Inference 3

Inference 3Wikipedia

Contents

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

2 Admissible rule 32.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Decidability and reduced rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Projectivity and unication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Bases of admissible rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5.1 Examples of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Semantics for admissible rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Structural completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.8 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Adverse inference 133.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Distributive property 144.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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4.5 Distributivity and rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.6 Distributivity in rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.7 Generalizations of distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.7.1 Notions of antidistributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Arbitrary inference 205.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Associative property 216.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Generalized associative law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.4.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.4.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.5 Non-associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.5.1 Nonassociativity of oating point calculation . . . . . . . . . . . . . . . . . . . . . . . . . 266.5.2 Notation for non-associative operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7 Biconditional elimination 297.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Biconditional introduction 318.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9 Biological network inference 329.1 Biological networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9.1.1 Transcriptional regulatory networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.1.2 Signal transduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.1.3 Metabolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.1.4 Protein-protein interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


10 Commutative property 35

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10.1 Common uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.2 Mathematical denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.3.1 Commutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.3.2 Commutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.3.3 Noncommutative operations in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . 3710.3.4 Noncommutative operations in mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 38

10.4 History and etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.5 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

10.5.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.5.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

10.6 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.7 Mathematical structures and commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.8 Related properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10.8.1 Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.8.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

10.9 Non-commuting operators in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

10.12.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.12.2 Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.12.3 Online resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

11 Commutativity of conjunction 4411.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.2 Generalized principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

12 Conjunction elimination 4612.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

13 Conjunction introduction 4813.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

14 Constraint inference 4914.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

15 Constructive dilemma 5015.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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15.2 Variable English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.3 Natural language example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5015.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

16 Contradiction 5216.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5216.2 Contradiction in formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

16.2.1 Proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.2.2 Symbolic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.2.3 The notion of contradiction in an axiomatic system and a proof of its consistency . . . . . . 54

16.3 Contradictions and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.3.1 Pragmatic contradictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.3.2 Dialectical materialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

16.4 Contradiction outside formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

17 Contraposition (traditional logic) 5817.1 Traditional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5817.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

18 Contrary (logic) 6018.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6018.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

19 Converse (logic) 6219.1 Implicational converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

19.1.1 Converse of a theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.2 Categorical converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

20 Correspondent inference theory 6420.1 Attributing intention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6420.2 Non-Common eects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6420.3 Low-Social desirability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520.4 Expectancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520.5 Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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20.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

21 Cut rule 6721.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6721.2 Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

22 De Morgans laws 6822.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

22.1.1 Substitution form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.1.2 Set theory and Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.1.3 Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7122.1.4 Text searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

22.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7122.3 Informal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

22.3.1 Negation of a disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7222.3.2 Negation of a conjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

22.4 Formal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7322.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7322.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7422.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7522.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

23 Deep inference 7623.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7623.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

24 Destructive dilemma 7724.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7724.2 Natural language example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7724.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.4 Example proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

25 Dictum de omni et nullo 7925.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

26 Disjunction elimination 81

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26.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8126.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

27 Disjunction introduction 8327.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

28 Disjunctive syllogism 8428.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8428.2 Natural language examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8528.3 Inclusive and exclusive disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8528.4 Related argument forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8528.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

29 Distributive property 8729.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8729.2 Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8729.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

29.3.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8829.3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8929.3.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

29.4 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8929.4.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8929.4.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

29.5 Distributivity and rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9029.6 Distributivity in rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9029.7 Generalizations of distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

29.7.1 Notions of antidistributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9129.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9129.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

30 Double negation 9330.1 Double negative elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

30.1.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9430.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9430.3 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9430.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

31 Downward entailing 9631.1 Strawson-DE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9631.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

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31.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

32 Existential generalization 9832.1 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9832.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9832.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

33 Existential instantiation 9933.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9933.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

34 Exportation (logic) 10034.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10034.2 Natural language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

34.2.1 Truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10034.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

34.3 Relation to functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10134.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

35 Grammar induction 10235.1 Grammar Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10235.2 Learning Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10235.3 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

35.3.1 Grammatical inference by trial-and-error . . . . . . . . . . . . . . . . . . . . . . . . . . 10335.3.2 Grammatical inference by genetic algorithms . . . . . . . . . . . . . . . . . . . . . . . . 10335.3.3 Grammatical inference by greedy algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 10335.3.4 Distributional Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10335.3.5 Learning of Pattern languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10435.3.6 Pattern theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

35.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10435.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10435.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10535.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

36 Hypothetical syllogism 10636.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10636.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10636.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10736.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

37 Superaltern 10837.1 Valid immediate inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

37.1.1 Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10837.1.2 Obverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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37.1.3 Contrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10837.2 Invalid immediate inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

37.2.1 Illicit contrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10937.2.2 Illicit subcontrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10937.2.3 Illicit subalternation (Superalternation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109


38 Implicature 11038.1 Types of implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

38.1.1 Conversational implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11038.1.2 Conventional implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

38.2 Implicature vs entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11138.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11138.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11138.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11238.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11238.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

39 Inductive functional programming 11339.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

40 Inductive probability 11440.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

40.1.1 Minimum description/message length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11540.1.2 Inference based on program complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 11540.1.3 Universal articial intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

