Inference 2Wikipedia

Contents

1 Adverse inference 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Arbitrary inference 22.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Biological network inference 33.1 Biological networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1.1 Transcriptional regulatory networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.2 Signal transduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1.3 Metabolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1.4 Protein-protein interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Constraint inference 64.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5 Contradiction 75.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 Contradiction in formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5.2.1 Proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.2.2 Symbolic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2.3 The notion of contradiction in an axiomatic system and a proof of its consistency . . . . . . 9

5.3 Contradictions and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3.1 Pragmatic contradictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3.2 Dialectical materialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.4 Contradiction outside formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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6 Contraposition (traditional logic) 136.1 Traditional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Contrary (logic) 157.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 Converse (logic) 178.1 Implicational converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8.1.1 Converse of a theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Categorical converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9 Correspondent inference theory 199.1 Attributing intention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.2 Non-Common eects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.3 Low-Social desirability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.4 Expectancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.5 Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 Deep inference 2210.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

11 Dictum de omni et nullo 2311.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2311.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

12 Downward entailing 2512.1 Strawson-DE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2512.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

13 Grammar induction 27

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13.1 Grammar Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.2 Learning Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.3 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

13.3.1 Grammatical inference by trial-and-error . . . . . . . . . . . . . . . . . . . . . . . . . . 2813.3.2 Grammatical inference by genetic algorithms . . . . . . . . . . . . . . . . . . . . . . . . 2813.3.3 Grammatical inference by greedy algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 2813.3.4 Distributional Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2813.3.5 Learning of Pattern languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2913.3.6 Pattern theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

13.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2913.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2913.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3013.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

14 Superaltern 3114.1 Valid immediate inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

14.1.1 Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3114.1.2 Obverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3114.1.3 Contrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

14.2 Invalid immediate inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3214.2.1 Illicit contrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3214.2.2 Illicit subcontrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3214.2.3 Illicit subalternation (Superalternation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

14.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3214.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

15 Implicature 3315.1 Types of implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

15.1.1 Conversational implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3315.1.2 Conventional implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

15.2 Implicature vs entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

16 Inductive functional programming 3616.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

17 Inductive probability 3717.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

17.1.1 Minimum description/message length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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17.1.2 Inference based on program complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.1.3 Universal articial intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

17.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4017.2.1 Comparison to deductive probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4017.2.2 Probability as estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4017.2.3 Combining probability approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

17.3 Probability and information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4117.3.1 Combining information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4117.3.2 The internal language of information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

17.4 Probability and frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.4.1 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4317.4.2 The frequentest approach applied to possible worlds . . . . . . . . . . . . . . . . . . . . . 4317.4.3 The law of total of probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4417.4.4 Alternate possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4417.4.5 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4517.4.6 Implication and condition probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

17.5 Bayesian hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4617.5.1 Set of hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

17.6 Boolean inductive inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717.6.1 Generalization and specialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717.6.2 Newtons use of induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717.6.3 Probabilities for inductive inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

17.7 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.7.1 Derivation of inductive probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.7.2 A model for inductive inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

17.8 Key people . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

18 Inference 5518.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

18.1.1 Example for denition #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5618.2 Incorrect inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5618.3 Automatic logical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

18.3.1 Example using Prolog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.3.2 Use with the semantic web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.3.3 Bayesian statistics and probability logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.3.4 Nonmonotonic logic[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

18.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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18.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

19 Inference engine 6119.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.2 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

20 Inference objection 6420.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6420.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

21 Inverse (logic) 6721.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6721.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

22 Logical hexagon 6822.1 Summary of relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6922.2 Interpretations of the logical hexagon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

22.2.1 Modal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6922.3 Further extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

23 Material inference 7123.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.2 Material inferences vs. enthymemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.3 Non-monotonic inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7223.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

24 Obversion 7324.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.2 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

25 Resolution inference 7525.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

26 Rule of inference 7726.1 The standard form of rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7726.2 Axiom schemas and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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26.3 Example: Hilbert systems for two propositional logics . . . . . . . . . . . . . . . . . . . . . . . . 7826.4 Admissibility and derivability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

27 Scalar implicature 8127.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8127.2 Examples of scalar implicature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8127.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8227.4 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8227.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

28 Solomonos theory of inductive inference 8428.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

28.1.1 Philosophical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8428.1.2 Mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

28.2 Modern applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8428.2.1 Articial intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8528.2.2 Turing machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

28.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8628.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8628.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8728.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

29 Square of opposition 8929.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8929.2 The problem of existential import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.3 Modern squares of opposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.4 Logical hexagons and other bi-simplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9329.5 Square of opposition (or logical square) and modal logic . . . . . . . . . . . . . . . . . . . . . . . 9329.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9429.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9429.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

30 Strong inference 9530.1 The single hypothesis problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9530.2 Strong Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9530.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9530.4 Strong inference plus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9530.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

31 Subalternation 9731.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

CONTENTS vii

32 Superaltern 9832.1 Valid immediate inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

32.1.1 Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9832.1.2 Obverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9832.1.3 Contrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

32.2 Invalid immediate inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.2.1 Illicit contrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.2.2 Illicit subcontrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.2.3 Illicit subalternation (Superalternation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

32.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

33 Type inference 10033.1 Nontechnical explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10033.2 Technical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10133.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10133.4 HindleyMilner type inference algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10233.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10233.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

34 Uncertain inference 10334.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10334.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10334.3 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10434.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10434.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

35 Veridicality 10535.1 Veridicality in semantic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

35.1.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10535.1.2 Nonveridical operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10635.1.3 Downward entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10635.1.4 Non-monotone quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10635.1.5 Hardly and barely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10635.1.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10735.1.7 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10735.1.8 Habitual aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10735.1.9 Generic sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10735.1.10 Modal verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10735.1.11 Imperatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10835.1.12 Protasis of conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10835.1.13 Directive intensional verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

viii CONTENTS

35.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10835.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 109

35.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10935.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11135.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Chapter 1

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]

1.1 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

1

Chapter 2

Arbitrary inference

In clinical psychology, arbitrary inference is a type of cognitive bias in which a person quickly draws a conclusionwithout the requisite evidence.[1] It commonly appears in Aaron Beck's work in cognitive therapy.

2.1 See also Aaron T. Beck Clinical Psychology Cognitive bias Cognitive therapy Jumping to conclusions

2.2 References[1] Sundberg, Norman (2001). Clinical Psychology: Evolving Theory, Practice, and Research. Englewood Clis: Prentice Hall.

ISBN 0-13-087119-2.

2

Chapter 3

Biological network inference

Biological network inference is the process of making inferences and predictions about biological networks.

