§ 34. I shall return hereafter to the consideration of this root-mistake, but it is better to begin with a statement of the truth. We are to omit the subject of probability in general, and confine ourselves to the particular instance of that which is called mathematical probability. And the point which first presents itself to our notice, is the necessity of limiting the possibilities. Before we can advance a single step we must have the whole of the chances before us. This exhaustive survey may rest on knowledge or on arbitrary assumption, but it is always presupposed. The calculation of chances, in a word, must be based on a disjunctive judgment, and the hypo thetical assertions, which represent the chances, take place within the bounds of that judgment. But disjunction, as we know (Chap. IV.), implies a categorical foundation. This basis of fact is the condition of our assertions about the chances.
§35. Take a simple instance. A die has been thrown without our knowledge, or is now about to be thrown before us. As a previous step to reckoning the chances we must make some categoric statements. We must be able to say, The die will fall (or has fallen), and will fall beside in a certain way. It must have one side up, and this, whatever else it is, will at least be not other than all these six sides. It must have a quality determined as what is common to the six, and not determined as what will be none of them. On this cate gorical foundation all the rest is based, and without it there is no possibility of advance.
This result has a most important application. There is no probability before all reality. There is none which does not stand on a basis of fact assumed or actual, and which is not a further development of that basis.
§ 36. We have seen the foundation of our disjunctive judgment. What is it that completes it? It is of course the setting out of exclusive alternatives. These alternative possi bilities are given us in the various hypothetical judgments which we are able to make as to the number on the face which we know is lying uppermost, or which will so lie. We have now a disjunctive judgment, enclosing an exhaustive statement of exclusive possibilities. But we have not yet got to mathe matical probability. To reach this a further step is to be made. We must take the possibilities all to be equal, or, if they are not equal, we must make them comparable.
§ 37. The possibilities must all be equally probable. What does this mean? It means that there is no more to be said for one than there is for another. The possibilities are each a hypothetical result from certain conditions; and these results are equal, when, in the first place, they follow each from no more than one single set of conditions, and when in the second place, I attach no more weight to any one set than I do to the others. When, in short, I have no more reason for making one hypothetical judgment than I have for making any other, they are possible alike and equally probable.
X must be a or b or c. X qualified by certain conditions would be a, if qualified by other conditions would be bf and so with c. If in my knowledge I have any ground 35 for taking X in one set of conditions rather than in another, then a, b, and c are not equally likely. If such a ground is absent, then they are equal. Again, if X will give a with a single set of con ditions, and b or c with more than one set, the chances are different in the different cases.36 Otherwise they are the same.
§ 38. If the separate alternatives are not found equal, then we must either give up our attempt to reckon chances, or must find some common unit of value. We must analyze one possi bility, and find, perhaps, that its final result is really two; or that, though the final result is one, it will follow from two or three sets of conditions, and hence can stand for two or three units. In these cases there were two hypothetical judgments which we joined in one. Again, if we can not divide the greater, we may join the smaller. By considering two or more alternatives as one, we raise the whole to a unit of higher value.
§ 39- Where we have a disjunction the alternatives of which are equally likely, or are reduced to alternatives which are equally likely, we can state the chances. Since we have the same ground to think every possibility true, the probability of each is just the same quantity. In our knowledge they divide the actual fact between them equally. The reality then we represent as unity, and each alternative possibility we represent by a fraction, of which the denominator is the number of equal alternatives, and the numerator is one. Against our belief in the general fact we have nothing to set. Against any one of its developments we have to set the whole of the others.
§ 40. Take the instance of the die. We know it will fall in a certain way. So much is categorical, and we have now to determine the further possibilities. What are the conditions from which in each case our hypothetical results proceed? They are first the general character of the fall, those positive and negative general conditions from which comes a fall with one of the six faces up, and no more than one. Do these furnish a ground for making one fall more likely than others? Clearly they do not.