40.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11740.2.1 Comparison to deductive probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11740.2.2 Probability as estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11740.2.3 Combining probability approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

40.3 Probability and information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11840.3.1 Combining information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11840.3.2 The internal language of information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

40.4 Probability and frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11940.4.1 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12040.4.2 The frequentest approach applied to possible worlds . . . . . . . . . . . . . . . . . . . . . 12040.4.3 The law of total of probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12140.4.4 Alternate possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12140.4.5 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12240.4.6 Implication and condition probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

40.5 Bayesian hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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40.5.1 Set of hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12340.6 Boolean inductive inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

40.6.1 Generalization and specialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12440.6.2 Newtons use of induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12440.6.3 Probabilities for inductive inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

40.7 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12640.7.1 Derivation of inductive probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12640.7.2 A model for inductive inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

40.8 Key people . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13040.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13040.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13140.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

41 Inference 13241.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

41.1.1 Example for denition #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13341.2 Incorrect inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13341.3 Automatic logical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

41.3.1 Example using Prolog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13441.3.2 Use with the semantic web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13441.3.3 Bayesian statistics and probability logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13441.3.4 Nonmonotonic logic[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

41.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13541.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13641.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13641.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

42 Inference engine 13842.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13842.2 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13942.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13942.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

43 Inference objection 14143.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14143.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

44 Inverse (logic) 14444.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14444.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

45 List of rules of inference 14545.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

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45.2 Rules for classical sentential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14545.2.1 Rules for negations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14545.2.2 Rules for conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14645.2.3 Rules for conjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14645.2.4 Rules for disjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14745.2.5 Rules for biconditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

45.3 Rules of classical predicate calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14845.4 Table: Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

45.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14945.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

45.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

46 List of valid argument forms 15146.1 Valid syllogistic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

46.1.1 Unconditionally valid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15146.1.2 Conditionally valid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

46.2 Valid propositional forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15146.2.1 Modus ponens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15146.2.2 Modus tollens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15246.2.3 Hypothetical syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15246.2.4 Disjunctive syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15246.2.5 Constructive dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

46.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

47 Logical hexagon 15447.1 Summary of relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15547.2 Interpretations of the logical hexagon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

47.2.1 Modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15547.3 Further extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15647.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15647.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15647.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

48 Material implication (rule of inference) 15748.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15748.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15748.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

49 Material inference 15949.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15949.2 Material inferences vs. enthymemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15949.3 Non-monotonic inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15949.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

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49.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

50 Modus non excipiens 161

51 Modus ponendo tollens 16251.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

52 Modus ponens 16352.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16352.2 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16452.3 Justication via truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16452.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16452.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16552.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16552.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

53 Modus tollens 16653.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16653.2 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16753.3 Relation to modus ponens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16753.4 Justication via truth table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16853.5 Formal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

53.5.1 Via disjunctive syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16853.5.2 Via reductio ad absurdum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

53.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16853.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16853.8 External link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

54 Negation as failure 16954.1 Planner semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16954.2 Prolog semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16954.3 Completion semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16954.4 Autoepistemic semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17054.5 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17154.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17154.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

55 Negation introduction 17255.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17255.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17255.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

56 Obversion 17356.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

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56.2 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17456.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

57 Resolution (logic) 17557.1 Resolution in propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

57.1.1 Resolution rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17557.1.2 A resolution technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

57.2 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17657.3 Resolution in rst order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

57.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17657.3.2 Informal explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

57.4 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17957.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17957.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17957.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17957.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

58 Resolution inference 18158.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18158.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

59 Rule of inference 18359.1 The standard form of rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18359.2 Axiom schemas and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18459.3 Example: Hilbert systems for two propositional logics . . . . . . . . . . . . . . . . . . . . . . . . 18459.4 Admissibility and derivability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18559.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18559.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

60 Rule of replacement 18760.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

61 Structural rule 18861.1 Common structural rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18861.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

62 Scalar implicature 18962.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18962.2 Examples of scalar implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18962.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19062.4 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19062.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

63 SLD resolution 192

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63.1 The SLD inference rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19263.2 The origin of the name SLD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19263.3 The computational interpretation of SLD resolution . . . . . . . . . . . . . . . . . . . . . . . . . 19363.4 SLD resolution strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19363.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19363.6 SLDNF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19463.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19463.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19463.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

64 Solomonos theory of inductive inference 19564.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

64.1.1 Philosophical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19564.1.2 Mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

64.2 Modern applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19564.2.1 Articial intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19664.2.2 Turing machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

64.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19764.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19764.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19864.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

65 Square of opposition 20065.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20065.2 The problem of existential import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20365.3 Modern squares of opposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20365.4 Logical hexagons and other bi-simplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20465.5 Square of opposition (or logical square) and modal logic . . . . . . . . . . . . . . . . . . . . . . . 20465.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20565.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20565.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

66 Strong inference 20666.1 The single hypothesis problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20666.2 Strong Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20666.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20666.4 Strong inference plus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20666.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

67 Structural rule 20867.1 Common structural rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20867.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

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68 Subalternation 20968.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