3.1 Biological networksIn a topological sense, a network is a set of nodes and a set of directed or undirected edges between the nodes. Manytypes of biological networks exist, including transcriptional, signalling and metabolic. Few such networks are knownin anything approaching their complete structure, even in the simplest bacteria. Still less is known on the parametersgoverning the behavior of such networks over time, how the networks at dierent levels in a cell interact, and how topredict the complete state description of a eukaryotic cell or bacterial organism at a given point in the future. Systemsbiology, in this sense, is still in its infancy.There is great interest in network medicine for the modelling biological systems. This article focuses on a necessaryprerequisite to dynamic modeling of a network: inference of the topology, that is, prediction of the wiring diagramof the network. More specically, we focus here on inference of biological network structure using the growing setsof high-throughput expression data for genes, proteins, and metabolites. Briey, methods using high-throughput datafor inference of regulatory networks rely on searching for patterns of partial correlation or conditional probabilitiesthat indicate causal inuence.[1][2] Such patterns of partial correlations found in the high-throughput data, possiblycombined with other supplemental data on the genes or proteins in the proposed networks, or combined with otherinformation on the organism, form the basis upon which such algorithms work. Such algorithms can be of use ininferring the topology of any network where the change in state of one node can aect the state of other nodes.

3.1.1 Transcriptional regulatory networks

Genes are the nodes and the edges are directed. A gene serves as the source of a direct regulatory edge to a targetgene by producing an RNA or protein molecule that functions as a transcriptional activator or inhibitor of the targetgene. If the gene is an activator, then it is the source of a positive regulatory connection; if an inhibitor, then it is thesource of a negative regulatory connection. Computational algorithms take as primary input data measurements ofmRNA expression levels of the genes under consideration for inclusion in the network, returning an estimate of thenetwork topology. Such algorithms are typically based on linearity, independence or normality assumptions, whichmust be veried on a case-by-case basis.[3] Clustering or some form of statistical classication is typically employed toperform an initial organization of the high-throughput mRNA expression values derived from microarray experiments,in particular to select sets of genes as candidates for network nodes.[4] The question then arises: how can the clusteringor classication results be connected to the underlying biology? Such results can be useful for pattern classication for example, to classify subtypes of cancer, or to predict dierential responses to a drug (pharmacogenomics). Butto understand the relationships between the genes, that is, to more precisely dene the inuence of each gene on theothers, the scientist typically attempts to reconstruct the transcriptional regulatory network. This can be done by dataintegration in dynamic models supported by background literature, or information in public databases, combined withthe clustering results.[5] The modelling can be done by a Boolean network, by Ordinary dierential equations or Linearregression models, e.g. Least-angle regression, by Bayesian network or based on Information theory approaches.[6]For instance it can be done by the application of a correlation-based inference algorithm, as will be discussed below,

3

4 CHAPTER 3. BIOLOGICAL NETWORK INFERENCE

an approach which is having increased success as the size of the available microarray sets keeps increasing [1][7][8]

3.1.2 Signal transductionSignal transduction networks (very important in the biology of cancer). Proteins are the nodes and directed edgesrepresent interaction in which the biochemical conformation of the child is modied by the action of the parent (e.g.mediated by phosphorylation, ubiquitylation, methylation, etc.). Primary input into the inference algorithm would bedata from a set of experiments measuring protein activation / inactivation (e.g., phosphorylation / dephosphorylation)across a set of proteins. Inference for such signalling networks is complicated by the fact that total concentrationsof signalling proteins will uctuate over time due to transcriptional and translational regulation. Such variation canlead to statistical confounding. Accordingly, more sophisticated statistical techniques must be applied to analyse suchdatasets.[9]

3.1.3 MetabolicMetabolite networks. Metabolites are the nodes and the edges are directed. Primary input into an algorithm wouldbe data from a set of experiments measuring metabolite levels.

3.1.4 Protein-protein interactionProtein-protein interaction networks are also under very active study. However, reconstruction of these networksdoes not use correlation-based inference in the sense discussed for the networks already described (interaction doesnot necessarily imply a change in protein state), and a description of such interaction network reconstruction is leftto other articles.

3.2 See also Cytoscape tool Bayesian probability Network medicine

3.3 References[1] Marbach D, Costello JC, Kner R, Vega NM, Prill RJ, Camacho DM, Allison KR, The DREAM5 Consortium, Kellis M,

Collins JJ, Stolovitzky G (2012). Wisdom of crowds for robust gene network inference. Nature Methods 9 (8): 796804.doi:10.1038/nmeth.2016. PMC 3512113. PMID 22796662.

[2] Sprites, P; Glymour, C; Scheines, R (2000). Causation, Prediction, and Search: Adaptive Computation and Machine Learn-ing (2nd ed.). MIT Press.

[3] Oates, C.J. and Mukherjee, S.; Mukherjee (2012). Network Inference and Biological Dynamics. To appear in Ann.Appl. Stat. arXiv 1112: 1047. arXiv:1112.1047. Bibcode:2011arXiv1112.1047O.

[4] Guthke, R et al. (2005). Dynamic network reconstruction from gene expression data applied to immune response duringbacterial infection.. Bioinformatics 21 (8): 162634. doi:10.1093/bioinformatics/bti226. PMID 15613398.

[5] Hecker, M et al. (2009). Gene regulatory network inference: Data integration in dynamic models - A review.. Biosystems96 (1): 86103. doi:10.1016/j.biosystems.2008.12.004. PMID 19150482.

[6] van Someren, E et al. (2002). Genetic network modeling.. Pharmacogenomics 3 (4): 507525. doi:10.1517/14622416.3.4.507.PMID 12164774.

[7] Faith, JJ et al. (2007). Large-Scale Mapping and Validation of Escherichia coli Transcriptional Regulation from a Com-pendium of Expression Proles. PLoS Biology 5 (1): 5466. doi:10.1371/journal.pbio.0050008. PMC 1764438. PMID17214507.

3.3. REFERENCES 5

[8] Hayete, B; Gardner, TS; Collins, JJ (2007). Size matters: network inference tackles the genome scale. Molecular SystemsBiology 3 (1): 77. doi:10.1038/msb4100118. PMC 1828748. PMID 17299414.

[9] Oates, C.J. and Mukherjee, S. (2012). Structural inference using nonlinear dynamics. CRiSM Working Paper 12 (7).

Chapter 4

Constraint inference

In constraint satisfaction, constraint inference is a relationship between constraints and their consequences. A set ofconstraints D entails a constraint C if every solution to D is also a solution to C . In other words, if V is a valuationof the variables in the scopes of the constraints in D and all constraints in D are satised by V , then V also satisesthe constraint C .Some operations on constraints produce a new constraint that is a consequence of them. Constraint compositionoperates on a pair of binary constraints ((x; y); R) and ((y; z); S) with a common variable. The composition of suchtwo constraints is the constraint ((x; z); Q) that is satised by every evaluation of the two non-shared variables forwhich there exists a value of the shared variable y such that the evaluation of these three variables satises the twooriginal constraints ((x; y); R) and ((y; z); S) .Constraint projection restricts the eects of a constraint to some of its variables. Given a constraint (t; R) itsprojection to a subset t0 of its variables is the constraint (t0; R0) that is satised by an evaluation if this evaluation canbe extended to the other variables in such a way the original constraint (t; R) is satised.Extended composition is similar in principle to composition, but allows for an arbitrary number of possibly non-binary constraints; the generated constraint is on an arbitrary subset of the variables of the original constraints. Givenconstraints C1; : : : ; Cm and a list A of their variables, the extended composition of them is the constraint (A;R)where an evaluation of A satises this constraint if it can be extended to the other variables so that C1; : : : ; Cm areall satised.

