The Logic of Chance/Contents
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TABLE OF CONTENTS^{[1]}.
PART I. | ||
PHYSICAL FOUNDATIONS OF THE SCIENCE OF PROBABILITY. Chh. I—V. | ||
CHAPTER I. | ||
THE SERIES OF PROBABILITY. | ||
§§1, 2. | Distinction between the proportional propositions of Probability, and the propositions of Logic. | |
3, 4. | The former are best regarded as presenting a series of individuals, | |
5. | Which may occur in any order of time, | |
6, 7. | And which present themselves in groups. | |
8. | Comparison of the above with the ordinary phraseology. | |
9, 10. | These series ultimately fluctuate, | |
11. | Especially in the case of moral and social phenomena, | |
12. | Though in the case of games of chance the fluctuation is practically inappreciable. | |
13, 14. | In this latter case only can rigorous inferences be drawn. | |
15, 16. | The Petersburg Problem. | |
CHAPTER II. | ||
ARRANGEMENT AND FORMATION OF THE SERIES. LAWS OF ERROR. | ||
§§1, 2. | Indication of the nature of a Law of Error or Divergence. | |
3. | Is there necessarily but one such law, | |
4. | Applicable to widely distinct classes of things? | |
5, 6. | This cannot be proved directly by statistics; | |
7, 8. | Which in certain cases show actual asymmetry. | |
9, 10. | Nor deductively; | |
11. | Nor by the Method of Least Squares. | |
12. | Distinction between Laws of Error and the Method of Least Squares. | |
13. | Supposed existence of types. | |
14—16. | Homogeneous and heterogeneous classes. | |
17, 18. | The type in the case of human stature, &c. | |
19, 20. | The type in mental characteristics. | |
21, 22. | Applications of the foregoing principles and results. | |
CHAPTER III. | ||
ORIGIN OR PROCESS OF CAUSATION OF THE SERIES. | ||
§1. | The causes consist of (1) 'objects,' | |
2, 3. | Which may or may not be distinguishable into natural kinds, | |
4—6. | And (2) 'agencies.' | |
7. | Requisites demanded in the above: | |
8, 9. | Consequences of their absence. | |
10. | Where are the required causes found? | |
11, 12. | Not in the direct results of human will. | |
13—15. | Examination of apparent exceptions. | |
16—18. | Further analysis of some natural causes. | |
CHAPTER IV. | ||
HOW TO DISCOVER AND PROVE THE SERIES. | ||
§1. | The data of Probability are established by experience; | |
2. | Though in practice most problems are solved deductively. | |
3—7. | Mechanical instance to show the inadequacy of any à priori proof. | |
8. | The Principle of Sufficient Reason inapplicable. | |
9. | Evidence of actual experience. | |
10, 11. | Further examination of the causes. | |
12, 13. | Distinction between the succession of physical events and the Doctrine of Combinations. | |
14, 15. | Remarks of Laplace on this subject. | |
16. | Bernoulli's Theorem; | |
17, 18. | Its inapplicability to social phenomena. | |
19. | Summation of preceding results. | |
CHAPTER V. | ||
THE CONCEPTION OF RANDOMNESS. | ||
§ 1. | General Indication. | |
2—5. | The postulate of ultimate uniform distribution at one stage or another. | |
6. | This area of distribution must be finite: | |
7, 8. | Geometrical illustrations in support: | |
9. | Can we conceive any exception here? | |
10, 11. | Experimental determination of the random character when the events are many: | |
12. | Corresponding determination when they are few. | |
13, 14. | Illustration from the constant π. | |
15, 16. | Conception of a line drawn at random. | |
17. | Graphical illustration. | |
PART II. | ||
LOGICAL SUPERSTRUCTURE ON THE ABOVE PHYSICAL FOUNDATIONS. Chh. VI—XIV. | ||
CHAPTER VI. | ||
MEASUREMENT OF BELIEF. | ||
§§ 1, 2. | Preliminary remarks. | |
3, 4. | Are we accurately conscious of gradations of belief? | |
5. | Probability only concerned with part of this enquiry. | |