69 Superaltern 21069.1 Valid immediate inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

69.1.1 Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21069.1.2 Obverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21069.1.3 Contrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

69.2 Invalid immediate inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21169.2.1 Illicit contrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21169.2.2 Illicit subcontrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21169.2.3 Illicit subalternation (Superalternation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211


70 Tautology (rule of inference) 21270.1 Relation to tautology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21270.2 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21270.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

71 Transposition (logic) 21471.1 Formal notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21471.2 Traditional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

71.2.1 Form of transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21571.2.2 Sucient condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21571.2.3 Necessary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21571.2.4 Grammatically speaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21571.2.5 Relationship of propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21571.2.6 Transposition and the method of contraposition . . . . . . . . . . . . . . . . . . . . . . . 21671.2.7 Dierences between transposition and contraposition . . . . . . . . . . . . . . . . . . . . . 216

71.3 Transposition in mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21671.4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21671.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21671.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21671.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21771.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

72 Type inference 21872.1 Nontechnical explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21872.2 Technical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21972.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21972.4 HindleyMilner type inference algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22072.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

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72.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

73 Uncertain inference 22173.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22173.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22173.3 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22273.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22273.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

74 Universal generalization 22374.1 Generalization with hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22374.2 Example of a proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22374.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22374.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

75 Universal instantiation 22575.1 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22575.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22575.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

76 Veridicality 22776.1 Veridicality in semantic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

76.1.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22776.1.2 Nonveridical operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22876.1.3 Downward entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22876.1.4 Non-monotone quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22876.1.5 Hardly and barely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22876.1.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22976.1.7 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22976.1.8 Habitual aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22976.1.9 Generic sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22976.1.10 Modal verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22976.1.11 Imperatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23076.1.12 Protasis of conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23076.1.13 Directive intensional verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

76.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23076.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 231

76.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23176.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23676.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Chapter 1

Absorption (logic)

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

P ! Q) P ! (P ^Q)

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

1.1 Formal notation

The absorption rule may be expressed as a sequent:

P ! Q ` P ! (P ^Q)

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

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

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

1.2 Examples

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

1

2 CHAPTER 1. ABSORPTION (LOGIC)

1.3 Proof by truth table

1.4 Formal proof

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

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

[3] Russell and Whitehead, Principia Mathematica

Chapter 2

Admissible rule

This article is about rules of inference in logic systems. For the concept in decision theory, see admissible decisionrule.

In logic, a rule of inference is admissible in a formal system if the set of theorems of the system does not changewhen that rule is added to the existing rules of the system. In other words, every formula that can be derived usingthat rule is already derivable without that rule, so, in a sense, it is redundant. The concept of an admissible rule wasintroduced by Paul Lorenzen (1955).

2.1 DenitionsAdmissibility has been systematically studied only in the case of structural rules in propositional non-classical logics,which we will describe next.Let a set of basic propositional connectives be xed (for instance, f!;^;_;?g in the case of superintuitionisticlogics, or f!;?;g in the case of monomodal logics). Well-formed formulas are built freely using these connectivesfrom a countably innite set of propositional variables pn. A substitution is a function from formulas to formulaswhich commutes with the connectives, i.e.,

f(A1; : : : ; An) = f(A1; : : : ; An)

for every connective f, and formulas A1, , An. (We may also apply substitutions to sets of formulas, making = {A: A }.) A Tarski-style consequence relation[1] is a relation ` between sets of formulas, and formulas, suchthat

1. A ` A;2. if ` A then ; ` A;3. if ` A and ; A ` B then ; ` B;

for all formulas A, B, and sets of formulas , . A consequence relation such that

1. if ` A then ` A

for all substitutions is called structural. (Note that the term structural as used here and below is unrelated to thenotion of structural rules in sequent calculi.) A structural consequence relation is called a propositional logic. Aformula A is a theorem of a logic ` if ? ` A .For example, we identify a superintuitionistic logic L with its standard consequence relation `L axiomatizable bymodus ponens and axioms, and we identify a normal modal logic with its global consequence relation `L axiomatizedby modus ponens, necessitation, and axioms.

3

4 CHAPTER 2. ADMISSIBLE RULE

A structural inference rule[2] (or just rule for short) is given by a pair (,B), usually written as

A1; : : : ; AnB

or A1; : : : ; An/B;

where = {A1, , An} is a nite set of formulas, and B is a formula. An instance of the rule is

A1; : : : ; An/B

for a substitution . The rule /B is derivable in ` , if ` B . It is admissible if for every instance of the rule, Bis a theorem whenever all formulas from are theorems.[3] In other words, a rule is admissible if, when added tothe logic, does not lead to new theorems.[4] We also write j B if /B is admissible. (Note that j is a structuralconsequence relation on its own.)Every derivable rule is admissible, but not vice versa in general. A logic is structurally complete if every admissiblerule is derivable, i.e., ` = j .[5]In logics with a well-behaved conjunction connective (such as superintuitionistic or modal logics), a ruleA1; : : : ; An/Bis equivalent to A1 ^ ^An/B with respect to admissibility and derivability. It is therefore customary to only dealwith unary rules A/B.

