In this chapter we continue our discussion of imperfect induction in formal logic, and more particularly a discussion of the principle that relative chance cannot happen permanently and uniformly, being the rational ground of the validity of imperfect induction.

    The Aristotelian principle rejects the uniform repetition of relative chance in a reasonable number of observations and experiments. Now suppose that such reasonable number is ten; then, the Aristotelian principle means that if there is no causal relation between a and b, and found a ten times, b would be absent once, at least among those ten times, for if b is related to a and those ten times it would mean that relative chance happens in ten times, and that is which the principle rejects. And when the principle shows that any two phenomena not causally related do not come together one time among the ten times, that principle does not specify the experiment in which the two phenomena do not relate; thus the principle involves a sort of knowledge of an indefinite rejection. There are in our ordinary state of affairs instances of knowledge of indefinite rejection: we may know that this sheet of paper is not black (and that is knowledge of definite rejection), but we may know only that the sheet cannot be black and white at the same time (and this is knowledge of indefinite rejection). The sort of knowledge which rejects something in an indefinite (or exact) way may be called indefinite knowledge, and the sort of knowledge which involves a definite rejection of something may be called definite knowledge in consequence, the Aristotelian rejection of relative chance is an instance of indefinite knowledge.

    We may easily explain how definite knowledge arises. If you say 'this sheet of paper is not black', this may depend on your seeing it. But if you say of a sheet of paper that you do not know its definite colour, and that it must not be black and white at the same time this means that one of the two colours is absent, and this is due to your not seeing the paper. For if you saw it clearly, you would have specified its colour, then you assert your indefinite knowledge as a result of the law that black and white cannot be attributed to one thing at the same time. Such indefinite knowledge arises in two ways.

    First, I begin with the impossibility of conceiving two things to be connected with each other, thus we have indefinite rejection, e.g., I exclude the blackness or whiteness to be predicated of a sheet of paper; this is a result of recognizing that black and white cannot come together in one thing [it can mix together to become grey colour for example, but then it won't be fully black or white which the author meant in the example, reader's note]. Secondly, one may not conceive the impossibility of two things to happen together, but only know that one of them, at least does not exist. Suppose you know that one of the books in your study is absent, but you did not specify the book; here you have knowledge of indefinite rejection; nevertheless there is not such impossibility among the books being put together as that impossibility of black and white being together. Thus our knowledge of indefinite rejection may depend on definite rejection (the loss of a book) without specifying it.

    We may now conclude that knowledge of indefinite rejection arises either from conceiving the impossibility of two things coming together, or from definite rejection without specifying it.

    The Aristotelian principle of rejecting relative chance, is now shown[?] to be due to a sort of knowledge of indefinite rejection. We have also previously shown that knowledge of indefinite rejection arises from impossibility or from unspecified possibility. Now, we may claim that the rejection of concomitance, at least, in one experiment is an indefinite knowledge on the basis of impossibility, that is, relative chance does not happen in one of those ten experiments. We may also claim that the rejection of concomitance in one experiment at least is an indefinite knowledge on the basis of unspecified possibility, that is, it is definite rejection in fact but unspecified to us. In what follows, we shall try to make clear our position in relation to that Aristotelian principle and deny that it is a rational a priori principle and thus not a logical ground of inductive inference.

    When there is no causal relation between a and b and bring out a in ten consequent experiments, the Aristotelian principle would assert that b is not concomitant with a at least once in those experiments if we take nine the maximum number for recurring relative chances. We maintain that indefinite knowledge of denying at least one relative chance is not explained on the ground of our conceiving impossibility between relative chances, that is, similar concomitance which do not occur owing to causal relation.

    For example, suppose we want to examine the effect of a certain drink and whether it causes a headache; we give the drink to a number of people and observe that they all have headache. Here we observe two things, the association of that drink with headache (this is something objective); and a random choice by the experiments (this is something subjective). If there is really a causal relation between the drink and headache, these two associations are natural result of that relation, and there is no relative chance. But if we know already that there is no causal relation, then there is relative chance; we then [???] whether relative chance apply to objective concomitance between drink and headache or subjective concomitance between random choice of instances and headache.

    It is possible that I consciously choose those persons susceptible for headache and subject them to experiment, and then I get a positive result which actually happened by relative chance. It is also possible that random choice is associated with headache. For suppose that relative chance would not be repeated ten times, the experimenter may choose randomly nine persons, but if so, he would be unable to choose randomly any of those persons since relative chance cannot occur ten times.

