The Logic of Chance, 3rd edition An Essay on the Foundations and Province of the Theory of Probability, With Especial Reference to Its Logical Bearings and Its — Key Ideas to Explore

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.

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.

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._

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.

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._

LOGICAL SUPERSTRUCTURE ON THE ABOVE PHYSICAL FOUNDATIONS. Chh. VI-XIV.

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.

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._

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?_

§§ 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.

CHANGE, 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_.)

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.

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.

§ 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._

VARIOUS APPLICATIONS OF THE THEORY OF PROBABILITY. Chh. XV-XIX.

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._

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?

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.

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 be 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._)

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 determinable _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._

_ON CERTAIN KINDS OF GROUPS OR SERIES AS THE FOUNDATION OF PROBABILITY._

1. It is sometimes not easy to give a clear definition of a science at the outset, so as to set its scope and province before the reader in a few words. In the case of those sciences which are more immediately and directly concerned with what are termed objects, rather than with what are termed processes, this difficulty is not indeed so serious. If the reader is already familiar with the objects, a simple reference to them will give him a tolerably accurate idea of the direction and nature of his studies. Even if he be not familiar with them, they will still be often to some extent connected and associated in his mind by a name, and the mere utterance of the name may thus convey a fair amount of preliminary information. This is more or less the case with many of the natural sciences; we can often tell the reader beforehand exactly what he is going to study. But when a science is concerned, not so much with objects directly, as with processes and laws, or when it takes for the subject of its enquiry some comparatively obscure feature drawn from phenomena which have little or nothing else in common, the difficulty of giving preliminary information becomes greater. Recognized classes of objects have then to be disregarded and even broken up, and an entirely novel arrangement of the objects to be made. In such cases it is the study of the science that first gives the science its unity, for till it is studied the objects with which it is concerned were probably never thought of together. Here a definition cannot be given at the outset, and the process of obtaining it may become by comparison somewhat laborious.

The science of Probability, at least on the view taken of it in the following pages, is of this latter description. The reader who is at present unacquainted with the science cannot be at once informed of its scope by a reference to objects with which he is already familiar. He will have to be taken in hand, as it were, and some little time and trouble will have to be expended in directing his attention to our subject-matter before he can be expected to know it. To do this will be our first task.

2. In studying Nature, in any form, we are continually coming into possession of information which we sum up in general propositions. Now in very many cases these general propositions are neither more nor less certain and accurate than the details which they embrace and of which they are composed. We are assuming at present that the truth of these generalizations is not disputed; as a matter of fact they may rest on weak evidence, or they may be uncertain from their being widely extended by induction; what is meant is, that when we resolve them into their component parts we have precisely the same assurance of the truth of the details as we have of that of the whole. When I know, for instance, that all cows ruminate, I feel just as certain that any particular cow or cows ruminate as that the whole class does. I may be right or wrong in my original statement, and I may have obtained it by any conceivable mode in which truths can be obtained; but whatever the value of the general proposition may be, that of the particulars is neither greater nor less. The process of inferring the particular from the general is not accompanied by the slightest diminution of certainty. If one of these 'immediate inferences' is justified at all, it will be equally right in every case.

But it is by no means necessary that this characteristic should exist in all cases. There is a class of immediate inferences, almost unrecognized indeed in logic, but constantly drawn in practice, of which the characteristic is, that as they increase in particularity they diminish in certainty. Let me assume that I am told that _some_ cows ruminate; I cannot infer logically from this that any particular cow does so, though I should feel some way removed from absolute disbelief, or even indifference to assent, upon the subject; but if I saw a herd of cows I should feel more sure that some of them were ruminant than I did of the single cow, and my assurance would increase with the numbers of the herd about which I had to form an opinion. Here then we have a class of things as to the individuals of which we feel quite in uncertainty, whilst as we embrace larger numbers in our assertions we attach greater weight to our inferences. It is with such classes of things and such inferences that the science of Probability is concerned.

3. In the foregoing remarks, which are intended to be purely preliminary, we have not been able altogether to avoid some reference to a subjective element, viz. the degree of our certainty or belief about the things which we are supposed to contemplate. The reader may be aware that by some writers this element is regarded as the subject-matter of the science. Hence it will have to be discussed in a future chapter. As however I do not agree with the opinion of the writers just mentioned, at least as regards treating this element as one of primary importance, no further allusion will be made to it here, but we will pass on at once to a more minute investigation of that distinctive characteristic of certain classes of things which was introduced to notice in the last section.

In these classes of things, which are those with which Probability is concerned, the fundamental conception which the reader has to fix in his mind as clearly as possible, is, I take it, that of a series. But it is a series of a peculiar kind, one of which no better compendious description can be given than that which is contained in the statement that it combines individual irregularity with aggregate regularity. This is a statement which will probably need some explanation. Let us recur to an example of the kind already alluded to, selecting one which shall be in accordance with experience. Some children will not live to thirty. Now if this proposition is to be regarded as a purely indefinite or, as it would be termed in logic, 'particular' proposition, no doubt the notion of a series does not obviously present itself in connection with it. It contains a statement about a certain unknown proportion of the whole, and that is all. But it is not with these purely indefinite propositions that we shall be concerned. Let us suppose the statement, on the contrary, to be of a numerical character, and to refer to a given proportion of the whole, and we shall then find it difficult to exclude the notion of a series. We shall find it, I think, impossible to do so as soon as we set before us the aim of obtaining accurate, or even moderately correct inferences. What, for instance, is the meaning of the statement that two new-born children in three fail to attain the age of sixty-three? It certainly does not declare that in any given batch of, say, thirty, we shall find just twenty that fail: whatever might be the strict meaning of the words, this is not the import of the statement. It rather contemplates our examination of a large number, of a long succession of instances, and states that in such a succession we shall find a numerical proportion, not indeed fixed and accurate at first, but which tends in the long run to become so. In every kind of example with which we shall be concerned we shall find this reference to a large number or succession of objects, or, as we shall term it, _series_ of them.

A few additional examples may serve to make this plain.

Let us suppose that we toss up a penny a great many times; the results of the successive throws may be conceived to form a series. The separate throws of this series seem to occur in utter disorder; it is this disorder which causes our uncertainty about them. Sometimes head comes, sometimes tail comes; sometimes there is a repetition of the same face, sometimes not. So long as we confine our observation to a few throws at a time, the series seems to be simply chaotic. But when we consider the result of a long succession we find a marked distinction; a kind of order begins gradually to emerge, and at last assumes a distinct and striking aspect. We find in this case that the heads and tails occur in about equal numbers, that similar repetitions of different faces do so also, and so on. In a word, notwithstanding the individual disorder, an aggregate order begins to prevail. So again if we are examining the length of human life, the different lives which fall under our notice compose a series presenting the same features. The length of a single life is familiarly uncertain, but the average duration of a batch of lives is becoming in an almost equal degree familiarly certain. The larger the number we take out of any mixed crowd, the clearer become the symptoms of order, the more nearly will the average length of each selected class be the same. These few cases will serve as simple examples of a property of things which can be traced almost everywhere, to a greater or less extent, throughout the whole field of our experience. Fires, shipwrecks, yields of harvest, births, marriages, suicides; it scarcely seems to matter what feature we single out for observation.[1] The irregularity of the single instances diminishes when we take a large number, and at last seems for all practical purposes to disappear.