2.2 Examples Classical propositional calculus (CPC) is structurally complete.[6] Indeed, assume that A/B is non-derivable

rule, and x an assignment v such that v(A) = 1, and v(B) = 0. Dene a substitution such that for everyvariable p, p = > if v(p) = 1, and p = ? if v(p) = 0. Then A is a theorem, but B is not (in fact, B is atheorem). Thus the rule A/B is not admissible either. (The same argument applies to any multi-valued logic Lcomplete with respect to a logical matrix whose all elements have a name in the language of L.)

The KreiselPutnam rule (aka Harrop's rule, or independence of premise rule)

(KPR) :p! q _ r(:p! q) _ (:p! r)

is admissible in the intuitionistic propositional calculus (IPC). In fact, it is admissible in every superin-tuitionistic logic.[7] On the other hand, the formula(:p! q _ r)! (:p! q) _ (:p! r)is not an intuitionistic tautology, hence KPR is not derivable in IPC. In particular, IPC is not structurallycomplete.

The rule

pp

is admissible in many modal logics, such as K, D, K4, S4, GL (see this table for names of modal logics).It is derivable in S4, but it is not derivable in K, D, K4, or GL.

The rule

p ^ :p?

is admissible in every normal modal logic.[8] It is derivable in GL and S4.1, but it is not derivable in K,D, K4, S4, S5.

2.3. DECIDABILITY AND REDUCED RULES 5

Lbs rule

(LR) p! pp

is admissible (but not derivable) in the basic modal logic K, and it is derivable in GL. However, LR isnot admissible in K4. In particular, it is not true in general that a rule admissible in a logic L must beadmissible in its extensions.

The GdelDummett logic (LC), and the modal logic Grz.3 are structurally complete.[9] The product fuzzylogic is also structurally complete.[10]

2.3 Decidability and reduced rulesThe basic question about admissible rules of a given logic is whether the set of all admissible rules is decidable.Note that the problem is nontrivial even if the logic itself (i.e., its set of theorems) is decidable: the denition ofadmissibility of a rule A/B involves an unbounded universal quantier over all propositional substitutions, hence aprioriwe only know that admissibility of rule in a decidable logic is01 (i.e., its complement is recursively enumerable).For instance, it is known that admissibility in the bimodal logics Ku and K4u (the extensions of K or K4 with theuniversal modality) is undecidable.[11] Remarkably, decidability of admissibility in the basic modal logic K is a majoropen problem.Nevertheless, admissibility of rules is known to be decidable in many modal and superintuitionistic logics. The rstdecision procedures for admissible rules in basic transitive modal logics were constructed by Rybakov, using thereduced form of rules.[12] A modal rule in variables p0, , pk is called reduced if it has the form

Wni=0

Vkj=0 :0i;jpj ^

Vkj=0 :1i;jpj

p0

;

where each :ui;j is either blank, or negation : . For each rule r, we can eectively construct a reduced rule s (calledthe reduced form of r) such that any logic admits (or derives) r if and only if it admits (or derives) s, by introducingextension variables for all subformulas in A, and expressing the result in the full disjunctive normal form. It is thussucient to construct a decision algorithm for admissibility of reduced rules.LetWni=0 'i/p0 be a reduced rule as above. We identify every conjunction 'i with the set f:0i;jpj ;:1i;jpj j j kgof its conjuncts. For any subset W of the set f'i j i ng of all conjunctions, let us dene a Kripke modelM = hW;R;i by

'i pj () pj 2 'i;

'iR'i0 () 8j k (pj 2 'i ) fpj ;pjg 'i0):Then the following provides an algorithmic criterion for admissibility in K4:[13]

Theorem. The ruleWni=0 'i/p0 is not admissible in K4 if and only if there exists a set W f'i j i ng such that1. 'i 1 p0 for some i n;2. 'i 'i for every i n;3. for every subset D of W there exist elements ; 2W such that the equivalences

pj if and only if ' pj ^pj for every ' 2 D pj if and only if pj and ' pj ^pj for every ' 2 D

hold for all j.


Similar criteria can be found for the logics S4, GL, and Grz.[14] Furthermore, admissibility in intuitionistic logic canbe reduced to admissibility in Grz using the GdelMcKinseyTarski translation:[15]

A jIPC B if and only if T (A) jGrz T (B):

Rybakov (1997) developed much more sophisticated techniques for showing decidability of admissibility, which applyto a robust (innite) class of transitive (i.e., extending K4 or IPC) modal and superintuitionistic logics, including e.g.S4.1, S4.2, S4.3, KC, Tk (as well as the above mentioned logics IPC, K4, S4, GL, Grz).[16]

Despite being decidable, the admissibility problem has relatively high computational complexity, even in simplelogics: admissibility of rules in the basic transitive logics IPC, K4, S4, GL, Grz is coNEXP-complete.[17] This shouldbe contrasted with the derivability problem (for rules or formulas) in these logics, which is PSPACE-complete.[18]

2.4 Projectivity and unicationAdmissibility in propositional logics is closely related to unication in the equational theory of modal or Heytingalgebras. The connection was developed by Ghilardi (1999, 2000). In the logical setup, a unier of a formula A in alogic L (an L-unier for short) is a substitution such that A is a theorem of L. (Using this notion, we can rephraseadmissibility of a rule A/B in L as every L-unier of A is an L-unier of B".) An L-unier is less general than anL-unier , written as , if there exists a substitution such that