4.1 See also Constraint satisfaction problem

4.2 References Dechter, Rina (2003). Constraint processing. Morgan Kaufmann. ISBN 1-55860-890-7 Apt, Krzysztof (2003). Principles of constraint programming. Cambridge University Press. ISBN 0-521-

82583-0 Marriott, Kim; Peter J. Stuckey (1998). Programming with constraints: An introduction. MIT Press. ISBN

0-262-13341-5

6

Chapter 5

Contradiction

For other uses, see Contradiction (disambiguation).In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs

when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions ofeach other. Illustrating a general tendency in applied logic, Aristotles law of noncontradiction states that One cannotsay of something that it is and that it is not in the same respect and at the same time.By extension, outside of classical logic, one can speak of contradictions between actions when one presumes that theirmotives contradict each other.

5.1 History

By creation of a paradox, Plato's Euthydemus dialogue demonstrates the need for the notion of contradiction. In theensuing dialogue Dionysodorus denies the existence of contradiction, all the while that Socrates is contradictinghim:

". . . I in my astonishment said: What do you mean Dionysodorus? I have often heard, and have beenamazed to hear, this thesis of yours, which is maintained and employed by the disciples of Protagoras andothers before them, and which to me appears to be quite wonderful, and suicidal as well as destructive,and I think that I am most likely to hear the truth about it from you. The dictum is that there is no suchthing as a falsehood; a man must either say what is true or say nothing. Is not that your position?"

Indeed, Dionysodorus agrees that there is no such thing as false opinion . . . there is no such thing as ignorance anddemands of Socrates to Refute me. Socrates responds But how can I refute you, if, as you say, to tell a falsehoodis impossible?".[1]

5.2 Contradiction in formal logic

Note: The symbol ? (falsum) represents an arbitrary contradiction, with the dual tee symbol > usedto denote an arbitrary tautology. Contradiction is sometimes symbolized by Opq", and tautology byVpq". The turnstile symbol, ` is often read as yields or proves.

In classical logic, particularly in propositional and rst-order logic, a proposition ' is a contradiction if and only if' ` ? . Since for contradictory ' it is true that ` '! for all (because? ! ), one may prove any propositionfrom a set of axioms which contains contradictions. This is called the "principle of explosion" or ex falso quodlibet(from falsity, whatever you like).In a complete logic, a formula is contradictory if and only if it is unsatisable.

7

8 CHAPTER 5. CONTRADICTION

This diagram shows the contradictory relationships between categorical propositions in the square of opposition of Aristotelian logic.

5.2.1 Proof by contradiction

Main article: Proof by contradiction

For a proposition ' it is true that ` ' , i. e. that ' is a tautology, i. e. that it is always true, if and only if :' ` ?, i. e. if the negation of ' is a contradiction. Therefore, a proof that :' ` ? also proves that ' is true. The use ofthis fact constitutes the technique of the proof by contradiction, which mathematicians use extensively. This appliesonly in a logic using the excluded middle A _ :A as an axiom.

5.2. CONTRADICTION IN FORMAL LOGIC 9

5.2.2 Symbolic representation

In mathematics, the symbol used to represent a contradiction within a proof varies. Some symbols that may beused to represent a contradiction include , Opq, )( , , , and ; in any symbolism, a contradiction may besubstituted for the truth value "false, as symbolized, for instance, by 0. It is not uncommon to see Q.E.D. or somevariant immediately after a contradiction symbol; this occurs in a proof by contradiction, to indicate that the originalassumption was false and that its negation must therefore be true.

5.2.3 The notion of contradiction in an axiomatic system and a proof of its consistency

A consistency proof requires (i) an axiomatic system (ii) a demonstration that it is not the case that both the formula pand its negation ~p can derived in the system. But by whatever method one goes about it, all consistency proofs wouldseem to necessitate the primitive notion of contradiction; moreover, it seems as if this notion would simultaneouslyhave to be outside the formal system in the denition of tautology.When Emil Post in his 1921 Introduction to a general theory of elementary propositions extended his proof of theconsistency of the propositional calculus (i.e. the logic) beyond that of Principia Mathematica (PM) he observed thatwith respect to a generalized set of postulates (i.e. axioms) he would no longer be able to automatically invoke thenotion of contradiction such a notion might not be contained in the postulates:

The prime requisite of a set of postulates is that it be consistent. Since the ordinary notion of consistencyinvolves that of contradiction, which again involves negation, and since this function does not appear ingeneral as a primitive in [the generalized set of postulates] a new denition must be given.[2]

Posts solution to the problem is described in the demonstration An Example of a Successful Absolute Proof of Con-sistency oered by Ernest Nagel and James R. Newman in their 1958 Gdel's Proof. They too observe a problemwith respect to the notion of contradiction with its usual truth values of truth and falsity. They observe that:

The property of being a tautology has been dened in notions of truth and falsity. Yet these notionsobviously involve a reference to something outside the formula calculus. Therefore, the procedure men-tioned in the text in eect oers an interpretation of the calculus, by supplying a model for the system.This being so, the authors have not done what they promised, namely, 'to dene a property of for-mulas in terms of purely structural features of the formulas themselves. [Indeed] . . . proofs ofconsistency which are based on models, and which argue from the truth of axioms to their consistency,merely shift the problem.[3]

Given some primitive formulas such as PMs primitives S1 V S2 [inclusive OR], ~S (negation) one is forced todene the axioms in terms of these primitive notions. In a thorough manner Post demonstrates in PM, and denes(as do Nagel and Newman, see below), that the property of tautologous as yet to be dened is inherited": ifone begins with a set of tautologous axioms (postulates) and a deduction system that contains substitution and modusponens then a consistent system will yield only tautologous formulas.So what will be the denition of tautologous?Nagel and Newman create two mutually exclusive and exhaustive classes K1 and K2 into which fall (the outcome of)the axioms when their variables e.g. S1 and S2 are assigned from these classes. This also applies to the primitiveformulas. For example: A formula having the form S1 V S2 is placed into class K2 if both S1 and S2 are in K2;otherwise it is placed in K1", and A formula having the form ~S is placed in K2, if S is in K1; otherwise it is placedin K1".[4]

Nagel and Newman can now dene the notion of tautologous: a formula is a tautology if, and only if, it falls in theclass K1 no matter in which of the two classes its elements are placed.[5] Now the property of being tautologousis described without reference to a model or an interpretation.

