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Mathematical and Practical Certainty.
once be admitted that no conclusion from inductive reasoning, i. e. from the observed to the unobserved, can enjoy more than a provisional security. If the unbroken experience of all observers, in innumerable instances, be really no ground for extending the conclusion to one unobserved instance admittedly parallel, then and in that case inductive argument should have no influence on human belief. But if, on the other hand, such large and uniform experience of the past is irresistibly felt to warrant a .conclusion as to the future, we should then confidently adopt that conclusion, though with a distinct perception and admission of a risk of error more or less infinitesimal, which we make up our minds to disregard. And it is thus that we come to rest in practical, as distinct from mathematical, certainty, in all physical inquiry, and in all the transactions of life.
It is to express the perception, and enable us to speak consistently, and at the same time definitely, concerning the amount, of this risk, that the term PROBABILITY has been invented a term having reference to our ignorance of the analysis of erents, and of the efficient causes which really necessitate the successive steps by which they arise; and that not generally, but with special and personal reference to the party using that term ; so that the same physical relation the same historical statement - the same future event
different degrees of probability in the eyes of parties differently informed of the circumstances, the causes in action, the reputation for veracity of the testifying authors, or their opportunities of knowing the facts related.
The scale of probability, as viewed in its greatest latitudle; obviously extends from the assured impossibility of the event contemplated to the certainty that it will happen. The total interval between these extremes, either of which is complete knowledge, is occupied by higher or lower degrees of expectation or belief, determined by the partial knowledge we happen to possess, and may be regarded as a natural unit susceptible of numerical subdivision into fractional parts — much as the interval from the freezing to the boiling point on the thermometric scale may be subdivided into aliquot parts or degrees. Properly speaking there is no natural numerical measure of a mental impression, any more than of a corporeal sensation; but in both cases we are sure that higher degrees in the numerical scale may well represent greater intensities of the impression, and in both there is proof that equal increments of a certain element, purely ideal in the one, though possibly substantial in the other, answer to equal numerical differences on the scale; and that the greater or less abundance of this element, in some way or other, determines the degree of intensity of the impression in question.
But the scale of probability plainly. admits of a much more precise graduation than that which would merely mark a general increment or decrease, inasmuch as it is obviously capable of an exact bisection, marked by a definite state of mind, — that, namely, where the mind is completely balanced between the expectation of the event happening and not happening; and this state is therefore indicated by assigning } as the measure of probability in its case. In fact the non-happening of an event is in itself an event; and in the case of a balanced state of mind this event is held to be as probable as the other; so that the unit of certainty must be taken as equally divided between them. In reference to this state of neutrality, then, the words probable’ and improbable' present a meaning. An event is probable' when its probability numerically estimated exceeds 3, improbable' when it falls short of that fraction.
The certainty of an event is not usually spoken of in common parlance as a probability, -as 0 is not commonly called “ a number,' nor the whole' "a part.' Continuity of mathematical language, however, obliges us to identify a probability having 1 for its measure with certainty. Yet there seems to be some psychological cause, some involuntary mental action, in the sort of leap which most men make from a high probability to absolute assurance, - bearing no remote analogy to the sudden consilience or springing into one (with an immediate sentiment of tangible reality) of the two images seen by binocular vision, when gradually brought within a certain proximity; or as some eminent authorities in the higher logic seem to have become impressed with a conviction of the necessary truth of certain physical axioms, which others continue to regard only as inductive propositions of very great generality. There is no doubt that minds differ materially in their readiness to make this spring, and to acquiesce in probable propositions as if certain.
Into the delicate and refined system of mathematical reasoning, now generally known as the · Calculus of Probabilities,' the metaphysical idea of Causation does not enter. The term Cause is used in these investigations without reference to any assumed power to effect a given result by inherent activity. It simply expresses the occasion for a more or less frequent occurrence of that result, and may consist quite as well in the removal of an impediment as in any direct agency. The distinction is that taken by metaphysicians between the efficient and formal cause. The result, itself too, is regarded not as a magnitude or phenomenon susceptible of varieties of degree
Probability distinguished from Chance.
according to the intensity of causation, but merely as an event which must either happen or not happen ; and which will happen more or less frequently, according to the facilities so afforded for its happening under the action of its proper but unknown physical or moral causes, be they what they may, or the impediments interposed to defeat them. Moreover, the sort of events contemplated in establishing the fundamental principles of this calculus are such as, in their simplicity, absolutely exclude one another without the possibility of compromise, or passing into each other by insensible gradation, Hence the frequency in its reasonings, of illustration by the drawing of balls of different colours, or otherwise differently marked, from urns; the distinction between the colours or marks in such cases being obviously absolute, and mutually exclusive. Such events are commonly said by writers on the subject to be contrary to each other. We should prefer the word complementary, as we should • hypothesis' or occasion' for 'cause,' and we think the subject would acquire an accession of clearness by this change in its nomenclature. The distinction itself is most important, and requires to be steadily borne in mind in all applications of this calculus, the chief delicacies in which depend on duly resolving any contemplated event into a determinate succession, or simultaneous combination, of other elementary events mutually exclusive and yet presenting equal facilities for their occurrence.