`L p$ pfor every variable p. A complete set of uniers of a formula A is a set S of L-uniers of A such that every L-unierof A is less general than some unier from S. A most general unier (mgu) of A is a unier such that {} is acomplete set of uniers of A. It follows that if S is a complete set of uniers of A, then a rule A/B is L-admissible ifand only if every in S is an L-unier of B. Thus we can characterize admissible rules if we can nd well-behavedcomplete sets of uniers.An important class of formulas which have a most general unier are the projective formulas: these are formulas Asuch that there exists a unier of A such that

A `L B $ Bfor every formula B. Note that is a mgu of A. In transitive modal and superintuitionistic logics with the nite modelproperty (fmp), one can characterize projective formulas semantically as those whose set of nite L-models has theextension property:[19] if M is a nite Kripke L-model with a root r whose cluster is a singleton, and the formula Aholds in all points of M except for r, then we can change the valuation of variables in r so as to make A true in r aswell. Moreover, the proof provides an explicit construction of a mgu for a given projective formula A.In the basic transitive logics IPC, K4, S4, GL, Grz (and more generally in any transitive logic with the fmp whoseset of nite frame satises another kind of extension property), we can eectively construct for any formula A itsprojective approximation (A):[20] a nite set of projective formulas such that

1. P `L A for every P 2 (A);2. every unier of A is a unier of a formula from (A).

It follows that the set of mgus of elements of (A) is a complete set of uniers of A. Furthermore, if P is a projectiveformula, then

P jL B if and only if P `L B

for any formula B. Thus we obtain the following eective characterization of admissible rules:[21]

A jL B if and only if 8P 2 (A) (P `L B):

2.5. BASES OF ADMISSIBLE RULES 7

2.5 Bases of admissible rulesLet L be a logic. A set R of L-admissible rule is called a basis[22] of admissible rules, if every admissible rule /Bcan be derived from R and the derivable rules of L, using substitution, composition, and weakening. In other words,R is a basis if and only if jL is the smallest structural consequence relation which includes `L and R.Notice that decidability of admissible rules of a decidable logic is equivalent to the existence of recursive (or recursivelyenumerable) bases: on the one hand, the set of all admissible rule is a recursive basis if admissibility is decidable. Onthe other hand, the set of admissible rules is always co-r.e., and if we further have an r.e. basis, it is also r.e., henceit is decidable. (In other words, we can decide admissibility of A/B by the following algorithm: we start in paralleltwo exhaustive searches, one for a substitution which unies A but not B, and one for a derivation of A/B from Rand `L . One of the searches has to eventually come up with an answer.) Apart from decidability, explicit bases ofadmissible rules are useful for some applications, e.g. in proof complexity.[23]

For a given logic, we can ask whether it has a recursive or nite basis of admissible rules, and to provide an explicitbasis. If a logic has no nite basis, it can nevertheless has an independent basis: a basis R such that no proper subsetof R is a basis.In general, very little can be said about existence of bases with desirable properties. For example, while tabularlogics are generally well-behaved, and always nitely axiomatizable, there exist tabular modal logics without a niteor independent basis of rules.[24] Finite bases are relatively rare: even the basic transitive logics IPC, K4, S4, GL, Grzdo not have a nite basis of admissible rules,[25] though they have independent bases.[26]

2.5.1 Examples of bases The empty set is a basis of L-admissible rules if and only if L is structurally complete. Every extension of the modal logic S4.3 (including, notably, S5) has a nite basis consisting of the single rule[27]

p ^ :p? :

Visser's rules

n^i=1

(pi ! qi)! pn+1 _ pn+2_ r

n+2_j=1

n^i=1

(pi ! qi)! pj_ r

; n 1

are a basis of admissible rules in IPC or KC.[28]

The rules

q !

n_i=1

pi_r

n_i=1

(q ^q ! pi) _ r; n 0

are a basis of admissible rules of GL.[29] (Note that the empty disjunction is dened as ? .)

The rules


(q ! q)!

n_i=1

pi_r

n_i=1

(q ! pi) _ r; n 0

are a basis of admissible rules of S4 or Grz.[30]

2.6 Semantics for admissible rulesA rule /B is valid in a modal or intuitionistic Kripke frame F = hW;Ri , if the following is true for every valuation

in F:

if 8x 2W (x A) for all A 2 , then 8x 2W (x B) .

(The denition readily generalizes to general frames, if needed.)Let X be a subset of W, and t a point in W. We say that t is

a reexive tight predecessor of X, if for every y in W: t R y if and only if t = y or x = y or x R y for some x inX,

an irreexive tight predecessor of X, if for every y in W: t R y if and only if x = y or x R y for some x in X.

We say that a frame F has reexive (irreexive) tight predecessors, if for every nite subset X of W, there exists areexive (irreexive) tight predecessor of X in W.We have:[31]

a rule is admissible in IPC if and only if it is valid in all intuitionistic frames which have reexive tight prede-cessors,

a rule is admissible in K4 if and only if it is valid in all transitive frames which have reexive and irreexivetight predecessors,

a rule is admissible in S4 if and only if it is valid in all transitive reexive frames which have reexive tightpredecessors,

a rule is admissible in GL if and only if it is valid in all transitive converse well-founded frames which haveirreexive tight predecessors.

