Chapter 5. Induction and Certainty
Subjective Role in Certainty
We have hitherto discussed the first phase of inductive inference, that is the deductive phase provided that we accept the axioms presupposed by the theory of probability. But there is a difference between deduction in the inductive process and deduction in purely axiomatic systems, such as we find in geometry. Such difference is clearly shown in that purely deductive systems prove the objectivity of mathematical truths, whereas induction in its deductive phase gives us higher degree of credibility.
This credibility is expressed by greater probability value arising from collecting a greater number of cases concerning the principle of causality. Now, the deductive conclusion in induction shows a degree of credibility of the statement, "A causes B", and not of the principle of causality itself. Such credibility would approximate, but does not reach, certainty. Thus, the deductive phase of induction does not give us certainty in causality or induction generalisation but gives us a greater value of credibility in causality and generalisation [of A causing B]. Now, we want to ask whether such value may reach certainty in a later step of inductive process.
Kinds of certainty
To answer the previous question, we should define the meaning of certainty. There are three kinds of certainty: logical, subjective and objective. We shall explain each in turn. First, by logical certainty, we mean the sense used in Aristotelian logic, the denial of which is self-contradictory. Logical certainty consists of two sorts of knowledge, one of which implies the other. When we say that x is a man' and that 'x is a great man', we say that the latter implies the former. Logical certainty does not only apply to statements but also to the terms of a categorical statement; for instance, straight line is the shortest distance between two points: the relation between the two terms is logical, and it is logically impossible to deny the predicate of the subject. Again, mathematical certainty is a sort of logical certainty because the former involves that a statement implies another.
Subjective certainty is another sort of certainty, which means knowing a statement to be true such that no doubt in it arises. You may deny such certainty without contradiction. I may see handwriting and recognise that it belongs to some friend, but it may really belong to another.
We now come to the third sort of certainty: objective certainty. We may first distinguish the statement said to be certain and the degree of credibility towards it. Suppose I knew that a friend of mine is dead, then the statement 'x died' is certain, but I have also a strong belief of his death; then credibility has degrees ranging from the slightest probability to certainty. In consequence, as to human knowledge, truth and falsity of a statement correspond to reality. But as to our credibility the statement may be true, nevertheless we do not feel certain about it. Suppose someone threw a piece of coin and believed it would rest on its head owing to his desire to be so, and this comes true, then both the statement and his belief are true; yet he would [could] be mistaken in his belief since he foresaw it a priori. This shows the distinction between subjective and objective certainty, the former is acquirement of the highest degree of credibility even if there be n[o] objective ground. The latter is the utmost credibility on object grounds. Thus, we may be in a situation in which there is subjective certainty without objective certainty and vice versa. For objective certainty is independent of subjective states of the mind.
Likewise, we may distinguish subjective from objective probability. The latter expresses a definite degree of probability in correspondence with factual data, whereas subjective probability expresses the degree of credibility owned by some individual whether it is consistent with facts or not. We want now to know the objective ground of certainty. A mathematical or logical statement is certain because it is deduced from prior statements; it is the same with objective certainty which is deduced from prior objective certainties. Further, as in formal deduction we start from unproved axioms, likewise in objective certainty, we assume basic or primitive or immediate beliefs. Thus, the objective basis of any degree of credibility presupposes an axiom, namely, that there are degrees of credibility immediately known to be objective. Thus we have two sorts of deduction, deduction of statements and deduction of the degree of credibility. The statement the internal angles of a triangle are equal to two right ones is a deduction of the former, while when I throw a piece of coin and I say it would rest on its head or tail I talk of the latter. Now we want to ask whether the value of probability involved in inductive [inclusion???conclusion] could be transformed into certainty, and to this we shall turn.
Certainty which is required for induction cannot be logical certainty because if we say that a causes b it is not logically impossible to suppose that a does not, or that b can have another cause than a; that a causes b cannot be deduced from our observations. Again, subjective certainty is not required here because the majority of men take it to be beyond doubt. It is objective certainty which is required for induction? Now, is there any justification for saying that inductive inference may be objectively certain?
Objective certainty require[s] an axiom
We have hitherto remarked that valid and objective degree of credibility can be deduced from other objective degrees of credibility but such objective degree can be immediately given; we have remarked also that objective credibility requires an axiom, namely, that there are degrees of objective credibility immediately given. These degrees being given are not deduced from other prior degrees of credibility but this does not mean that degrees of credibility not deduced from others are primitive and immediate, because we may find certain subjective degrees of credibility, as opposed to objective credibility that cannot be deduced from prior assumptions.
