russell, Stuart; Norvig, Peter (2003) [1995]. Artificial Intelligence: A Modern Approach (2nd ed.). Prentice Hall. p. 333.

ISBN 978-0137903955.

[2] Tao, Jiao; Sirin, Evren; Bao, Jie; McGuinness, Deborah L. (2010). Integrity constraints in OWL. Proc. AAAI.

[3] OWL Web Ontology Language Reference

174

Chapter 144

Unsatisfiable core

In mathematical logic, given an unsatisfiable Boolean propositional formula in conjunctive normal form, a subset ofclauses whose conjunction is still unsatisfiable is called an unsatisfiable core of the original formula.Many SAT solvers can produce a resolution graph which proves the unsatisfiability of the original problem. This canbe analyzed to produce a smaller unsatisfiable core.An unsatisfiable core is called a minimal unsatisfiable core, if every proper subset (allowing removal of any arbitraryclause or clauses) of it is satisfiable. Thus, such a core is a local minimum, though not necessarily a global one. Thereare several practical methods of computing minimal unsatisfiable cores.[1][2]

Aminimum unsatisfiable core contains the smallest number of the original clauses required to still be unsatisfiable. Nopractical algorithms for computing the minimum core are known. Algorithms for Computing Minimal UnsatisfiableSubsets. Notice the terminology: whereas minimal unsatisfiable core was a local problem with an easy solution, theminimum unsatisfiable core is a global problem with no known easy solution.

144.1 References[1] N. Dershowitz, Z. Hanna, and A. Nadel, A Scalable Algorithm for Minimal Unsatisfiable Core Extraction

[2] Stefan Szeider, Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

175

Chapter 145

Vagrant predicate

Vagrant predicates are logical constructions that exhibit an inherent limit to conceptual knowledge.[1] Such predi-cates can be used in general descriptions but are self-contradictory when applied to particulars. For instance, thereare numbers which have never been mentioned but no example can be given as this would contradict its definition.Vagrant predicates have been proposed and studied by Nicholas Rescher.F is a vagrant predicate iff ( ∃ u)Fu is true while nevertheless Fu0 is false for each and every specifically identifiedu0.[2]

When infinity is thought as number greater than any given, a similar idea is conceived. However vagrancy needsnot to be monotonous and occurs also within bounds. Rescher has used vagrant predicates to solve the vaguenessproblem.[1][2]

145.1 References[1] Rescher N., Unknowability, Lexington books, 2009

[2] Rescher N., Informal Logic, Vol. 28, No.4 (2008), pp. 282-294

176

Chapter 146

Valentino Annibale Pastore

Valentino Annibale Pastore (November 13, 1868 - February 27, 1956) was an Italian philosopher and logician.Pastore was born in Orbassano.He studied literature at the University of Turin under Arturo Graf. His thesis La vita delle forme letterarie (Thelife of literary forms) was published in 1892 in Turin. Pastore then turned to philosophy, influenced by the worksof Pasquale d'Ercole, Friedrich Kiesow, Antonio Garbasso, and Giuseppe Peano, publishing his own thesis Sopra leteorie della scienza: logica, matematica, fisica (On the theories of science: logic, mathematics, physics) in 1903.He was professor in Turin from 1913 until 1939, leading a laboratory of “experimental logic”. He eventually focusedon logical aspects and procedures in science.Pastore died in Turin.

146.1 Works

• Sopra la teoria della scienza: Logica, matematica e fisica on Internet Archive, 1903

• Logica formale dedotta dalla considerazione dei modelli meccanici on Internet Archive, 1906

• Del nuovo spirito della scienza e della filosofia, 1907

• Sillogismo e proporzione, 1910

• Dell'essere e del conoscere, 1911

• Il pensiero puro, 1913

• Il problema della causalitá, con particolare riguardo alla teoria del metodo sperimentale, 1921

• Il solipsismo, 1923

• La logica del potenziamento, 1936

• Logica sperimentale, 1939

• L'acrisia di Kant, 1940

• La filosofia di Lenin, 1946

• La volontá dell'assurdo. Storia e crisi dell'esistenzialismo, 1948

• Logicalia, 1957

• Dioniso, 1957

• Introduzione alla metafisica della poesia, 1957

177

178 CHAPTER 146. VALENTINO ANNIBALE PASTORE

146.2 External links• Biography

• Picture

Chapter 147

Vampire (theorem prover)

Vampire is an automatic theorem prover for first-order classical logic developed in the School of Computer Scienceat the University of Manchester by Andrei Voronkov together with Kryštof Hoder and previously with AlexandreRiazanov. So far it has won the “world cup for theorem provers” (the CADE ATP System Competition) in the mostprestigious CNF (MIX) division eleven times (1999, 2001–2010).[1][4]

147.1 Background

Vampire’s kernel implements the calculi of ordered binary resolution and superposition for handling equality. Thesplitting rule and negative equality splitting can be simulated by the introduction of new predicate definitions anddynamic folding of such definitions. ADPLL-style algorithm splitting is also supported. A number of standard redun-dancy criteria and simplification techniques are used for pruning the search space: tautology deletion, subsumptionresolution, rewriting by ordered unit equalities, basicness restrictions and irreducibility of substitution terms. Thereduction ordering used is the standard Knuth-Bendix ordering.A number of efficient indexing techniques are used to implement all major operations on sets of terms and clauses.Run-time algorithm specialisation is used to accelerate forward matching.Although the kernel of the system works only with clausal normal forms, the preprocessor component accepts aproblem in the full first-order logic syntax, clausifies it and performs a number of useful transformations beforepassing the result to the kernel. When a theorem is proven, the system produces a verifiable proof, which validatesboth the clausification phase and the refutation of the conjunctive normal form.Along with proving theorems, Vampire has other related functionalities such as generating interpolants.Executables can be obtained from the system website [5] A somewhat outdated version is available under the GNULesser General Public License as part of Sigma KEE.[6]

147.2 References[1] Riazanov, A.; Voronkov, A. (2002). “The design and implementation of VAMPIRE”. AI Communications 15 (2-3/2002):

91–110. ISSN 0921-7126.