The general conditions, which we have considered so far, are known to exist. The fact must take place in such a way that these conditions will be realized. But, beside this known * Wolff has expressed the principle very well, " Probabilior est propositio, si subjecto predicatum tribuitur ob plura requisita ad veritatem, quam si tribuitur ob pauciora."
element, there are a number of circumstances about which we are in doubt. The particular throw must be the result of one particular position of the die, the contraction of particular muscles in the thrower, and the character of the surface which receives the fall. The number of different sets of conditions which would lead to the result, is very great, and in part per haps unknown.37 Still this makes no difference. They are all at least known or assumed to be compatible with the reality, and they lead indifferently to any one of the six results. With respect to each face we have exactly as much reason to think it uppermost, as we have to think any other face uppermost. The chances are equal; and since they are six, and since they divide the sphere of a single unity, they are each one-sixth. We have a certain reason to expect one face, say for instance four, but we have the same reason five times over not to look for four.
§ 41. Now suppose one face loaded. The final possibilities are still six in number, but their value is not equal. There are more sets of conditions, which would lead to the loaded face being downwards, than sets which would bring the opposite face into the same position. I have thus more reason to look for one than I have to expect the rest. My task is now to get a fresh unit by breaking up some or all of the possibilities. If I succeed in this, the whole will again be divided into fractions expressing the respective chances, but these fractions will be unequal. The units of reason to look for each face will be more in one case and less in another.
§ 42. The above is, I think, the entire foundation of the doctrine of chances. It is perfectly simple and entirely rational. It need not appeal as a warrant for its existence to those splendid successes which make it indispensable. Rightly understood its principles by themselves are abundantly clear and beyond all controversy.
We have no cause and no right to follow the theory even into its first and most simple applications, but we can not pass over an important point. Where we can not determine numerically the conditions of different possibilities in a way that is direct, we can proceed indirectly. For example, in the case of a loaded die, I may have no data for calculating the chances, since I may not have accurate knowledge of the conditions. But I can go to the result in another way. I can throw the die a number of times, and, setting down the numbers for every face, can then in view of an unknown throw state the fractions in accordance with the relations of these numbers. But this inverse process implies no appeal to a different principle.
Let us perceive its nature. I assume that I have no reason whatever to think the unknown throw, which I wish to deter mine, different from the rest. I therefore take it as simply the same. But I can not take it as the same as any one,38 for then it must be different from others. It is therefore the same in its general character, with possible alternatives which fall within the data supplied by the actual series. It remains to reduce these possibilities to fractions.
We are obliged to reason from effect to cause. If a known cause A would produce a given effect, and if we have no reason whatever to believe in any other cause,39 we assume we can go from the effect to A. The effect we are considering is a certain series, and the question is, Do we know the one cause which would produce that series?
I hardly think we do. However long and however regular the series may be, we can never say that there is one and but one disposition of elements, which leads and must lead to the series we have seen. And if we could say this, and assume beside that the unknown throw will follow from this deter minate cause, then there would no longer be any probability in the case. The whole thing would be understood and cer tain. But we obviously do not know this one special cause which would produce our series. We can determine no more than its general character. It must be such a cause as would give a series possessing certain numerical relations. And we assume that an arrangement of which we can say, " It is the real possibility, with respect to any throw, of chances disposed in those numerical relations," is such a cause. It is therefore probable that the series is the effect of this cause. And since (by another assumption) we have no reason to believe in any other cause, it is certain that the series has resulted from this cause. And since again we assume that the unknown throw has a general character the same as that possessed by the series, we proceed without any further hesitation to reckon its chances directly.