What distinguishes those objective degrees of credibility immediately given is that they are consistent with each other, such that the least inconsistency among them shows that they include something subjective. Thus one way of finding out that a certain degree of credibility is not immediate axiom is to show that it is contradictory with some objective degrees which have general acceptance. For example, suppose someone believes that the book I want to buy is not the one missing in my own study, then I can show him that the degree of his belief is subjective not objective, that his degree is not primitive and immediate, and that it is contradictory with some other beliefs, such that he strongly believes that there is a book missing and I want to buy another copy but what to buy is not the missing one. Now, his belief that any book I want to buy is not the missing one does contradict with his belief that there is one book missing in my study. Therefore his degree of credibility is subjective not objective.
If we are concerned with induction in this light, we find that the degree of credibility required for the inductive conclusion is objective, being derived from other objective degrees provided that such degree is always less than certainty. The inductive conclusion cannot reach certainty because there is always a value of probability which makes the difference between the probable and the certain. Now, for induction to reach the highest degree of objectivity, we must assume that there is a degree of credibility, immediately given, and such assumption is needed for any deductive process, which depends on an assumption, namely, that there are certain objective degrees of credibility, not deduced from prior degrees. Although we accept this assumption, we cannot prove it, nor can we prove any similar assumption. We cannot even prove that highest degree of believing the law of non-contradiction is one that is immediately given. And if we approve this assumption, we have three points to consider.
First, we must give a precise formula of the assumption, for induction to reach the highest degree of objectivity, that is, certainty. Secondly, we must specify the necessary conditions required for the assumption to be valid and avoid falsity. Finally, we must be sure that these conditions are to be fulfilled in the objects and concepts hitherto studied in the first deductive phase of induction, and thus, it is possible for induction to proceed to its second phase, i.e., to reach certainty.
The formulation of the postulate
The postulate presupposed by induction in its second phase is concerned not with objective reality, but with human knowledge itself, and can be stated as follows.
When a great number of probable values are reached in a specific inquiry and a larger value is obtained, this is transformed into certainty. For human knowledge is so constituted that it does not satisfy with small probable values. That is, the postulate assumes that neglecting smaller values in favour of larger ones which would come to certainty is a natural inclination of human knowledge.
When we move from higher probabilities to certainty, we do not rely on psychological factors as optimism or pessimism. The probability of the death of a person about to have a surgical operation could move to certainty as a result of pessimism on his part, but such certainty comes to an end if the person in question gets rid of his pessimistic state. Whereas the certainty involved in our postulate is one that cannot be an illusion.
We have already claimed that probability values are always connected with an indefinite knowledge, and that any probability value is one of a member of an indefinite knowledge; thus when the impact of probability values in some inquiry comes to the degree of certainty we face an indefinite knowledge which absorbs most of those values. Now, we may ask about the limit of the greater value which could become certainty and the limit of smaller values which would be ignored. People have transformed probability to certainty. Some think that they get certainty when probability value in a certain inquiry reaches a certain degree, whereas others do not think such degree satisfactory. But it is not necessary for the postulate required for induction to determine the degree which is a sign of certainty, but it is sufficient for the postulate to state the principle that the increasing number of probability values in a certain inquiry indicate the transformation of probability to certainty, and that the required degree is involved in successful inductions.
Conditions of the Postulate
The postulate under discussion stated that when probability values reach a certain degree and involve a greater number of cases, these values absorb smaller values and transform probability to certainty. But there is a [principal???principle] condition for the postulate to work, namely, that the passing away of lesser values should not to be with it the passing away of higher values. Take for example the case, of the missing book in a whole library containing 100,000 books. Here we have an indefinite knowledge that there is a book missing in this whole library, this knowledge involving bundled thousand probability values, and that each value is equal to 1/100,000, that is, the probability that anyone of these books is missing. Now, if we take anyone of these books we find that the values of presence of book equal all the values connected with all other books present in the library.
It follows that such book will be the beginning of our inquiry into the probability values of the present books, save one. But such greater value does not transform probability to certainty because the impact of greater values is nothing but an expression of the greater part of our knowledge of a missing book, and any book that is supposed present has value equal to the value of the probability of the missing one. In this case, the lesser value did not pass away in favour of the greater values of the present books; if it does, either this leads to the passing away of the smaller probability value opposite to all other values, or opposite to some values only.
Thus we maintain that it is impossible for the postulate to be acceptable. In consequence, the higher values cannot be transformed to certainty, since we presuppose one indefinite knowledge otherwise we always face probability values equal to the number of the terms of such knowledge. Now, if we want to make the postulate acceptable, we stipulate two sorts of indefinite knowledge. This takes two forms, which we shall presently state in detail.