[2] http://riazanov.webs.com/Riazanov_PhD_thesis.pdf (PhD thesis of Dr. Alexandre Riazanov on the implementation ofVampire)

[3] Urban, J.; Hoder, K.; Voronkov, A. (2010). “Evaluation of Automated Theorem Proving on the Mizar MathematicalLibrary”. Mathematical Software – ICMS 2010. Lecture Notes in Computer Science 6327. p. 155. doi:10.1007/978-3-642-15582-6_30. ISBN 978-3-642-15581-9.

[4] Voronkov, A. (1995). “The anatomy of vampire”. Journal of Automated Reasoning 15 (2): 237–265. doi:10.1007/BF00881918.

[5] http://www.vprover.org

[6] http://sigmakee.cvs.sourceforge.net/viewvc/sigmakee/Vampire

179

Chapter 148

Van Gogh fallacy

The Van Gogh Fallacy is an example of a logical fallacy. It is a type of fallacy wherein the conclusion is affirmedby its consequent (fallacy of affirming the consequent) instead of its antecedent (modus ponens). [1][2]

Its name is derived from a particular case that argues:

Van Gogh was misunderstood and living in poverty, but later on, he is recognized as one of the world’sgreatest artist. I am misunderstood and living in poverty. Therefore, I am going to be recognized as oneof the world’s greatest artists.

Although the argument itself sounds promising and provides hope to struggling artists and the like, it is invalid andshould not be taken as it is. The Van Gogh Fallacy is problematic as it promotes wishful thinking. More often thannot, it leads to unpleasant consequences.[1]

There are far more people in the “misunderstood” and unrecognized category than there those who are great. Havingsome common unimportant attributes together with a person does not instantiate that the samewill happen for oneself.In the case of the Van Gogh fallacy, sharing the misunderstood and poor attribute with Van Gogh does not equate toan individual sharing the same fate (i.e., getting recognized as a great artist). Such a case will only be true if thereis a one to one correlation between the two factors; this is rarely the case with correlation. (See correlation does notimply causation). The only thing that is guaranteed is the fact that being misunderstood and living in poverty doesnot rule out the possibility of greatness and recognition.[1]

The use of parody is a good way of demonstrating how weak the Van Gogh Fallacy is. [1] Some examples would be:

Albert Einstein has a face and he is well-known physicist. I also have a face and therefore, I too will bea well-known physicist.

When it is put this way, the ridiculousness of the argument is much clearer. It relies on a very weak analogy. Simplysharing similar superficial and trivial characteristics with a great person does not mean one shares all of the sameother attributes that made him or her great in the first place. [1]

148.1 References[1] Nigel Warburton, Thinking from A to Z, Routledge, 2000 -

[2] Joel Feinberg and Russ Shafer-Landau, “Reason and Responsibility”, Wadsworth,2013

180

Chapter 149

Vienna Summer of Logic

Logo of the scientific event

The Vienna Summer of Logic is a scientific event planned for the summer of 2014, combining 12 major confer-ences and several workshops from the fields of mathematical logic, logic in computer science, and logic in artificial

181

182 CHAPTER 149. VIENNA SUMMER OF LOGIC

intelligence.[1] The meetings will take place from July 9 to 24, 2014, and are expected to attract more than 2500scientists and researchers.[2]

The event is organized by the Kurt Gödel Society at Vienna University of Technology.[3] Participating meetingsinclude:[4]

In the Logic in Computer Science stream (representing the Federated Logic Conference (FLoC)):

• International Conference on Computer Aided Verification (CAV)

• IEEE Computer Security Foundations Symposium (CSF)

• International Conference on Logic Programming (ICLP)

• International Joint Conference on Automated Reasoning (IJCAR)

• Conference on Interactive Theorem Proving (ITP)

• Joint meeting of the EACSL Annual Conference on Computer Science Logic (CSL) and the ACM/IEEESymposium on Logic in Computer Science (LICS)

• International Conference on Rewriting Techniques and Applications (RTA) joint with the International Con-ference on Typed Lambda Calculi and Applications (TLCA)

• International Conference on Theory and Applications of Satisfiability Testing (SAT)

• more than 70 FLoC workshops

• FLoC Olympic Games (system competitions)

• SAT/SMT Summer School

In theMathematical Logic stream:

• Logic Colloquium 2014 (LC)

• Logic, Algebra and Truth Degrees 2014 (LATD)

• Workshop on Compositional Meaning in Logic (GeTFun 2.0)

• The Infinity Workshop (INFINITY)

• Workshop on Logic and Games (LG)

• Workshop on Nonclassical Proofs: Theory, Applications and Tools (NCPROOFS)

• Kurt Gödel Fellowship Competition

In the Logic in Artificial Intelligence stream:

• International Conference on Principles of Knowledge Representation and Reasoning (KR)

• International Workshop on Description Logics (DL)

• International Workshop on Non-Monotonic Reasoning (NMR)

• International Workshop on Knowledge Representation for Health Care 2014 (KR4HC)

149.1 References[1] “Basic Logic Research Crucial for Computer, Software Engineering”. Scientific Computing. June 3, 2014. Retrieved 13

June 2014.

[2] Felser, Rudolf (10 December 2013). “Vienna Summer of Logic 2014”. Computerwoche. Retrieved 30 December 2013.

[3] http://vsl2014.at/organization/''. Missing or empty |title= (help);

[4] http://vsl2014.at/''. Missing or empty |title= (help);

149.2. EXTERNAL LINKS 183

149.2 External links• Vienna Summer of Logic

Chapter 150

Vivid knowledge

Vivid knowledge refers to a specific kind of knowledge representation.The idea of a vivid knowledge base is to get an interpretation mostly straightforward out of it – it implies theinterpretation. Thus, any query to such a knowledge base can be reduced to a database-like query.

150.1 Propositional knowledge base

Apropositional knowledge baseKB is vivid iff KB is a complete and consistent set of literals (over some vocabulary).[1]

Such a knowledge base has the property that it as exactly one interpretation, i.e. the interpretation is unique. Acheck for entailment of a sentence can simply be broken down into its literals and those can be answered by a simpledatabase-like check of KB.

150.2 First-order knowledge base

A first-order knowledge base KB is vivid iff for some finite set of positive function-free ground literals KB+,

KB = KB+ ∪ Negations ∪ DomainClosure ∪ UniqueNames,

whereby

Negations ≔ { ¬p | p is atomic and KB ⊭ p },DomainClosure ≔ { (cᵢ ≠ c ) | cᵢ, c are distinct constants },UniqueNames ≔ { ∀x: (x = c1) ∨ (x = c2) ∨ ..., where the cᵢ are all the constants in KB+ }.