Note that apart from a few trivial cases, frames with tight predecessors must be innite, hence admissible rules inbasic transitive logics do not enjoy the nite model property.

2.7 Structural completenessWhile a general classication of structurally complete logics is not an easy task, we have a good understanding ofsome special cases.Intuitionistic logic itself is not structurally complete, but its fragmentsmay behave dierently. Namely, any disjunction-free rule or implication-free rule admissible in a superintuitionistic logic is derivable.[32] On the other hand, the Mintsrule

(p! q)! p _ r((p! q)! p) _ ((p! q)! r)

2.8. VARIANTS 9

is admissible in intuitionistic logic but not derivable, and contains only implications and disjunctions.We know the maximal structurally incomplete transitive logics. A logic is called hereditarily structurally com-plete, if every its extension is structurally complete. For example, classical logic, as well as the logics LC and Grz.3mentioned above, are hereditarily structurally complete. A complete description of hereditarily structurally completesuperintuitionistic and transitive modal logics was given by Citkin and Rybakov. Namely, a superintuitionistic logicis hereditarily structurally complete if and only if it is not valid in any of the ve Kripke frames[9]

Similarly, an extension of K4 is hereditarily structurally complete if and only if it is not valid in any of certain twentyKripke frames (including the ve intuitionistic frames above).[9]

There exist structurally complete logics that are not hereditarily structurally complete: for example, Medvedevs logicis structurally complete,[33] but it is included in the structurally incomplete logic KC.

2.8 VariantsA rule with parameters is a rule of the form

A(p1; : : : ; pn; s1; : : : ; sk)

B(p1; : : : ; pn; s1; : : : ; sk);

whose variables are divided into the regular variables pi, and the parameters si. The rule is L-admissible if everyL-unier of A such that si = si for each i is also a unier of B. The basic decidability results for admissible rulesalso carry to rules with parameters.[34]

A multiple-conclusion rule is a pair (,) of two nite sets of formulas, written as

A1; : : : ; AnB1; : : : ; Bm

or A1; : : : ; An/B1; : : : ; Bm:

Such a rule is admissible if every unier of is also a unier of some formula from .[35] For example, a logic L isconsistent i it admits the rule

?;

and a superintuitionistic logic has the disjunction property i it admits the rule

p _ qp; q

:

Again, basic results on admissible rules generalize smoothly to multiple-conclusion rules.[36] In logics with a variantof the disjunction property, the multiple-conclusion rules have the same expressive power as single-conclusion rules:for example, in S4 the rule above is equivalent to

A1; : : : ; AnB1 _ _Bm :


Nevertheless, multiple-conclusion rules can often be employed to simplify arguments.In proof theory, admissibility is often considered in the context of sequent calculi, where the basic objects are sequentsrather than formulas. For example, one can rephrase the cut-elimination theorem as saying that the cut-free sequentcalculus admits the cut rule

` A; ; A ` ; ` ; :

(By abuse of language, it is also sometimes said that the (full) sequent calculus admits cut, meaning its cut-freeversion does.) However, admissibility in sequent calculi is usually only a notational variant for admissibility in thecorresponding logic: any complete calculus for (say) intuitionistic logic admits a sequent rule if and only if IPC admitsthe formula rule which we obtain by translating each sequent ` to its characteristic formulaV! W .2.9 Notes

[1] Blok & Pigozzi (1989), Kracht (2007)

[2] Rybakov (1997), Def. 1.1.3

[3] Rybakov (1997), Def. 1.7.2

[4] From de Jonghs theorem to intuitionistic logic of proofs

[5] Rybakov (1997), Def. 1.7.7

[6] Chagrov & Zakharyaschev (1997), Thm. 1.25

[7] Prucnal (1979), cf. Iemho (2006)

[8] Rybakov (1997), p. 439

[9] Rybakov (1997), Thms. 5.4.4, 5.4.8

[10] Cintula & Metcalfe (2009)

[11] Wolter & Zakharyaschev (2008)

[12] Rybakov (1997), 3.9

[13] Rybakov (1997), Thm. 3.9.3

[14] Rybakov (1997), Thms. 3.9.6, 3.9.9, 3.9.12; cf. Chagrov & Zakharyaschev (1997), 16.7

[15] Rybakov (1997), Thm. 3.2.2

[16] Rybakov (1997), 3.5

[17] Jebek (2007)

[18] Chagrov & Zakharyaschev (1997), 18.5

[19] Ghilardi (2000), Thm. 2.2

[20] Ghilardi (2000), p. 196

[21] Ghilardi (2000), Thm. 3.6

[22] Rybakov (1997), Def. 1.4.13

[23] Mints & Kojevnikov (2004)

[24] Rybakov (1997), Thm. 4.5.5

[25] Rybakov (1997), 4.2

[26] Jebek (2008)

2.10. REFERENCES 11

[27] Rybakov (1997), Cor. 4.3.20

[28] Iemho (2001, 2005), Rozire (1992)

[29] Jebek (2005)

[30] Jebek (2005,2008)

[31] Iemho (2001), Jebek (2005)

[32] Rybakov (1997), Thms. 5.5.6, 5.5.9

[33] Prucnal (1976)

[34] Rybakov (1997), 6.1

[35] Jebek (2005); cf. Kracht (2007), 7

[36] Jebek (2005, 2007, 2008)

2.10 References W. Blok, D. Pigozzi, Algebraizable logics, Memoirs of the American Mathematical Society 77 (1989), no. 396,

1989.