The first form of the postulate
We suppose that we have two sorts of indefinite knowledge and that probability values focus in one direction. We may state this [section negatively by negating a definite in term in the indefinite knowledge 1???], and positively by affirming another term of the same sort of knowledge. But those values associated in one direction belong to the indefinite knowledge 2. Thus, the association and the direction of it do not belong to one sort of knowledge but to two. Let us apply such form of the postulate to the proposition "A causes B" in view of two sorts of indefinite knowledge. First, the indefinite knowledge which determines the a priori probability that A causes B. If we assume that we have already known that B has a cause, that this is either A or C, then such knowledge includes two terms, let us call it knowledge 1. Secondly, the indefinite knowledge just stated may be taken as a ground for establishing the probability of causality thus such knowledge involves all successful cases where in C is probably a cause, let this be called knowledge 2.
If we get ten successful experiments, then we should have 1024 cases, being the terms of knowledge 2; one case of these is indifferent to the two terms of knowledge 1, the remaining cases favour one of the two terms in knowledge 1, that A causes B.
This means that knowledge contains 1024 probability values, that 1023 1/2 values constitute a positive grouping in a certain direction, namely, that A causes B, one of the two terms knowledge. This grouping gives the causal relation concerned higher probability. Now, we may validly apply our postulate to such a case. We postulate that such grouping of cases gives us certainty as to the causality of A to C, and the passing away of the contrary value. Such application involves no contradiction because this grouping expresses the greater part of knowledge 2. Now, we get the first form of applying our postulate: when a number of probability values of an indefinite knowledge increases outside limits, and leads to the passing away of the contrary value.
Such application involves no contradiction because this grouping expresses the greatest part of knowledge 2. Now, we get the first form of applying our postulate: when a number of probability values of an indefinite knowledge increase outside its limits, and leads to the passing away of one value, then this latter does not belong to the knowledge connected with the great number of values.
But there are two conditions to be satisfied for this application to hold. First, the proposition, expressing the vanishing of a probability value as opposed to the greater number of values, should not be concomitant to one of the terms of knowledge 2 to which belong those values. That is, if 'A causes B' is concomitant to the occurrence C in all successful experiments, and we know that if C occurs in all cases, then A is not the cause of B necessarily. This makes the application of the postulate difficult, because 'A causes B' would become one of the terms of knowledge 2. If the terms included in knowledge 2 lead to the vanishing of the value of causality, then they naturally absorb the probability value of C's occurring in all cases. And then the postulate is faced with the problem of certain values absorbed in favour of other values without justification.
For the application of the postulate to succeed, we must assume that the proposition, expressing the absorption of a certain value, is not concomitant to one of the terms of knowledge 2, as this in fact is the case of refuting 'A causes B'. Such refutation is not connected with the occurrence of C in all experiments. For C may occur in all experiments and yet B is caused by A.
The second condition is this, that the grouping of probability values must not be arbitrary; by grouping we mean that values are not related immediately to one proposition but some values are related to a proposition while others are related to another, and from these two propositions stands. This third proposition is called arbitrary. For example, suppose we put a very heavy stone on the top of a pillar under which some one is sitting. If the stone is properly put on the top it does not fall to the ground, but if improperly put, then we have an indefinite knowledge that the top has, say, thousand points. Such knowledge includes 999 values involving the fall of the stone and the death of the person underneath, and only one value that it does not. Now in view of the conditions stipulated for the postulate we may know of the great number of other values because all belong to one sort of knowledge. Here comes the arbitrary grouping which is supposed to overcome the difficulty. We may explain the certainty required by grouping the probabilities according to our postulate without the assuming that the one value referred to above will cease to stand.
Instead of assuming the certainty that the stone will fall we suggest another proposition, namely, that it is probable that the person in question dies from a heart attack not owing to the fall of the stone.
Then we find a third proposition, namely, that either the stone falls or the person dies from heart attack. Thus we falsely get the certainty that the person should die. If we do this, we cannot apply the postulate without contradiction, because if we suppose that the probability of the death of person from heart attack is equal to that of the person being kept alive. In such case we have two propositions having the same probability value: first, that either events would occur, the fall of stone or death from heart attack; the second is that either events would occur: the fall of the stone or that the person will not die from heart attack. Theses two propositions are equally probable, and this proves that the greater probability value arising from the grouping of values in the first proposition cannot be transformed to certainty, for if it can, then this would be without justification; and if the values of both propositions become certain, this means that we are certain of the falling of the stone, but we assumed this not to be so.
And if we assume that the value of the probability that the man would die out of heart attack is greater than the value not of his death, some events may possibly occur such that each one has an equal probability of the man's death out of heart attack. This may substantiated as follows:
1) Either the stone falls or the man dies out of heart attack.