For example, given a formula such as ~S1 V S2 and an assignment of K1 to S1 and K2 to S2 one canevaluate the formula and place its outcome in one or the other of the classes. The assignment of K1 to~S1 places ~S1 in K2, and now we can see that our assignment causes the formula to fall into class K2.Thus by denition our formula is not a tautology.

10 CHAPTER 5. CONTRADICTION

Post observed that, if the system were inconsistent, a deduction in it (that is, the last formula in a sequence of formulasderived from the tautologies) could ultimately yield S itself. As an assignment to variable S can come from eitherclass K1 or K2, the deduction violates the inheritance characteristic of tautology, i.e. the derivation must yield an(evaluation of a formula) that will fall into class K1. From this, Post was able to derive the following denition ofinconsistency without the use of the notion of contradiction:

Denition. A system will be said to be inconsistent if it yields the assertion of the unmodied variable p[S in the Newman and Nagel examples].

In other words, the notion of contradiction can be dispensed when constructing a proof of consistency; what replacesit is the notion of mutually exclusive and exhaustive classes. More interestingly, an axiomatic system need notinclude the notion of contradiction.

5.3 Contradictions and philosophyAdherents of the epistemological theory of coherentism typically claim that as a necessary condition of the justi-cation of a belief, that belief must form a part of a logically non-contradictory (consistent) system of beliefs. Somedialetheists, including Graham Priest, have argued that coherence may not require consistency.[6]

5.3.1 Pragmatic contradictionsA pragmatic contradiction occurs when the very statement of the argument contradicts the claims it purports. Aninconsistency arises, in this case, because the act of utterance, rather than the content of what is said, underminesits conclusion.[7] For examples, arguably, Nietzsche's statement that one should not obey others, or Moores paradox.Within the analytic tradition, these are seen as self-refuting statements and performative contradictions. Other tra-ditions may read them more like zen koans, in which the author purposes makes a contradiction using the traditionalmeaning, but then implies a new meaning of the word which does not contradict the statement.

5.3.2 Dialectical materialismIn dialectical materialism, contradiction, as derived by Karl Marx from Hegelianism, usually refers to an oppositioninherently existing within one realm, one unied force or object. This contradiction, as opposed to metaphysicalthinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not can-celling each other out, but actually dening each others existence. According to Marxist theory, such a contradictioncan be found, for example, in the fact that:

(a) enormous wealth and productive powers coexist alongside:(b) extreme poverty and misery;(c) the existence of (a) being contrary to the existence of (b).

Hegelian and Marxist theory stipulates that the dialectic nature of history will lead to the sublation, or synthesis, of itscontradictions. Marx therefore postulated that history would logically make capitalism evolve into a socialist societywhere the means of production would equally serve the exploited and suering class of society, thus resolving theprior contradiction between (a) and (b).Mao Zedongs philosophical essay furthered Marx and Lenins thesis and suggested that all existence is the result ofcontradiction.[8]

5.4 Contradiction outside formal logicColloquial usage can label actions and/or statements as contradicting each other when due (or perceived as due) topresuppositions which are contradictory in the logical sense.Proof by contradiction is used in mathematics to construct proofs.

5.5. SEE ALSO 11

Contradiction on Graham's Hierarchy of Disagreement

5.5 See also

Auto-antonym

Contrary (logic)

Double standard

Doublethink

False (logic)

Irony

Logical truth

Oxymoron

Paraconsistent logic

Paradox

Tautology (logic) (for symbolism of logical truth)

Truth

TRIZ

12 CHAPTER 5. CONTRADICTION

5.6 Footnotes[1] Dialog Euthydemus from The Dialogs of Plato translated by Benjamin Jowett appearing in: BK 7 Plato: Robert Maynard

Hutchins, editor in chief, 1952, Great Books of the Western World, Encyclopdia Britannica, Inc., Chicago.

[2] Post 1921 Introduction to a general theory of elementary propositions in van Heijenoort 1967:272.

[3] boldface italics added, Nagel and Newman:109-110.

[4] Nagel and Newman:110-111

[5] Nagel and Newman:111

[6] In Contradiction: A Study of the Transconsistent By Graham Priest

[7] Stoljar, Daniel (2006). Ignorance and Imagination. Oxford University Press - U.S. p. 87. ISBN 0-19-530658-9.

[8] ON CONTRADICTION

5.7 References Jzef Maria Bocheski 1960 Prcis of Mathematical Logic, translated from the French and German editions

by Otto Bird, D. Reidel, Dordrecht, South Holland.

Jean van Heijenoort 1967 From Frege to Gdel: A Source Book in Mathematical Logic 1879-1931, HarvardUniversity Press, Cambridge, MA, ISBN 0-674-32449-8 (pbk.)

Ernest Nagel and James R. Newman 1958 Gdels Proof, New York University Press, Card Catalog Number:58-5610.

5.8 External links Hazewinkel, Michiel, ed. (2001), Contradiction (inconsistency)", Encyclopedia of Mathematics, Springer,

ISBN 978-1-55608-010-4

Hazewinkel, Michiel, ed. (2001), Contradiction, law of, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Contradiction entry by Laurence R. Horn in the Stanford Encyclopedia of Philosophy

Chapter 6

Contraposition (traditional logic)

In traditional logic, contraposition is a form of immediate inference in which from a given proposition anotheris inferred having for its subject the contradictory of the original predicate, and in some cases involving a changeof quality (armation or negation).[1] For its symbolic expression in modern logic see the rule of transposition.Contraposition also has distinctive applications in its philosophical application distinct from the other traditionalinference processes of conversion and obversion where equivocation varies with dierent proposition types.

6.1 Traditional logicIn traditional logic the process of contraposition is a schema composed of several steps of inference involvingcategorical propositions and classes.[2] A categorical proposition contains a subject and predicate where the exis-tential impact of the copula implies the proposition as referring to a class with at least one member, in contrast tothe conditional form of hypothetical or materially implicative propositions, which are compounds of other proposi-tions, e.g. If P, then Q, where P and Q are both propositions, and their existential impact is dependent upon furtherpropositions where in quantication existence is instantiated (existential instantiation).Conversion by contraposition is the simultaneous interchange and negation of the subject and predicate, and is validonly for the type A and type O propositions of Aristotelian logic, with considerations for the validity an E typeproposition with limitations and changes in quantity. This is considered full contraposition. Since in the process ofcontraposition the obverse can be obtained in all four types of traditional propositions, yielding propositions withthe contradictory of the original predicate, contraposition is rst obtained by converting the obvert of the originalproposition. Thus, partial contraposition can be obtained conditionally in an E type proposition with a changein quantity. Because nothing is said in the denition of contraposition with regard to the predicate of the inferredproposition, it can be either the original subject, or its contradictory, resulting in two contrapositives which are theobverts of one another in the A, O, and E type propositions.[3]

By example: from an original, 'A' type categorical proposition,

All residents are voters,

which presupposes that all classes have members and the existential import presumed in the form of categoricalpropositions, one can derive rst by obversion the 'E' type proposition,

No residents are non-voters.