    It is not the number of relative chances that is important, but their comprehension of all the instances which belongs to one of the two phenomena. When we have two phenomena a and b and observe[d] that all the instances belonging to a are concomitant with b, it is impossible that the concomitance between b and a is by chance. But if we observed that a limited number of instances belonging to a is concomitant with b, it is not impossible to have connected by chance.

    We may face three phenomena a, b and c; when all instances of c are concomitant with b which are at the same time members of a, but we know nothing of the concomitance of other instances of a with b, then if you suppose that c is not a cause of b, we may conclude that a is cause of b, and say: all a is connected with b. Now, we may get an explanation of inductive inference under two conditions:

    (a) Complete concomitance in the sense that we add c to a and b, and that the observed instances of b would be all instances of c, but not all instances of a.

    (b) Previous knowledge that c is not causally related to b. When these conditions are fulfilled, we have two alternatives either a is cause of b and then no chance of b, and then c and b are concomitant by chance. But our discussion excludes complete chance, thus, a is cause of b.

    In every instance which involves incompatible things, we may utter hypothetical statement, namely, even if all factors for those things are coexist, they never do so by reason of their incompatibility. Suppose a room is too small to gather ten persons, then even if all of them are to enter that room, they could not. Now, concerning the possible repetition of relative chance, we are certain that such chance cannot recur uniformly. If you randomly choose a number of persons and give them a drink, we are sure that they would have headache by chance, but at the same time we cannot apply the previous hypothetical statement.

    Now, though we believe that relative chances do not occur regularly and uniformly, we cannot assert that they should not occur. Thus our assurance that the concomitance between having a certain drink and headache cannot be repeated uniformly does not arise from the incompatibility of such concomitances.

    We try to show in this objection that the indefinite knowledge on which the Aristotelian principle is based does not depend on probability. So, we must recognise that any indefinite knowledge is a result of the occurrence of a positive or a negative fact, but that indefinite knowledge of such fact depends on our confusing a fact with another. For example, if we are told by a trustworthy person that someone is dead and called his name but I could not hear the name clearly; in such a case we have an indefinite knowledge that at least one person died, that such knowledge is related to the fact of a certain death but the fact is said vaguely. Thus indefinite knowledge, resting on hesitation or unclear information, is related to a definite fact referred to vaguely, and any doubt about it causes such knowledge vanish.

    Now, taking notice of what formal logic says of relative chance and that it cannot recur consistently through time, we find that indefinite knowledge of this is not related to denying any chance in fact, and this means that the indefinite knowledge, that at least one instance of relative chances did not occur, does not rest on hesitation or probability. Chance happening which can be referred to vaguely is not a ground of indefinite knowledge, while the event of death which is referred to vaguely is a ground of the indefinite knowledge that someone is dead. Thus, we think that indefinite knowledge of the non-occurrence of at least one chance does not vanish even if we doubt in any chance referred to vaguely.

    Here we try to reject the idea of a priori indefinite knowledge based on analogy and hesitation. That is, we try to argue that the knowledge of the non-occurrence of chance at least one out of ten times is not an a priori indefinite knowledge. To begin with, we wish to define a priori science for formal logic. There are two sorts of a priori science in formal logic; ultimate rational sciences including ultimate beginnings of human knowledge, and rational sciences derived from those, and deduced from them. [a priori science or a priori knowledge; and, is it primary rational knowledge vs. secondary knowledge??? Translation problems]

    Both have a common basis, namely, that the predicate is attached to subject of necessity. It is not sufficient, in order for a science to be a priori, to attribute something to a subject but they must be attributed necessarily.

    This necessity is either derived from the nature of the subject or issued from a cause of the relation between subject and predicate. In the former, the statement is ultimate, and our knowledge of it is a priori of the first sort. If the terms are causally related, the statement is deduced, and our knowledge of it is a priori of the second sort. And the cause is called by formal logic the middle term. For example, the indefinite knowledge that a headache cannot occur By chance at least once in ten cases cannot be a priori knowledge, as formal logic is ready to claim. Such indefinite knowledge, if it rests on analogy and hesitation, is related to a chance in fact. We know that something really happened but we are unable to specify it.

    Now, we may argue that such knowledge is not a priori since we do not know whether this chance did not happen or it is necessary not to occur. If such knowledge means just the non -occurrence of the chance happening, then it is not a priori knowledge, since this involves a necessity between its terms. Whereas if such knowledge means the necessity of its non-occurrence, then such necessity is out of place in a table of chance. If we know that someone who had a drink, had a headache ten regular times, then we have no reason to deny that he got headache in any one of these ten times. But we supposed his feeling of headache for no sufficient reason, we believe that headache had not occurred to him in one of those ten times. Thus the knowledge of the non-occurrence of headache in some cases does not arise of a priori idea of the cause, just because we do not know the causes of headache.

