TRANSLATION EDITED BY THOMAS JEFFERSON
The process recommended in this aphorism is a species of proof to which we submit the projected operation. It is very useful to avoid error, for if the judgment we examine is found in opposition to anterior ones which are just, or necessarily connected with false consequences, it is evidently necessary to reject it; but this same process does not lead us directly and necessarily to truth, for it may be that no determining motive for the affirmative may result from the research.
In a case in which we want decisive reasons to determine us, no other resource is left us but to endeavour to obtain new lights, that is to say, to introduce new elements into the idea which is the subject of the judgment we are to form. This can be done in two ways only, either by seeking to collect new facts, or by endeavouring to make of those already known combinations which had not previously occurred to us, and thence to draw consequences which we had not before remarked.
The advice contained in this aphorism, is only the developement of the first part of aphorism 9th, and it can be nothing else; for when we are assured that we are not sufficiently acquainted with a subject to judge of it, there is no other resource but to study it more.
Finally, when the motives of determination fail us invincibly, we should know how to remain in complete doubt, and to suspend absolutely our judgment, rather than rest it on vain and confused appearances, since in these we can never be sure that there are not some false elements.
Remark and conclusion.
This is the last and most essential of logical principles; for in following it we may possibly remain in ignorance, but we can never fall into error; all our errors arising always from admitting into that which we know elements which are not really there, and which lead us to consequences which ought not to follow from those that are there effectively.
In effect, if from our first impressions the most simple to our most general ideas, and their most complicated combinations, we have never recognized in our successive perceptions but what is there, our last combinations would be as irreproachable as the first act of our sensibility. Thus, in logical rigour, it is very certain that we ought never to form a judgment but when we see clearly that the subject includes the attributes: that is to say, that the judgment is just.
But at the same time it is also very certain that in the course of life we seldom arrive at certitude, and are frequently obliged, nevertheless, to form a resolution provisionally; to form none being often to adopt one of the most decisive character, without renouncing the principle we have just laid down, or in any manner derogating from it. It is now proper to speak of the theory of probability. It is a subject I encounter with reluctance. First, because it is very difficult, and as yet very little elucidated; next, because one cannot hope to treat it profoundly when one is not perfectly familiar with the combinations of the science of quantities, and of the language proper to them. Finally, because even with these means the nature of the subject deprives us of the hope of arriving at almost any certain result, and leaves us only that of a good calculation of chances. Let us, however, endeavour to form to ourselves an accurate and just idea of it; this will perhaps be already to contribute to its progress.
The science of probability is not a part of logic, and ought not even to be regarded as forming a supplement to it. Logic teaches us to form just judgments, and to make series of judgments: that is to say, of reasonings which are consequent. Now, properly speaking, there are no judgments or series of judgments which are probable. When we judge that an opinion or a fact is probable, we judge it positively; and this judgment is just, false, or presumptuous, according as we have perfectly or imperfectly observed the principles of the art of logic. But it will be said, that the science of probability in teaching us to estimate this probability of an opinion, teaches us to judge justly whether this opinion is or is not probable. I admit it: but it produces this effect as the science of the properties of bodies, physics, teaches us to form the judgment that such a property appertains to such a body; as the science of extension teaches us to form the judgment that such a theorem results from the properties of such a figure; as the science of quantity teaches us that such a number is the result of such a calculation; finally, as all the sciences teach us to form sound judgments of the objects, which belong to their province. Nevertheless we cannot say, and we do not say, that they are but parts of logic, nor even that they are supplements to it. They all on the contrary throw light on the subjects of which they treat only in consequence of the means and processes with which they are furnished by sound logic. This is useful to all the sciences; but none of them either aid it immediately, supply its place, make a part of it, or are supplements to it. The science of probability has in this respect no particular privileges under this aspect; it is a science similar to all the others.
But I go further; the science to which we have given the name of the science of probability, is not a science: or to explain myself more clearly, we comprehend erroneously under this collective and common name a multitude of sciences or of portions of sciences quite different among themselves, strangers to one another, and which it is impossible to unite without confounding them all. In effect, that which is called commonly the science of probability comprehends two very distinct parts, of which one is the research, and the valuation of data, the other is the calculation, or the combination of these same data.
