Now it is one thing to say " Whatever I judge true holds good of reality," and another thing to say " What I fail to judge true is absent from reality." And there is this very great difference between them. In the first case we assume that, whatever else may be, at least so much is true. In the second we go so far as to say that what we have in our minds is co-extensive with reality. But, if we hold to this, we ought to go further. What the real does not exclude is not possible, it is actual and necessary (pp. 118-19). And if we shrink from this assertion, ought we to maintain that X is even possible?
§ 30. The mistake is apparent. A privative judgment (as we saw in Chapter III.) is not true of a subject, if that subject is confined to something without the sphere of the predicate. It then becomes obviously frustrate and unmeaning. You can not predicate absence unless you predicate the positive space from which the absent is lacking (Chap. III.). We shall find that this holds good of ultimate reality. To say of it, " It is without the rejection of X," is to say of it something which has no meaning unless, so to speak, the place left empty by this mere privation is occupied by a positive attribute. We ought to be able to say There is a quality the presence of which guarantees, or goes to guarantee, the absence of the CHAP. VII THE MODALITY OF JUDGMENTS 215 exclusion of X. But this quality would obviously be either the presence or compatibility of X. It is on the ground of this presence or compatibility that we ought to assert the possibility of X. For otherwise we fall into circular argument.
I will give an illustration. Suppose I were to say that an isosceles triangle with three unequal angles is certainly pos sible, and possible because it is not impossible. The universal triangle, so far as I am supposed to know it, tells me nothing about the nature of the isosceles.32 On the privative judgment that the universal triangle does not reject my idea,, I call it possible. Is not this absurd? It is absurd, because a privative judgment, where the subject is left entirely undetermined in respect of the suggestion, has no kind of meaning. Privation gets a meaning, where the subject is determined by a quality or an environment which we have reason to think would give either the acceptance or the rejection of X. But, if we keep entirely to the bare universal, we can not predicate absence, since the space we call empty has no existence.
§ 31. Or if our privative judgment has a meaning, then it has a false meaning (Chap. III.). It rests on a confusion between the universal and its psychological existence. We take the idea, as we find it existing within our minds as a psychical event, and then confound the determination it so gets with its logical qualities. We say Here is a fact, and we can not find that it does reject X. But the answer is simple. In the first place we have the reductio ad absurdum. Since the real has a quality on the ground of which it must accept or decline every possible suggestion (Chap. V.); and since the real here ex hyp. does not decline, it therefore must accept. X is not possible, it is actual and necessary. In the next place we directly deny the premise. In your experiment you have not got the reality, and you ought to know that you have not got it. If you wish to determine your empty universal so as to get an answer in regard to X, you have nothing to do with the psychological setting of this universal. The psychical environment is not the space which, in respect to X, must be full or empty. It is quite irrelevant and must be discarded. You must fill out your idea by adding to its content. When the content is supplied to such an extent that, in saying, " Re jection of X is still absent/* you mean that some of the condi- 2l6 THE PRINCIPLES OF LOGIC BOOK I tions of X are already present — when you mean that there are qualities which do affect the prospects of X, that a part of that attribute, which when complete will accept or reject X, is already there and that part is favourable — then I admit you may found possibility on your privative judgment.33 The com plaint I make is that your proceeding is frivolous. You have in your hands the positive ground on which your judgment is based directly, and you choose to proceed in a way which is indirect and in this case circular (Chap. V. §28).
We should never trust a privative judgment until we have seen its negative form. We should never trust a negative judgment until we have seen its affirmative ground. We should not take our impotence as a test of truth, until we at least have tried to discover the positive counterpart of that failure. The observance of these rules might preserve us from errors which sometimes are dangerous.
The relation of necessity and impossibility to our mental impotence is a subject which would carry us beyond the present volume. We shall add some remarks in our concluding Book. In the present chapter we have yet to see how modality is the passage from judgment to reasoning. But, before we indicate that transition, we must rapidly deal with a most important application of modality, so far at least as to show its connec tion with our general view.
§ 32. If Logic professed to supply a method for the dis covery of truth, the logician could not mention the theory of Probability34 without shame and confusion. The fruitful results of the modern rival would offer themselves in damag ing contrast with the sterility of the old and privileged veteran. And, where a true view of the claims of logic makes this con trast impossible, the logician, it may seem, has no right to trespass within the limits of another science. The objection is heightened when the writer on logic confesses himself un acquainted with mathematics. He may appear in this case to be talking about things of which he knows nothing.
But the objection rests on a misunderstanding. The prin ciples on which probabilities are reckoned, the actual basis and foundation of the theory, are not themselves mathematical. Before mathematics can deal with the subject some assump- CHAP. VII THE MODALITY OF JUDGMENTS 2I/ tions are necessary; and, though these assumptions can be justified by their results, it is desirable to examine them simply by themselves, to see what they are and whether they are true. An enquiry of this sort, by whomsoever it is made, is a logical enquiry.
§ 33. Probability, we know, has to do with possibilities. And starting from this, at the point we have reached, we can go at once to an important result. No statement we make about probabilities can, as such, be true of the actual facts. This is half the truth, and we must not forget it. But it is not more than half, nor is it even the half best worth remembering. It is just as true that an assertion about chances does make an affirmation about reality. Every hypothetical judgment, we have seen, must rest upon some categorical basis. The con clusions we have adopted enable us to say without further enquiry, Any theory which calls the doctrine of chances merely " objective," or merely " subjective," is certainly false. It is a vicious alternative which, if it were sound, would upset general results we have found to be true, and which is con trary to the special facts of the case.
§ 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, 2l8 THE PRINCIPLES OF LOGIC BOOK I 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 CHAP. VII THE MODALITY OF JUDGMENTS 219 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 pro- positio, si subjecto predicatum tribuitur ob plura requisita ad veritatem, quam si tribuitur ob pauciora."
22O THE PRINCIPLES OF LOGIC BOOK I 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 CHAP. VII THE MODALITY OF JUDGMENTS 221 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 222 THE PRINCIPLES OF LOGIC BOOK I 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 in- CHAP. VII THE MODALITY OF JUDGMENTS 223 correct. 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 224 THE PRINCIPLES OF LOGIC BOOK I 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.
CHAP. VII THE MODALITY OF JUDGMENTS 225 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?