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Chapter 3: InductionAnd Empiricism

The rationalistic theory of induction and its justification had been sketched in the previous chapters. Now, we turn to discuss the empiricist theory of induction, and its solution to the problems involved from a logical point of view. By empiricism is meant the assertion that experience is the source of all human knowledge and the refusal of any a priori knowledge independently of sensible experience.Inductive inference faces, as has already been noted, three main problems; (1) why should we suppose a cause of (b), excluding absolute chance for its occurrence? (2)if there is a cause of (b), why should we suppose that (a) is its cause being concomitant with it, and not supposing that (b) is connected with (c) by relative chance? (3) If we could make sure, by inductive process, that (a) is the cause of (b), on what ground can wegeneralise the conclusion that alla's would be causes ofb's ? Formal logic solved the first and third questions by appealing to certain a priori principles on rationalistic lines, and solved the second problem by supposing another a priori principle denying the systematic repetition of relative chance.

Since empiricism refuses any a priori principles and any rationalistic interpretation of causality, it was mainly concerned with giving different answers to the first and third questions, giving the second question a minor importance. We may make the empiricist theory clear if we distinguish three attitudes: certainty attitude, probability attitude, and psychological attitude.

Certainty Attitude

J.S. Mill, one of the greatest British empiricists, may be a representative of this attitude because he held the view that inductive conclusions are certainly true. His views on induction may be clarified if wesummarise his positions on the first and third questions, then on the second question, already referred to.

On the First and Third Questions

Certainty attitude holds that inductive inference has two notions as its ground, namely, causality and uniformity in nature, these are first premises. Like causes have like effects. It agrees also with formal logic in regarding inductive inference as a syllogism, the minor premise of which expresses particular instances, and its major premise expresses those two beliefs in causality and uniformity. It will be found for example that the extension of iron is, in all cases, concomitant with the occurrence of heat, and it is then concluded that when a phenomenon occurs under certain conditions, it occurs all the time in these conditions.

Further, empiricism differs from rationalism in considering causality. For whereas the latter takes causality as a priori principle, the former reduces it to sensible experience, that is why Mill maintains that our belief in causality is the outcome of widespread inductions in the physical world. We derive our concept of cause from induction, but once so derived, this concept becomes our basis of any subsequent inductive generalization.

Empiricism differs also from rationalism in detecting the meaning of cause; the rationalistic meaning of cause involves a necessary relation between certain phenomena, such as that when a phenomenon produces another, then the first is the cause, the second the effect. Whereas empiricism means by causal principle that every event has a cause in nature, but there is here no necessity [or production] since these go beyond experience. Causality means, for Mill, nothing more than a certain succession between two events. But not every temporal succession is causal; causal succession requires that succession should be uniform.

We may now compare rationalistic conception of cause to empiricist one. First, succession between cause and effect is temporal in the latter, whereas it is ontological in the former. Secondly, rationalism takes causal relation as uniform concomitance between two events, resulting from the existence of a cause and its production by necessity of the effect, this being deduced from its cause. But empiricism refuses causal relation to be uniform concomitance and holds it to be a relation of another kind, namely, the observation and not deduction of temporal succession.

Discussion

Our comments on certainty attitude concerning induction are as follows. First, the author differs from both rationalistic and empiricist answers to the second question, namely, whether induction needs causality as a necessary postulate. Both schools, though different, answer that question in the positive, while the author will say no, owing to what will be maintained in the sequel. Secondly, we agree with certainty attitude that causal principle is itself reached by induction, and thus hold that induction needs no a priori postulates. But the impasse of empiricism in ourview, is that it holds that induction is grounded upon causality postulate while it holds at the same time that causal principle is itself an inductivegeneralisation .

If we reach causality by induction, why need we any a priori postulate to vindicate induction? Finally, when we say that induction is the basis of our idea of causality without any a priori postulate, we understand causality in the rationalistic sense as expressing a relation between cause and effect; otherwise, we cannot argue that causality is a result of empirical generalization. For the principal condition on which induction depends is, in our view, to conceive causality in the rationalistic sense, and if this is not the case, inductive inference would be incapable of reaching any generalization whatsoever, even in the probabilistic level.

On the second question

It is now possible to state the empiricist answer to the second problem of induction, which was central to formal logic as we have seen. For empiricism maintains our belief in the uniformity of nature, that is, if two certain events succeed each other under certain circumstances, we believe succession to occur in the future. It is meant of course, not that the belief is maintained if these two events happened once or twice, but that those events should have occurred countless times; and then we reach the same rationalistic position that relative chance cannot recur uniformly, with the difference that the latter position depends on a priori statement while empiricism rests on inductive inference in reaching uniformity.

Answer to that question

J.S. Mill has given us in his inductive logic four methods to discover the causal relation between any two phenomena. These methods are concerned with the second problem of induction, rejecting relative chance in nature.

These methods are stated as follows: (1) The method of agreement: If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree is the cause (or effect) of the given phenomenon”[3] . For example, if the phenomenon to the explained (b), is preceded or succeeded in the first instance by the circumstances (a), (c), (d), in the second by (e), (f), (a), and in the third by (x), (a), (y), then the only circumstance in common (a) it is cause of (b).

If we wish to discuss this method deeply, we may discover clearly that it deals in fact with the problem of the probability of relative chance. In the first instance of the phenomenon (b), we find that (b) succeeds (a), but there is still the probability that (b) is caused by (a), or by (c), or (d), where as we find in the other instances that the probability is greater in the connection of (a) and (b) than otherwise.

Thus the role of this first method is confined to facing the second problem of induction, and enables us to minimize relative chance.

(2) The method of difference:“If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former; the circumstance in which alone the two instances differ is the effect, or the cause, or an indispensable part of the cause, of the phenomenon”[ 4]

Such method, like the first, is designed for solving the second problem and decreasing the probability of relative chance. For when we encounter the first instance of the phenomenon (a), and try a variety of circumstances, we cannot ascribe causality to only one of those circumstances; so that we cannot say A is the cause. And when we come to another circumstance, we get the same result. So we tend to make [A and a] causally related, and no relative chance involved.