[2]

All interpretations of a vivid first-order knowledge base are isomorphic.[3]

150.3 See also

• Closed world assumption

150.4 References[1] Knowledge Representation and Reasoning / Ronald J. Brachman, Hector J. Levesque / page 337

184

150.4. REFERENCES 185

[2] Knowledge Representation and Reasoning / Ronald J. Brachman, Hector J. Levesque / page 337

[3] Knowledge Representation and Reasoning / Ronald J. Brachman, Hector J. Levesque / page 339

186 CHAPTER 150. VIVID KNOWLEDGE

150.5 Text and image sources, contributors, and licenses

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Paxse, Gregbard, Cydebot, Addbot, Yobot, ZéroBot, Brison09 and Anonymous: 3• Accident (fallacy) Source: https://en.wikipedia.org/wiki/Accident_(fallacy)?oldid=666504366 Contributors: Bryan Derksen, Rfc1394,

Taak, Aequo, Guanabot, Silence, Themindset, Knucmo2, BD2412, Rjwilmsi, YurikBot, Reedy, CompuHacker, Stevage, Cybercobra,Andeggs, Ollj, ArglebargleIV, BlackMilk, JHP, Clan-destine, Isilanes, Shinju, Lechatjaune, Synthebot, Fadesga, XLinkBot, Addbot, Firi-Bot, Omnipaedista, Paine Ellsworth, Machine Elf 1735, MastiBot, Nomdecrayon, EmausBot, 478jjjz, ZéroBot, 1l2, Northamerica1000,Ihaveacatonmydesk and Anonymous: 9

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150.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 187

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188 CHAPTER 150. VIVID KNOWLEDGE

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• Herbrand interpretation Source: https://en.wikipedia.org/wiki/Herbrand_interpretation?oldid=635276419Contributors: CALR, Linas,Zero sharp, CBM, Gregbard, JaGa, Cyborg1, Hans Adler, Addbot, Proofreader77, Waheedghumman, Omnipaedista, Blas3nik andAnonymous: 3

• Hybrid logic Source: https://en.wikipedia.org/wiki/Hybrid_logic?oldid=634445606 Contributors: Zeno Gantner, Charles Matthews,Siroxo, AlexG, Chalst, Sandius, Qwertyus, Oo64eva, Algebraist, Dbtfz, Modallogic, CBM, Gregbard, Julian Mendez, Valeria.depaiva,PL290, Algebran, BrideOfKripkenstein and Anonymous: 3

• Idempotency of entailment Source: https://en.wikipedia.org/wiki/Idempotency_of_entailment?oldid=666497440 Contributors: GT-Bacchus, Hyacinth, Chalst, Wood Thrush, Melaen, RJFJR, MadMax, SmackBot, Gregbard, Fadesga, Addbot, Erik9bot, ChuispastonBot,Spockticus and Anonymous: 4

• Illicit major Source: https://en.wikipedia.org/wiki/Illicit_major?oldid=666497405Contributors: BryanDerksen, DocWatson42, Eequor,Taak, HorsePunchKid, Silence, Susvolans, Bobrayner, Tevildo, SmackBot, Bluebot, Andeggs, GrahamHardy, Maxim, Gorank4, Fadesga,Addbot, Logicchecker, Machine Elf 1735, YFdyh-bot and Anonymous: 10

• Illicit minor Source: https://en.wikipedia.org/wiki/Illicit_minor?oldid=666497418Contributors: BryanDerksen, DocWatson42, Eequor,Taak, Cacycle, Silence, Susvolans, DenisHowe, Xaa, Garybel, Bluebot, Andeggs, Emptymountains, Fadesga, Addbot, Machine Elf 1735,Muon, YFdyh-bot and Anonymous: 7

• Inclusion (logic) Source: https://en.wikipedia.org/wiki/Inclusion_(logic)?oldid=609028697Contributors: Michael Hardy, Jpbowen, Smack-Bot, Mathman1550, SMasters, Gregbard, KConWiki, Interchange88, Ironholds, Yobot and Here today, gone tomorrow

• Instantiation principle Source: https://en.wikipedia.org/wiki/Instantiation_principle?oldid=666497489Contributors: FranksValli, Nicholasink,Jaraalbe, Tavilis, SmackBot, Monkeycheetah, Radagast83, Gregbard, Shogun Luis, Alphachimpbot, Magioladitis, Philosopher123, Steevven1,Fadesga, Brokenplates\, AnomieBOT, Sae1962, Utility Monster, GoingBatty, Yiosie2356, ChrisGualtieri, Malina47 and Anonymous: 5

• Interpretability Source: https://en.wikipedia.org/wiki/Interpretability?oldid=612104705 Contributors: Ahoerstemeier, Dysprosia, Van-ished user 1234567890, Kntg, EmilJ, PWilkinson, Oleg Alexandrov, MrShamrock, CBM, Gregbard, Singularity, Matthew Yeager, Des-olateReality, Hans Adler, Qwfp, Lightbot, Yobot, BrideOfKripkenstein and Anonymous: 1

• Interval temporal logic Source: https://en.wikipedia.org/wiki/Interval_temporal_logic?oldid=664864923 Contributors: Sander~enwiki,BRW,OlegAlexandrov, Jpbowen, SmackBot, CBM,Alaibot, RainbowCrane, David Eppstein, R'n'B, Addbot, Lightbot, Gf uip,Wbm1058,Wikicauitl and Anonymous: 1

• Inverse (logic) Source: https://en.wikipedia.org/wiki/Inverse_(logic)?oldid=670844559 Contributors: Tarquin, Heron, Ryguasu, MichaelHardy, Dominus, Tonsofpcs, BenFrantzDale, Gku, Amerindianarts, Awis, Cmadler, Bhny, Aftermath, Doncram, SMcCandlish, Mhss,Kcordina, Byelf2007, Amniarix, Gregbard, Synergy, VoABot II, DGG, Mathman72, Kmhkmh, SieBot, ClueBot, TheOldJacobite, Ad-dbot, Sully111, Luckas-bot, Yobot, Martnym, MastiBot, JSquish, BG19bot and Anonymous: 21