2) Either the stone falls or rain falls.
3) Either the stone falls or the temperature increases.
Those propositions are equally probable assuming the equality of the probability of death, rain and temperature. This shows that the probability value of the first proposition cannot be transformed into certainty. And we know that the value of the occurrence of any of the three events is greater than not occurring, we may sometimes get a value higher than the value of occurring, by means of grouping the values of not occurring. All this shows that the application of the first form of the postulate in an arbitrary direction is self-contradictory, for the postulate to be acceptable it must be applied in a definite direction, and by this is meant a proposition that directs the probability values to confirmation such as the proposition that A causes B in the above example.
Objections and Answers
1. Is causality a term in indefinite Knowledge
An objection may be raised as to the application of the first form of postulate to causality, namely, that the example of causal relation, referred to above, does not fulfil the necessary condition of the postulate, that the proposition expressing improbability in favour of grouping a greater number of probability values must not be a term in the indefinite knowledge concerned, because such proposition must refute that A causes B, but this refutation is itself a term of that knowledge thus it cannot be absent in the latter.
This objection depends on the rule of multiplication, represented by the principle of inverse probability, and is irrelevant if we have in mind the rule of dominance which is an application of the third additional postulate, explained in a previous chapter this rule says that knowledge 2 is the sole ground of all the values which the improbability of A causing B, and the refutation of causality is not included in that knowledge.
The answer to this objection, provided we use the rule of multiplication instead of dominance rule, is as follows. The appearance of knowledge 3, resulting from multiplying the members of knowledge 1 in those of knowledge 2 depends on keeping in itself all the members of the other two pieces of knowledge. In such a case multiplication includes a number of probable instances which form the members of knowledge 3, and the negation of causality is substantiated in these members. But the postulate we suggest for induction in its second phase assumes that probability values grouped in knowledge 2 are inconsistent with the probability that A is not cause of B. Such postulate led to the passing away of the improbability of c causality, then knowledge 1 no longer includes both members to be multiplied in the members of knowledge 2. Thus knowledge 3 would not arise.
2. Attempt to deny our knowledge of causality
Another objection may be raised against our certainty about causality arising from the grouping of probabilities, according to our postulate. This objection is meant to argue that the postulate is false.
Let us make the objection clear. When we know that something is the case and we doubt in something else, then we may affirm what we know, whatever we say of what we doubt. For example, if we know that rain in fact falls, and doubt whether there is eclipse, this means that we are sure of the former, and our doubt of the latter does not affect the fact of raining. Our knowledge of the rain fall involves that of two hypothetical statements, namely, knowing that if there is eclipse, the rain falls, and knowing that if there is no eclipse rain also falls. That is, rain falls whether there is eclipse or no, and if we do not know these two statements we cannot know that rain falls.
In this light, if we analyse the conclusions we arrived at in the previous application of the postulate, we find that we have in fact got the knowledge that A causes B, and that we have got the probability of the occurrence of C, because such probability, however lower its value, cannot be ignored being a term of knowledge 2. If those conclusions were true and we are certain that A causes B with a doubt in the occurrence of C, it would be necessary that our; knowledge of causality involves two hypothetical statements, as has been already stated. Knowledge 2 is clearly not existing, that is, we do not know that A causes B provided that C has occurred in all experiments.
Thus, had we in fact observed C in all experiments, we would not have been certain that A causes B. This means that the refutation of causality is probable, provided that C always occurred, and since this hypothesis is probable the refutation of causality is probable. We may answer this objection as follows. Certainty about some fact may arise when we prove that it is the case of when we group probability values according to the postulate of inductive inference. The first sort of certainty affirms the fact, whether the other events occur in fact or not; thus it is impossible to group something certain together with our doubt in it. The second sort of certainty, arising from grouping a great number of probability values is not strict certainty, even if we assume the falsity of one or more values, for such assumption involves the falsity of some of those grouped values.
Now, certainty about causality, being a result of the grouping of a greater number of probability values, cannot be affirmation of causality if we assume that such values were false and that C occurred in all experiments. Hence, any inductive certainty about some fact, resulting from greater probability values, cannot be certain knowledge if it involves doubtful values; thus we cannot prove that causality is inductively not certain, if we suppose that C occurs in all successful experiments.