(3) Method of concomitant variations:“Whatever phenomenon varies in any manner, wherever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation”[ 5] .. If we have two phenomena, and studied one of them in various circumstances, then we find that it occurs in different degrees, and when in studying the other phenomenon we find that the variations happened to it correspond to those happened to the former phenomenon, then there is causal relation between the two phenomena.

This method, we notice, is nothing but a complex form of the method of agreement, for the third method involves a circumstance in common among various instances. Yet, the third method adds that such common circumstance has different degrees.

(4) Method of residues:“Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents” .

This method, it is said, enabled astronomers to discover, theoretically, the planet“Neptune” . For they held the theory of gravitation to be true, and observed some diversion in the orbit of the planet,“Uranus” , contrary to what gravitational theory prescribes. Such difference between theory and fact needed an explanation, so Liverier put forward the hypothesis that such diversion in the orbit of Uranus is due to interference of some yet unknown planet. Later, it was discovered and called Neptune.

Although this method is badly formulated, we may put itaright with application to the discovery of Neptune; this may be put as follows. When astronomers observed the diversion of Uranus from its normal orbit according to gravitational theory, they provided two alternatives to explain this: either to suppose the existence of a new planet which causes such diversion if the theory of gravitation is true, or that no new planet and then the theory is defective. Astronomers preferred the former, on the ground that very many other phenomena have confirmed the theory of gravitation, and then we tend to rule put the supposition that the diversion of Uranus' orbit happened by chance.

We may conclude that Mill's four methods are intended to consider the second of the three problems of induction, in opposition to relative chance. Formal logic put forward the principle“chance cannot occur permanently and consistently” , while Mill provided his four canons to oppose complete chance. Nevertheless, Mill had not succeeded in rejecting chance occurrence of phenomena in all respects, [x] since he made probable other circumstances to produce the effect other than the assigned cause.

Probability Attitude

This is the second attitude of theempiricistic theory of induction. Such attitude suggests that inductivegeneralisation needs certain assumption and postulates which can be confirmed independently of induction itself; but it maintains also that such confirmation is not possible on rationalistic lines since it rejects any a priori principles, not is it possible in accordance with certainty attitude which believed such postulates to be considered as results of previous inductions. And since those postulates cannot possibly be confirmed, inductive conclusions cannot be certain, but only probable: any more observation or experiment helps to increase probability of the conclusion.

It may be useful to quote from ProfessorZaki Naguib , a proponent of probability attitude, the following:

“The majority of those interested in induction, including a rational principle not derived from sense experience, as our ground ofgeneralising (scientific judgments). Even if you are enthusiast empiricist, you have to confess that there is something not derived from experience, namely, that what applies to some instances of a kind equally applies to all the instances belonging to it; hence our generalizations. Thus, Russell holds that we are obliged at the end to rest in induction to an unempirical basis, the so-called principle of induction. (Those who consider induction as the only scientific method think that all logic is empirical, and it is not expected of them to hold that induction requires a logical principle improvable by induction itself; such principle must be a priori).

Now, most logicians, including Russell, maintain that experience alone is not sufficient and then we either accept the principle of induction asindebatable assumption, or seek in vain for a justification of predicating future phenomena from the present”

“The question naturally arises, how can we judge the validity of inferring future events from past ones without recourse to any rational principle” such as the principle of induction suggested by Russell? Or is there a justification that new experiment be similar to past ones? Indefence of empiricism, we [main turn???] ask: what is meant by rational justification? This may mean that the conclusion is certainly true, or that induction is to be considered as deduction the conclusion of which be implied in its premises Induction in this sense has no rational justification, for induction is no deduction”.

“But such meaning is accepted neither in the sciences nor in ordinary way of speaking. If I was told one day that (a) will play chess with (b), and all that I know about them is that both had played six times in the past, and (a) won the game in four times out of six, while (b) won twice, then I am justified in saying that the probability ofa's winning is expected. Similarly, it is more probable that a falling body may come to the ground, the sun will rise tomorrow [than not] [???]. That is probability not certainty; but that is what the sciences are prepared to accept, because certainty is not expected save in mathematical propositions'[6]

Discussions

(1) We maintain that induction paves the way to scientificgeneralisations , but we wish to regard it not as deduction, but another sort of inference which proceeds from particular to general without need of any a priori principles. This topic will be considered at length in a later chapter, let it suffice now to say that we shall never prove that a normal person know by induction a great number ofgeneralisations ; we can have no proof to convince such person of this knowledge. How can we convince someone that if he eats he becomes hungry no longer, when he denies such process? Such person is similar to the idealist who denies the existence of the external world or any objective reality outside his own concepts. We cannot convince the idealist that he approves of the objective reality of his family, even if we are sure of that since he lives with them and other people. It is the same with the person who denies his knowledge of ordinarygeneralisations .

We may distinguish three sorts of certainty: logical, subjective and objective; the first concerns deductive inference; the second is a personal affair, while objective certainty concerns induction, and the last certainty is not proved but only postulated and explained. Further, it is absurd to find a rational justification of induction if by this is meant deductive process involving the law [of] non-contradiction. For inductive conclusion cannot be logically certain.

On the other hand, if it is meant by justification the claim that the negation of inductive conclusion is probable, then this is an important claim since it involves that inductive conclusion cannot be a postulate. We shall later be concerned with the condition of reasonable postulates.

(2) If we follow theempiricistic line of thought we shall not reject only inductive science but also any degree of probability for inductive inference. The representative quotation mentioned above shows that such probability depends on probability calculus, but it will be shown later that such calculus does not lead to increasing probability of inductive conclusion unless it does also lead to confirming the rationalistic conception of causality, but empiricism rejects such conception. Thus, empirical logic faces a dilemma, either it leaves theempiricistic conception of causality and adopts the rationalistic conception, or it excludes the rationalistic conception of causality and insists on theempiricistic conception, but then it will be unable to explain the probability of induction.

Psychological Attitude

We mean by this attitude, anempiricistic one that deprives induction of any objective validity but connects it with habit; and David Hume is a clear representative. ModernBehaviourism ,a great modern school of psychology, later come to continue the Human tradition and transfer it from the field of philosophy to that of psychology.