6. | Difficulty of measuring our belief; | |
7. | Owing to intrusion of emotions, | |
8. | And complexity of the evidence. | |
9. | And when measured, is it always correct? | |
10, 11. | Distinction between logical and psychological views. | |
12—16. | Analogy of Formal Logic fails to show that we can thus detach and measure our belief. | |
17. | Apparent evidence of popular language to the contrary. | |
18. | How is full belief justified in inductive enquiry? | |
19—23. | Attempt to show how partial belief may be similarly justified. | |
24—28. | Extension of this explanation to cases which cannot be repeated in experience. | |
29. | Can other emotions besides belief be thus measured? | |
30. | Errors thus arising in connection with the Petersburg Problem. | |
31. 32. | The emotion of surprise is a partial exception. | |
33, 34. | Objective and subjective phraseology. | |
35. | The definition of probability, | |
36. | Introduces the notion of a 'limit', | |
37. | And implies, vaguely, some degree of belief. | |
CHAPTER VII. | ||
THE RULES OF INFERENCE IN PROBABILITY. | ||
§ 1. | Nature of these inferences. | |
2. | Inferences by addition and subtraction. | |
3. | Inferences by multiplication and division. | |
4—6. | Rule for independent events. | |
7. | Other rules sometimes introduced. | |
8. | All the above rules may be interpreted subjectively, i.e. in terms of belief. | |
9—11. | Rules of so-called Inverse Probability. | |
12, 13. | Nature of the assumption involved in them: | |
14—16. | Arbitrary character of this assumption. | |
17, 18. | Physical illustrations. | |
CHAPTER VIII. | ||
THE RULE OF SUCCESSION. | ||
§ 1. | Reasons for desiring some such rule: | |
2. | Though it could scarcely belong to Probability. | |
3. | Distinction between Probability and Induction. | |
4, 5. | Impossibility of reducing the various rules of the latter under one head. | |
6. | Statement of the Rule of Succession; | |
7. | Proof offered for it. | |
8. | Is it a strict rule of inference? | |
9. | Or is it a psychological principle? | |
CHAPTER IX. | ||
INDUCTION. | ||
§§ 1—5. | Statement of the Inductive problem, and origin of the Inductive inference. | |
6. | Relation of Probability to Induction. | |
7—9. | The two are sometimes merged into one. | |
10. | Extent to which causation is needed in Probability. | |
11—13. | Difficulty of referring an individual to a class: | |
14. | This difficulty but slight in Logic, | |
15, 16. | But leads to perplexity in Probability: | |
17—21. | Mild form of this perplexity; | |
22, 23. | Serious form. | |
24—27. | Illustration from Life Insurance. | |
28, 29. | Meaning of 'the value of a life'. | |
30, 31. | Successive specialization of the classes to which objects are referred. | |
32. | Summary of results. | |
CHAPTER X. | ||
CHANCE, CAUSATION AND DESIGN. | ||
§ 1. | Old Theological objection to Chance. | |
2—4. | Scientific version of the same. | |
5. | Statistics in reference to Free-will. | |
6—8. | Inconclusiveness of the common arguments here. | |
9, 10. | Chance as opposed to Physical Causation. | |
11. | Chance as opposed to Design in the case of numerical constants. | |
12—14. | Theoretic solution between Chance and Design. | |
15. | Illustration from the dimensions of the Pyramid. | |
16, 17. | Discussion of certain difficulties here. | |
18, 19. | Illustration from Psychical Phenomena. | |
20. | Arbuthnott's Problem of the proportion of the sexes. | |
21—23. | Random or designed distribution of the stars. | |
(Note on the proportion of the sexes.) | ||
CHAPTER XI. | ||
MATERIAL AND FORMAL LOGIC. | ||
§ 1, 2. | Broad distinction between these views; | |
2, 3. | Difficulty of adhering consistently to the objective view; | |
