§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 226 THE PRINCIPLES OF LOGIC BOOK I 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 CHAP. VII THE MODALITY OF JUDGMENTS 227 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 one- sixth 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 228 THE PRINCIPLES OF LOGIC BOOK I 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 tauto- logous. 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.
23O THE PRINCIPLES OF LOGIC BOOK I 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 CHAP. VII THE MODALITY OF JUDGMENTS 231 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.
§ 56. (ii) But why, it may be asked, does the length of the series increase this probability? Does the greater length add any new ground to those we have for believing in the correspondence of the events with the chances? No, it does not add any. Does it decrease any ground we had before for thinking the opposite? Yes, it does do that; and it does it, I think, in the following way. The unknown residuum in the cause of each throw was assumed to be indifferent. But it was not at all assumed to be passive. It supplies the determining element in the cause. It decides for one face, though we do not know for which face it decides. Now how does it de cide? Does it act regularly and in strict rotation, or is it ir regular? That we do not know; but, taking the possibilities, we believe that those in favour of irregularity are more than those in favour of rotation. It is therefore most probable that our series will turn out to be irregular. But, since we know no reason to prefer any one face, we can not say that any pro portion other than strict equality is the most probable. How are these assertions to be reconciled? Very easily in this way. Owing to the assumed indifference of the causal residue the faces will probably appear in their right numbers; but, because of its irregularity, their appearance will probably be irregular, and irregular to an extent to which we can assign no limit. To combine these attributes it is necessary to sup pose that the whole series will be most probably regular, but will contain periodic irregularities. The greater the irregu larity becomes, the less grows the chance of a final regu larity, unless the series is proportionately lengthened. There fore, since we can fix no limit to the irregular sequence of the faces, we conclude that, the longer the series becomes, the greater becomes the probability of a regular result. And this is a rational, and necessary conclusion from our imperfect data.
232 THE PRINCIPLES OF LOGIC BOOK I § 57. It is true that, if you make a series longer, you de crease the chance of irregularity. It is true that, if per impossibile the series were so long that, in comparison with its length, every possible abnormal run was a period which other periods might easily balance in the completed cycle — if, I say, per impossibile this phantom could be real, it is true that the above chance of irregularity would vanish. If we as sume that what we do know gives us reason to believe in a series correspondent to our fractions; if we next assume, by virtue of a fiction, that the unknown residue gives no reason to believe in an unbalanced irregularity, then on these assump tions we may go to a conclusion, and we have no ground to disbelieve the statement that the series will exhibit the relations of the chances. But the first assumption is based on ignorance, and the second is based on a known impossibility.50 If we mean to speak about a series of events that could ever happen, we can say but this. It is certain there will be a series, each throw of which will give a single face. It is possible that in a series of any length but one single face should appear throughout. No arrangement is impossible. It is most prob able that the events will answer to the fractions, but against that probability there still remains another consideration, the chance arising from the possible irregularity of one part of the causal elements. This fraction is diminished by each increase of the series, but it does not disappear and it can not disappear.
§ 58. We do not know that in the long run the events will correspond to the probabilities. We do not know that, if we go on long enough, every chance will be realized. It is mere superstition which leads us to believe in the reality of the fiction which gives birth to these chimaeras. When I see the demonstrations, offered to gamblers against a bank, which prove to them that in the long run they can not but lose, I say to myself, On which side do I see the darker illusion? And I answer, On both sides the illusion is the same. For what is the root of the gambler's " system "? Is it not his belief that independent events are affected by each other? But this belief is a strict deduction from the premises offered him. If he really must lose, if there really is a cycle in which the chances must all be realized, then, let him observe the begin ning of the cycle, and mark the irregularities, and he surely CHAP. VII THE MODALITY OF JUDGMENTS 233 must win. Since to equalize the numbers the end of the cycle must balance the beginning, he can speculate on that balance and his " system " is right. " Oh, but it is wrong, for the series is not finite. It is only after an infinite duration of play that the balance is struck. It is absurd to say he can be sure of winning." But is it not then equally absurd to say that he is sure to lose? If you mean he must have lost by the end of his life, you have just admitted your assertion to be false. If you mean he must have lost when he has got to the end of infinite time, confess that your meaning is something like nonsense, and that the gambler is right in imagining that you, as a rational man, must mean something else. The truth is that your common assumption is false. There is no must about it. The chances consist of grounds for belief in the nature of a series no event of which is known. And all they tell us is this: that we have more reason to expect one thing than we have to expect another, and that the increased length of the series proportionately decreases a reason for doubt, which never quite vanishes.