Hume tries to clarify the problem of induction as follows. All inference concerning matters of fact is based on causal relation, and this relation is the sole relation which goes beyond the senses and informs us of entities that we do not perceive. If you asksome one about the cause of his belief in any absent matter of fact, he would justify such belief by means of his knowledge of another fact causally connected withit, he would say that he believes that x is sick because he saw a doctor visiting him. Or that if he was going yesterday to throw himself into fire, we are justified to say that he may be burnt, because there is a causal relation between thrown into fire and being burnt. We may now ask, how do we know such relation? The source of such knowledge is experience which enables us to observe the concomitance between both events. We may further ask, how do we know that actual concomitance between two events will happen in the future? We are in need of a justification of the principle that the future will be similar to the past. That was Hume's formulation of the problem. He solves the problem by saying that the justification of uniformity in nature is not logical but psychological and this can be done by giving analysis of cause-effect relation.

Perceptions are either impressions or ideas for Hume, and these are distinguished by virtue of the degree of vividness connected with impressions. And impressions include all sensations, emotions and sentiments.Whereas ideas are fading copies of impressions when the object of perception is absent. For instance, when we look at the sea, we perceive it vividly and clearly and here we get an impression, but when we turn our back to it, we get an idea, a copy of such impression. Hume then maintains that impressions are prior to ideas, that every idea, simple or complex, has, for its origin, an impression.

Impressions forHume, are of two kinds, impressions of sense and those of reflection. When we see a lion we obtainan impressions of it, which is clear and vivid, and when the lion disappears, the mind is able to keep an idea of the beast. Such idea produces in the mind fear and aversion, and these may be called impressions. And the operation by which we recall our firstimpressions is memory, our second impressions is imagination. Ideas of memory differ from ideas of imagination in clarity and vividness as well as their being literal copies of the original impressions, whereas imagination is free. However, freedom of imagination does not involve invention but is somehow derived from a previous impression; it is free in the sense that it is able to manipulate various ideas and construct out of them what it will.

Among those ideas produced by impression, there are certain relationswhich make the mind proceeds from an idea to another, and such is called association. Relations of association are three; likeness, contiguity in place or time, and causality, the last being the most important. For it involves only one of the two terms of the relation; for instance, when I put water on fire, the causal relation stirs in me the idea of the heat, though I get no impression of such heat; I get an impression of only one of its terms, i.e., water being on fire. The case is different with relation of likeness and contiguity, because they make the mind pass fromon idea to another like it or contiguous to it.

Now, we may ask, what gives rise in our mind to the idea of cause and effect? What impression gives rise to our idea of cause? And Hume replies that the idea of cause means not merely the spatial or temporal contiguity of two phenomena, but necessity. But from what impression can we get the idea of necessity. Suppose we saw the event concomitant with another event (b) once, then we cannot say with certainty that there is a relation between them.

But suppose that such concomitance has happened several times,then we are justified in claiming a causal relation. Thus repetition and coexistence is the source of our idea of necessity. Hume clarifies his position by saying that repetition and coexistence is not the source of necessity, but necessity comes from an extra impression, that is, readiness of the mind to pass from one thing to another usually accompanying it.

Such is the nature of necessity involved in the causal relation; it is something in mind not in things; however, we are disposed to apply it to things outside us and think that all events do have such relation among them. Now, Hume could explain inductive inference and its jump from particular to universal in a subjective manner based on, mental habit and psychological necessity, not objectively in term of external reality. This does not mean that he doubted inductive conclusions and in proposition based on experience; on the contrary, he believed in them. But what does he mean by belief? Belief is an idea involving vividness and force. As has already been said, Hume makes impressions more vivid and clearer than ideas. Now, certain ideas may obtain such vividness and clarity and thus become beliefs; the main difference between belief and imagination is that the former is an idea which acquires the vividness and force of impression, while the latter is an idea which does not.

Such vividness and force of belief depend on two things: (a) there being a vivid impression of something (or an idea of memory which has enjoyed the vividness of impressions); (b) the concomitance of this something with some other thing, then the mind passes from one to the other.

Examination of psychological attitude

(1) Belief

Hume explains what is meant by belief in two ways:

(A) The difference between idea and belief is not in content but in our way ofrecognising them. If there is an idea and we judge that the object of such idea exists, then we have two different things, the difference is not known byrecognising elements in the second case which are absent in the first, thus the idea of existence is not different from something existing. Hence existence is not an attribute added to other attributes of an object of perception. And our belief in the existence of something has nothing added to the mere existence of it. Nevertheless, there is clear distinction between identifying an idea of an object in my mind and my belief that such object exists. Since this distinction is not a constituent of the contents of the idea, then it is a product of the way of identifying it.

(B) This distinction is due to the advent of idea in our mind in a forceful and vivid way; if the idea is feeble it is not a belief. Thus belief may be defined as a vivid and forceful idea.

In discussion of Hume's theory of belief, we observe two things. First, we agree with Hume that a belief differs from an idea not in having an element of existence to its content, but in our way of identifying it. But we disagree in justifying such difference. In claiming that existence is not a characteristic of belief, we see that it may be an element of the content of idea as well; if we do not believe that there is a bird with two heads,then we may have an idea of it, and may further conceive its existence without our believing it. Therefore, the element of existence may be included in both idea and object, and we have to find out a characteristic which distinguishes them, it is our way of perceiving an object which gives belief. Hume arrived at the same conclusion from a different premise, namely, that existence is not an element added to the properties of the object of perception. This is based on his principle that an idea is necessarily a copy of an impression, and since there is no impression from which the idea of existence is derived, Hume has to say that existence is not a distinct idea.

Secondly, we observe that certain ideas may be vivid and effective without being beliefs, such as that which we get from illusion; a stick the lower part of which is immersed in water is observed as if it were broken. It may be objected that our belief in the straightness of stick is due to discovery of visual illusion and with the help of actual impressions: thus such belief is nothing but an idea possessed of vitality and force. The objection does not alter our position that the idea of straightness of the stick is vivid without being a belief.

Further, in explaining belief as vivid idea, Hume supposes that this vividness is derived from an impression, either directly as copy, or indirectly as causally related; and this means that any idea not derivedform impression is not a belief. But this does not accord with reality, because we may have a number of beliefs without there being copies of impressions. How could Hume explain someone's belief that a ghost frightens him, so long as he had not received an impression of ghost? We must distinguish two things in matters of belief, explanation and evaluation. To explain it we have to distinguish an idea from a belief. And we should give an explanation which applies to all beliefs, regardless of judging it true or false.