3. Misapplication of inductive postulate
We may imagine a third objection stating that we sometimes give an application contrary to the postulate itself. This comes out when we get probability values contrary to the phenomena under examination but equal to the favoured values, e.g. the examination whether A causes B or not. In such a case it is impossible that negative values (negation of causality) would be superseded owing to their lesser degree. For then the result that A causes B would have no justification since the values, positive and negative, are equal. If C occurs concomitant to B in all experiments then it is not probable that A causes B, and if C does not occur in all experiments then it is not the cause of B. Thus we reach the absurd conclusion that knowledge 2 has superseded the probability value of one of its terms i.e., the probability that C occurs in all experiments.
We may simplify the previous objection as follows. We assume first that causality is verified inductively; we assume secondly one of two alternatives: either C did not happen even once, or that A is not cause of B. In both assumptions, we have values in the one equal to those in the other, and both belong to knowledge 2. Now if we suppose that this knowledge involves certainty about the first assumption without the second, then the supposition is a probability without justification. And if we suppose that knowledge 2 involves certainty about both assumptions, this means that we are certain that C does not occur in all experiments, thus knowledge 2 has superseded one of its terms.
It may seem that we answer the objection is terms of the second condition stipulated for the application of the postulate, namely, that the objective of the inquiry should be real not arbitrary. We may say that we encounter the refutation of causality with an arbitrary objective which is a complex event, that we encounter the affirmation of causality with an arbitrary objective which is choosing an alternative, as in the example of a stone falling from the top of the pillar on a person underneath.
But the postulate depends on a real objective, that is, affirmation or negation of causality. But when we have done this, we have proved that such application is self contradictory.
The present objection tries to include the real aim of inquiry as arbitrary, and show that applying the postulate to both real and arbitrary inquiries leads to contradiction. Now, to get rid of these contradictions, the postulate must be concerned with the real, not the arbitrary, inquiry. This we do in what follows.
Indefinite knowledge 2, with all its probability values, positive and negative, is directed towards our certainty in causality. If this knowledge fulfils such certainty, then causality is affirmed, that is, there would be no probability that C occurs with B in all experiments, together with the non-occurrence of A. It has been shown that part of this complex event does not happen; in consequence, the probability of Cs occurrence in all experiments is associated with the probability of the concomitance of A and B. Thus, the arbitrary inquiry would be part of knowledge 2; and in this case the postulate does not apply, so long as we stipulated that the postulate cannot apply to those cases in which indefinite knowledge supersedes some values, in favour of others.
We now realise that applying the postulate to the real investigation expels the arbitrary one out of its domain because the former becomes one of the terms of indefinite knowledge itself. But if we suppose the application of the postulate to the arbitrary investigation, this would not extract the real investigation out of its own domain.
For the application of arbitrary investigation leads to a knowledge that the complex event does not occur-the event consisting of the occurrence of C in all experiments and the concomitance of A and B. That A causes B, and that C uniformly occurs with B cannot happen at the same time. And that is the true course of application. For the certainty that A's causing B together with C's occurring do not happen and this does not make us believe that the denial of causality is connected with the uniform occurrence of C, otherwise we would believe the hypothetical statement: 'if C uniformly occurs then A is not cause of B' but we are not certain of the truth of this statement.
The outcome of all this is that the postulate can explain the real investigation of causality, and cannot explain arbitrary investigations.
4. Indefinite Probability
We may state the last objection to applying our postulate. The postulate supposes that the grouping of many probability values contrary to the investigation concerned rules out the latter's value. Suppose also the probability that A is not cause of B to be ruled out owing to the lesser values favouring it. But if we add all causal relations which we inductively arrived at, and observe the probability that at least one of these relations, may not occur in fact, then we find this latter case more probable that the denial of all relations.
For the assumption that at least one causal relation cannot occur is not ruled out by the affirmation of many causal relations, but it can be ruled out by multiplying those values. Thus we find that the probability that at least one causal relation does not occur, this probability remains as probable as indefinite knowledge. In this case it is impossible to apply the postulate to all domains of induction, because if it is applied to all domains it would rule out the probability that a certain value does happen, whereas we assumed that such probability cannot be ruled out according to the postulate. And [11???] the postulate be applied to some, but not all, causal relations, the application is. without justification.
Answer. The probability value here is a consequence of the addition of all values refuting causality, and the postulate is capable of ruling out such values.
The Second Form of the Postulate
In considering the first form of the application of the postulate, we have confronted two sorts of knowledge, the first absorbs the probability value of one of its terms (knowledge 1), the other is the cause of this absorption in virtue of the grouping of a great number of values in a single investigation, and thus we avoid the supposition that a sort of knowledge absorbs or rules out one of its values being equal.
