< The Logic of Chance

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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?

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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.

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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?

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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.
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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.
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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.

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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.
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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:

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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

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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.)
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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.


  1. ↑ Chapters and sections which are nearly or entirely new are printed in italics.
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