Now the success of the research and valuation of data, if the question is on the probability of a narration, consists in a knowledge of the circumstances, proper to the fact in itself, and to all those who have spoken of it:—thus it depends on and forms a part of the science of history. If the question is on the probability of a physical event, this research of data consists in acquiring a knowledge of anterior facts and of their connection:—thus it appertains to physics. If the question is on the probable results of a social institution, or of the deliberations of an assembly of men, the anterior facts are the details of the social organization, or of the intellectual dispositions and operations of these men:—thus it depends on social and moral science, or on ideology. Finally, when it is only to foresee the chances of the play of cross and pile, the data would be the construction of the piece, the manner of resistance of the medium in which it moves, that of the bodies against which it may strike, the motion proper to the arm which casts it, and which are more or less easy to it. Thus these data would still depend on the physical constitution of animate and inanimate bodies. Then as to the research of data, and to the fixation of their importance, the pretended science of probability is composed of a multitude of different sciences, according to the subject on which it is employed; and consequently it is not a particular science.
As to the combination of the data once established, the science of probability is nothing, when we employ calculation therein, but the science of quantity or of calculation itself; for the difficulty does not consist in giving to abstract unity any concrete value whatever, and sometimes one and sometimes another, but in knowing all the resources which perfect calculation furnishes to make of this unity and of all its multiplied combinations the most complicated, and to connect them regularly without losing their clue.
We see then that neither in regard to the research and valuation of data, nor in regard to the combinations of these same data, the pretended science of probability is not a particular science distinct from every other.
We might rather consider it either as a branch of the science of quantities, and as an employment which we make of it in certain parts of several different sciences which are susceptible of this application, or as the reunion of scattered portions of many sciences, strangers the one to the other, which have only so much in common as to give place to such questions as can only be resolved by a very learned and very delicate employment of the admirable means of calculation furnished by the science of quantities in the state of perfection which it has at this time attained; but this is not seeing the theory of probability in its full extent, for we cannot always employ calculation in the estimation of probability. Nevertheless this manner of considering and decomposing what is called the science of probability explains to us already many of the things which concern it, and puts us in the way of forming to ourselves an accurate and complete idea of it.
We see first why it is the mathematicians who have had the idea of it, and who have, if we may so say, created and made it entirely. It is because such as they have conceived it, it consists principally in the employment of a powerful agent which was at their disposal; they have been able to push to a great length speculations which other men have been obliged to abandon in consequence of a want of means to pursue them.
We also see why these mathematicians principally and almost entirely employed themselves on subjects of which the data are very simple, such as the chances of games of hazard, and of lotteries, or the effects of the interest of money lent; it is because their principal advantage consisting in their great skill in calculation, they have with reason preferred the objects where this art is almost every thing, and where the choice and valuation of data present scarcely any difficulty; and it is in fact in cases of this kind that they have obtained a success both curious and useful.
We moreover see why it is that all the efforts of these mathematicians, even the most skilful, when they have undertaken to treat in the same manner subjects of which the data were numerous, subtile and complicated, have produced little else than witty conceits which may be called difficiles nugae, learned trifles. It is because the farther they have pursued the consequences resulting from the small number of data which they have been able to obtain, the farther they have departed from the consequences which these same data would have produced, united with all those often more important, which they have been obliged to neglect from inability to unravel and appreciate them. This is the cause why we have seen great calculators, after the most learned combinations, give us forms of balloting the most defective, not having taken into account a thousand circumstances, inherent in the nature of men and of things, attending only to the circumstance of the number of the one and of the other. It is the reason why Condorcet himself, when he undertook to apply the theory of probabilities to the decisions of assemblies, and particularly to the judgments of tribunals, either has not ventured to decide any thing on actual institutions, and has confined himself to reasoning on imaginary hypothesis, or has often been led to expedients absolutely impracticable, or which would have inconveniences more serious than those he wished to avoid.
Whatever respect I bear to the great intelligence and high capacity of this truly superior and ever to be regretted man, I do not fear to pass so bold a sentence on this part of his labours, for I am in some measure authorized to do it by himself. The title of Essay which he has given to his treatise, and the motto which he has prefixed to it, prove how much he doubted of the success of such an enterprise, and what confirms it is, that in his last work, composed on the eve of an unfortunate death, in which he has traced with so firm a hand the history of the progress of the human mind, and in which he has assigned to the theory of probabilities so great a part in the future success of the moral sciences, he uses with all the candour which characterises him these expressions, page 362—“This application, notwithstanding the happy efforts of some geometricians, is still, if I may so say, but in its first elements, and it must open to following generations a source of intelligence truly inexhaustible.” Yet he had then made not only the learned essay of which we are speaking, but also a work greatly superior, the elements of the calculation of probabilities and of its application to games of hazard, to lotteries and to the judgments of men, which were not published till the year 1805.