4. | Especially in the case of Hypotheses. | |
5. | The doubtful stage of our facts is only occasional in Inductive Logic. | |
6—9. | But normal and permanent in Probability. | |
10, 11. | Consequent difficulty of avoiding Conceptualist phraseology. | |
CHAPTER XII. | ||
CONSEQUENCES OF THE DISTINCTIONS OF THE PREVIOUS CHAPTER. | ||
§§ 1, 2. | Probability has no relation to time. | |
3, 4. | Butler and Mill on Probability before and after the event. | |
5. | Other attempts at explaining the difficulty. | |
6—8. | What is really meant by the distinction. | |
9. | Origin of the common mistake. | |
10—12. | Examples in illustration of this view, | |
13. | Is Probability relative? | |
14. | What is really meant by this expression. | |
15. | Objections to terming Probability relative. | |
16, 17. | In suitable examples the difficulty scarcely presents itself. | |
CHAPTER XIII. | ||
ON MODALITY. | ||
§ 1. | Various senses of Modality; | |
2. | Having mostly some relation to Probability. | |
3. | Modality must be recognized. | |
4. | Sometimes relegated to the predicate, | |
5, 6. | Sometimes incorrectly rejected altogether. | |
7, 8. | Common practical recognition of it. | |
9—11. | Modal propositions in Logic and in Probability. | |
12. | Aristotelian view of the Modals; | |
13, 14. | Founded on extinct philosophical views; | |
15. | But long and widely maintained. | |
16. | Kant's general view. | |
17—19. | The number of modal divisions admitted by various logicians. | |
20. | Influence of the theory of Probability. | |
21, 22. | Modal syllogisms. | |
23. | Popular modal phraseology. | |
24—26. | Probable and Dialectic syllogisms. | |
27, 28. | Modal difficulties occur in Jurisprudence. | |
29, 30. | Proposed standards of legal certainty. | |
31. | Rejected formally in English Law, but possibly recognized practically. | |
32. | How, if so, it might be determined. | |
CHAPTER XIV. | ||
FALLACIES. | ||
§§ 1—3. | (I.)Errors in judging of events after they have happened. | |
4—7. | Very various judgments may be thus involved. | |
8, 9. | (II.)Confusion between random and picked selections. | |
10, 11. | (III.)Undue limitation of the notion of Probability. | |
12—16. | (IV.)Double or Quits: the Martingale. | |
17, 18. | Physical illustration. | |
19, 20. | (V.)Inadequate realization of large numbers. | |
21—24. | Production of works of art by chance. | |
25. | Illustration from doctrine of heredity. | |
26—30. | (VI.)Confusion between Probability and Induction. | |
31—33. | (VII.)Undue neglect of small chances. | |
34, 35. | (VIII.)Judging by the event in Probability and in Induction. | |
PART III. | ||
VARIOUS APPLICATIONS OF THE THEORY OF PROBABILITY. Chh. XV—XIX. | ||
CHAPTER XV. | ||
INSURANCE AND GAMBLING. | ||
§§ 1, 2. | The certainties and uncertainties of life. | |
3—5. | Insurance a means of diminishing the uncertainties. | |
6, 7. | Gambling a means of increasing them. | |
8, 9. | Various forms of gambling. | |
10, 11. | Comparison between these practices. | |
12—14. | Proofs of the disadvantage of gambling:— (1) on arithmetical grounds: | |
15, 16. | Illustration from family names. | |
17. | (2) from the 'moral expectation'. | |
18, 19. | Inconclusiveness of these proofs. | |
20—22. | Broader questions raised by these attempts. | |
CHAPTER XVI. | ||
APPLICATION OF PROBABILITY TO TESTIMONY. | ||
§§ 1, 2. | Doubtful applicability of Probability to testimony. | |
3. | Conditions of such applicability. | |
4. | Reasons for the above conditions. | |
5, 6. | Are these conditions fulfilled in the case of testimony? | |
7. | The appeal here is not directly to statistics. | |
8, 9. | Illustrations of the above. | |
10, 11. | Is any application of Probability to testimony valid? | |
CHAPTER XVII. | ||
