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Volume 7, 1874
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On Probability.—(Part I.)

By the Rev. R. Kidd, LL.D.

[Read before the Auckland Institute, 1st June, 1874.]

1.

There are two portions of the science of Probability which it is the aim of the following observations in some degree to elucidate. Our present inquiry will be as to the exact nature and relations of what we term Probability, a subject which is, I think, partially involved in confusion. On a subsequent occasion we may consider a certain application of the calculus, which appears to me to merit some development, and which has been hitherto unnoticed in almost every treatise on Probability.
2.

(Popular signification of Probability.)—In common discourse we ascribe probability to a proposition only when we regard it as not certainly true, but more likely to be true than to be false. The term “ probability,” as thus used, is equivalent to likelihood or verisimilitude. We find, accordingly, that this is not the primary signification in that language, the Latin, from which the word is derived; and it will not be irrelevant to advert for a moment to its earliest import. Indeed the primary meaning of the word will be found to illustrate very appositely our determination of its present significance. The root prob denoted approbation; to be probable, probabilis, was to be worthy of being approved. In the classical Latin writers the word has both of these significations, viz., approvable and likely; while the primary meaning alone pertains to the cognate words probus, probitas, as to our English probity, approbation. We may perceive how the secondary sense flowed from the primary, when we consider that occasions of deliberation respecting the choice of one or other course of action would be, in primitive states of society, almost the only occasions of attempting to prove a proposition; and that the most approvable course is that from which a good result is the most probable. Demonstrations or proofs were not demanded, in primitive times, for history, theology, jurisprudence, or any science; but in all eras of human affairs the meed of approbation is awarded to “sage counsel in cumber.”
3.

(Scientific Signification.)—In technical phraseology yet another application of the term Probability has become established. In this wider signification probability is recognized as pertaining to every proposition of which we do not know either the truth or the falsity. The grades of philosophic probability include downwards all that we consider possible; and they range up to all of which we do not consider the contrary to be impossible. And certainly it is fitting to have some designation which shall be thus widely generic. Between the lowest possibility and the state of equipoise of evidence,

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and again between this and the highest moral certainty, the successive differences are merely differences of degree, while the shades of graduation may be innumerable. We may refer for an example to the probable duration of life. In the “English Life Tables,” calculated by Dr. Farr from the census of 1841, with other data, and published in conjunction with the Reports of the Registrar-General, we find that of the men resident in England, and aged 30 years, one-half is, very nearly, the ratio of those who attain to the age of 66 years. This implies that in the case of an individual Englishman, aged 30, and of whom we do not know that he is other than an average specimen, the probability of his attaining to the age of 66 is equal to that of his dying previously. That he will complete his 65th year is a little more likely; and that he will complete his 67th year is, by a small difference, less probable. But these three probabilities, or fractions of probability, are manifestly homogeneous, the variation, in each instance, being merely a difference of degree, and not a diversity of kind. And we might similarly go on, year by year, through the still diminishing probabilities of survival, to the age of 90 or 100; while on the other hand we might in like manner ascend with constantly increasing likelihood, from the expectation that the individual in question may survive the 66th year, till we should come to the year, the month, the day, through which he is now passing. Although in such a series of propositions the lower grades are not probable, in the sense of being likely to be realized, they have, however, each of them a fraction of probability. In the absence of a term more appropriate, this word “probability” is therefore used with an extended signification, so as to include all the grades of moral certainty, likelihood, unlikelihood, and mere possibility.
4.