The contrapositive of the original proposition is then derived by conversion to another 'E' type proposition,

No non-voters are residents.

The process is completed by further obversion resulting in the 'A' type proposition that is the obverted contrapositiveof the original proposition,

13

14 CHAPTER 6. CONTRAPOSITION (TRADITIONAL LOGIC)

All non-voters are non-residents.

The schema of contraposition:[4]

Notice that contraposition is a valid form of immediate inference only when applied to A and O propositions. Itis not valid for I propositions, where the obverse is an O proposition which has no converse. The contrapositionof the E proposition is valid only with limitations (per accidens). This is because the obverse of the E propositionis an A proposition which cannot be validly converted except by limitation, that is, contraposition plus a change inthe quantity of the proposition from universal to particular.Also, notice that contraposition is a method of inference which may require the use of other rules of inference.The contrapositive is the product of the method of contraposition, with dierent outcomes depending upon whetherthe contraposition is full, or partial. The successive applications of conversion and obversion within the process ofcontraposition may be given by a variety of names.The process of the logical equivalence of a statement and its contrapositive as dened in traditional class logic is notone of the axioms of propositional logic. In traditional logic there is more than one contrapositive inferred from eachoriginal statement. In regard to the A proposition this is circumvented in the symbolism of modern logic by therule of transposition, or the law of contraposition. In its technical usage within the eld of philosophic logic, the termcontraposition may be limited by logicians (e.g. Irving Copi, Susan Stebbing) to traditional logic and categoricalpropositions. In this sense the use the term contraposition is usually referred to by transposition when applied tohypothetical propositions or material implications.

6.2 See also

6.3 Notes[1] Brody, Bobuch A. Glossary of Logical Terms. Encyclopedia of Philosophy. Vol. 5-6, p. 61. Macmillan, 1973. Also,

Stebbing, L. Susan. AModern Introduction to Logic. Seventh edition, p.65-66. Harper, 1961, and Irving Copis Introductionto Logic, p. 141, Macmillan, 1953. All sources give virtually identical denitions.

[2] Irving Copis Introduction to Logic, pp. 123-157, Macmillan, 1953.

[3] Brody, p. 61. Macmillan, 1973. Also, Stebbing, p.65-66, Harper, 1961, and Copi, p. 141-143, Macmillan, 1953.

[4] Stebbing, L. Susan. A Modern Introduction to Logic. Seventh edition, p. 66. Harper, 1961.

6.4 References Blumberg, Albert E. Logic, Modern. Encyclopedia of Philosophy, Vol.5, Macmillan, 1973. Brody, Bobuch A. Glossary of Logical Terms. Encyclopedia of Philosophy. Vol. 5-6, p. 61. Macmillan,

1973. Copi, Irving. Introduction to Logic. MacMillan, 1953. Copi, Irving. Symbolic Logic. MacMillan, 1979, fth edition. Prior, A.N. Logic, Traditional. Encyclopedia of Philosophy, Vol.5, Macmillan, 1973. Stebbing, Susan. A Modern Introduction to Logic. Cromwell Company, 1931.

Chapter 7

Contrary (logic)

Contrary is the relationship between two propositions when they cannot both be true (although both may be false).Thus, we can make an immediate inference that if one is true, the other must be false.The law holds for the A and E propositions of the Aristotelian square of opposition. For example, the A proposition'every man is honest' and the E proposition 'no man is honest' cannot both be true at the same time, since no one canbe honest and not honest at the same time. But both can be false, if some men are honest, and some men are not. Forif some men are honest, the proposition 'no man is honest' is false. And if some men are not honest, the proposition'every man is honest' is false also.

7.1 See also Contradiction

7.2 External links Article on contradiction in the Stanford Encyclopedia of Philosophy

15

16 CHAPTER 7. CONTRARY (LOGIC)

This diagram shows the contrary relationship between categorical propositions in the square of opposition of Aristotelian logic.

Chapter 8

Converse (logic)

In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. For theimplication P Q, the converse is Q P. For the categorical proposition All S is P, the converse is All P is S.In neither case does the converse necessarily follow from the original statement.[1] The categorical converse of astatement is contrasted with the contrapositive and the obverse.

8.1 Implicational converseLet S be a statement of the form P implies Q (P Q). Then the converse of S is the statement Q implies P (Q P).In general, the verity of S says nothing about the verity of its converse, unless the antecedent P and the consequent Qare logically equivalent.For example, consider the true statement If I am a human, then I am mortal. The converse of that statement is IfI am mortal, then I am a human, which is not necessarily true.On the other hand, the converse of a statement with mutually inclusive terms remains true, given the truth of theoriginal proposition. Thus, the statement If I am a bachelor, then I am an unmarried man is logically equivalent toIf I am an unmarried man, then I am a bachelor.A truth table makes it clear that S and the converse of S are not logically equivalent unless both terms imply eachother:Going from a statement to its converse is the fallacy of arming the consequent. However, if the statement S and itsconverse are equivalent (i.e. if P is true if and only if Q is also true), then arming the consequent will be valid.

8.1.1 Converse of a theorem

In mathematics, the converse of a theorem of the form P Q will be Q P. The converse may or may not be true.If true, the proof may be dicult. For example, the Four-vertex theorem was proved in 1912, but its converse onlyin 1998.In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken asestablishing context. That is, the converse of Given P, if Q then R will be Given P, if R then Q. For example, thePythagorean theorem can be stated as:

Given a triangle with sides of length a, b, and c, if the angle opposite the side of length c is a rightangle, then a2 + b2 = c2.

The converse, which also appears in Euclids Elements (Book I, Proposition 48), can be stated as:

Given a triangle with sides of length a, b, and c, if a2 + b2 = c2, then the angle opposite the side oflength c is a right angle.

17

18 CHAPTER 8. CONVERSE (LOGIC)

8.2 Categorical converseIn traditional logic, the process of going from All S are P to its converse All P are S is called conversion. In the wordsof Asa Mahan, The original proposition is called the exposita; when converted, it is denominated the converse.Conversion is valid when, and only when, nothing is asserted in the converse which is not armed or implied in theexposita.[2] The exposita is more usually called the convertend. In its simple form, conversion is valid only forE and I propositions:[3]

The validity of simple conversion only for E and I propositions can be expressed by the restriction that No termmust be distributed in the converse which is not distributed in the convertend.[4] For E propositions, both subjectand predicate are distributed, while for I propositions, neither is.For A propositions, the subject is distributed while the predicate is not, and so the inference from an A statement toits converse is not valid. As an example, for the A proposition All cats are mammals, the converse All mammalsare cats is obviously false. However, the weaker statement Some mammals are cats is true. Logicians deneconversion per accidens to be the process of producing this weaker statement. Inference from a statement to itsconverse per accidens is generally valid. However, as with syllogisms, this switch from the universal to the particularcauses problems with empty categories: All unicorns are mammals is often taken as true, while the converse peraccidens Some mammals are unicorns is clearly false.In rst-order predicate calculus, All S are P can be represented as 8x:S(x) ! P (x) .[5] It is therefore clear that thecategorical converse is closely related to the implicational converse, and that S and P cannot be swapped in All S areP.