• Inverse resolution Source: https://en.wikipedia.org/wiki/Inverse_resolution?oldid=498574129 Contributors: Fresheneesz, Gregbard,Blaisorblade and SchreiberBike

• Invincible ignorance fallacy Source: https://en.wikipedia.org/wiki/Invincible_ignorance_fallacy?oldid=624720360 Contributors: Mr-wojo, Rpyle731, Rjwilmsi, Cassowary, Kvn8907, Tevildo, Sadads, Cybercobra, Gregbard, Marek69, Niceguyedc, Addbot, HRoestBot,EmausBot, ZéroBot, Helpful Pixie Bot and Anonymous: 8

• Issue trees Source: https://en.wikipedia.org/wiki/Issue_trees?oldid=661371555 Contributors: Hu12, PamD, Magioladitis, Tgeairn, Wil-helmina Will, JimVC3, FrescoBot, Callanecc, Joie67, Borkificator, Nombizarreetpascommun, DoctorKubla and Anonymous: 3

• Lambert ofAuxerre Source: https://en.wikipedia.org/wiki/Lambert_of_Auxerre?oldid=667961690Contributors: Gregbard, VolkovBot,Ontoraul, Henry Delforn (old), Dthomsen8, Addbot, Rubinbot, ZéroBot, Philip J.1987qazwsx and KasparBot

• Lemma (logic) Source: https://en.wikipedia.org/wiki/Lemma_(logic)?oldid=666501088 Contributors: Michael Hardy, Hyacinth, RichFarmbrough, Stemonitis, David Haslam, Calréfa Wéná, RussBot, Byelf2007, Grumpyyoungman01, Gregbard, Al Lemos, R'n'B, Fadesga,Addbot, Yobot, AnomieBOT, Erik9bot, RockMagnetist and Anonymous: 3

• Limitation of size Source: https://en.wikipedia.org/wiki/Limitation_of_size?oldid=635377439 Contributors: Charles Matthews, RyanReich, Salix alba, Trovatore, Arundhati bakshi, SmackBot, Steve Byrne, JRSpriggs, Hans Adler, 1ForTheMoney, Citation bot and Brirush

• Lindenbaum’s lemma Source: https://en.wikipedia.org/wiki/Lindenbaum’{}s_lemma?oldid=635377540Contributors: CharlesMatthews,Filemon, Nortexoid, SmackBot, CBM, Richhoncho, David Eppstein, Quux0r, Pit-trout, Addbot, Doremo, WikitanvirBot, Deltahedron,Brirush and Anonymous: 2

• Literal (mathematical logic) Source: https://en.wikipedia.org/wiki/Literal_(mathematical_logic)?oldid=660586491Contributors: ObradovicGoran, Kbdank71, Mhss, Tsca.bot, CBM, Simeon, Gregbard, Cydebot, Thijs!bot, Egriffin, R'n'B, Mikhail Dvorkin, Matěj Grabovský,Termininja, Mikolasj, Krassotkin, Lagenar, Tijfo098, Tiago de Jesus Neves and Anonymous: 8

• Logic Spectacles Source: https://en.wikipedia.org/wiki/Logic_Spectacles?oldid=666654570 Contributors: DavidBrooks, JoshuacUK,Hathawayc, Malcolma, Cadillac, SmackBot, LessHeard vanU, Gregbard, MarshBot, Phil Bridger, Fadesga, Josve05a and Anonymous: 1

• Logical constant Source: https://en.wikipedia.org/wiki/Logical_constant?oldid=666654591 Contributors: Timrollpickering, Nortexoid,BD2412, Reinis, SmackBot, The great kawa, Frap, Lambiam, Dbtfz, CBM, Gregbard, Nick Number, SieBot, Randomblue, Fadesga,MystBot, Addbot, Luckas-bot, AnomieBOT, RibotBOT, Undsoweiter, FrescoBot, Hriber, Tijfo098, Frietjes, Masssly, Camila CavalcantiNery, ZX95 and Anonymous: 6

• Logical cube Source: https://en.wikipedia.org/wiki/Logical_cube?oldid=602609068 Contributors: Pierre de Lyon, Gregbard, Pjoef, Rc-sprinter123, Masssly, Theopolisme, Scuns Dotus and PIerre.Lescanne

190 CHAPTER 150. VIVID KNOWLEDGE

• Main contention Source: https://en.wikipedia.org/wiki/Main_contention?oldid=666654615 Contributors: Diego Moya, BD2412, Johnhartley, Byelf2007, Grumpyyoungman01, Gregbard, Steel, Al Lemos, Coin945, Cnilep, MiNombreDeGuerra, Fadesga, EmausBot,BG19bot, Pmccaff and Anonymous: 3

• Material nonimplication Source: https://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

• MaxEnt school Source: https://en.wikipedia.org/wiki/MaxEnt_school?oldid=537260375 Contributors: Kudpung, Addbot, Dawynn andLibb Thims

• McNamara fallacy Source: https://en.wikipedia.org/wiki/McNamara_fallacy?oldid=675900105Contributors: Bender235, AnmaFinotera,Zakksez, SmackBot, Blantant, Hut 8.5, Magioladitis, KConWiki, Arms & Hearts, CorenSearchBot, Addbot, PMLawrence, AnomieBOT,EmausBot, TheMysterious ElWillstro, ZéroBot, HandsomeFella, FeatherPluma, Justincheng12345-bot, Cup o' Java, Ihaveacatonmydesk,BU Rob13 and Anonymous: 9

• Middle term Source: https://en.wikipedia.org/wiki/Middle_term?oldid=647024362Contributors: Taak, PWilkinson,Malcolma, Grumpyy-oungman01, Gregbard, Cydebot, Alaibot, Shinju, Maxim, Emptymountains, SchreiberBike, Addbot, Adeliine, Thehelpfulbot, FrescoBot,FraterDamocles, Vieque and Anonymous: 5

• Models And Counter-Examples Source: https://en.wikipedia.org/wiki/Models_And_Counter-Examples?oldid=473492561 Contribu-tors: El Pantera and Palosirkka

• Monadic Boolean algebra Source: https://en.wikipedia.org/wiki/Monadic_Boolean_algebra?oldid=623204166 Contributors: MichaelHardy, Charles Matthews, Kuratowski’s Ghost, Oleg Alexandrov, Trovatore, Mhss, Gregbard, R'n'B, Safek, Hans Adler, Alexey Muranov,Addbot, Tijfo098, JMP EAX and Anonymous: 4