But in the second form of application, we will suppose that the sort of knowledge, which rules out the lesser value in favour of the greater value in one single investigation, is the same knowledge which rules out the probability value of one of its terms. That is, though our knowledge absorbs the value of this, we are not led to wipe out knowledge itself or to rule out some values in favour of others, both being equal, without justification. For, in the present application we assume that the part of knowledge, the value of which is ruled out, is not equal to the standing part, but smaller than other values. Thus, the application of the postulate is not confronted with any difficulty such as we have established something without justification or the vanishing of knowledge itself. It is possible to suppose the postulate to rule out our knowledge of the lesser values without falling into reaching a conclusion without justification because the justification of ruling it out is its being small in content.
Still we have an essential point to explain, namely, how do probability values in a certain sort of knowledge differ from each other though the importance of such knowledge lies in the various values being equal, as we have seen in the theory of probability? Such difference in the values involved in an indefinite knowledge must be understood in virtue of another indefinite knowledge.
Let the former be called indefinite knowledge 1, and the latter knowledge 2. This last knowledge must offer an unequal distribution of the values of knowledge 1; and this is done by one of two ways which we shall illustrate in two examples.
We suppose first the occurrence of events, let them be three events. We notice inductively that the cases of the occurrence of each event are more frequent than not.
Then the probability that the three events would not occur is lesser than other values. For example, suppose we find in the newspaper that the cases of true news is twice the cases of false ones. Suppose we have before us three news, then two of them are supposedly true and the third is false. Then we get two sorts of indefinite knowledge, knowledge 1 and knowledge 2. The former includes eight probabilities concerning those three news and their truth and falsehood, one probability of which would be the falsehood of the three news. Within this sort of knowledge the values of these eight probabilities are equal, namely, the value of each is 1/8. But knowledge 2 would include nine probabilities, three concern the truth and falsehood of each of the three news, on the supposition that the cases of truth are twice those of falsity. That is, knowledge 2 includes the being of a case in each of the nine probabilities, thus we have 27 truth values, one of which involving the falsity of all cases, and the rest involve the truth of at least one value. Thus knowledge 2 changes the probability values of knowledge 1, then making the values unequal.
There is another way of applying the postulate. We may suppose a group of events, the occurrence of which is equal to its non-occurrence, thus we get equal values of all the possible probabilities of the occurrence of non-occurrence. These probability would include the indefinite knowledge. Yet some of these probabilities are correlated with one opposite case, included in knowledge 2, such that the latter case is lesser in value than the other cases. For example, if we throw a piece of coin ten times, we find that it is probable that the coin is on its head or is not. The two probabilities are equal, by multiplying them in each throw we get 1024 probabilities, and these constitute the indefinite knowledge 1. Within this knowledge, the values are equal. For instance, the occurrence of the coin on its head in the first, fourth, ninth and ten times, and on its tail in all times must be equal.
Yet it is known that the occurrence of the second case is strange enough, while the occurrence of the first case is not. This means that there is one factor which makes the second case less in value than the other case. Such factor shows the importance of knowledge 2.
Now, what is this factor? It is this factor which made ancient formal logic believe that it is impossible to regard the uniform occurrence of certain events as chances. Formal logic denies that chances consistently recur in a great number of experiments; for instance we do not expect a piece of coin to rest on its head in thousand throws.
However, formal logic wrongly explains why this is impossible. Aristotelians rejected chances on a priori principles, but recurrent chances may well be explained in another way.
If we have before us an experiment, say throwing a piece of coin, in a number of cases, and compare them, we shall find that these cases have more differences among each other according to the circumstances belonging to each than the circumstances they have in common; instances of the latter are the direction of air, the position of the hand, and other circumstances which may interfere and direct the experiment. Now, if we suppose the coin to rest on its head a number of successive times by chance, this means that the items of circumstance which are permanent are the cause of what happens.
Then when we observe the recurrence of some factor we explain this by the permanence of the circumstances belonging of the coin. Such explanation have a very small value, because the changeable circumstances are more numerous, and each of these may be a factor in directing the experiment.
Take another example. Suppose we invite fifty persons for dinner and predict beforehand the colour of costumes they will wear, we will find that the probability of their coming wearing their costumes in one colour very small. For the choice of the costume for each person differ according to their personal circumstances being vastly different. If it happened that they all come with costumes having one common colour, this means that the few circumstances they have in common are the cause of what happened. Thus the probability is very small if they wore costumes having one and the same colour.
The truth of the matter in both examples is there being an indefinite knowledge that the cause of the coin's resting on its head, or choosing yellowish costumes, is either one of the various circumstances involved. And the items of this knowledge (which we call knowledge 2) are more numerous than those of knowledge 1, because the latter derives its items from the various forms of probable situations. But knowledge 2 derives its items from the number of circumstances related to the first situation multiplied with those related to the second situation and so on. Thus we find in knowledge 2 a great number of probability values which oppose the supposition of consistent chances.