(2) Causality and Reason

Hume maintains that causal principle does not arise in pure reason, and cannot be deduced from the law of non-contradiction, it is known to us through experience and not a priori. If something as cause has not happened in connection with some other thing as effect in experience, we cannot perceive causal relation. “If we supposeAdamto have perfect mental acts” Hume says, “he could not have deduced from liquidity and transparence of water that he would suffocate when he sinks in it”[ 7]. We have to distinguish causal principle from causal relations among events. By the former, we mean that every event has a cause, by the latter the relations of heat to extension, boiling to evaporation, eating to nourishment. Rationalism claims that causal principle is known a priori, and that causal relation is perceived a priori. Aristotelians, among rationalists, maintain that our knowledge of the principle is a priori and not derived from experience, while our knowledge that heat is the cause of extension of iron is so derived.

Thus, concerning causal relations, Hume and Aristotelians agree. Let us now discuss Hume position of our knowledge of causal principle. We agree with Hume that this principle is not deduced from the law of non-contradiction, for there is no contradiction in the occurrence of an event without any cause. Now there are two attempts to defend the view of medieval Aristotelians that causal principle is unempirical.

The first attempt may be stated as follows. All events are contingent; by contingency is meant that existence and non-existence are equally possible. Then, for an event to occur, there must be something having the power of giving existence to it the event, rather than not, and this something is the cause. This argument, if discussed thoroughly, is merely a deduction of cause from itself, thus it is a petty principle or that the argument presupposes causality.

The second attempt way formulated thus.

(a) Every essence is possible by itself, and does not exist unless something pushed its existence by necessity.

(b) Every possible essence must not exist except by virtue of an external cause, becauseits possible existence means that its existence and non-existence are equally acceptable. By necessity is meant that its existence is more probable than non-existence.

Therefore, since necessity of possible essence does not arise except by virtue of an external cause, then it cannot come into existence except by means of a cause. This argument, like the former, is formally invalid because it uses causality as premise, which is meant to be proved.

We suggest that if rationalism is to defend the causal principle as a priori, it should claim that the principle is an ultimate proposition in the mind, instead of saying that it is logically deduced from ultimate principles, and thus it becomes impossible to deduce an ultimate proposition from another. Naturally, Hume would reject the suggestion but we shall have occasion later to defend it.

(3) Causality and Experience

When Hume maintained that causal relations cannot be deductively inferred one from another, he claimed also that they cannot be empirically inferred. For all knowledge of the external world is derived from sensible impressions, and we get no impression of causal relation as necessary, we never get an impression of something that can be deduced from impression of anther. All that is acquired from experience is that the effect follows its cause as a matter of fact: if moving billiard-ball is seen to come across another at rest, this is seen to move, so that our sight is affected in such a way that a moving ball is succeeded by another movingball[ 8] . Thus, Hume concludedhat causal relation can only be given a psychological, not a logical empirical, explanation: no necessity between food and nutrition, but constant connection between our ideas of both.

Hume's rejection of causality as involving necessity raises two questions, namely, how can we understand the idea of causality as involving necessity since every idea is to be a copy of impression?and how do we believe in causality as objective relation between any two events independently of experience? Hume answered the former by showing that our idea of causality derives from an impression preceding another. And he admitted that the latter involves real problem and claimed that the causal relation is subjective not objective, that is, it lies in our mind as a relation between two ideas not between two events in the world.

If we accept Hume's answer of the first question, then we admit our having the concept of causality. We may then ask whether causality has an objectivereality? Although we get, in Hume's opinion, the idea of cause from an impression of connection, there is no reason preventing us of asking whether such idea has objective reality.

Suppose, [???]with Hume, that we cannot get the idea of causality by pure reason, we cannot prove that pure reason refutes causality; that is to say, it is probable that every event must have a cause, since it cannot be confirmed or refuted by reason[9] . On the other hand, we wish to ask whether there is any empirical evidence for the probability of the proposition about causality in objective reality. Hume maintained that there is none, but we shall have occasion later to show such probability.

(4) Concept of Causality

For Hume, every simple idea is a copy of an impression, and when he tries in vain to find out an impression of causality which involves logical necessity, he supposed an impression derived from a connection among certain ideas of succession, and saw that the succession of such ideas stirs in the mind a certain impression of expectation, when we get an impression of the first event we expect the other to occur. It is noticed here that such mental expectation is inferred from Hume's dictum that impressions are prior to ideas, but we see that this dictum is based on induction.

For all simple ideas, in Hume's opinion, are similar to simple impression, and though the idea of causality is not reached by induction, it must have been reached, as idea, by induction. Now, if inductive inference has no objective value, as Hume claims, so is causality. But Hume's application of induction to our idea of cause is invalid, because successfulgeneralisation should not apply to those kinds of things which have specific differences with other kinds. For example, if we find by induction that all metals except gold extend by heat, we cannot include gold in ourgeneralisation for its specific difference from iron, copper etc. Our idea of causality is similar to gold in this respect: if we find that all ideas are preceded by impressions, we cannot say this of our idea of cause which is specifically different from other ideas.

(5) Belief in causality

Let us first state Hume's theory of belief. It is an idea having a high degree of vividness and strength which is derived from a vivid impression or another idea.

When we have two ideas involving causal relation, the former being vivid is belief, and when we move from the idea of cause to that of effect, this requires a similar one once idea is related to an impression it becomes a belief, and if an idea is not so related it requires two things in order to become a belief: (a) a certain relation with another idea which enables us to proceed from the one to the other; such mental habit results from repeated concomitance between any two events in experience; (b) that the other idea should be also vivid. Now, we may criticize such theory by bringing the following points.

(a) Belief in causality involves two propositions, one of which is categorical, e.g., iron extends in heat, the other is hypothetical such as iron extends if exposed to heat. But whereas Hume's theory of belief explains our belief in the former proposition, it does not explain our belief in the latter. For which idea could be taken to be belief in the hypothetical proposition? Is it our idea of the extension of iron or that of heat as cause of extension? Hume cannot have chosen the first answer, because our idea of the extension of iron cannot be a belief unless it acquires a high degree of vividness from its relation to our idea of heat, but our idea of heat has no such degree since it did not occur in fact. The idea of cause, in the case of hypothetical proposition, is not vivid but a mere hypothesis and thus cannot be a belief.