A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Logic Guides vol. 35, Oxford University Press, 1997.ISBN 0-19-853779-4

P. Cintula and G. Metcalfe, Structural completeness in fuzzy logics, Notre Dame Journal of Formal Logic 50(2009), no. 2, pp. 153182. doi:10.1215/00294527-2009-004

A. I. Citkin, On structurally complete superintuitionistic logics, Soviet Mathematics Doklady, vol. 19 (1978),pp. 816819.

S. Ghilardi, Unication in intuitionistic logic, Journal of Symbolic Logic 64 (1999), no. 2, pp. 859880. ProjectEuclid JSTOR

S. Ghilardi, Best solving modal equations, Annals of Pure and Applied Logic 102 (2000), no. 3, pp. 183198.doi:10.1016/S0168-0072(99)00032-9

R. Iemho, On the admissible rules of intuitionistic propositional logic, Journal of Symbolic Logic 66 (2001),no. 1, pp. 281294. Project Euclid JSTOR

R. Iemho, Intermediate logics and Vissers rules, Notre Dame Journal of Formal Logic 46 (2005), no. 1, pp.6581. doi:10.1305/ndj/1107220674

R. Iemho, On the rules of intermediate logics, Archive for Mathematical Logic, 45 (2006), no. 5, pp. 581599.doi:10.1007/s00153-006-0320-8

E. Jebek, Admissible rules of modal logics, Journal of Logic and Computation 15 (2005), no. 4, pp. 411431.doi:10.1093/logcom/exi029

E. Jebek, Complexity of admissible rules, Archive for Mathematical Logic 46 (2007), no. 2, pp. 7392.doi:10.1007/s00153-006-0028-9

E. Jebek, Independent bases of admissible rules, Logic Journal of the IGPL 16 (2008), no. 3, pp. 249267.doi:10.1093/jigpal/jzn004

M. Kracht, Modal Consequence Relations, in: Handbook of Modal Logic (P. Blackburn, J. van Benthem, andF. Wolter, eds.), Studies of Logic and Practical Reasoning vol. 3, Elsevier, 2007, pp. 492545. ISBN 978-0-444-51690-9

P. Lorenzen, Einfhrung in die operative Logik undMathematik, Grundlehren der mathematischen Wissenschaftenvol. 78, SpringerVerlag, 1955.


G. Mints and A. Kojevnikov, Intuitionistic Frege systems are polynomially equivalent, Zapiski Nauchnyh Sem-inarov POMI 316 (2004), pp. 129146. gzipped PS

T. Prucnal, Structural completeness of Medvedevs propositional calculus, Reports on Mathematical Logic 6(1976), pp. 103105.

T. Prucnal,On two problems ofHarvey Friedman, Studia Logica 38 (1979), no. 3, pp. 247262. doi:10.1007/BF00405383 P. Rozire, Rgles admissibles en calcul propositionnel intuitionniste, Ph.D. thesis, Universit de Paris VII, 1992.

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V. V. Rybakov, Admissibility of Logical Inference Rules, Studies in Logic and the Foundations of Mathematicsvol. 136, Elsevier, 1997. ISBN 0-444-89505-1

F. Wolter, M. Zakharyaschev, Undecidability of the unication and admissibility problems for modal and de-scription logics, ACM Transactions on Computational Logic 9 (2008), no. 4, article no. 25. doi:10.1145/1380572.1380574PDF

Chapter 3

Adverse inference

Adverse inference is a legal inference, adverse to the concerned party, drawn from silence or absence of requestedevidence. It is part of evidence codes based on common law in various countries.According to Lawvibe, the 'adverse inference' can be quite damning at trial. Essentially, when plaintis try to presentevidence on a point essential to their case and cant because the document has been destroyed (by the defendant),the jury can infer that the evidence would have been adverse to (the defendant), and adopt the plaintis reasonableinterpretation of what the document would have said... [1]

The United States Court of Appeals for the Eighth Circuit pointed out in 2004, in a case involving spoliation (de-struction) of evidence, that "...the giving of an adverse inference instruction often terminates the litigation in that it is'too dicult a hurdle' for the spoliating party to overcome. The court therefore concluded that the adverse inferenceinstruction is an 'extreme' sanction that should 'not be given lightly'.... [2]

This rule applies not only to evidence which is destroyed, but also to evidence which exists but the party refuses toproduce, and to evidence which the party has under his control, and which is not produced. See Notice to produce.This adverse inference is based upon the presumption that the party who controls the evidence would have producedit, if it had been supportive of his/her position.It can also apply to a witness who is known to exist but which the party refuses to identify or produce.After a change in the law in 1994 the right to silence under English law was curtailed because the court and jury wereallowed to draw adverse inference from such a silence.[3] Under English law when the police caution someone they sayYou do not have to say anything. But it may harm your defence if you do not mention, when questioned, somethingwhich you later rely on in court. because under English law the court and jury can draw an adverse inference fromfact that someone did not mention a defence when given the chance to do so when charged with an oence.[3][4]