(Ambiguity.)—The important difference between the popular and the scientific applications of the word “Probability” may occasion an injurious ambiguity by ministering to one or other of the twin fallacies of credulity and over-scepticism; each of which is based upon ignoring the distinctions between mere possibility and likelihood, or between lower probabilities and moral certainty. That a speculation almost desperate may eventuate in success, is possible; there is some degree of probability, in the scientific sense, in favor of the expectation: but such a result is improbable, unlikely; and if important interests are involved, to incur such a hazard is unwise. Again, the most cogent circumstantial evidence may possibly mislead; it is, therefore, only a conclusion of probability, in the generic application of the term, that A. B., who is arraigned for murder upon such evidence, is guilty; and, in cases of this kind, it sometimes happens that, in despite of data affording a moral certainty, the moral cowardice of jurors occasions a failure of justice. A like fallacy has been incurred with reference to the tenet, that Mind does perceptibly preside over the operations and evolutions of Nature. I have heard

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persons of extensive information, but apparently unacquainted with the nature of Probability, ridicule what they termed “belief in a probable God.” Now, whatever may be the merits or demerits of the old theory, that “the invisible things” of Creative Agency are clearly inferrible from their effects, such criticism, at all events, is fallacious. Probability, in the extended application of the term, includes moral certainty; and, furthermore, the probability, in any case, as we shall presently have occasion to notice, belongs to the evidence adduced, and is not at all an attribute of the object of the thought or belief. Probability, it has been well observed, as distinguished from demonstrative proof, is the rule of human life. There are innumerable matters of practical importance, as to which we cannot reasonably look for a direct knowledge, or that species of certainty which is termed absolute. In all such cases the question to be considered is, What is the degree of the so-called probability? Is it of so high a grade as to warrant a rational assurance, and therefore to demand a practical recognition?
5.

We now proceed to determine the basis of Probability. It is evident, in the first place, that any supposed event or thing, about which we may cogitate, is either real or unreal, and that there are no degrees of reality. But in ascribing a probability we do not either affirm or deny reality; and the degrees of probability are innumerable. Probability, therefore, is not a quality of things in themselves, or objectively considered. In the proposition, Hannibal will probably become master of Italy, or, the Duke of Normandy will probably conquer England, it must be allowed that, notwithstanding the grammatical construction, there is no assertion made as to the mode or character of the achievement. New Zealand is probably a remnant of a submerged continent: in saying this, we do not characterize in any way either the submergence of one part or the survival of another; but we intimate that we have much evidence in harmony with that conclusion, while our knowledge of the events is incomplete. To allege that a stated event is probable, in whatever degree, is merely a mode of expression whereby we designate the extent of our knowledge upon the subject as compared with the extent of the statement; and the probability varies according to the varying data of our knowledge. When, by an ominous accident, the eminent statesman, Mr. Huskisson, was killed on the occasion of the first-constructed railway being opened, his death was a matter of certainty to the inhabitants of Liverpool, at the same time that his being alive was a supposition of high probability to a person in London or Kent. The basis of a probability is, then, the amount of our knowledge of the determining circumstances; and this knowledge has respect either to the intrinsic circumstances of the individual case itself, or in general to the attributes of a class of things to which the case in question belongs, or to both of these sources of inference.
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6.

(Probability implies imperfect knowledge and rationality.)—The perception of probability, therefore, pertains to minds that have only a partial knowledge of the thing in question, and are rational. These two conditions must be combined in order to the perception of probability. To a being omniscient of the relations pertaining to any object of thought, there is, with respect to that object, a full certainty of direct knowledge, so that there is no place for probability. Such, for example, is the case of our own knowledge with reference to abstract relations of number and magnitude. On the other hand, a being totally irrational is incapable of the mental pondering that a perception of probability implies. In a mind devoid of reflection there is no distinction between vivid impression and belief. The perception of probability implies a combination of imperfect knowledge with rationality.
7.

(Probability is a property of Propositions.)—We have seen that Probability is not an attribute of the events themselves respecting which we may inquire. It would be idly pedantic to condemn, or on all occasions to refrain from, the usual phraseology, whereby we speak of probable events, facts, etc.; but it is to be noted that these expressions are authorized only by custom and convenience. Of what, then, is Probability a property? When we say that something is probable, what is it that we thus assert to be worthy of our acceptance, or to have some degree of acceptibility?