It requires also to be dwelt on with some emphasis in another point of view, as establishing a chain of relation between the province of this branch of science and that of Physics, which concerns itself with efficient causes, on the one hand, and with Natural Theology, which refers phenomena to final ones, on the other. So considered, it lies at the root of all philosophical inquiry: Chance, indeed, is admitted into its reasonings as the expression of our ignorance of agents, arrangements, and motives, but with the express view to its exclusion from their results. We speak of it as opposed to human certainty, not as opposed to Providential design. * And, as the first step towards narrowing its domain, we endeavour to form a correct estimate of its extent. Among all the applications of this calculus by far the niost important are those which come directly in aid of physical, social, and moral inquiry, by enabling us to measure either the d gree of rational reliance we may place on numerical data (the fundamental elements of Physical science), or the decisiveness with which we are justified in pronouncing the existence of a formal
• De Morgan, Encyc. Metropol., art. Probabilities.
cause or determining condition, from the records of a succession of phenomena. Such conditions once placed in evidence and rendered matter of practical certainty, we hand them over to reasoners of another kind, to discover by appropriate inquiries or experiments in what they consist, and what other offices they may fulfil in the great arrangements of creation.
It is matter of familiar observation and experience that a single occurrence of an event, accompanied by any circumstance then for the first time noticed, is enough to raise a considerable amount of expectation that a recurrence of the same circumstance will issue in the reproduction of the same event. The one becomes indissolubly associated with the other, and is connoted with it; that is to say, set down in memory as one of its distinctive marks. A man with a black crape over his face presented a pistol at me yesterday, alone and at nightfall, and demanded my purse. I shall never see a craped face in future (especially if alone and at dusk) without expecting also to see the pistol and hear the unwelcome demand. The unusual event and the unusual circumstance become associated in imagination, never after to be disunited; and even when further experience may have shown that they often occur disjoined, the occurrence of what has been once set down as a mark or sign of a highly painful or pleasurable incident continues to agitate us with a feeling we cannot shake off, however condemned by reason. In infancy or early youth, when all phenomena are new and striking, and all pains and pleasures vivid, these earliest connotations make a deep and indelible impression, and become either the germs of knowledge or the roots of prejudice. Now it may be worth while to inquire what account the theory of Probabilities gives of this impression, apart from all metaphysical considerations. What is the numerical measure of the expectation (derived from a simple consideration of equipossible combinations) that, of two well characterised events, each of which has been once, and once only, observed, and then in connexion with the other, the next appearance
of the one will be accompanied with that of the other? The happening of one event (A) (no matter which may be considered as equivalent to inserting the hand into an urn containing no other than black and white balls, at least one of each, but without any further restriction as to their numbers, absolute or relative; while the coincident happening of the other event (B) may be assimilated to the drawing thence a ball of the one or the other colour, the opposite colour being held thenceforward to denote its not happening. The second happening of the event (A) will therefore come to be assimilated to a second insertion
Probability the Measure of Expectation.
of the hand into the same urn, the ball first drawn not being replaced *, and a second happening of (B) will be expressed by the drawing thence of a ball of the same colour as the first; its not happening by the contrary colour. Under these circumstances, an exact analysis of all the possible combinations assigns for the probability antecedent to the first drawing that the second drawing will produce the same colour as the first; or, as commonly expressed, there are two chances to one in favour of such a result. It is never without its instruction to trace this sort of parallel between mental impressions and abstract numerical relations. As in the theory of sound, we are led to perceive that the uninstructed ear, in a manner unknown to us, feels out the exact coincidence of numerical ratios, and the sense is delighted with such coin, cidence; so here we find that a sentiment arises in the uninstructed mind - we know not how, yet irresistibly -- to which exact science enables us to trace a parallel, if not to see a reason, in the numerical preponderance of favourable over unfavourable cases, in an indefinite and absolutely unknown multitude of combinations.
As Probability 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. Expectation is merely a belief in the future t, and differs in no way, so far as the measure of its degree is concerned, from that in the past. It may be more difficult to weigh the credibility of human testimony than to reason on contingencies in passing events; but the difficulty exists only in making the estimation, not in the mode of calculating on it when made. Numerically speaking, a certain percentage of every man's assertions is incorrect; and the way in which overwhelming probabilities may arise from the accumulation of such imperfect statements on the one hand, or in which all reasonable reliance may be destroyed by successive hearsay transmission on the other, is not among the least interesting subjects of consideration in this calculus.
* This is essentially involved in the conditions. Though we may presume, or guess, that a combination which has once happened may happen a second time, we are not sure that it can. There
may impossibility in the very nature of the events that it should. If we replace the ball first drawn, we leave no room for the contingency that the supposed dependent event may be unique in its kind, and having once happened can never happen again.
† Brother Jonathan applies the word expect' indiscriminately to past, present, and future.