8.3 See also

8.4 References[1] Robert Audi, ed. (1999), The Cambridge Dictionary of Philosophy, 2nd ed., Cambridge University Press: converse.

[2] Asa Mahan (1857), The Science of Logic: or, An Analysis of the Laws of Thought, p. 82.

[3] William Thomas Parry and Edward A. Hacker (1991), Aristotelian Logic, SUNY Press, p. 207.

[4] James H. Hyslop (1892), The Elements of Logic, C. Scribners sons, p. 156.

[5] Gordon Hunnings (1988), The World and Language in Wittgensteins Philosophy, SUNY Press, p. 42.

8.5 Further reading Aristotle. Organon. Copi, Irving. Introduction to Logic. MacMillan, 1953. Copi, Irving. Symbolic Logic. MacMillan, 1979, fth edition. Stebbing, Susan. A Modern Introduction to Logic. Cromwell Company, 1931.

Chapter 9

Correspondent inference theory

Correspondent inference theory is a psychological theory proposed by Edward E. Jones and Keith Davis (1965)that systematically accounts for a perceivers inferences about what an actor was trying to achieve by a particularaction. [1] The purpose of this theory is to explain why people make internal or external attributions. People comparetheir actions with alternative actions to evaluate the choices that they have made, and by looking at various factorsthey can decide if their behaviour was caused by an internal disposition. The covariation model is used within this,more specically that the degree in which one attributes behavior to the person as opposed to the situation. Thesefactors are; does the person have a choice in the partaking in the action, is their behavior expected by their social role,and is their behavior consequence of their normal behavior?

9.1 Attributing intentionThe problem of accurately dening intentions is a dicult one. For every observed act, there are a multitude ofpossible motivations. If a person buys someone a drink in the pub, he may be trying to curry favour, his friend mayhave bought him a drink earlier, or he may be doing a favour for a friend with no cash.The work done by Jones and Davis only deals with how people make attributions to the person; they do not deal withhow people make attributions about situational or external causes.Jones and Davis make the assumption that, in order to infer that any eects of an action were intended, the perceivermust believe that (1) the actor knew the consequences of the actions (e.g., the technician who pushed that button atChernobyl did not know the consequences of that action), (2) the actor had the ability to perform the action (couldLee Harvey Oswald really have shot John Kennedy?), and (3) the actor had the intention to perform the action.

9.2 Non-Common eectsThe consequences of a chosen action must be compared with the consequences of possible alternative actions. Thefewer eects the possible choices have in common, the more condent one can be in inferring a correspondent dis-position. Or, put another way, the more distinctive the consequences of a choice, the more condently you can inferintention and disposition.Suppose you are planning to go on a postgraduate course, and you short-list two colleges - University College Londonand the London School of Economics. You choose UCL rather than the LSE. What can the social perceiver learnfrom this? First there are a lot of common eects - urban environment, same distance from home, same examsystem, similar academic reputation, etc. These common eects do not provide the perceiver with any clues aboutyour motivation. But if the perceiver believes that UCL has better sports facilities, or easier access to the UniversityLibrary then these non-common or unique eects which can provide a clue to your motivation. But, suppose you hadshort-listed UCL and University of Essex and you choose UCL. Now the perceiver is faced with a number of non-common eects; size of city; distance from home; academic reputation; exam system. The perceiver would then bemuch less condent about inferring a particular intention or disposition when there are a lot of non-common eects.The fewer the non-common eects, the more certain the attribution of intent.

19

20 CHAPTER 9. CORRESPONDENT INFERENCE THEORY

9.3 Low-Social desirability

People usually intend socially desirable outcomes, hence socially desirable outcomes are not informative about apersons intention or disposition. The most that you can infer is that the person is normal - which is not sayinganything very much. But socially undesirable actions are more informative about intentions & dispositions. Supposeyou asked a friend for a loan of 1 and it was given (a socially desirable action) - the perceiver couldn't say a great dealabout your friends kindness or helpfulness because most people would have done the same thing. If, on the otherhand, the friend refused to lend you the money (a socially undesirable action), the perceiver might well feel that yourfriend is rather stingy, or even miserly.In fact, social desirability - although an important inuence on behaviour - is really only a special case of the moregeneral principle that behaviour which deviates from the normal, usual, or expected is more informative about apersons disposition than behaviour that conforms to the normal, usual, or expected. So, for example, when peopledo not conform to group pressure we can be more certain that they truly believe the views they express than peoplewho conform to the group. Similarly, when people in a particular social role (e.g. doctor, teacher, salesperson, etc.)behave in ways that are not in keeping with the role demands, we can be more certain about what they are really likethan when people behave in role.

9.4 Expectancies

Only behaviours that disconrm expectancies are truly informative about an actor. There are two types of expectancy.Category-based expectancies are those derived from our knowledge about particular types or groups of people. Forexample, if you were surprised to hear a wealthy businessman extolling the virtues of socialism, your surprise wouldrest on the expectation that businessmen (a category of people) are not usually socialist.Target-based expectancies derive from knowledge about a particular person. To know that a person is a supporter ofMargaret Thatcher sets up certain expectations and associations about their beliefs and character.

9.5 Choice

Another factor in inferring a disposition from an action is whether the behaviour of the actor is constrained by situ-ational forces or whether it occurs from the actors choice. If you were assigned to argue a position in a classroomdebate (e.g. for or against Neoliberalism), it would be unwise of your audience to infer that your statements in thedebate reect your true beliefs - because you did not choose to argue that particular side of the issue. If, however,you had chosen to argue one side of the issue, then it would be appropriate for the audience to conclude that yourstatements reect your true beliefs.Although choice ought to have an important eect on whether or not people make correspondent inferences, researchshows that people do not take choice suciently into account when judging another persons attributes or attitudes.There is a tendency for perceivers to assume that when an actor engages in an activity, such as stating a point of viewor attitude, the statements made are indicative of the actors true beliefs, even when there may be clear situationalforces aecting the behaviour. In fact, earlier, psychologists had foreseen that something like this would occur; theythought that the actor-act relation was so strong - like a perceptual Gestalt - that people would tend to over-attributeactions to the actor even when there are powerful external forces on the actor that could account for the behaviour.

9.6 See also

Edward E. Jones

Attribution theory

Revealed preferences

9.7. REFERENCES 21

9.7 References[1] Berkowitz, Leonard (1965). Advances in Experimental Social Psychology Vol 2, p.222. Academic Press, . ISBN 978-0-12-

015202-5.

9.8 External links Gilbert, D. T. (1998). Speeding with Ned: A personal view of the correspondence bias. In J. M. Darley &

J. Cooper (Eds.), Attribution and social interaction: The legacy of E. E. Jones. Washington, DC: APA Press.PDF.

Chapter 10

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 rst calculus ofdeep inference was proposed by Kurt Schtte,[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.