To ascribe probability to a supposed event is to allege, that the supposition of the event's occurrence is a probable supposition; that it is, in other words, a supposition having a certain amount of claim to be accepted as in accordance with fact. A. supposition is itself a fact, a mental phenomenon; but its value usually consists in its relation to another, an objective, fact, whether this be mental or otherwise. Imaginations may be vain, thoughts may be erroneous; and they are so in proportion as they are inconsistent with objective reality. To designate a probability is to estimate the claim of a supposition or hypothesis as to its being in accordance with fact. To say that the event A is highly probable, is to indicate that the supposition of the occurrence of A is to be provisionally received as having an evidence approaching to certainty: to assert that the event B is morally certain, is to declare that the evidence in favor of that conclusion is practically equivalent to opportunity of absolute knowledge. Probability, then, is an attribute of suppositions as compared with the known data of objective reality. Now, all suppositions are, or are expressible in, propositions—i.e., declarations or statements; and they are unsusceptible of full consideration except as so expressed. We may, therefore, assume the following principle, that Probability, like Truth, is a quality of propositions, and of propositions only. By saying that a supposed event is probable, in whatsoever degree, we mean that this degree of probability pertains to the supposition of the event, or to the proposition in which that hypothesis or judgment is stated.

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8.

(Fact, Event.)—We may observe, in passing, that while the word “fact,” like the word “ event,” is, according to its etymology, applicable only to a change in things, it may, however, be allowably used to signify any reality. It is, for example, allowable to speak of the brightness of the star Sirius as a fact. Any question, indeed, respecting a fact, in this wide sense of the term, is resolivible into a question respecting an event or events. For any evidence, whatsoever it may be, of an existence must be evidence of the existent thing having been, directly or mediately, perceived; and a perception is an event. We may, therefore, in treating of proofs and probabilities, employ ad libitum the occurrence of events as representing all reality of things.
9.

(Numerical Notation of Probability.)—In the mathematical calculation of probabilities, the number 1 or unity is employed to signify that the data necessitate the truth of the given proposition; 0 or zero signifies that the data necessitate its falsity; and the intermediate fractions are used to denote the various amounts of probability. Thus, if a probability be stated as 1/2, this denotes that the truth and the falsity of the proposition are equally probable; if we estimate the probability of a proposition to be 2/3, we estimate the probability of its falsity to be 1/3; and so on.

Now, to employ an arithmetical fraction in any real computation implies that we contemplate something or other as divisible into so many equal parts as there are units in the “ denominator,” and that of those equal parts we take so many as there are units in the “ numerator.” Thus, if we speak of 3/4 of a mile, or of a pound, or of X, we mean that if the mile, the pound, or X, be divided into 4 equal parts, then of those 4 our proposition pertains to 3. What, then, it behoves us to inquire, are the equal parts which a fraction of probability denominates?

10.

(Probability and Belief.)—It has become usual to assume that Probability is the same as Belief; and that, accordingly, the amount of a proposition's probability is neither more nor less than the amount of belief given to that proposition. Thus Professor De Morgan says (Formal Logic, chap. 9): “By degree of probability we really mean, or ought to mean, degree of belief. It is true that we may, if we like, divide probability into ideal and objective, and that we must do so, in order to represent common language.” And he adds: “ I throw away objective probability altogether, and consider the word as meaning the state of the mind with respect to an assertion, a coming event, or any other matter on which absolute knowledge does not exist.” And he broadly avers, that even when an error of computation is incurred in deducing the amount of a probability, still the conclusion erroneously arrived at constitutes the probability to the computator; so that precisely the same data may render different probabilities. A similar defi-

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  • nition is given by Sir John Herschel (Edinburgh Review, July, 1850; reprinted in “ Essays,” p. 376): “ As Probability,” he says, “ is the numerical measure of our expectation that an event will happen, so it is also that of our belief that one has happened, or that any proposed proposition is true.” In a very recent work of repute, Dr. Bain's “ Logic,” published in 1870, we read (III. ix. 6): “ Probability expresses a state of the mind, and also a situation among objective facts. As a state of the mind, it is a grade or variety of belief. The highest degree of belief is called Certainty; the inferior degrees are degrees of Probability.”