I believe, then, that I have advanced nothing rash in observing that in subjects difficult by the number, subtility, complexity and intimate connexion of the circumstances to be considered, without the omission of any of them, the great talent of well combining those, not sufficiently numerous, which have been perceived, has not been sufficient to preserve the most skilful calculators from important errors and great misreckonings. We perceive that that was to be expected. But new I must go further, and all this leads me to a last reflection, which flows from the nature of things, like those which have just been read, which confirms several important principles established in the preceding volumes, which far from annihilating the great hopes of Condorcet tends to assure and realise them, by restraining them within certain limits; but which appear to me to show manifestly, how far the calculation of probabilities is from being the same thing with the theory of probability. Observe in what this observation consists.
The principal object of the theory of probability and its great utility, is in setting out from the reunion of a certain number of given causes, to determine the degree of the probability of the effects which ought to follow; and setting out from the reunion of a certain number of known effects, to determine the degree of the probability of the causes, which have been able to produce them. We may even say that all the results of this theory are but different branches of this general result, and may be traced to be nothing more than parts of it.
Now we have previously seen, and on different occasions, that for beings of any kind, to be successfully submitted to the action of calculation, it is necessary they should be susceptible of adaptation to the clear, precise and invariable divisions of the ideas of quantity, and to the series of the names of numbers and of cyphers, which express them. This is a condition necessary to the validity of every calculation from which that which has probability for its object, cannot be any more exempt, than that which conducts to absolute certainty.
Hence it rigorously follows, that there is a multitude of subjects of which it would be absolutely impossible to calculate the data, if even (which is not always the case) it should be possible to collect them all without overlooking any.
Assuredly the degrees of the capacity, of the probity of men, those of the energy and the power of their passions, prejudices and habits, cannot possibly be estimated in numbers. It is the same as to the degrees of influence of certain institutions, or of certain functions, of the degrees of importance of certain establishments, of the degrees of difficulty of certain discoveries, of the degrees of utility of certain inventions, or of certain processes. I know that of these quantities, truly inappreciable and innumerable in all the rigour of the word, we seek and even attain to a certain point, in determining the limits, by means of number, of the frequency and extent of their effects; but I also know that in these effects which we are obliged to sum and number together as things perfectly similar, in order to deduce results, it is almost always and I may say always impossible to unravel the alterations and variations of concurrent causes, of influencing circumstances, and of a thousand essential considerations, so that we are necessitated to arrange together as similar a multitude of things very different, to arrive only at those preparatory results which are afterwards to lead to others which cannot fail to become entirely fantastical.
Is an example desired, very striking, drawn from a subject which surely does not present as many difficulties of this kind as moral ideas? Here is one. Certainly none of those who have undertaken to estimate the effort of the muscles of the heart, have erred against the rules of calculation, nor, what is more, against the laws of animated mechanics, the certainty of which should still preserve them from many errors. Yet some have been led to estimate this effort at several thousands of pounds, and others only at some ounces; and nobody knows with certainty which are nearest to truth. What succour then can we derive from calculation, when even availing ourselves properly of it we are subject to such aberrations and to such prodigious incertitude?
It is then true, and I repeat it, that there is a multitude of things to which the calculation of probabilities like every other calculation is completely inapplicable. These things are much more numerous than is generally believed, and even by many very skilful men, and the first step to be taken in the science of probability is to know how to distinguish them. It is for the science of the formation of our ideas, for that of the operations of our intelligence, in a word for sound ideology, to teach us the number of these things, to enable us to know their nature, and to show us the reasons why they are so refractory. And it is a great service which it will render to the human mind, by preventing it in future from making a pernicious use of one of its most excellent instruments. It already shows us that the science of probability is a thing very distinct from the calculation of probability with which it has been confounded, since it extends to many objects to which the other cannot attain. This is what I principally proposed to elucidate.