10.1 Notes[1] Kurt Schtte. Proof Theory. Springer-Verlag, 1977.

10.2 Further reading Kai Brnnler, Deep Inference and Symmetry in Classical Proofs (Ph.D. thesis 2004) , also published in book

form 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.

22

Chapter 11

Dictum de omni et nullo

In Aristotelean logic, dictum de omni et nullo (Latin: the maxim of all and none) is the principle that whatever isarmed or denied of a whole kind K may be armed or denied (respectively) of any subkind of K. This principle isfundamental to syllogistic logic in the sense that all valid syllogistic argument forms are reducible to applications ofthe two constituent principles dictum de omni and dictum de nullo.[1]

Dictum de omni (sometimes misinterpreted as universal instantiation) is the principle that whatever is universallyarmed of a kind is armable as well for any subkind of that kind.Example:

(1) Dogs are mammals.(2) Mammals have livers.Therefore (3) dogs have livers.

Premise (1) states that dog is a subkind of the kind mammal.Premise (2) is a (universal armative) claim about the kind mammal.Statement (3) concludes that what is true of the kind mammal is true of the subkind dog.Dictum de nullo is the related principle that whatever is denied of a kind is likewise denied of any subkind of thatkind.Example:

(1) Dogs are mammals.(4) Mammals do not have gills.Therefore (5) dogs do not have gills.

Premise (1) states that dog is a subkind of the kind mammal.Premise (4) is a (universal negative) claim about the kind mammal.Statement (5) concludes that what is denied of the kind mammal is denied of the subkind dog.Each of these two principles is an instance of a valid argument form known as universal hypothetical syllogism in rst-order predicate logic. In Aristotelean syllogistic, they correspond respectively to the two argument forms, Barbaraand Celarent.

11.1 See also Aristotle Syllogism Term logic Class (philosophy) Class (set theory)

23

24 CHAPTER 11. DICTUM DE OMNI ET NULLO

Natural kind Type (metaphysics) Downward entailing Monotonic function

11.2 References Aristotle, Prior Analytics, 24b, 28-30.

11.3 Notes[1] John Stuart Mill (15 January 2001). System of Logic Ratiocinative and Inductive: Being a Connected View of the Principles

of Evidence and the Methods of Scientic Investigation. Elibron.com. p. 114. ISBN 978-1-4021-8157-3. Retrieved 6March 2011.

11.4 External links Logical Form (Stanford Encyclopedia of Philosophy)

Chapter 12

Downward entailing

In linguistic semantics, a downward entailing (DE) propositional operator is one that denotes a monotone decreasingfunction. A downward entailing operator reverses the relation of semantic strength among expressions. An expressionlike run fast is semantically stronger than the expression run since run fast is true of fewer things than the latter.Thus the proposition John ran fast entails the proposition John ran.Examples of DE contexts include not, nobody, few people, at most two boys. They reverse the entailmentrelation of sentences formed with the predicates run fast and run, for example. The proposition Nobody ranentails that Nobody ran fast. The proposition At most two boys ran entails that At most two boys ran fast.Conversely, an upward entailing operator is one that preserves the relation of semantic strength among a set ofexpressions (for example, more). A context that is neither downward nor upward entailing is non-monotone, suchas exactly.Ladusaw (1980) proposed that downward entailment is the property that licenses polarity items. Indeed, Nobodysaw anything is downward entailing and admits the negative polarity item anything, while * I saw anything isungrammatical (the upward entailing context does not license such a polarity item). This approach explains many butnot all typical cases of polarity item sensitivity. Subsequent attempts to describe the behavior of polarity items relyon a broader notion of nonveridicality.

12.1 Strawson-DEDownward entailment does not explain the licensing of any in certain contexts such as with only:

Only John ate any vegetables for breakfast.

This is not a downward entailing context because the above proposition does not entail Only John ate kale forbreakfast (John may have eaten spinach, for example).Von Fintel (1999) claims that although only does not exhibit the classical DE pattern, it can be shown to be DE ina special way. He denes a notion of Strawson-DE for expressions that come with presuppositions. The reasoningscheme is as follows:

1. P Q

2. [[ only John ]] (P) is dened.

3. [[ only John ]] (Q) is true.

4. Therefore, [[ only John ]] (P) is true.

Here, (2) is the intended presupposition. For example:

1. Kale is a vegetable.

25

26 CHAPTER 12. DOWNWARD ENTAILING

2. Somebody ate kale for breakfast.

3. Only John ate any vegetables for breakfast.4. Therefore, only John ate kale for breakfast.

Hence only is a Strawson-DE and therefore licenses any.Giannakidou (2002) argues that Strawson-DE allows not just the presupposition of the evaluated sentence but just anyarbitrary proposition to count as relevant. This results in over-generalization that validates the use if any' in contextswhere it is, in fact, ungrammatical, such as clefts, preposed exhaustive focus, and each/both:

* It was John who talked to anybody.* JOHN talked to anybody.* Each student who saw anything reported to the Dean.* Both students who saw anything reported to the Dean.

12.2 See also Entailment (pragmatics) Veridicality Polarity item

12.3 References Ladusaw, William (1980). Polarity Sensitivity as Inherent Scope Relations. Garland, NY. Von Fintel, Kai (1999). NPI-Licensing, Strawson-Entailment, and Context-Dependency. Journal of Seman-

tics (16): 97148.

Giannakidou, Anastasia (2002). Licensing and sensitivity in polarity items: from downward entailment tononveridicality. In Maria Andronis; Anne Pycha; Keiko Yoshimura. CLS 38: Papers from the 38th AnnualMeeting of the Chicago Linguistic Society, Parasession on Polarity and Negation. Retrieved 2011-12-15.

Chapter 13

Grammar induction

Grammar induction, also known as grammatical inference or syntactic pattern recognition, refers to the process inmachine learning of learning a formal grammar (usually as a collection of re-write rules or productions or alternativelyas a nite state machine or automaton of some kind) from a set of observations, thus constructing a model whichaccounts for the characteristics of the observed objects. More generally, grammatical inference is that branch ofmachine learning where the instance space consists of discrete combinatorial objects such as strings, trees and graphs.There is now a rich literature on learning dierent types of grammar and automata, under various dierent learningmodels and using various dierent methodologies.

13.1 Grammar Classes

Grammatical inference has often been very focused on the problem of learning nite state machines of various types(see the article Induction of regular languages for details on these approaches), since there have been ecient algo-rithms for this problem since the 1980s.More recently these approaches have been extended to the problem of inference of context-free grammars and richerformalisms, such as multiple context-free grammars and parallel multiple context-free grammars. Other classes ofgrammars for which grammatical inference has been studied are contextual grammars, and pattern languages.

13.2 Learning Models

The simplest form of learning is where the learning algorithm merely receives a set of examples drawn from thelanguage in question, but other learning models have been studied. One frequently studied alternative is the casewhere the learner can ask membership queries as in the exact query learning model or minimally adequate teachermodel introduced by Angluin.