§ 43. We may notice in passing that, if we had to suppose that the series might arise from some other cause, beside the one we have already mentioned, a further complication would be at once introduced.40 But this we need not consider; for the most simple case of inverse or inductive probable reasoning proceeds as above, and is sufficient to show the principle employed. And we may notice again that there are assump tions involved, which we shall have to discuss in a following section. We may here remark that, if we are not satisfied with a probable conclusion, if we go on to assert that the series has actually been produced by a cause of a certain character, which will operate again in the unknown throw, our assump tion is doubtful, if it is not false. But, to resume, however this point may be decided hereafter, the nature of our reason ing on chances is the same in inductive as it is in deductive probability. The chances of the new throw represent the pro portion of our grounds for belief. The fact that these grounds have been supplied by a series, and the reduction of that series to its actual or probable cause, makes no difference to the principle. What grounds have we got for determining the throw that is to take place? Those grounds which as causes have determined the known series. What are those grounds?
They are those from which we go to the series in hypothetical judgments. What is the nature of these? We do not know them exactly, but, so far as known, we can arrange them as units, and groups of units, which stand to one another in certain relations. But grounds for belief, which stand to one another in numerical relations, are what we mean by the chances of the throw.
§ 44. From this hurried account of the general nature of what has been called the Logic of Chance, we pass to the removal of erroneous ideas. It is evident, in the first place, that probability does not affirm about the fact as such. The event may be past and absolutely fixed, but our alternatives continue to be truly asserted. But, on the other hand, if the chances are not facts, are they nothing at all but our belief about facts? Is probability simply the quantity of the belief we happen to possess? No, that once more would be incorrect. We need not trouble ourselves to discuss the mean ing assignable to "quantity of belief," for the whole idea must be banished at once. The amount of our belief is psychological, the probability of a fact is always logical. No matter what it is we happen to believe in, whether it exist or do not exist, our belief itself is unaffected. But an asser tion about chances must be true or false. It depends on fact and refers to that, though it is not true or false of the special fact in question.
§45. We have not contradicted ourselves. Probability tells us what we ought to believe, what we ought to believe on certain data. These data are assertions about reality, and the conclusion as to what we ought to believe results from a com parison of our grounds for belief. Since these grounds are the conditions of hypothetical judgments, the judgments again must be true or false, and they rest upon categorical bases. In these two points, (i) the general ground of the disjunction, and (ii) the special grounds of the alternatives, probability is true or false of reality. We call it " objective."
On the other hand probability is " subjective." If I say "The probability of S — P is y1^," this may be true although S — P is impossible. It is true to-day, and to-morrow it is true that the chance is -jfo, and the next day |. The belief must change with my varying information, and it is true throughout these variations, and is true though every one of them is an error. How can this be " objective "? It seems to lack the very differentia of truth.
The solution is obvious. Within the probability41 what is true or false is not the premises but the conclusion I draw from them. Given certain assumptions, there is only one way of stating the chances. Given certain grounds for belief or disbelief, there is only one correct inference to the fractional result. This result is neither " subjective " nor " relative," if those phrases mean that it might be different with different men. From certain data there is but one conclusion, and, if this is different in different heads, then one or both of these heads is mistaken. Probability is no more " relative " and " subjective " than is any other act of logical inference from hypothetical premises. It is relative to the data with which it has to deal, and is not relative in any other sense. It starts with certain assumptions about the nature of the fact, and it tells us what, if we are ready to take these assumptions as true, we ought to believe in consequence. If this is not to be " objective " and necessary, then farewell for ever to both these phrases.
Probability as such is not true of the fact, but it always has a reference to fact. It is concerned with certain special deductions from the basis of propositions which are true or false in fact.42 It certainly is confined to those deductions. But it possesses, when kept within its own limits, truth abso lute and unquestionable and that never can vary.
§ 46. Probability is neither simply " subjective " nbr yet simply " objective." This vicious alternative is the first of the errors we have to dismiss. It is allied to another elemen tary mistake, which must next engage us.
It is mere misunderstanding which supposes that chance involves a series, and that the logic of probability is essentially concerned with statistical frequency. It is mere error which finds the necessary meaning of " The probability of S — P is i," in " Once in a series of four events S — P will be true." This mistaken theory contains some truth, but has taken one part of the truth for the whole.