    Formal logic is mistaken in claiming that indefinite knowledge of regular recurrence of chance is a priori knowledge. For it says it of indefinite knowledge that if there is no causal relation between (a) and (b), then there is uniform concomitance between them. Suppose such concomitance to be ten successive occurrences, we may conclude that (a) is cause of (b) if ten times succession is fulfilled. For example, if (a) is a substance supposed to increase headache, (b) the increase of headache, and ten headached-persons got the treatment and they got more pain, we conclude that regular relation between (a) and (b) is causal and not by chance. Suppose we later discovered that one of the ten persons had got a tablet of aspirin, without our knowing it; this discovery will falsify our test and our experiment was made really on nine persons only. And if ten experiments are the minimum of reaching an inductive conclusion, then we have got no knowledge of causal relation in that experiment.

    Thus, any experiment will be insignificant if we realise that besides (a) and (b) (supposed to be causally related) there is some other factor which we had not taken notice of during the experiment. Thus, formal logic fails to explain these facts within its theory of justifying induction, which presupposes indefinite knowledge that chance cannot recur uniformly. For if such a priori indefinite knowledge were the basis of inductive inference and discovering a causal relation between (a) and (b), our knowledge of causality would not have been doubted by our discovering a third factor with (a) and (b). This discovery denotes the occurrence of one chance only, and this does not refute our a priori knowledge, supposed by formal logic, that chance cannot recur regularly in the long run.

    The only correct explanation of such situation is that knowledge that chance does not happen at least once is a result of grouping a number of probabilities: the probability of the non-occurrence of chance in the first example, in the second,... etc. If one of these probabilities is not realised, i.e., if we discover a chance happening even once, we no longer have knowledge of such probabilities. And this means that this knowledge is not a priori.

    When we start an experiment to produce (a) and (b), and think of the sort of relation between them; we are either sure that (c) does not occur as cause, or we think that its occurrence or non-occurrence is indifferent to the production of (b). Concerning the first probability, formal logic is convinced of (a) being the cause of (b), since (c) does not occur. Then we need not, for formal logic, repeat the experiment. On the other hand, we may find that our knowledge of causality in this case depends on repeating the experiment and find the causal relation between (a) and (b). The reason for this is to make sure of the effect of (c); that is, the more (c) occurs, the less (a) is believed to be the cause, and vice versa.

    This means that inductive inference of the causal relation between (a) and (b) is inversely proportional to the number of cases in which (c) occurs. Thus, unless we have a priori knowledge that (b) has a different cause in nature, we tend to confirm the causal relation of [???] and (b). For the probability of the occurrence of (c) is low. The connection between inductive inference to causal relation and the number of the probabilities of (c) occurring in many experiments cannot be explained by formal logic. For if induction is claimed to be a result of an a priori ultimate knowledge that there is no relative chance, then the more we get concomitance between two events, we conclude the causal relation between them, minimizing the effect of the occurrence or non-occurrence of (c).

    If we assume that the long run, in which we claim that relative chance does not recur, is represented by ten successful experiments, then the concomitance between drink and feeling of headache in nine successive experiments is probable, but not probable if the concomitance happens in ten successive experiments.

    Now, we try to argue that such knowledge is not an immediate datum given a priori. First every a priori rational knowledge of something necessarily implies a priori knowledge of its consequence. Secondly, if it is true that relative chance cannot uniformly recur rational statement. The problem of the probability of absolute chance is overcome by assuming the principle of causality. The problem of the probability of relative chance is overcome by denying its uniform recurrence in the long run. The problem of doubling uniformity in nature is finally overcome by assuming a statement derived from causality, namely, like cases give like results.

    Such situation may be summarised in two points. First, formal logic maintains that inductive inference requires three postulates to meet its three problems, thus acquires the desired generalisation. If these postulates are shaken, inductive science collapses. Second, formal logic maintains that the principle of causality, the denial of the recurrence of relative chance, and the statement that like cases give like results are all a priori rational statements independent of experience. Hence, its postulates are accepted.

    Our previous discussion was confined so far to only one of those three statements, namely, the denial of relative chance. We have concluded that such statement is not a priori; it cannot work as a postulate of induction. In our view, formal logic is mistaken not only in regarding such statement a priori, but also in claiming that inductive inference needs a priori postulates. We shall later see in this book that induction may work without any a priori postulates, that postulates, given by formal logic may themselves be acquired by induction.