Now, that there are degrees of belief cannot be questioned; but how are they to be enumerated, measured, or weighed? How come we at the denominator in the supposed fraction of belief? And if we assume a denominator, how are we to determine the numerator? Consciousness merely indicates, without much nicety of discrimination, that we know or are ignorant; believe, or doubt, or disbelieve; are more or less certain or dubious. This with respect to each person's own mental phenomena; while as to those of our neighbour we have no direct knowledge whatever. The alleged fractions of belief are, in fact, ascertained, not from definitive investigation into the state of men's minds, but from examination of the objective data to which the belief does or ought to correspond. The quantum of credence, in any given case, ought to be approximately proportional to the quantity and quality of the data; but whether it actually is so or not, is a question as to a matter of fact totally distinct from that which is asserted in the given proposition. Probability and belief are not, therefore, identical; but they are, or ought to be, correlative; and the interpretation of the degrees of probability is not to be sought or found in the mental conditions of the investigator. Right belief is correspondent to probability; wrong belief is at variance with probability; and the probability itself is determined by the data. To attempt evolving the logic or calculus of Probability from the metaphysics of Belief, is to set out in the wrong direction; and it would be, at the best, to seek the solution of a problem comparatively easy by substituting what is obscure. The substitution, however, is gratuitous. The probability of a proposition is not constituted by the belief that is actually given to it, whether the belief of an individual, or the average amount of belief entertained by the aggregate of judges, or whatever other standard of actual belief we may select; but, as even the etymology of the word “probability” indicates, it is proportional to the degree of belief or acceptance that ought to be given, according to the data of knowledge. Which is equivalent to saying, that probability is determined by evidence; and accordingly as a testimony is trustworthy, or an argument cogent, a proposition so supported is proportionally probable or certain, however scantily or extensively the value of the evidence may be recognized.

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Of the assumption that Probability is identical with Belief, a further and more demonstrative refutation will present itself, after we shall have referred somewhat more particularly to the numerical notation of Probability.

11.

(Probability is neither wholly objective nor wholly subjective.)—In the foregoing remarks it was requisite, first, as is usually done, to disentangle the subject from the current forms of expression, which would imply, prima facie, that probability is a quality of things in themselves, or as objectively considered. We also find it necessary to obviate the other extreme, whereby it has been assumed that Probability is simply Belief. Being aware that Probability could not be identified with the objective, writers of deservedly high repute have unwarily adopted the opposite and less obvious fallacy of making Probability to be merely subjective. Dr. Bain, in the passage quoted above, appears to have contemplated the untenableness of wholly identifying Probability with Belief; but he has recourse to an expedient not less untenable, when he assumes that there are two kinds of Probability, the one resident in the mind of the investigator, and the other in the objective facts. This assumption is made tacitly, and is entirely unsupported. The simple truth is, that Probability is neither wholly objective, nor wholly subjective, inasmuch as it is a claim to acceptance constituted by a relation between the two. Probability is not, on the one hand, a quality of the things themselves respecting which we inquire; nor, on the other hand, does it consist in any mere condition of the mind of the inquirer. The probability of a proposition is the value of its claim to acceptance as being true, or in accordance with fact; and this value is estimated by comparing that which is asserted in the proposition with the data that we know. The knowledge is in the mind of the reasoner, and the proposition which states the probability, and which ought to be in accordance with that knowledge, is framed in his mind; but the data thus known are objective, and the probability of the proposition is its claim to acceptance as constituted by the relation of those data to the matter treated of in the proposition.
12.

(Basis of the Numerical Notation.)—It is now requisite for us to analyse some easy examples of numerical probability, in order that we may be enabled to designate more definitively the principles involved.

Let us suppose that we know an event, which we may call A, to have occurred on some day in the month of June, and that this is the whole of the data: we would estimate the probability of the following proposition, The event A occurred before the 21st day of June. As June consists of 30 days, all equal to one another, the case is distinguishable into 30 alternatives having equal claims upon our acceptance; and of these 30 alternatives the proposition in question is affirmed by 20, representing two decades of days. The whole case, therefore, is appropriately expressed in the following three alternative propositions:—

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(1.)