On the other hand, Hume cannot accept the answer that the belief which we possess, in the case of giving a hypothetical proposition, is that heat is the cause of extension. For Hume denies that there are causal relations in objective reality and claims only that such relations are among our ideas. Then when we supply a hypothetical proposition involving causality, we mean, for Hume, the mental habit which helps to proceed from the idea of heat to that of extension.

When we reflect on this, we find that we talk not about the future of events in reality, but the future of our mental habits. Thus Hume cannot on his theory explain our belief in causality in the course of future events. In other words, if we have the right to suppose future to be similar to the past and present, we may apply his supposition to objective reality. But if, as Hume insists, we are not justified in our belief in uniformity, we have no right to talk about the future of mental habits. We may conclude that Hume's theory fails to explain inductive inference which provides hypothetical as well as categorical statements.

(b) Hume gives a ground to establish his theory of inductive inference and causality. He asks, why we need innumerable cases to reach a conclusion without being satisfied with only a small number of cases; and he gives his answer. Whereas the conclusion which the mind reaches from contemplating one circle are the same as when we see a number of circles, we cannot conclude from seeing only one body moving by impulse that all bodies, move by impulse. For in the latter case we require repetition of concomitance between those objects, thus we acquire a habit of inferring the one from the other so that all inferences in experience are effects of habit not byreasoning[ 10] .

But we may explain the validity of inductive inference including our beliefs in causality and uniformity without recourse to Hume's theory. Suppose we observe an event (a) followed once by another event (b), we may say that their concomitance happened by chance, and that (b) is caused by a yet unknown event (c). But if (a) is always followed by (b) in such a way that wherever (a) occurs, (b) follows, then chance is eliminated; repetition and absence of exception is a basis of inductive inference, not in terms of mental habits but in terms of objective reality, as known in probability calculus.

(c) Suppose someone tried to find out the effect of a certain drug on people having a certain disease, and found that this drug gives rise to some physiological phenomenon, then he would conclude that the drug is the cause of that phenomenon. Suppose, further, that such a person discovered that his partner intended to mislead him by providing those cases susceptible for such phenomenon,then our person would give up claiming a causal relation between the drug and that phenomenon. Here we are entitled to admit the objective reality of cause relations in the course of events apart from our ideas, because our experiment involves that certain phenomena should have as yet undiscovered causes. And Hume-ian habit fails to explain such cases.

(d) If belief expresses a vivid idea, how does Hume explain our doubt in a proposition when its truth or falsehood on a par? He may answer by saying that our idea of the existence and the non-existence of its objective referenceis not vivid. If we are in doubt whether rain fell yesterday, then our idea of falling idea or not-falling are faint, and thus no belief. Further, Hume's criterion of belief does not work when we have no doubt but probability of an occurrence.

Now, we may provide some criteria for the probability of rain's falling yesterday, namely, clouds or bad weather etc. If these are noticed to occur in most cases when rain falls, then there is probability that rain falls when such conditions occur gain; then the idea of rain's falling is probable not a belief.

Probability is of two kinds: (a) that which depends on frequency such as the high frequency of rain's falling in the example mentioned above, (b) that which depends on a logical basis. For instance, suppose we are told of the death of only one person among the passengers of anaeroplane ; suppose further that all the passengers were three men, then the probability of the death any of them is 1/3, the death of one among two of them is 2/3 Now, Hume's theory of probability fits with the first kind but not with the second.

Physiological Explanation of Induction

We have hitherto discussed Hume's psychological treatment of induction. We now turn to the physiological treatment ofinduction, that is, explaining it in terms of conditioned reflex provided byBehaviourism . Such theory regards inductive inference as a sort of correlation between a conditioned stimulus and certain reaction, instead of a correlation between two ideas in the mind in Hume's theory. The law of conditioned reflex is theBehaviouristic starting point which may be stated as follows. When an event leads to a certain reaction, the former is a stimulus, the latter a reaction, and if such event frequently occurs together with something, this something is said to be sufficient to give rise to that reaction. This law applies equally to man and animal. The traditional example of the law is Pavlov's experiment of a horse which is found to have more saliva on seeing food; when we condition the appearance of food with ringing a bell, it is found that saliva increases as the bell is heard even if there is no food. Ringing the bell becomes a condition of a certain reaction.

Now,Behaviourism claims that inductive inference could be explained in terms of some form of that law; for example, if (b) being a stimulus leads to certain reaction, and if (a) is frequently found to occur with (b), (a) becomes a conditioned stimulus giving rise to that reaction. Thus, we know the existence of (b) whenever we see (a).

In comment, two points may be stated. First, is the reaction stimulated by (b) what we mean when we say we perceive (b), or we mean that the perception is represented in a psychological element by such physiological reaction? Second, is it possible to explain induction merely as discovering a conditioned stimulus for it? It is the latter question that concerns us.

Induction cannot be explained in terms of stimulus reaction, because we mean by inductive inference either that (b) occurs when (a) does, or that whenever we see (a) we see (b). The former case is a particularone, the latter is so general that goes beyond mere observation and experiment. If, for the sake of argument, physiological explanation fits inductive conclusions in particular cases, it does not fit with general hypothetical inductive statements, that is, if (a) happens, (b) does,For generalisation is not a reaction to stimuli but something new. Further, we do not use induction merely to make clear that (a) is succeeded by (b), but we use it also to prove the existence of the external world. It will be argued in the final chapter that the real ground of our belief in the external world is induction, and that inductive conclusions are not merely frequency of occurrence in the field of perception but they have some novelty differing from all previous reactions. Therefore, inductive inference is to be distinguished from [law-] conditioned reflex.

Induction

Part 1: Induction and Epistemology

Chapter 1: Aristotelian Induction

Meanings of Induction

Induction, as has been said in the Introduction, is a sort of inference proceeding from particular proposition to general ones; the former being based on observation and experiment. By observation [is meant] one's attention to a certain natural phenomenon as actually occurring, to discover its causes and relations to other phenomena. By experiment is meant one's interference and effort to produce such a phenomenon in a variety of circumstances, to discover those causes and relations. The difference between observation and experiment is that between observing lightning, for instance, as it naturally occurs, and actively producing it in a certain way in the laboratory. Thus, inductive inference begins with observing a certain phenomenon or actively producing it in many cases, and then establishing a general conclusion suggested by these observations and experiments.