13.3 Methodologies

There are a wide variety of methods for grammatical inference. Two of the classic sources are Fu (1977) and Fu(1982). Duda, Hart & Stork (2001) also devote a brief section to the problem, and cite a number of references. Thebasic trial-and-error method they present is discussed below. For approaches to infer subclasses of regular languagesin particular, see Induction of regular languages. A more recent textbook is de la Higuera (2010) [1] which coversthe theory of grammatical inference of regular languages and nite state automata. D'Ulizia, Ferri and Grifoni [2]provide a survey that explores grammatical inference methods for natural languages.

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13.3.1 Grammatical inference by trial-and-error

The method proposed in Section 8.7 of Duda, Hart & Stork (2001) suggests successively guessing grammar rules(productions) and testing them against positive and negative observations. The rule set is expanded so as to be able togenerate each positive example, but if a given rule set also generates a negative example, it must be discarded. Thisparticular approach can be characterized as hypothesis testing and bears some similarity to Mitchels version spacealgorithm. The Duda, Hart & Stork (2001) text provide a simple example which nicely illustrates the process, butthe feasibility of such an unguided trial-and-error approach for more substantial problems is dubious.

13.3.2 Grammatical inference by genetic algorithms

Grammatical Induction using evolutionary algorithms is the process of evolving a representation of the grammar ofa target language through some evolutionary process. Formal grammars can easily be represented as a tree structureof production rules that can be subjected to evolutionary operators. Algorithms of this sort stem from the geneticprogramming paradigm pioneered by John Koza. Other early work on simple formal languages used the binary stringrepresentation of genetic algorithms, but the inherently hierarchical structure of grammars couched in the EBNFlanguage made trees a more exible approach.Koza represented Lisp programs as trees. He was able to nd analogues to the genetic operators within the stan-dard set of tree operators. For example, swapping sub-trees is equivalent to the corresponding process of geneticcrossover, where sub-strings of a genetic code are transplanted into an individual of the next generation. Fitness ismeasured by scoring the output from the functions of the lisp code. Similar analogues between the tree structured lisprepresentation and the representation of grammars as trees, made the application of genetic programming techniquespossible for grammar induction.In the case of Grammar Induction, the transplantation of sub-trees corresponds to the swapping of production rulesthat enable the parsing of phrases from some language. The tness operator for the grammar is based upon somemeasure of how well it performed in parsing some group of sentences from the target language. In a tree representationof a grammar, a terminal symbol of a production rule corresponds to a leaf node of the tree. Its parent nodescorresponds to a non-terminal symbol (e.g. a noun phrase or a verb phrase) in the rule set. Ultimately, the root nodemight correspond to a sentence non-terminal.

13.3.3 Grammatical inference by greedy algorithms

Like all greedy algorithms, greedy grammar inference algorithms make, in iterative manner, decisions that seem tobe the best at that stage. These made decisions deal usually with things like the making of a new or the removing ofthe existing rules, the choosing of the applied rule or the merging of some existing rules. Because there are severalways to dene 'the stage' and 'the best', there are also several greedy grammar inference algorithms.These context-free grammar generating algorithms make the decision after every read symbol:

Lempel-Ziv-Welch algorithm creates a context-free grammar in a deterministic way such that it is necessaryto store only the start rule of the generated grammar.

Sequitur and its modications.

These context-free grammar generating algorithms rst read the whole given symbol-sequence and then start to makedecisions:

Byte pair encoding and its optimizations.

13.3.4 Distributional Learning

A more recent approach is based on Distributional Learning. Algorithms using these approaches have been appliedto learning context-free grammars and mildly context-sensitive languages and have been proven to be correct andecient for large subclasses of these grammars.[3]

13.4. APPLICATIONS 29

13.3.5 Learning of Pattern languagesAngluin denes a pattern to be a string of constant symbols from and variable symbols from a disjoint set. Thelanguage of such a pattern is the set of all its nonempty ground instances i.e. all strings resulting from consistentreplacement of its variable symbols by nonempty strings of constant symbols.[note 1] A pattern is called descriptivefor a nite input set of strings if its language is minimal (with respect to set inclusion) among all pattern languagessubsuming the input set.Angluin gives a polynomial algorithm to compute, for a given input string set, all descriptive patterns in one variablex.[note 2] To this end, she builds an automaton representing all possibly relevant patterns; using sophisticated argumentsabout word lengths, which rely on x being the only variable, the state count can be drastically reduced.[4]

Erlebach et al. give a more ecient version of Angluins pattern learning algorithm, as well as a parallelized version.[5]

Arimura et al. show that a language class obtained from limited unions of patterns can be learned in polynomialtime.[6]

13.3.6 Pattern theoryPattern theory, formulated by Ulf Grenander,[7] is a mathematical formalism to describe knowledge of the world aspatterns. It diers from other approaches to articial intelligence in that it does not begin by prescribing algorithmsand machinery to recognize and classify patterns; rather, it prescribes a vocabulary to articulate and recast the patternconcepts in precise language.In addition to the new algebraic vocabulary, its statistical approach was novel in its aim to:

Identify the hidden variables of a data set using real world data rather than articial stimuli, which was com-monplace at the time.

Formulate prior distributions for hidden variables and models for the observed variables that form the verticesof a Gibbs-like graph.

Study the randomness and variability of these graphs. Create the basic classes of stochastic models applied by listing the deformations of the patterns. Synthesize (sample) from the models, not just analyze signals with it.

Broad in its mathematical coverage, Pattern Theory spans algebra and statistics, as well as local topological and globalentropic properties.

13.4 ApplicationsThe principle of grammar induction has been applied to other aspects of natural language processing, and have beenapplied (among many other problems) to morpheme analysis, and even place name derivations. Grammar inductionhas also been used for lossless data compression and statistical inference via MML and MDL principles.

13.5 See also Articial grammar learning Syntactic pattern recognition Inductive inference Straight-line grammar Kolmogorov complexity Automatic distillation of structure Inductive programming

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13.6 Notes[1] The language of a pattern with at least two occurrences of the same variable is not regular due to the pumping lemma.

[2] x may occur several times, but no other variable y may occur

13.7 References[1] de la Higuera, Colin (2010). Grammatical Inference: Learning Automata and Grammars. Cambridge: Cambridge Univer-

sity Press.

[2] DUlizia, A., Ferri, F., Grifoni, P. (2011) A Survey of Grammatical Inference Methods for Natural Language Learning,Articial Intelligence Review, Vol. 36, No. 1, pp. 1-27.

[3] Clark and Eyraud (2007) Journal of Machine Learning Research, Ryo Yoshinaka (2011) Theoretical Computer Science

[4] Dana Angluin (1980). Finding Patterns Common to a Set of Strings (PDF). Journal of Computer and System Sciences21: 4662. doi:10.1016/0022-0000(80)90