CREDIBILITY OF EXTRAORDINARY STORIES. | ||
§ 1. | Improbability before and after the event. | |
2, 3. | Does the rejection of this lead to the conclusion that the credibility of a story is independent of its nature? | |
4. | General and special credibility of a witness. | |
5—8. | Distinction between alternative and open questions, and the usual rules for application of testimony to each of these. | |
9. | Discussion of an objection. | |
10, 11. | Testimony of worthless witnesses. | |
12—14. | Common practical ways of regarding such problems | |
15. | Extraordinary stories not necessarily less probable. | |
16—18. | Meaning of the term extraordinary, and its distinction from miraculous | |
19, 20. | Combination of testimony. | |
21, 22. | Scientific meaning of a miracle. | |
23, 24. | Two distinct prepossessions in regard to miracles, and the logical consequences of these. | |
25. | Difficulty of discussing by our rules cases in which arbitrary interference can be postulated. | |
26, 27. | Consequent inappropriateness of many arguments. | |
CHAPTER XVIII. | ||
ON THE NATURE AND USE OF AN AVERAGE, AND ON THE DIFFERENT KINDS OF AVERAGE. | ||
§ 1. | Preliminary rude notion of an average, | |
2. | More precise quantitative notion, yielding (1) the Arithmetical Average, | |
3. | (2) the Geometrical | |
4. | In asymmetrical curves of error the arithmetic average must he distinguished from, | |
5. | (3) the Maximum Ordinate average, | |
6. | (4) and the Median. | |
7. | Diagram in illustration. | |
8—10. | Average departure from the average, considered under the above heads, and under that of | |
11. | (5) The (average of) Mean Square of Error. | |
12—14. | The objects of taking averages. | |
15. | Mr Galton's practical method of determining the average, | |
16, 17. | No distinction between the average and the mean. | |
18—20. | Distinction between what is necessary and what is experimental here. | |
21, 22. | Theoretical defects in the determination of the 'errors'. | |
23. | Practical escape from these. | |
(Note about the units in the exponential equation and integral.) | ||
CHAPTER XIX. | ||
THE THEORY OF THE AVERAGE AS A MEANS OF APPROXIMATION TO THE TRUTH. | ||
§§ 1—4. | General indication of the problem: i.e. an inverse one requiring the previous consideration of a direct one. | |
[I. The direct problem:—given the central value and law of dispersion of the single errors, to determine those of the averages. §§ 6—20.] | ||
6. | (i) The law of dispersion may be determinate à priori, | |
7. | (ii) or experimentally, by statistics. | |
8, 9. | Thence to determine the modulus of the error curve. | |
10—14. | Numerical example to illustrate the nature and amount of the contraction of the modulus of the average-error curve. | |
15. | This curve is of the same general kind as that of the single errors; | |
16. | Equally symmetrical, | |
17, 18. | And more heaped up towards the centre. | |
19, 20. | Algebraic generalization of the foregoing results. | |
[II. The inverse problem: given but a few of the errors to determine their centre and law, and thence to draw the above deductions. §§ 21—25.] | ||
22, 23. | The actual calculations are the same as before, | |
24. | With the extra demand that we must determine how probable are the results. | |
25. | Summary. | |
[III. Consideration of the same questions as applied to certain peculiar laws of error. §§ 26—37.] | ||
26. | (i) All errors equally probable, | |
27. 28. | (ii) Certain peculiar laws of error. | |
29, 30. | Further analysis of the reasons for taking averages. | |
31—35. | Illustrative examples. | |
36, 37. | Curves with double centre and absence of symmetry. | |
38, 39. | Conclusion. |
- ↑ Chapters and sections which are nearly or entirely new are printed in italics.