Aristotle did not distinguish between observation and experiment, and considered induction as any inference based on enumerating particular instances, consequently, he classified induction into perfect and imperfect, if the conclusion refers to all the particulars in question, induction is perfect, if it includes reference to some particular instances only, induction is imperfect[1] .

Aristotle has considered perfect induction in a way different from his consideration of imperfect induction. Induction cannot be divided, in our view, into perfect and imperfect because induction in factproceeds from particular to universal, whereas perfect induction does not do so, but its premises are general like its conclusion. Thus, we regard perfect induction as deduction not induction; and it is imperfect induction that is induction proper.

Aristotle's perfect induction

Perfect induction was of great logical value for Aristotle being as rigorous as syllogism. When syllogism predicates major terms of minor term by virtue of a middle term, its conclusion is certain; similarly, the conclusion of perfect induction relates a predicate to a subject by means of enumerating all instances of that subject, thus the certainty of such conclusion. Further, Aristotle considers perfect induction a basis of recognizing the ultimate premises of syllogistic reasoning.

We reach those premises not by syllogism but by perfect induction. For, in syllogism we predicate the major term to the minor term by means of the middle term, this being a predicate of the minor term and subject of the major term; and if we try to prove syllogistically that the major term is asserted of the middle term, or that the middle term is asserted of the minor term, we have to find out the middle term between them, and then we go on until we reach ultimate premises wherein we relate predicate to subject without any medium.

And as we cannot get a syllogism without a middle term, the only way for Aristotle to reach such ultimate premises of syllogism is by perfect induction. Later on, medieval logicians did not give such a great value to perfect induction, but they still regarded it as an important means of arriving at ultimate premises.

Criticism of perfect induction

Our comments on Aristotelian perfect induction are as follows:

(1) We are concerned in this book with induction proceeding from particular to universal, thus perfect induction lies outside our interest, since it is a sort of deduction the premises of which are also universal, and the principle of non contradiction is sufficient to show the truth of its conclusion.

(2) We may ask, what is the use of the conclusion of perfect induction for us? Two Aristotelian answers are expected, (i ) the conclusion asserts a logical or causal relation between its two terms. When we say John, Peter and Smith are all the individuals of the human species; John, Peter and Smith eat; therefore every man eats. It may here be said that the conclusion asserts a causal relation between humanity and eating, (ii) Aristotle may not insist on regarding the conclusion as asserting a causal relation, but show the fact that men eat, by complete enumeration of all individuals.

Let us discuss these answers. Aristotle would be mistaken if he thought that perfect induction gives a causal relation between the terms of the conclusion otherwise this conclusion would give new information not included in the premises; and then the inductive reasoning loses its logical validity and cannot be explained by the law of non - contradiction alone. Further, if we take the conclusion of perfect induction as giving a fact about its terms and not a certain relation between them, such a conclusion would indeed be valid since it is contained in the premise, but then perfect induction would not be a proof in Aristotle's sense. He conceived proof as giving a logically certain relation between the terms of the conclusion, and this certainty arises from our discovery of the true cause of that relation. Such a cause may be the subject itself and the predicate may be either an essential attribute or not; if essential attribute, then the conclusion is an ultimate premise, but if not, the conclusion would be demonstrated only in a secondary sense.

Now, if the conclusion of perfect induction just states that men eat, without asserting that humanity is a cause of eating, then it is not a demonstrative proposition, and a fortiori, induction is proof no longer. And if perfect induction is unable to give demonstrative statements, then there is no way to establish ultimate premises of proof.

(3) Perfect induction gives us a judgment about, at most, actually observed instances but not instances which may exist in the future. We may observe, theoretically speaking, all the instances of man in the past and present and see that they eat, but cannot now observe men that may come in the future. Thus perfect induction cannot give us a strictly universal conclusion. And it makes no difference to make induction dealing with particulars, e.g. John,Peter .. and to arrive at a general conclusion such as every man eats, or dealing with species such as man, horse, lion to judge that all animals die.For a species or genus does not include individuals or species actually existed and observed only, but a species may have other individuals, and genus other species.

(4) Perfect induction has recently been criticized not only as a proof in the Aristotelian sense, but also as a proof in any sense. Suppose I arrived at the conclusion, all matter is subject to gravitation, after a long series of experiments in a great number of instances. Induction maybe formulated thus:

a1' a2' a3'…an are subject to gravitation.

a1' a2' a3'…an are all the kinds of matter that exist.

.all matter is gravitational.

When I see a piece of stone, I judge that it is subject to this law, not because I give a newjudgement , for stones are among the kinds under experiment, but because when I come across some instance not included in my experiments, I judge that the conclusion applies to the new instance as well.

This objection may be retorted on Aristotelian lines. In perfect induction, we do not intend to say that this piece or that piece of stone is subject to gravitation, but that all pieces of matter are so.

Aristotle distinguished syllogism from induction, the former predicates the major term to the minor term by means of the middle term, whereas the latter predicates the major term to the middle term by means of the minor term. Thus, the conclusion that this or that piece of stone has gravitational property is reached not by induction but by a syllogism, formulated thus: these instances have gravitational property; these instances are all matter thatexists all matter has gravitational property,

Further, it should be remarked that the statement all pieces of iron extend by heat is not merely enumerating particular statements expressing the fact this and that piece extend by heat, but it is a different statement from all those particular ones. For the statement all pieces of iron extend by heat is reached by induction in two steps. First, we collect all pieces of iron in the world, separating them from all other species of matter and conclude that these are all iron that exists. Secondly, we turn to every piece of iron and show that each extends by heat. [Only then perfect induction could be properly asserted, reader's note]

Recapitulation

The results reached so far are as follows, (a) The subject of perfect induction does not concern those who consider induction in the modern sense; (b) Perfect inductioncan not be regarded as a proof in the Aristotelian sense for it is unable to discover the cause; (c) Perfect induction is formally a valid inference and (d) General statements in science cannot be reached through this sort of induction.