Finally, as I have before announced, this observation does not destroy the great hopes which the piercing genius of Condorcet had made him conceive from the employment of calculation in general, and from that of probability in particular, in the advancement of the moral sciences; for if the different shades of our moral ideas cannot be expressed in numbers, and if there are many other things relative to social science, which are equally incapable of being estimated and calculated directly, these things depend on others which often render them reducible to calculable quantities, if we may use the expression. Thus for example, the degrees of the value of all things useful and agreeable, that is to say, the degrees of interest we attach to their possession cannot be noted directly by figures, but all those which can be represented by quantities of weight or extension of a particular thing, become calculable and even comparable the one with the other; in like manner the energy and durability of the secret springs which cause and preserve the action of the organs constituting our life are not susceptible of direct appreciation, but we judge of them by their effects. Time and different kinds of resistance and waste are susceptible of very exact divisions. This is sufficient for us, and we derive thence a great multitude of results and of valuable combinations; now there is an infinity of things in the moral sciences which offer us similar resources; but there are also many which offer none, and once more it is of great importance to discriminate perfectly between them: For first, in respect to these latter, every employment of calculation is abusive; and moreover there are often species of quantities presented which appear calculable, but which are inextricably complicated by mixture with those other species of quantities which I permit myself to call refractory, and then if calculation be applied thereto, the most skilful mathematicians are inevitably led into enormous errors; against this in my opinion they have not always been sufficiently on their guard. As to these two latter cases we may say of calculation what has been said of the syllogistic art as to all our reasonings whatsoever; that is, that it conducts our mind much less correctly than the simple light of good sense aided by sufficient attention.
This is all I had to observe on the science and calculation of probability, and I draw from it the following consequences: The theory of probability is neither a part of nor a supplement to logic. This theory moreover is not a science separate and distinct from all others. All sciences have a positive and a conjectural part. In all of them the positive part consists in distinguishing the effects which always and necessarily follow certain causes, and the causes which always and necessarily produce certain effects. In all of them also the conjectural part consists in proceeding from the reunion of a certain number of given causes to determine the degrees of probability of the effects which ought to follow from them, and in proceeding from the reunion of a certain number of known effects to determine the degree of probability of the causes which have been able to produce them. In these two parts, when the ideas compared are not of a nature to comport with the application of the names of numbers and of figures, we can only employ the ordinary instruments of reasoning, that is to say our vulgar languages, their forms, and the words which compose them. In these two parts equally when the ideas compared by the clearness, constancy, and precision of their subdivisions are susceptible of adaptation to the divisions of the series of the names of numbers, and of figures, we can employ with great advantage, instead of the ordinary instruments of reasoning, the instruments proper to the science of the ideas of quantity, that is to say, the language of calculation, its formulas, and its signs. It is this which constitutes in respect to the conjectural part the calculation of probability. It is necessary to distinguish it carefully from the science of probability; for the one is of use in all cases in which the object is a likelihood of any kind whatsoever; it is properly the conjectural part of all other sciences, whereas the other calculation has place only in those cases in which we can employ the language of calculation; it is but an instrument, of which unhappily the science of probability cannot always avail itself.
The science of probability consists in the talent and sagacity necessary to know the data, to chuse them, to perceive their degrees of importance, to arrange them in convenient order, a talent to which it is very difficult to prescribe precise rules, because it is often the product of a multitude of unperceived judgments. On the contrary, the calculation of probability, properly so called, consists only in following correctly the general rules of the language of calculation in those cases in which it can be employed.
This calculation is often extremely useful and extremely learned; but it is necessary carefully to distinguish the occasions on which we can avail ourselves of it, for however little the ideas which we attempt to calculate are mingled with those which I have named refractory, and which are truly incalculable, we are inevitably led into the most excessive misreckonings. It is what I think has happened but too frequently to skilful men, who by their knowledge, and even by their mistakes, have put us into the way of discovering their cause.
I will limit myself to this small number of results. I perceive that it is to diffuse but little direct light on a subject, which is so much the more important and the more extensive, as unfortunately certitude is for the most part far from us. But if I have contributed to the formation of a just and clear idea of it I shall not have been useless. I have much more reason than Condorcet for saying “I have not thought that I was giving a good work, but merely a work calculated to give birth to better ones, &c.”*
Not wishing to occupy myself longer with the conjectural part of our knowledge, and not believing it necessary to add to the small number of principles which I have established before this long digression, and which embrace in my opinion every thing of importance in the logical art, such as it proceeds from true logical science; it only remains for me to endeavour to make a happy application of this art to the study of our will and its effects. It is this I am going to undertake, with a hope that my instruments being better, I may better succeed than perhaps men more skilful but not so well armed.