The event A occurred in the first decade of June; or,
(2.)

It occurred in the second decade; or else,
(3.)

It occurred in the third decade.

That the event occurred on a day of June is assumed to be known: the numerical representative of this proposition is, therefore, the integer 1; and, consequently, as each of the three alternatives constituting the case is equally probable, the probability of each is represented by the fraction 1/3. But the proposition in question, viz., that the event occurred before the 21st day of June, is true, if either of the first two alternatives be true; while it is false, if the third alternative be the true one. The probability of the proposition, therefore, as furnished by those data, is 2/3.

The following familiar example is noticeable as presenting an easy problem, in the solution of which a celebrated mathematician erred, and as illustrating, accordingly, the fundamental distinction that there is between Probability and Belief. Let us suppose a coin to be taken such that, when it is thrown, either side of the coin is equally likely to fall uppermost. The probability of the obverse side falling uppermost in any throw is 1/2; for the case is comprised in two equal alternatives, one of which affirms, and the other negatives, the proposition in question. Let us now further ask, What is the probability of the obverse being shown in one or other of two throws? This case consists of the following four alternatives:—

(1.)

Both throws show reverse; or
(2.)

The first throw shows reverse, and the second obverse; or
(3.)

The first throw shows obverse, and the second will show reverse; or else
(4.)

The first throw shows obverse, and the second also will show obverse.

These alternatives being all equally probable, and their number being 4, while it is assumed that one or other of them is certain to occur, the probability of each alternative is 1/4. But any one of the last three alternatives affirms the proposition in question; so that the probability of the proposition is 3/4. This implies, that if the experiment were repeated a large number of times, the obverse would be shown by one or other of two throws in three-fourths of the instances or thereabouts; and such experiments have actually been made with results correspondent to the theory. But the eminent encyclopedist of the last century, D'Alembert, inferred from the same data, that the resultant probability is 2/3. He did not take into consideration that, in order to obtain the accurate estimate of a probability, there must be equal quantities represented by the units of the denominator. We may, of course, divide the case into only the three most obvious alternatives, viz., those of reverse in each throw, reverse in the first throw and obverse in the second, and obverse in the first throw; but these alternatives do not present equal claims upon our acceptance.

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The last of the three is, as we have seen, distinguishable into two, each of which has the same amount of probability as the first or second alternative.

One other example we may adduce, which also was miscalculated by a notable personage, the eminent philosopher of our own day, John Stuart Mill. Suppose that a thing, which we shall call T, is a member of the class A, and that of the members of this class just two-thirds have the attribute X; also, that the same thing T is a member of another class, viz., B, and that in this class the same attribute X pertains to just three-fourths of the members—the membership in the one class being assumed to be unconnected with that in the other: what is the resultant probability that the thing T possesses the attribute X? In the earlier editions of the “ System of Logic “ this question was answered erroneously; but subsequent editions gave the correction. The discussion occupies several pages of that work, and is rather abstruse; but by our having recourse at once to the fundamental principle, that of dividing the case into equal alternatives, the solution becomes easy. Two-thirds of the class A is the portion possessing the attribute X: we may, therefore, consider the class A as consisting of sets each composed of three members; and of each triad let the first two possess the attribute in question, and the third want it. The members of each triad we will designate as A1, A2, A3. Similarly, the class B consists of quaternions, in each of which we designate the members as B1, B2, B3, B4; and of these let the fourth alone be without the attribute X. The whole case, then consists of the following 7 alternatives:—

(1.)

T is A1 and B1, or
(2.)

T is A1 and B2, or
(3.)

T is A1 and B3.

These are the first three alternatives; and we have now exhausted A1, because T cannot be A1 and B4; inasmuch as A1 has the attribute X, and B4 wants it. The remaining alternatives are, therefore, as follows:—

(4.)

T is A2 and B1, or
(5.)

T is A2 and B2, or
(6.)