Aristotle's imperfect induction

The Problem of induction

If you ask an ordinary man to explain how we proceed from particular statements to a general inductive conclusion, his answer may be that we face two phenomena in all experiments such as between heat and extension of iron, and since the extension of iron has a natural cause, we naturally conclude from constant relation between heat and extension that heat is the cause, and if so, we have right to make the generalization that when iron is subjected to heat it extends. But this explanation does not satisfy the logician for many reasons. (A) Induction should first establish the causal law [which is an a priori principle in rationalistic epistemology, but not in theempiricistic epistemology, which considers empirical observation to be the only source of knowledge, reader's note] among natural phenomena, otherwise extension of iron has probably no cause and may happen spontaneously, and hence another piece of iron may not extend by heat in the future. (B) If induction has got to establish causality in nature, it suggests that the extension of iron has a cause, but has no right to assert off band that the cause is heat just because heat is connected with extension. Extension of iron must have a cause but it may be something other than heat, heat might have been concomitant with the extension of iron without being its cause [since observation of two adjacent phenomena doesn't necessarily mean that one is the cause of the other, for example in the case of morning following night, nobody says night is the cause of morning, reader's note]. Induction should therefore establish that heat any other is thecause[ ?]. (C) If induction could establish the principle of causality among natural phenomena, and could also argue that a is the cause of b, it still has to prove that such causal relation will continue to exist in the future, and in all the yet unobserved instances, otherwise the general inductive statement is baseless [the most it could generalize is that heat causes extension in the piece(s) of iron under observation and for that piece(s) of iron only, reader's note].

Aristotelian logic has an answer on logical ground to the second question only; as to the first and the third, it is satisfied with the answers given in the Aristotelian rationalistic epistemology. Rationalism involves the causal principle (every event has acause ) , independently of sensible experience. Rationalism involves also the principle that“like causes have like effects” this being a principle deduced from causal principle, and would be the ground of the third question mentioned above. It is the second question only that the Aristotelian logic has got to face and solve, that is, how can we infer the causal relation between any two phenomena that have mere concomitance and not reduce such concomitance to mere chance? To overcome this, Aristotelian logic offers a third rationalistic principle that we now turn to state in detail.

Formal logic and the problem

When ageneralisation is through induction, we either apply it to instances which are different in some properties from those we have observed, or apply it to instances that are exactly like those we have observed; the formergeneralisation , for Formal logic, is logically invalid, because we have no right to infer a general conclusion from premises some of which state some properties unlike the properties stated in other premises. Suppose we observed all animals and found that they move the lower part of their mouth in eating, we cannotgeneralise this phenomenon to sea animals, since these have different properties from the animals already observed.[2]

But inductivegeneralisation is logically valid, when applied to like unobserved instances which are similar to instances observed. Validity here is not based on mere enumeration of instances, for this does not prove that there is causal relation between any two phenomena. Formal logic has found a way to assert causal relation in inductivegeneralisations , if we add, to the observation of instances, a rational a priori principle, that is, chance cannot be permanent or repetitious, or between any two phenomena not causally related, concomitance cannot happen all the time or most of the time. Such principle may take a syllogisticform : a and b have been observed together many times, when two phenomena are observed to be severally connected, one is a cause of the other; therefore a is cause of b. This syllogism proceeds from general to particular, and not vice versa, thus not induction.

We then observe that the role played by imperfect induction, for formal logic, is producing a minor premise of a syllogism. This inductive inference involving a sort of syllogism is called by formal logicians an experience, and this is considered a source of knowledge. The difference between experience and imperfect induction is that the latter is merely an enumeration of observed instances, while the former consists of such induction plus the a priori principle already stated.

Consequently, it may be said that formal logic regards imperfect induction as a ground of science, if experience as previously defined, is added; that is if we add, to observation of severalinstances , the a priori principle that chance cannot happen permanently and systematically.

Misunderstanding of formal logic

Some modern thinkers mistakenly thought that formal logic rejects inductivegeneralisations and is interested only in perfect induction. But formal logic showed, as we have seen, that imperfect induction can give logically validgeneralisation if we collected, several instances and added a rational principle, such that we reach a syllogism proving causality, and that is called experience [which is also a] [xand a] source of knowledge.

Further, some commentators of formal logic have understood the distinction between imperfect induction and experience in a certain way. Perfect induction is based on passive observation while experience needs active observation. An example of the former is that when we observe a great number of all swans are black. An example of the latter is that when we heat iron and observe that iron extends and conclude that iron extends by heat. This attempt to distinguish induction from experience anticipates the modern conception of experience and makes imperfect induction similar to systematic observation. But this explanation is mistaken, for experience is meant by formal logicians no more than imperfect induction plus the construction of a syllogism, the minor premise of which is based on induction, while the major premise states a rational principle rejecting the repetition of chance happenings.

Aristotelian epistemology and induction

The formal logical view of introducing a priori principles in induction is related to rationalistic epistemology which includes that reason independently of sense experience is a source of knowledge. And this theory of knowledge is opposed to the empiricist theory which insists on sense experience as the only source of human knowledge. If we maintain that chance cannot be permanent or repetitious this must be established by induction, and thus that principle is nothing but an empiricalgeneralisation , thus it cannot be regarded as the logical foundation of validgeneralisation .

Although we are enthusiast about rationalistic epistemology, as will be shown later, we think that Aristotle's principle (chance cannot be permanent andrepetitious ) is not an a priori principle, but a result of inductive process.

Formal logic and chance

Let us make clear how chance is defined by formal logicians. We may first clarify,“chance” , by making clear its opposite, i.e., necessity. Necessity is either logical or empirical. Logical necessity is a relation between two statements or two collections of statements, such that if you deny one of them, then they become contradictory, e.g., logical necessity between Euclidean postulates and theorems. On the other hand, empirical necessity is a causal relation between two things such as between fire and heat, heat and boiling, poison and death; and causality has nothing to do with logical necessity, in the sense that it is not contradictory to assert that fire does not produce heat, and so on. There is a great difference between the statement' the triangle has not three side's and the statement“heat is not a cause of boiling water” ,The former is self contradictory while the latter is not; necessity between heat and boiling is a matter of fact not a matter of logic.