• Monotonicity of entailment Source: https://en.wikipedia.org/wiki/Monotonicity_of_entailment?oldid=665481074 Contributors: Hy-acinth, Joyous!, Chalst, Smmurphy, Josh Parris, Helvetius, Intgr, Open2universe, SmackBot, JanusDC, Sabik, Floridi~enwiki, Gregbard,Thijs!bot, Meredyth, Addbot, Constructive editor and Anonymous: 9

• Multimodal logic Source: https://en.wikipedia.org/wiki/Multimodal_logic?oldid=636552298Contributors: Guppyfinsoup, Physis, VaughanPratt, Gregbard, Alaibot, Berolschaeffer, Trappist the monk, Tijfo098, Helpful Pixie Bot, Metadox and Anonymous: 2

• Multiple-conclusion logic Source: https://en.wikipedia.org/wiki/Multiple-conclusion_logic?oldid=405064210 Contributors: BD2412,Salix alba, Rick Norwood, SmackBot, Dbtfz, CBM and Westerdundrun

• Neighborhood semantics Source: https://en.wikipedia.org/wiki/Neighborhood_semantics?oldid=421458980Contributors: Bearcat, Nor-texoid, Malcolma, Mr Stephen, CBM, Mere Interlocutor and Monohipinhimer

• Non-wellfoundedmereology Source: https://en.wikipedia.org/wiki/Non-wellfounded_mereology?oldid=559103897Contributors: MichaelHardy, Crasshopper, Gregbard and Tassedethe

• Nonfirstorderizability Source: https://en.wikipedia.org/wiki/Nonfirstorderizability?oldid=525081734Contributors: TheAnome,MichaelHardy, P0lyglut, Ben Standeven, Nortexoid, SmackBot, MrDrBob, Dbtfz, CBM, Asenine, DOI bot, Citation bot, Omnipaedista, Bride-OfKripkenstein, Citation bot 1, Tkuvho, Tijfo098 and Anonymous: 6

• Normal form (natural deduction) Source: https://en.wikipedia.org/wiki/Normal_form_(natural_deduction)?oldid=532304588 Con-tributors: Paul August, Chalst, RxS, Hairy Dude, SmackBot, Addbot, Piano non troppo, Erik9bot, ClueBot NG and Anonymous: 2

• Normal modal logic Source: https://en.wikipedia.org/wiki/Normal_modal_logic?oldid=676529352 Contributors: Charles Matthews,Lady Tenar, Sirmob, Rich Farmbrough, Chalst, EmilJ, Nortexoid, Oleg Alexandrov, Tizio, YurikBot, SmackBot, Mhss, Zero sharp,CBM, David Eppstein, Heyitspeter, ClueBot, Addbot, HanielBarbosa, GrouchoBot, Erik9bot and Anonymous: 6

• OBJ3 Source: https://en.wikipedia.org/wiki/OBJ3?oldid=570455145 Contributors: Pnm, Stan Shebs, Rich Farmbrough, RxS, Jpbowen,Dreadstar, Gregbard, Cydebot, Skier Dude, Jerryobject, Joswig, Addbot, Dawynn, Gf uip, BabbaQ, Hmainsbot1 and Anonymous: 2

• Objection (argument) Source: https://en.wikipedia.org/wiki/Objection_(argument)?oldid=666654835Contributors: Piotrus, Rich Farm-brough, SmackBot, Byelf2007, Grumpyyoungman01, Neelix, Gregbard, Al Lemos, CommonsDelinker, Haikon, Newbyguesses, Kivaan,Denisarona, Fadesga, Addbot, PranksterTurtle, Ptbotgourou, Erik9bot, Pollinosisss, Wikielwikingo, Mjbmrbot and Anonymous: 12

• One-sided argument Source: https://en.wikipedia.org/wiki/One-sided_argument?oldid=642988496Contributors: EdPoor, Ixfd64, Rpyle731,Edcolins, Kappa, PullUpYourSocks, -Ril-, Tabletop, Dangerous Angel, Cool3, Bluebot, Gregbard, RickardV, Mild Bill Hiccup, Erkin-Batu, Addbot, Piano non troppo, Machine Elf 1735, Gamewizard71, Pollinosisss, Buddy23Lee, Josve05a, Gwen-chan, ClueBot NG,Joefromrandb, Ugncreative Usergname, 069952497a, HunterGaylor, Ihaveacatonmydesk, Coconutporkpie and Anonymous: 14

• Otter (theoremprover) Source: https://en.wikipedia.org/wiki/Otter_(theorem_prover)?oldid=661152480Contributors: Dwheeler, Dcoet-zee, Greenrd, Stephan Schulz, Tobias Bergemann, Kate, Qutezuce, Remuel, RussBot, Ott2, SmackBot, Thumperward, Michael Kinyon,CBM, Simeon, Cydebot, David Eppstein, Maghnus, El Pantera, DOI bot, Yobot, MattTait, Bamyers99, Palosirkka, Monkbot and Anony-mous: 5

• Pars destruens/pars construens Source: https://en.wikipedia.org/wiki/Pars_destruens/pars_construens?oldid=658755728Contributors:Gregbard, Gökhan, Gwern, Anarchia, RickardV, Sanya3, Addbot, Yobot, Nirbhai1699 and Anonymous: 4

• PhoX Source: https://en.wikipedia.org/wiki/PhoX?oldid=543964661 Contributors: Greenrd, Tobias Bergemann, Mitsukai, CBM, Cyde-bot, David Eppstein, Jerryobject, Addbot, Palosirkka and Anonymous: 5

• Polysyllogism Source: https://en.wikipedia.org/wiki/Polysyllogism?oldid=589155122 Contributors: Silverfish, Charles Matthews, Taak,Kwamikagami, TheParanoidOne, RuM, Alexb@cut-the-knot.com, SmackBot, NickShaforostoff, Lambiam, Rigadoun, CBM, Neelix,Gregbard, Cydebot, Deflective, MER-C, Marks2222, Dorftrottel, TXiKiBoT, Addbot, Icanhasedit, Luckas-bot, AdjustShift, Erik9bot,Tryphaena, Shanghainese.ua, 478jjjz, ZéroBot, Helpful Pixie Bot, BG19bot, Brad7777, Jochen Burghardt, Begadkepat and Anonymous:8