T is A2 and B3, or else, lastly,
(7.)

T is A3 and B4.

Of these seven equal alternatives, one or other of which must be true, the first six affirm the proposition in question, viz., that T possesses the attribute X, and the last alternative alone negatives it; so that the resultant probability of the proposition is, at the most, six-sevenths. Mr. Mill mistakenly inferred the resultant to be eleven-twelfths, until set right by a mathematical friend. It would occupy too much of our time and attention to exhibit here the manner in which the error was incurred.

To assign, then, a fraction of probability implies that the case presents a certain objective quantity, of whatsoever kind it may be. This quantity is

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considered as divisible into so many equal parts as there are units in the denominator; and the proposition in question predicates respecting so many of those parts as there are units in the numerator. In the first of the three instances that we have examined, the denominated quantity is the days of June, considered as 30 in number, and all equal to each other; of which 30 the proposition in question predicates respecting two-thirds. In the second instance, the unit of the denominated quantity is two throws of a coin; and the denominator is the number of the modes in which these two throws may be varied. In the last of our examples the unit is a member of each of the classes A and B; the denominator, 7, is the smallest number of instances that; can exhibit an average specimen; and the numerator, 6, is the number of the instances to which the proposition in question applies.

I have referred thus explicitly to the quantities which furnish the fraction of probability, for the purpose of its being rendered palpably plain that the basis of the probability is objective. For general purposes the fraction of probability is sufficiently interpreted by our saying, that the denominator of the fraction is the number of equally probable alternatives into which the case is considered as distinguishable, and that the numerator is the number of those alternatives which affirm the proposition in question.

13.

(Hypothetic estimate of Alternatives.)—The supposed simple cases of probability which we have taken as examples for analysis, belong to that class of probabilities in which the alternatives can be distinctly perceived to be equal. But in the actual affairs of life this condition is less frequently realized. In such cases, nevertheless, the numerical notation of probability is sometimes employed for the purpose of more convenient discussion, especially in the combination of probabilities. In the formation of such hypothetic estimates the import of the probability, and of the fraction representing it, is essentially similar to the foregoing. Let us, for example, suppose the proposition in question to be, that the author of the Letters of Junius was Sir Philip Francis; and let us suppose that we estimate the probability of this proposition as being adequately represented by the fraction four-fifths. We cannot, it may be assumed, distinctly assign five equal alternatives, as constituting the case; but supposing that we have found four-fifths in favor of the given proposition to be a fitting representative of the probability, we consider that if the case, as known to us, were distinguished into equal alternatives, then about four-fifths of such alternatives would affirm the given proposition. In passing from a case of numerically definite quantities in the basis of the alternatives, to a case not susceptible of a like distinctness of enumeration, we lose the categorical precision in the data; but it is still the mutual relation of the same two things that determines the probability of the proposition, viz., the relation of what is asserted in the proposition in question to the data that are known.
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14.

(Inference of Belief.)—It is now desirable, in conclusion, to revert briefly to the question as to the relation of Probability to Belief. The subject has, in several of its aspects, so much of importance or of interest as to render it worth while to have investigated it closely. It is, then, to be observed, that if the statement of a probability asserted simply the existence of a belief, or of a fraction of belief, there could be no such thing as a demonstrative deduction of probabilities, even from data absolutely assumed. But all agree that from assumed data there are demonstrative deductions of probabilities.

Let us advert to any case whatever of inference of the simplest possible kind. Employing the usual symbolic syllogism—Every M is P, S is M, therefore S is P—we may attach to these alphabetic symbols whatever meaning we please. If the data were merely, that each of those two premises is believed by a given individual, whom we may call X. Y., we could not infer absolutely that X. Y. believes the conclusion. He may not have put the premises together; or he may be so unreasonable as not to accept the consequence. If we assume that he has combined the premises, this is an assumption additional to that of the two beliefs. Let this additional assumption be made, and then we may infer, as a high probability, that X. Y. believes the conclusion; because, in the great majority of instances of believing and comparing the premises of a syllogism, the conclusion also is believed. And it is morally certain that if a hundred persons were experimented upon, especially if at all a favourable specimen of rationality, then, in most or all of the hundred instances, the putting together of believed premises would be accompanied by a belief of the conclusion. But these are inferences from an induction of facts not given in the original premises; they are not demonstrative inferences, nor inferences from the mere beliefs of X. Y. or his fellows.