3.1 References[1] Virgin Gets Hammered by Adverse Inference, LawVibe.com, April 4, 2007.

[2] Morris v. Union Pacic R. R., 373 F.3d 896, 900 (8th Cir.2004)

[3] Baksi, Catherine (24 May 2012), Going no comment": a delicate balancing act, Law Society Gazette

[4] CPP (26 September 2014), Adverse Inferences, Crown Prosecution Service

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Chapter 4

Distributive property

Distributivity redirects here. It is not to be confused with Distributivism.

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive lawfrom elementary algebra. In propositional logic, distribution refers to two valid rules of replacement. The rulesallow one to reformulate conjunctions and disjunctions within logical proofs.For example, in arithmetic:

2 (1 + 3) = (2 1) + (2 3), but 2 / (1 + 3) (2 / 1) + (2 / 3).

In the left-hand side of the rst equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the1 and the 3 individually, with the products added afterwards. Because these give the same nal answer (8), it is saidthat multiplication by 2 distributes over addition of 1 and 3. Since one could have put any real numbers in place of2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes overaddition of real numbers.

4.1 DenitionGiven a set S and two binary operators and + on S, we say that the operation

is left-distributive over + if, given any elements x, y, and z of S,

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

is right-distributive over + if, given any elements x, y, and z of S:

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

is distributive over + if it is left- and right-distributive.[1]

Notice that when is commutative, the three conditions above are logically equivalent.

4.2 MeaningThe operators used for examples in this section are the binary operations of addition ( + ) and multiplication ( ) ofnumbers.There is a distinction between left-distributivity and right-distributivity:

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4.3. EXAMPLES 15

a (b c) = a b a c (left-distributive)(a b) c = a c b c (right-distributive)

In either case, the distributive property can be described in words as:To multiply a sum (or dierence) by a factor, each summand (or minuend and subtrahend) is multiplied by this factorand the resulting products are added (or subtracted).If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity impliesright-distributivity and vice versa.One example of an operation that is only right-distributive is division, which is not commutative:

(a b) c = a c b cIn this case, left-distributivity does not apply:

a (b c) 6= a b a cThe distributive laws are among the axioms for rings and elds. Examples of structures in which two operations aremutually related to each other by the distributive law are Boolean algebras such as the algebra of sets or the switchingalgebra. There are also combinations of operations that are not mutually distributive over each other; for example,addition is not distributive over multiplication.Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of asum with each summand of the other sums (keeping track of signs), and then adding up all of the resulting products.

4.3 Examples

4.3.1 Real numbersIn the following examples, the use of the distributive law on the set of real numbers R is illustrated. When multi-plication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point ofview of algebra, the real numbers form a eld, which ensures the validity of the distributive law.

First example (mental and written multiplication)

During mental arithmetic, distributivity is often used unconsciously:

6 16 = 6 (10 + 6) = 6 10 + 6 6 = 60 + 36 = 96

Thus, to calculate 6 16 in your head, you rst multiply 6 10 and 6 6 and add the intermediate results. Writtenmultiplication is also based on the distributive law.

Second example (with variables)

3a2b (4a 5b) = 3a2b 4a 3a2b 5b = 12a3b 15a2b2

Third example (with two sums)

(a+ b) (a b) = a (a b) + b (a b) = a2 ab+ ba b2 = a2 b2= (a+ b) a (a+ b) b = a2 + ba ab b2 = a2 b2

Here the distributive law was applied twice and. It does not matter which bracket is rst multiplied out.

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Fourth Example Here the distributive law is applied the other way around compared to the previous examples.Consider12a3b2 30a4bc+ 18a2b3c2 :

Since the factor 6a2b occurs in all summand, it can be factored out. That is, due to the distributive law one obtains12a3b2 30a4bc+ 18a2b3c2 = 6a2b(2ab 5a2c+ 3b2c2) :

4.3.2 MatricesThe distributive law is valid for matrix multiplication. More precisely,

(A+B) C = A C +B Cfor all l m -matrices A;B and m n -matrices C , as well as

A (B + C) = A B +A Cfor all l m -matrices A and m n -matrices B;C . Because the commutative property does not hold for matrixmultiplication, the second law does not follow from the rst law. In this case, they are two dierent laws.

4.3.3 Other examples1. Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive.2. The cross product is left- and right-distributive over vector addition, though not commutative.3. The union of sets is distributive over intersection, and intersection is distributive over union.4. Logical disjunction (or) is distributive over logical conjunction (and), and conjunction is distributive over

disjunction.5. For real numbers (and for any totally ordered set), the maximum operation is distributive over the minimum

operation, and vice versa: max(a, min(b, c)) = min(max(a, b), max(a, c)) and min(a, max(b, c)) = max(min(a,b), min(a, c)).

6. For integers, the greatest common divisor is distributive over the least common multiple, and vice versa: gcd(a,lcm(b, c)) = lcm(gcd(a, b), gcd(a,

Inference 3

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