• Post disputation argument Source: https://en.wikipedia.org/wiki/Post_disputation_argument?oldid=641411446 Contributors: Mrwojo,Oddharmonic, Cobaltbluetony, SmackBot, Iridescent, PamD, Wikidudeman, Acetrouble8, Addbot, Yobot, ChrisGualtieri and Anony-mous: 1

150.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 191

• Predicate logic Source: https://en.wikipedia.org/wiki/Predicate_logic?oldid=668159355 Contributors: Toby Bartels, Michael Hardy,Andres, Hyacinth, Robbot, MathMartin, Giftlite, Leonard G., Mindmatrix, Thekohser, Eubot, Chobot, Sharkface217, Jpbowen, Tomisti,SmackBot, Mhss, Cybercobra, Nakon, Byelf2007, Wvbailey, Bjankuloski06en~enwiki, George100, CBM, Gregbard, Naudefj, Thijs!bot,EdJohnston, JAnDbot, Hypergeek14, Stassa, Vanished user g454XxNpUVWvxzlr, Policron, Dessources, JohnBlackburne, AnonymousDissident, Gerakibot, Soler97, Kumioko, DesolateReality, Xiaq, ClueBot, Taxa, Djk3, TimClicks, Addbot, Jayde239, Yobot, AnomieBOT,Materialscientist, RandomDSdevel, ESSch, Keri, Logichulk, Xnn, EmausBot, WikitanvirBot, Mayur, ClueBot NG, Satellizer, ChesterMarkel, MerlIwBot, Helpful Pixie Bot, Virago250, Brad7777, Jochen Burghardt, BoltonSM3, Tomajohnson and Anonymous: 34

• Principle of nonvacuous contrast Source: https://en.wikipedia.org/wiki/Principle_of_nonvacuous_contrast?oldid=666502212 Contrib-utors: Charles Matthews, Stemonitis, Filipem, Gregbard, Ludvikus, Fadesga, Addbot, Yobot, AnomieBOT, ClueBot NG and Anonymous:1

• Principles of Mathematical Logic Source: https://en.wikipedia.org/wiki/Principles_of_Mathematical_Logic?oldid=655691626 Con-tributors: Michael Hardy, MakeRocketGoNow, Oleg Alexandrov, Rjwilmsi, Mathbot, PhS, Jon Awbrey, Gregbard, Eleuther, Phe-bot,Addbot, Omnipaedista, Suslindisambiguator and Anonymous: 7

• Probabilistic proposition Source: https://en.wikipedia.org/wiki/Probabilistic_proposition?oldid=636858557Contributors: Michael Hardy,Poccil, Dcfleck, Malcolma, Bluebot, CBM, Gregbard, Alaibot, David Eppstein, Melcombe, AnomieBOT, Erik9bot and Anonymous: 3

• Problem ofmultiple generality Source: https://en.wikipedia.org/wiki/Problem_of_multiple_generality?oldid=675001204Contributors:Charles Matthews, Aetheling, Fenice, Chalst, TheParanoidOne, Oleg Alexandrov, Rick Norwood, Sectryan, Mhss, Cybercobra, Vina-iwbot~enwiki, Tomlee2060, Zero sharp, Gregbard, Cydebot, Egriffin, Philogo, Eferrier, Newbyguesses, Fadesga, Addbot, Yobot, John ofReading, Helpful Pixie Bot, Solomon7968, Knife-in-the-drawer and Anonymous: 9

• Proof net Source: https://en.wikipedia.org/wiki/Proof_net?oldid=615489541Contributors: Michael Hardy, Silverfish, CharlesMatthews,Kaustuv, Chalst, Oleg Alexandrov, WoodenTaco, Ott2, SmackBot, Physis, CBM, Gregbard, Magioladitis, A3nm, Selinger, Safulop,Addbot, 9258fahsflkh917fas, Greatfermat, Frietjes, ChrisGualtieri, Soujak and Anonymous: 3

• Proof-theoretic semantics Source: https://en.wikipedia.org/wiki/Proof-theoretic_semantics?oldid=650356165 Contributors: Edward,Michael Hardy, Chalst, Velvetsmog, Porcher, Trovatore, Mhss, Gregbard, Cydebot, John254, Nick Number, Hjoole, Unara, The Wikighost, Cerabot~enwiki, Leftarrow and Anonymous: 4

• Propositional variable Source: https://en.wikipedia.org/wiki/Propositional_variable?oldid=635471524Contributors: Giftlite, Creidieki,Aisaac, Woohookitty, Kbdank71, Mitsukai, Trovatore, Mhss, Foxjwill, Jon Awbrey, Mets501, CBM, Gregbard, AndrewHowse, Cydebot,Julian Mendez, Thijs!bot, Pomte, Pdabrowiecki, Addbot, Amirobot, Tijfo098, Brirush and Anonymous: 2

• Prototype Verification System Source: https://en.wikipedia.org/wiki/Prototype_Verification_System?oldid=637556645 Contributors:Dwheeler, Zeno Gantner, Greenrd, Langec, Neutrality, Chalst, TheParanoidOne, RxS, Feydey, Wavelength, Neilbeach, Jpbowen, Smack-Bot, Eskimbot, Disavian, HenningThielemann, Gregbard, Cydebot, Skier Dude, Free Software Knight, Addbot, Ptbotgourou, GoingBatty,Pvs user and Anonymous: 3

• Provability logic Source: https://en.wikipedia.org/wiki/Provability_logic?oldid=664876084 Contributors: Edward, Charles Matthews,Kntg, Chalst, EmilJ, Nortexoid, PWilkinson, SLi, Trovatore, Black Falcon, Nahaj, Chris the speller, OneSixOne, CBM, Gregbard, DavidEppstein, VanishedUserABC, DumZiBoT, Addbot, D'ohBot, ZéroBot, Mogism, Brirush, ProvLog and Anonymous: 4

• Proving a point Source: https://en.wikipedia.org/wiki/Proving_a_point?oldid=601827604 Contributors: Pigman, SmackBot, ColonelWarden, Bearian, Debresser, Yobot, FredZ, Thetruthisnegotiable and Anonymous: 1