Thus we see that, even with respect to the plainest cases of necessary sequence, we cannot from mere belief demonstrate belief. And why? Because every act of belief is a distinct event; and, as in the case of other matters of fact, we do not know the machinery of the causation so well as to reason absolutely. We know absolutely that if every M be P, and S be M, then S is P; but we do not know absolutely, with reference to any person whatever, that if he believe those premises, he also believes this conclusion.

And if a person's belief of the premises of a syllogism in “ Barbara” does not of itself enable us to infer his belief of the conclusion, much less would his belief of two separate probabilities warrant our inferring his belief of their resultant. D'Alembert believed that if a given coin were thrown twice, the probability of the obverse side falling uppermost would be in each throw 1/2; but the great mathematician failed to believe the necessary resultant, that the probability of an obverse in one or other of the two throws is 3/4. Reasoning

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from data almost as simple, the profound philosopher Mill arrived at a resultant probability widely different from the demonstrable conclusion. Allowing, as all do, that the sequence of numerically expressed probabilities is a matter of rigorous demonstration, we cannot consistently deny that Probability is something else than Belief.

The relation of Probability to Belief is more correctly delineated in the older writers than in the most recent. “Probability,” says Hume (Essays), “arises from a superiority of chances on any side; and, according as this superiority encreases and surpasses the opposite chances, the probability receives a proportionate encrease, and begets a still higher degree of belief or assent to that side in which we discover the superiority.” The real relations are here indicated: the “chances,” i.e. alternatives, determine the probability; and this probability, so far as we “discover” it, “begets" a correspondent belief. The definition of Probability given by the chief writer on the philosophy of Induction, Mr. Mill, is also essentially correct. The “probability to us,” he observes (System of Logic, III. xviii. 1), “means the degree of expectation which we are warranted in entertaining by our present evidence.”

The distinction between the actual belief and due belief, which is slurred over by some writers who identify Probability with Belief, is fundamental; and the resultant diversity is great. To ascertain actual beliefs, we must interrogate men's minds; to estimate the degrees of belief that are warranted, we must look to the objective evidence of facts. As the eye does not cause or constitute the rays of light, nor their abundance or paucity; and as the ear does not produce the acoustic vibrations of the atmosphere; so neither does the recipience of the mind create the evidence that is presented to it. And as visibility, therefore, differs from seeing, and audibility from hearing, so probability differs from belief.

15. In concluding these remarks on the nature and basis of Probability, permit me to summarize in a few words the principal results at which we have arrived, in so far as they appear to be peculiar to this paper.

First, Probability is a property of suppositions or propositions, and of these only.

Secondly, the most appropriate definition of Probability appears to be simply this: The probability of a proposition is its claim to be accepted as true; and the value of the claim is determined by the relation of what is predicated to the data. As Truth and Falsity, strictly so called, are properties of propositions only, and Probability is the claim to be true, hence, as above mentioned, it is only to propositions that Probability also belongs.

Thirdly, when a numerical value is assigned to a probability, the denominator of the fraction represents some quantity, of whatsoever kind it may be, which is assumed to be an object of thought, and to be divisible into equal

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parts. The numerator is the number of those parts to which the proposition in question applies.

Fourthly, the identification of Probability with Belief, to which recent writers have been prone, is untenable. It is an extreme reaction against that which is implied in some usual forms of expression, whereby probability is attributed to things in themselves, or as objectively considered.

Fifthly, and lastly, Probability is neither wholly objective nor wholly subjective, being constituted by a relation between the two. The knowledge of the data, and the hypothesis of what is not in the data, are in the mind of the investigator; while the data known, and the circumstances of the data, are objective.

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