In consequence, we see in good light the Aristotelian principle that "chance does not recur uniformly and regularly". This principle is in our view, not a priori or logical rules, but a grouping of probability values such that the probability of uniform chance is very low.
Reformulation of Aristotle's principle
We may now reformulate this principle after ruling out its so called a priori character in the following way:
(1) We are aware of a great number of varieties between a certain point of time and other points, and between any given state in nature and others;
(2) We are also aware of a small number of resemblances between any two points of time or any two physical states.
(3) Such awareness makes the values of these varieties very great and significant;
(4) If the first point of time, or the first state of a given object, leads to an event which we cannot at the time know its cause, then our expectation of the next point of time or state to bring the same event by chance is much less than having a different event. Let us call the principle reformulated "the rule of irregularity".
We must notice that we presuppose that the interference of changeable factor producing a certain event involves variation and difference from state to another; thus we regard the value of the event occurring regularly equal to the value of having the permanent, not changeable factors which produce the event.
Such presupposition may be confirmed inductively: the difference between two things is connected with the difference between the conclusions. The inductive confirmation of the presupposition is to suppose it certain.
This means that we have applied the first form of the postulate in its second phase, by means of which we arrived at this inductive statement. But in order to explain the difference between the values of the items of knowledge 1, it is not necessary to obtain the inductive confirmation of this statement as certain. It is sufficient to confirm it with higher probability. That is, the supposition of the effect of the variations and changeable factors on the event implies the non-recurrence of this event uniformly in every case.
Now, in disclosing the Aristotelian principle rightly formulated as the grouping of probabilities, we can explain certain vague points in the way of applying such principle. First, the principle of irregularity involves that the uniform recurrence of chance is improbable, provided that the so called regularity is real not artificial. By real regularity, I mean the regularity which shows a common cause, as when the coin rests on its head two successive times; this means that the circumstances common to both cases are the same.
And since the differing circumstances are more numerous than the common ones, the probability of the uniform recurrence of chance occurrences is very low. By artificial regularity, I mean the regularity which does not involve a common cause. Suppose that when we throw a piece of coin, someone expects randomly that it will rest on its head in the first throw, on its tail in the next throw, on its head in the third and on its tail in the fourth [...???] and so on till the tenth throw.
In this case there is an artificial regularity because we do not suppose the cause of the coin's resting on its head in the first throw is the same cause of what happened in the second throw. Thus the probability of the truth of this expectation by mere chance all the time is not lower than any other probability. We notice in fact that the probability of the truth and falsity of the expectation are equal.
Secondly, the rule of irregularity doubles the probability value of chance repetition in case of real regularity, as we have already seen; and so long as regularity between supposed chances are clearer, the rule of irregularity is more successful. Suppose we are told that someone (x) made ten trips in the course of ten months, and in each trip he happened to have a road accident; this would be strange. But if we are told that he made ten trips in one month and in each trip he had an accident, this would be stranger. Again, if x invited ten friends and it happened by chance that all of them were ill then none come. This would be stranger than supposing that he invited ten friends in ten months time but one friend could not come in the first invitation, another friend in the second invitation and so on. In both cases, we have regular chances that the first is the more regular than the second, on the ground that all happened at the same time. This means that when real regularity of chances are more consistent and uniform, then the probability of their occurrence all over is low. For the circumstances which each invited has are naturally different according to the physical, psychological and social factors each of them has.
Likewise, differences in circumstances of the ten persons are much clearer than the agreements in their states. Consequently, it is strange to judge that all are caught with a headache at the same time in spite of the great variation in their circumstances. But it is less strange to judge that a friend was ill in this month and another was ill in the last month and so on.
We may recapitulate. The second form of applying the postulate of induction presupposes two sorts of indefinite knowledge; knowledge 1 includes all probabilities of occurrence and non-occurrence in respect of certain group of events. These probabilities have equal values in that knowledge. Knowledge 2 helps to change those values in two ways. First, knowledge 2 makes the Probability of the non-occurrence of any event in the group less than that in knowledge 1 provided that the probability of the occurrence of each event is more than its non-occurrence. Second, knowledge 2 makes the occurrence of all events in the group the least probable, provided that these events are really regular, then by means of the rule of irregularity, the probability of regular events in each time is very low, and when there is no such equality of values involved in knowledge 1, knowledge 2 can take the least value a centre of probability for the contrary, and this will be ruled out according to the inductive postulate.