• Regularmodal logic Source: https://en.wikipedia.org/wiki/Regular_modal_logic?oldid=450911891Contributors: Nortexoid, Spug, Simeon,Addbot and AvicAWB

• Robinson’s joint consistency theorem Source: https://en.wikipedia.org/wiki/Robinson’{}s_joint_consistency_theorem?oldid=671837343Contributors: Giftlite, Waltpohl, Tillmo, RDBury, CBM, Gregbard, DavidCBryant, Addbot, Amirobot, Stefan.vatev, Trappist the monkand Helpful Pixie Bot

• Rule of replacement Source: https://en.wikipedia.org/wiki/Rule_of_replacement?oldid=674435285Contributors: Michael Hardy, ENeville,Arthur Rubin, Gregbard, Cliff, Legobot, SwisterTwister, FrescoBot, Olexa Riznyk, IfYouDoIfYouDon't, Helpful Pixie Bot, BG19bot,Dooooot, Jochen Burghardt and Anonymous: 5

• Rules of passage (logic) Source: https://en.wikipedia.org/wiki/Rules_of_passage_(logic)?oldid=622119350Contributors: Michael Hardy,BD2412, SmackBot, Myasuda, Gregbard, Palnot and Anonymous: 1

• Sacrifice of the intellect Source: https://en.wikipedia.org/wiki/Sacrifice_of_the_intellect?oldid=596956567 Contributors: Ezhiki, RichFarmbrough, Xezbeth, Pigman, Cesium 133, Wwallacee, Gregbard, Addbot, Mapswhets2 and Anonymous: 2

• Sanctioned specialisation Source: https://en.wikipedia.org/wiki/Sanctioned_specialisation?oldid=530812301 Contributors: Bearcat,GregorB, BD2412, Malcolma, Andthu, Alaibot, Addbot and LilHelpa

• Second-order predicate Source: https://en.wikipedia.org/wiki/Second-order_predicate?oldid=637847504Contributors: Awaterl, MichaelHardy, Oliver Pereira, Dori, Alex S, Elwikipedista~enwiki, Oleg Alexandrov, GregorB, Graham87, Fram, SmackBot, CBM, Gregbard,David Eppstein, Erik9bot and Anonymous: 4

• Second-order propositional logic Source: https://en.wikipedia.org/wiki/Second-order_propositional_logic?oldid=620964980 Contrib-utors: Michael Hardy, Jason Quinn, Chalst, Meloman, Gregbard, Epsilon0 and Jochen Burghardt

• Self-reference puzzle Source: https://en.wikipedia.org/wiki/Self-reference_puzzle?oldid=534581049 Contributors: Gregbard, Styro-foam1994, SharkD, Dinnertimeok, Chaotic iak and Anonymous: 2

• Self-verifying theories Source: https://en.wikipedia.org/wiki/Self-verifying_theories?oldid=606600583Contributors: CharlesMatthews,TravelingDude, Anville, Karnan, Metahacker, Chalst, Mairi, Cohesion, Oleg Alexandrov, Porcher, Trovatore, BranStark, CBM, Gregbard,Acroterion, Joeoettinger, Thehotelambush, Emk (ja), Addbot, Unzerlegbarkeit and Anonymous: 4

• Sentence (logic) Source: https://en.wikipedia.org/wiki/Sentence_(logic)?oldid=670287163Contributors: Michael Hardy, Silverfish, CharlesMatthews, Giftlite, Spayrard, Rgdboer, Oleg Alexandrov, Linas, BD2412, Qwertyus, Mkehrt, 4C~enwiki, Ihope127, Rick Norwood,Trovatore, Bbaumer, Maksim-e~enwiki, Bigbluefish, Mhss, Mets501, CBM, Gregbard, Skier Dude, Anonymous Dissident, Philogo,Ctxppc, DesolateReality, Addbot, TaBOT-zerem, Citation bot, MastiBot, Masssly, MerlIwBot, Helpful Pixie Bot and Anonymous: 4

192 CHAPTER 150. VIVID KNOWLEDGE

• Slothful induction Source: https://en.wikipedia.org/wiki/Slothful_induction?oldid=653418608 Contributors: The Anome, John, DavidEppstein, A.Ou, Addbot, FrescoBot, Solomonfromfinland, ZéroBot, Tolly4bolly, DemonicPartyHat, Helpful Pixie Bot, MrBill3, Sol1,Lesser Cartographies, Jerodlycett, PaulBustion87 and Anonymous: 6

• Specialization (logic) Source: https://en.wikipedia.org/wiki/Specialization_(logic)?oldid=418789112Contributors: TheAnome, Patrick,MartinHarper, Snoyes, Saforrest, Sj, Cambyses, Christopherlin, Utcursch, Jeshii, Twinxor, RoyBoy, Bobo192, Rd232, Andrew Gray,ZeiP, Suruena, RainbowOfLight, Defixio, FlaBot, Fram, DVD R W, SmackBot, KnowledgeOfSelf, Cazort, Can't sleep, clown will eatme, Celarnor, Richard001, 16@r, Adambiswanger1, Chrislk02, Clamster5, Catgut, Mmustafa~enwiki, Infrangible, J.delanoy, Rehrer, So-liloquial, Jamelan, Frank Romein~enwiki, Seanust, Latics, JL-Bot, ClueBot, Excirial, Hegsbfgszd, Zwasqx, Awehrfjkashkfjd, La Pianista,Jadtnr1, Addbot, Luckas-bot, Erik9bot, EdoBot and Anonymous: 40

• Strict logic Source: https://en.wikipedia.org/wiki/Strict_logic?oldid=666155622 Contributors: Charles Matthews, Jitse Niesen, AndrewEisenberg, Uberjivy, SmackBot, Maksim-e~enwiki, WilyD, Fplay, CBM, Synthebot, Paraconsistent, Erik9bot and Anonymous: 3

• Syllogistic fallacy Source: https://en.wikipedia.org/wiki/Syllogistic_fallacy?oldid=675805018 Contributors: Bryan Derksen, Mrwojo,Gracefool, Taak, Silence, Bluebot, Andeggs, George100, Gregbard, FJPB, Djhmoore, Addbot, Yobot, E235, Machine Elf 1735, Ihavea-catonmydesk and Anonymous: 4

