Alhassanain(p) Network for Heritage and Islamic Thought

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.