§ 47. Is the series real or is it imaginary? Let us first take it as real, as something that exists, has existed, or will exist. Must the judgment " The chance of S — P is J," refer always and essentially to an actual series? The assertion would be preposterous. The event S — P may be hypothetical. It may have a probability of J on the ground of assumptions which we know are not true. Where is then the real series? The event again may be unique. The chance of my dying before I am forty is, say, ^. Does this mean that if I die three times, one case will realize the possibility? The event once more need not be an event. It need be nothing which ever could happen in time, and we should deceive ourselves if we gave it that name. " It is even chances that the soul is noth ing but a function of the body ": the probability is J. " It is one to two that God is a person ": the probability is -J-. " It is one to ninety-nine that the will is free ": the probability is you-* * Of course I do not mean these fractions as an expression of my opinion.
It may be said, no doubt, that the figures are illusory, and that we can not find any unit of value; but I hardly think this objection can stand. Admit that the case is highly improbable, it still is possible that in the mind of some man the grounds, present for and against such judgments as these, might be reduced to a common denominator. How can we deny it? and, if we do not deny it, what becomes of our series?
§ 48. The series clearly can not be real. Let us take it as imaginary. The question is then, Is such a fictitious imagi nary series the proper way in which to represent probability? Can we say, It is my meaning, or the only true way in which to render my meaning? This, I think, would be an absurdity. It will not stand a serious examination.
Probability can indeed be always represented by a fictitious series. " It is two to one he is guilty " may be rendered by saying, " Two times out of three a verdict on such evidence as this would be right." Even when the possibility is unique, we yet can abstract from that quality and say, " Men such as I am would die before forty two times out of three." Nay, even when we leave events altogether behind us, we still can keep up this mode of expression by a fictitious series. Imaginary judgments here become the events. " It is even chances the soul is a bodily function " may be translated by " In making such judgments as this a man would be wrong through one half of the series and right through the other half."
But is such a way of putting our meaning the real and essential idea we entertain? When we wish to be correct, are we forced so to speak? It always is possible, but is it always necessary? Is it always even natural? And then there re mains a question in reserve, Is it not incorrect?
§49. Let us begin with its possibility. Why can we always express the chances by making use of a fictitious series? For this reason. When the grounds from which we reckon are considered as causes, we are accustomed to suppose that their issue in a series of phenomena will exhibit the same numerical proportions that our fractions possess. If so, then on one side the causes (or cause) of the series and, on the other side, the series itself will answer to each other. We say what we have to say of the cause, indifferently, either by stating its effects, or by setting out the reasons it gives us to expect one effect and not another. This is natural enough where the fictitious series is imagined to be real. It is not so natural with unique events, where the series strikes us as specially manufactured to express the chance. It is still less natural where the possibility itself is not an event, and the series is nothing but the series of judgments. But even here it still is possible. Since psychologically the grounds are causes (p. 545), since, in other words, the logical reasons which necessitate the result are what produces the fact of the judgment, I can imagine, if I please, a series of judgments, and say, Since these numerically answer to the reasons I have, therefore such a numerical part will be true. The expression by a series is here quite unnatural, but it still is possible.
§ 50. The issuing of a certain series is only one way of putting probability. It is sometimes a natural way; it is sometimes a not unnatural way; it is sometimes most un natural. But it is never the right way; it is never more than a manner of statement; it is never the real meaning and in tent. Even when I start from an actual series, I must leave it before I can get to probability. I must go to its cause by what is called a method of reduction, by an inductive hy pothesis. And I can not simply define this cause as that which either has issued, or will issue, in a certain series. I can not do the first, for that would be certainty and not probability. And I can not do the second without an assumption which I am unable to justify.