Let us now turn to chance. To say that something happens by chance is to say that it is neither logically not empirically necessary to happen. Chance is either absolute or relative. Absolute chance is the happening of something without any cause, as the boiling of water without a cause; whereas relative chance is the occurrence of an event as having a cause, but it happens that it is connected with the occurrence with another event by chance, for example, when a Kettle full of water under heat boils, but a glass of water under the zero point freezes; thus it happened by chance that the Kettle boiled at the same time when the glass freezes. Chance here is relative because both boiling and freezing have causes( not by chance ) but their concomitance is by chance. Thus, absolute chance is the occurrence of an event without any necessity, logical or empirical - without any cause;where as relative chance is the concomitance of two events without any causal relation between them.

Now, absolute chance forAristotle, is impossible, for this sort of chance is opposed to the causal principle. Thus, in rejecting absolute chance, Aristotelian epistemology and other sort of rationalism establish the causal principle, and consider it the basis of the answer to the first of our three questions related to the problem of induction; and goes with this the answer to the third question which is deduced from the causal principle. But, for Aristotelian rationalism, relative chance is not impossible, because it is not opposed to causality.

The concomitance between frozen water and boiled water by chance does not exclude that freezing or boiling has a cause. We have in that instance three sorts of concomitance: frozen water and boiled water, freezing, and heat to the zero point, boiling and heat in high temperature; the first being by chance, the latter two are causally related. There is a great difference between concomitance by virtue of causal relation and concomitance by relative chance; the former is uniform and repetitious, such as between the concomitance between heat and boiling, or lightning and thunder. The latter is neither uniform nor recurrent, for example, you for many times, when you go out, you meet a friend, but this does not happen uniformly.

Formal logic takes the previous view as a ground of the principle that chance does not happen permanently or uniformly, considers it a priori principle, and by chance is meant relative chance.

Need of definite formulation

Despite clear exposition previouslystated, the principle that chance does not happen permanently and uniformly has to be clarified. We ought to know precisely whether the rejection of relative chance applies to all time past, present and future, or is confined to the field of experiments made by some person in a definite stretch of time.

In the former, it follows that relative chance does not recur in all time, but that is impossible since we cannot observe all natural phenomena in the past and future. And if meant by the principle that we reject uniform repetition in the field of experiments made by some person, it follows that the principle seeks to show that relative chance does not recur in a reasonable number of observations and experiments. But the Aristotelian principle has to specify the reasonable number of experiments required. Can we formulate the principle thus: relative chance does not recur in ten or hundred or thousand experiments? Suppose we specified the reasonable number by ten, then if we put some water in a low temperature and it freezes, we cannot discover the causal relation from doing the experiment only once; we have to repeat the experiment ten times, in this case we have right to discover the causal relation.

The crucial point of difference

We differ from formal logic on the principle that chance cannothappen uniformly mainly not its truth but its character. We accept the principle but refuse its being a priori and rational nature. Formal logic regards that principle as independent of all sensible experience and then is considered a ground of all inductive inferences; for if it is considered an empirical principle and derived from experience, it cannot be a principle of induction but itself an inductivegeneralisation . Such principle is, in our view, a result of induction, arrived at through a long chain of observations. Now, the question arises, what evidence formal logic has to maintain that such principle is a priori?

In fact, there is no evidence, and formal logic considers the principle as among primitive and primary principles and these do not need evidence or proof. Formal logic divides our knowledge into two sorts; primary and secondary; former is intuitively perceived by the mind such as the law of non-contradiction; but secondary knowledge is deduced from the primitive one, such as the internal angles of a triangle are equal to two right ones. Primitive knowledge needs no proof but secondary sort of knowledge does. But since formal logic regards experience as one of the sources of knowledge,than[ ?] empirical propositions are primitive.

Since formal logic regards empirical statements as primitive statements, and claims that the principle about chance is primitive,then such principle needs no demonstration, exactly as the principle of non -contradiction need not. Since we have known the definite concept of the principle which rejects relative chance for formal logic, it is now easy to reject that principle. If this Aristotelian principle asserts the impossibility of recurrence of relative chance, as the law of non -contradiction asserts the impossibility of contradiction, we can easily claim that the former principle is not found in us, because we all distinguish the law of non - contradiction from the principle of non -recurrence of relative chance. For, whereas we cannot conceive a contradiction in our world, we can conceive the uniformity of relative chance, though it does not really exist [spurious correlations in social sciences, for example, between number of fire trucks sent to rescue and the destruction caused by the fire. The more the fire trucks, it appears more the fire damage as observed in the recurring events. So is the larger number of fire truck responsible for larger destruction? There is a third variable that actually explains the cause and that is the hugeness of fire. The massive the fire, the more trucks needed every time, and the massive the fire, the more chances of destruction every time]. And if the Aristotelian principle rejects the recurrence of relative chance in our world together with admitting that it is possible to recur,then the principle is not a rational a priori principle independent of experience, because a priori principles are either necessary or impossible, if it is only possible, how can we reject it independently of sense experience? We have said enough to conclude that the principle of rejecting relative chance is not among a priori principles. In the following chapter we shall give a detailed refutation of the a priori character of the principle.

Chapter 2: Criticism of Aristotelian Induction

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.

Indefinite Knowledge

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.

Genesis 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 definitecolour , and that it must not be black and white at the same time this means that one of the twocolours is absent, and this is due to your not seeing the paper. For if you saw it clearly, you would have specified itscolour , 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 greycolour 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.

Aristotelian principle and indefinite knowledge

The Aristotelian principle of rejecting relative chance, is nowshown[ ?] 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.

First Objection

When there is no causal relation between a and b and bring outa 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 toa 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 toa 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 toa 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 eithera 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.

Second objection

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.

Third Objection

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 mustrecognise 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 vaguelyis 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.

Fourth Objection

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 adrink, 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 sufficientreason, 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.

Fifth Objection

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 tenheadached -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 werealise 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 notrealised , 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.

Sixth Objection

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).

Seventh Objection

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. Firstevery a priori rational knowledge of something necessarily implies a priori knowledge of its consequence. Secondly, if it is true that relative chance cannot uniformlyrecur 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 besummarised in two points. First, formal logic maintains that inductive inference requires three postulates to meet its three problems, thus acquires the desiredgeneralisation . 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.