• T-schema Source: https://en.wikipedia.org/wiki/T-schema?oldid=544243364 Contributors: Charles Matthews, Adam78, Frencheigh,Chalst, PWilkinson, Kzollman, Marudubshinki, Trovatore, Mhss, Physis, Mets501, CBM, Gregbard, Cydebot, Kenneth M Burke, Gram-marmonger, Philogo, Addbot, Ht686rg90, Hriber, Tijfo098, Philofet, Helpful Pixie Bot and Anonymous: 6

• Tacit assumption Source: https://en.wikipedia.org/wiki/Tacit_assumption?oldid=558358688 Contributors: Rednblu, Adam78, Mecan-ismo, Cask05, SmackBot, DMS, JHunterJ, CmdrObot, Egriffin, X96lee15, Lenticel, Goblinman, MironGainz and Anonymous: 1

• Takeuti’s conjecture Source: https://en.wikipedia.org/wiki/Takeuti’{}s_conjecture?oldid=627015510 Contributors: Michael Hardy,TakuyaMurata, Charles Matthews, Chalst, Mairi, CambridgeBayWeather, ArglebargleIV, Gregbard, David Eppstein, Cobi, Tradereddy,AlptaBot, Anne Bauval, Omnipaedista, FrescoBot, Proof Theorist and Anonymous: 3

• Tee (symbol) Source: https://en.wikipedia.org/wiki/Tee_(symbol)?oldid=630094104 Contributors: Hyacinth, Rpyle731, EmilJ, DePiep,Cybercobra, Gregbard, Cydebot, Egriffin, Rumping, Plastikspork, LittleWink, SporkBot, L'ami Iami, JPaestpreornJeolhlna and Anony-mous: 2

• TheGame ofLogic Source: https://en.wikipedia.org/wiki/The_Game_of_Logic?oldid=674832113Contributors: Fadesga, Attila.lendvai,Suslindisambiguator, ChrisGualtieri, MrNiceGuy1113 and Anonymous: 1

• Third-cause fallacy Source: https://en.wikipedia.org/wiki/Third-cause_fallacy?oldid=641454910 Contributors: Heron, Mrwojo, Banno,BD2412, NeonMerlin, Pelago, SmackBot, Jews, Copysan, Dollarback,MarshBot, PrestonH, Chaos5023, Hadrian89, Leuma1234, Timelezz,Ihaveacatonmydesk and Anonymous: 4

• Transparent Intensional Logic Source: https://en.wikipedia.org/wiki/Transparent_Intensional_Logic?oldid=629619944 Contributors:Jayjg, Physis, AntOnTrack, CBM, Gregbard, Widefox, Ontoraul, Synthebot, Duz48, Hamoudaldosari and Devypt

• Triangle of opposition Source: https://en.wikipedia.org/wiki/Triangle_of_opposition?oldid=625020331Contributors: Gregbard, Yobot,Theopolisme and Couscousliar

• Truth condition Source: https://en.wikipedia.org/wiki/Truth_condition?oldid=629461605 Contributors: The Anome, Vaganyik, Ixfd64,GPHemsley, Banno, Gentgeen, RedWolf, Burschik, Rich Farmbrough, Ceyockey, Zinnyard, Salix alba, YurikBot, Dmharvey, Tomisti,Dbtfz, Gregbard, Cydebot, Letranova, Andrewaskew, Fadesga, Legobot, Aaron Kauppi, Erik9bot, Gamewizard71 and Anonymous: 9

• Two-variable logic Source: https://en.wikipedia.org/wiki/Two-variable_logic?oldid=662183365Contributors: Rpyle731, BD2412, A3nm,Wgolf and G S Palmer

• Unique name assumption Source: https://en.wikipedia.org/wiki/Unique_name_assumption?oldid=621641792 Contributors: RainerWasserfuhr~enwiki, RHaworth, MacTed, Qwertyus, SmackBot, StephenReed, Cydebot, Hqb, Jens Lehmann 2008, ThorbjoernHansen,Yobot and AvicAWB

• Unsatisfiable core Source: https://en.wikipedia.org/wiki/Unsatisfiable_core?oldid=654295495 Contributors: Edward, Michael Hardy,Jok2000, DBeyer, Salmar, LouScheffer, Gregbard, Alaibot, D123488, EgoWumpus and Anonymous: 8

• Vagrant predicate Source: https://en.wikipedia.org/wiki/Vagrant_predicate?oldid=442106129 Contributors: Rpyle731, Drbreznjev, Ael2 and Yobot

• Valentino Annibale Pastore Source: https://en.wikipedia.org/wiki/Valentino_Annibale_Pastore?oldid=659605596 Contributors: Mag-nus Manske, Rich Farmbrough, Aldux, SmackBot, Paxse, SummerWithMorons, Fadesga, Addbot, FrescoBot, Cit vësco, VIAFbot andKasparBot

• Vampire (theorem prover) Source: https://en.wikipedia.org/wiki/Vampire_(theorem_prover)?oldid=625278032 Contributors: Rbrwr,Kwertii, Stephan Schulz, Tobias Bergemann, SLi, NavarroJ, Baccala@freesoft.org, Jabencarsey, SmackBot, Bluebot, Michael Kinyon,Ezrakilty, Gregbard, Cydebot, David Eppstein, R'n'B, Duncan.Hull, BSoD, FrescoBot, Σ, Chanmichael, Chricho, Sboosali and Anony-mous: 21

• Van Gogh fallacy Source: https://en.wikipedia.org/wiki/Van_Gogh_fallacy?oldid=666502130 Contributors: Mandarax, Gregbard, Mal-colmxl5, Fadesga, Omnipaedista, Claritas, H3llBot, Joydeep, Kostlivec, HunterIV4, Scott Rollans, Merylmm and Anonymous: 1

• Vienna Summer of Logic Source: https://en.wikipedia.org/wiki/Vienna_Summer_of_Logic?oldid=635877610 Contributors: Bearcat,Stephan Schulz, Rpyle731, Tim!, Thpani, Kateshortforbob, Armbrust, Josve05a and BG19bot

• Vivid knowledge Source: https://en.wikipedia.org/wiki/Vivid_knowledge?oldid=646953751 Contributors: Michael Hardy, Largoplazo,Albertzeyer, John of Reading and Anonymous: 2

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194 CHAPTER 150. VIVID KNOWLEDGE

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