It is obvious, in the first place, that to take a series, and to say " The cause which has produced this series — has pro duced this series " is merely frivolous. On the other hand, if I add " will produce this very same series on other occasions," that is not frivolous, but is either irrelevant or else unjustifi able. If it means " In another case where the conditions are not discrepant, the same cause will be followed by the same effect," that assertion is true but is quite irrelevant, because merely hypothetical. For in an actual fresh case I do not know the fresh conditions, and, if I did, I do not know what the old cause specially is. I do not know the actual cause (or causes) of the former series. I do not know that these are present again in the unknown case. I do not know what conditions the fresh case brings; and, if I did, I might be unable to deduce the result from the complication of elements. In short I can not go from a given series to an unknown series or an unknown case. To reason directly is of course impossible, and I can not reason indirectly through the cause, because I do not know the actual cause in one case or the other. Its general character, to a certain limit, I do know in one case, and assume in the other, but this general character does not imply a series, and the individual cause itself I do not know and so can not use.
The upshot of this is that within probability you really have not got the effects on one side and the cause on the other. If then you give as the essence of probability the produc tion of a series with certain marks, you go beyond what your data will warrant. For your actual series has now 43 ceased to be taken as a series of events produced in time. It has degene rated into a set of conflicting reasons, possibilities as to an event of a certain sort, which in default of detailed information I use in order to determine my judgment. My probabilities do not represent a series as such. I now have nothing what ever but conflicting grounds for belief and expectation, grounds for belief as to any fresh case or number of cases that have the general character of my series. And these fractional reasons, which are all I can work with, are the same in any one new instance as in any number of new instances. Thus the sup posed differentia of an imagined series, in the first place, would add nothing to the probability which already exists apart from the idea of any series. But, in the second place, if it does add, and if it goes on to say that the series must have a character answering to the expectation, then it adds what is false.
§ 51. And with this we come to an obstinate illusion. There is a common idea that, if you know the chances of any set of events, you really know the character of the actual events which are to take place. It is supposed that the series will correspond to the fractions. For instance, if we take the case of a die, the chance of any one face is J-, and from this we argue, " In a series of throws each face will be seen in onesixth of the run." But we have no right to any such assertion. Not knowing the cause, knowing only a part while part is hidden, we can say no more than that onr information leads us to expect a certain result. It is monstrous to argue that therefore that certain result must happen. It is false reason ing a priori, and a posteriori the facts confute it. It is not found in experiment that actual runs do always, or often,44 correspond exactly to the fractions of the chances. That cor respondence is after all the most probable event, but to make it more is a fundamental error.
§ 52. I shall return to the truth contained in this error, but at present we must try to get rid, if we can, of the error itself. We may expect an objection. " Experiment," it will be said, " does not disprove the assertion that is made. That assertion is not that in a finite series the numbers will come right. They will come right only if we go on long enough, and in the long run." But what is this " long run "? It is an ambiguity or else a fiction. Does it mean a finite time? Then the assertion is false. Does it mean a time which has no end, an infinite time? Then the assertion is nonsense. An infinite series is of course not possible. It is self-contradic tory; it could not be real. And to say that something will certainly happen under impossible conditions, is far removed from asserting its reality. The affirmation that an event may be assumed to take place in an infinite series, and not outside it, would, in the mouth of any one who knew what he meant, be a suggestion that the event may not take place at all.45 § 53. I hope I need not protest that I am hardly so foolish as to attempt to offer an ignorant objection to the use of infinities and infinitesimals within the sphere of mathematics.40 I would rather say nothing at all on this matter than appear as presuming to doubt the validity 'of processes employed by the greatest men in the exactest of sciences. But I shall not so be misunderstood. An objection to the use within cer tain sciences of certain ideas must be taken within the limits of those sciences. But the use of these ideas outside their science carries with it no authority, and, so long as the general meaning is understood, may be criticized by men who are igno rant of the science in which the ideas give brilliant results. It is so with infinity. Outside mathematics an infinite number is an idea that attempts to solder elements which are abso lutely discrepant. It could not exist until the world, as known in our experience, was utterly shattered and transmuted from CHAP. VII THE MODALITY OF JUDGMENTS 22Q the roots. I could not find an illustration I would sooner use to express impossibility. And it is this idea which, out side mathematics, is presented to us in the error we are combating. Mr. Venn, for whose powers I feel great respect, and from whose Logic of Chance we all can learn, holds that in the long run every chance will be realized. This •" long run," he tells us, is an infinite series (p. 146), and (unless I very much misunderstand him) he goes on to call it a " physi cal fact" (p. 163). His book is much injured by this terrible piece of bad metaphysics. He has translated a mathematical idea into a world where it becomes an absurdity.