The first way of applying the postulate does not work and is insufficient. For instance, if we suppose that the motives of saying truth is double those of telling lies, then 2/3 of the news are true and 1/3 are false. Suppose we randomly collected a thousand items of news, which form knowledge 1 having all news true and false so far, the probability of the falsity of all news is equal to any other probability. But if we introduce knowledge 2 we get all possible probabilities involving the motives of truth and falsity. The items of knowledge 2 are more numerous than those of knowledge 1, because any probable event in knowledge 1 corresponds to three probable events in knowledge 2, provided that every piece of news has three motives, and the probability of getting the motive of lie as one of the three motives relative to each piece of our news, includes contrary values larger those included in knowledge 1. Thus, the probability of the falsity of all false news is the lowest probability included in knowledge 1.
All this is acceptable, but it is not regarded as a true application of the inductive postulate and rules out small probability values of the falsity of all news, nor does it give rise to the certainty that some of them at least are true, though the application does not contradict knowledge 1, since the probabilities meant to be ruled out is lower than any other probability. But the application is inconsistent with another sort of knowledge, that is, we usually know there being of thousand false news in the whole body of news we have- this may be expressed in an indefinite knowledge 3. The items of this knowledge include every group consisting of 1000 news the truth or falsity of which we do not know. It involves the equality between any false group and any other.
In consequence, if we randomly choose 1000 piece of news out of the whole news, and by virtue of knowledge 2 including the determination of the values of knowledge 1, made the probability of the falsity of all news in its lowest degree, this does not justify the application of the postulate and reach certainty in justify the application of the postulate and reach certainty in opposition to that probability. For if we apply the postulate to the first thousand news only, this is without justification; and if we apply it to the whole news we have, then we face an indefinite knowledge of 1000 false news. Thus it is impossible to apply the postulate according to the first way of introducing knowledge 2 because this is inconsistent with knowledge 3, and this leads to certainty without justification or ruling out the values of knowledge 3.
If we replace knowledge 3 by a relatively greater degree of probability, we reach the same conclusion. For then we suppose we do not know there being thousand false news, but we have only reasonable probability. Such probability is an indefinite probability. If we apply the postulate to all groups of thousands we rule out this probability though it is of reasonable degree, thus we cannot rule it out according to the postulate.
It may be mentioned in this context that indefinite probability differs from the probability referred to in our fourth objection to the first form of applying the postulate, i.e., the probability that at least one causal relation is not constant. For such probability is a result of grouping the probabilities of non-causality. But here the probability replacing knowledge 3 is not a result of such false probabilities; it is the ratio of falsity of the news to truth. Therefore the first way of introducing knowledge 2 is insufficient for reasonable application of the postulate. But the second way of application is more sufficient.
Objection and Answer
It may be objected to this second way of applying the postulate that introducing the change of values of knowledge 1 and disappearance of the lowest of these values, give rise to the disappearance of the value of one of the items of knowledge 2. For example, if there is no probability that a piece of coin rests on its head in thousand successive throws; this means that knowledge 2 will also lose one of its items which is the supposition of something in common in the course of different throws and this something would be the factor determining the coin's head or tail. Thus the application of the postulate is self-contradictory because knowledge 2 negates one of its probability values which are equal. In consequence, knowledge 2 rules out either one of the equal values or all its values.
In answer to this objection, we argue that the disappearance of the small value in knowledge 1 is made by virtue of the effect of the number of value in this sort of knowledge, not by means of the opposite values in knowledge 2. Now, the values superseding the probability of the coin's resting on its head in thousand successive throws within knowledge 1 would be sufficient and of greater value. But here there is an obstacle, namely, that the value of the coin's rest on its head all the time equals any other probability value within knowledge 1; and there is no justification in this knowledge ruling out such value. Thus the postulate requires knowledge 2 to lessen the value of the coin's appearing on its head all the time, and then overcome the obstacle.
Finally there is a further objection, namely, that it is possible to prove that if there is any value the frivolity of which is presupposed by the inductive postulate, by reason of its unimportance, we can find equal values which are not frivolous though unimportant, and this shows that the unimportance of a certain value does not and itself in value that it is ruled out. The argument adduced by the objection is stated as follows. In any value the ruling out of which is presupposed by the postulate, we can suppose an indefinite knowledge consisting of a great number of items so that the division of certainty number on these items equals the indefinite value supposed to be ruled out. For we know that the value of any of these items cannot be ruled out however we increase or obtain a value which cannot be ruled out.
There is an answer to such argument. When we have two equal probability values it is impossible to rule out one of them and keep the other, for this would be without justification. But if the grouping of data relative to a certain value belongs to an indefinite knowledge while the data relative to another equal value belongs to a different knowledge, then it is possible to rule out one of the values in favour of the other.