§ 5Q.51 I must not be suspected of a desire to intrude into mathematics if, in this connection, I venture to remark on a well-known paradox. I am to toss a coin, and to go on toss ing so long as I throw heads and nothing but heads. I am to receive £2 if I throw head once; if I throw head twice I am to win £4; for three successive heads I get £8, and so on accordingly. The series is supposed to have no limit except the appearance of a tail. And the question arises, how much am I to pay for the privilege of one single trial? The answer given is, An infinite sum; for it is possible I may throw an infinite series of nothing but heads (vid. De Morgan, Proba bilities, p. 99). The reasoning on which this conclusion seems to rest is exceedingly simple, and I need hardly say that I do not doubt its perfect validity within mathematics. And I think I see that no other answer can possibly be given. Unless an arbitrary limit is fixed, I may be allowed to say in all humility that I think I understand that, if this possibility has any value at all, then the worth of my chance is either incalculable or else is infinite. If this answer is given me by a special science, I dutifully receive it as true— within that science.
But if I am told that in actual fact the result is true, I 234 THE PRINCIPLES OF LOGIC BOOK I must be allowed to protest. I must be permitted to remark that the reasoning is absurd and the result is nonsense. I do not mean merely that it is absurd if we take it as a practical precept, because a man can not live for ever, and all the money in the world is finite. I mean that it is a theoretical absurdity. It is not true ideally any more than really. Since an infinite sum is an impossibility, the infinite series can not possibly be thrown. There is no chance whatever. There is no fraction at all. It is nothing I could win. It is nothing I can expect. It is nothing for which I can reasonably pay. The result is a deduction from premises known to be false and impossible.
It is idle to answer that the problem is " stated in the ideal form " (Venn, ibid. p. 137). There is a difference surely between ideals which as such do not exist, because they are abstractions, and ideals which are downright self-contradic tions. It is one thing to say, " There is a connection between abstract elements, so that when one of these is found as a real quality we shall have the other," and another thing to continue this assertion, when we know that the first of these elements is self-contradictory and could not possibly be any quality of reality. In this latter case what is true of fact can not be the consequence of an impossibility, but only the basis of the hypothetical judgment. Neither antecedent nor con sequent is taken as real or even as possible. But in a com mon abstract judgment the antecedent is taken as at least a possible quality of the world.52 Mr. Venn perhaps would question this difference between an abstraction and an im possibility, and would perhaps assert that an infinite series is really possible. In any case I must be allowed to protest against the invasion of logical reason by mathematical fic tions. If an infinite series is thought possible, we should be told how it can be possible. If it is not thought possible, it should not be offered us as if it were.
§ 60. There are other points in the theory of chances which have logical interest, but we have no space to discuss them here. We have said enough to make clear the relation in which that theory stands to our general principles. We have to avoid the fiction of the infinite long run, and the vicious alternative of " objective " and " subjective," and the CHAP. VII THE MODALITY OF JUDGMENTS 235 false assumption that the essence of chance involves a series of events in time. If we keep clear of these pitfalls, the truth is by no means difficult to reach, and we hope above to have stated it clearly in its general form.* § 61. There is an aspect of modality we have neglected to notice. The omission was intentional, and the mention of this aspect has been reserved for the present place. There is an old doctrine which connects universality with necessity, and that doctrine is true. The necessary we saw was the ideal consequent, and such a consequent can not come except from an ideal antecedent. You never can say " B follows from A," " is because of A," " must be, given A," unless A is present in a determinate form. A must be a content without any mixture of mere sensuous conditions.53 It must be ideal, abstract, and so universal. If the ancient doctrine on its logical side may suffer some loss, since necessity becomes for logic hypothetical, yet it stands all the firmer. The " because " can not couple anything but universals.
§ 62. We may notice an error which creeps in with this truth.5* The antecedent in necessity must be universal, but it need not be more universal than the consequent. Where we say " because " we do not always appeal to anything more abstractly general than that which follows from our reason. " A must be equal to B, because C is equal to both B and A," " A must be removed by one foot from C, since B, which touches both in a certain manner, is one foot long." The consequence is not less general than the antecedent, and we deceive ourselves in thinking it always must be so.
No doubt in the cases where you say " because " you may find what we call the principle of the sequence, and that of course must be more abstract than the actual consequent. But the principle is not the antecedent itself. It is the base of the general connection, not the sufficient reason of the particular consequent. There is no more need for the con sequent to be more concrete than the antecedent, than there is for the effect to be more special than the cause. These ideas are nothing but kindred illusions (Book III. Chap. II.).