§ 54. We must everywhere protest against the introduction of such fictions into logic, and protest especially where the ideas are not offered in the shape of fictions. The formula of the " long run " must be banished from logic, and must carry with it a kindred illusion in the imbecile phrase, " if you go on long enough." " The event," we are told, " will answer to the chances." But it does not answer. "Oh, it will, if you only will go on long enough. You toss a coin and, the chances being equal, if you only go on long enough, the number of heads and tails will be the same." But this is ridiculous. If I toss the coin until the numbers are equal, of course they will be equal. If I toss it once more then, by the hypothesis, they become unequal I might just as well say, " If I only go on long enough the events will certainly not answer to the chances." 4T Your formula is false or else tautologous. If it means " Suppose the numbers are equal, and sup pose I then stop, the numbers will be equal," that is surely tautologous. But if it means the numbers will turn out equal in an infinite series, then that is false, for such a series is im possible.* § 55. But let us turn from the error and see the truth which lies hid beneath it. It is false that the chances must be realized in a series. It is however true that they most probably will be, and true again that this probability is in creased, the greater the length we give to our series. What * Cf. Lotze, Logik, 437. I may remark that if the formula meant, " The series is sure to cross and re-cross the point of equality," then, in the first place it would be false, since there is no certainty; and, in the second place, such an oscillation is not equality.
reason have we for holding these two beliefs? (i) Why do we think that the series will probably answer to the fractions? (ii) Why do we think that in a longer series the correspondence is more likely?
(i) Probability, we have seen, is not essentially concerned with any series. It is based upon grounds which, even if we consider them as real, may not be causal in the sense of pro ductive of events in time. They may be causes cognoscendi and not essendi.48 It is when our grounds are grounds for belief as to the nature of an agency, which is to produce events in time, that we are able to consider them as causal elements. And this is the case we have to suppose.
We know that a series is to be thrown with a single die. Let us first take one throw. That will have a cause, and the cause is only partially known. We know that it is complex and consists of many elements. Of these elements, so far as they are distinctly known, five parts are hostile to any single face and but one part favourable. The unknown residue, so far as it determines the case, is quite unknown; and, though it is not indifferent and though it can not be so, yet within our knowledge we must take it as indifferent. In the cause of the single throw there are therefore, beside the unknown factors, one sixth part of the agencies favourable to each face.
Now take the whole series. That series, before I throw it, is as certain and fixed as though I had thrown it already. But here again I do not know the causes. About one part I know nothing in detail, and so I must take it as being in different, although I am sure it is not so in reality. Of the rest of the agencies, which I suppose, one sixth is favourable to each face, and five sixths hostile. What conclusion can I draw as to the nature of the series? Will one agency pro duce that result which we suppose it would produce, did the others not intervene? Will in each case of the series the sup posed majority of agents prevail? We have no means of knowing. The series, absolutely fixed, is fixed by what we do not comprehend. We must take the possibilities, and the possibility for which there is most ground is the likeliest. There is less ground to think that in a series of six throws one face will be absent, and one twice present, than that all should show once.49 In the latter case we do but make ignorance a ground for complete indifference. In the former case we give a preference without any kind of warrant. It is not that each face has any sort of claim to come uppermost once. It is that no face has more claim than another to show itself twice. This is why we think the most likely series, or the least unlikely, will be that which corresponds to our fractions.