Human Knowledge And Probability
Chapter 1. Classes of Statements
After having considered in detail our new interpretation of inductive inference, we come now to study the theory of knowledge and its main topics. We shall take, as our ground, intuitionism in a certain sense to be later specified. We shall start this task with a brief exposition of Aristotelian theory of knowledge.
Principles of demonstration
Formal logic claims that the objects proper of human knowledge are those which involve certainty, and by certainty is meant by Aristotle knowing a statement beyond doubt. Certain statements are of two kinds :
(a) statements which are conclusions of prior certain ones;
(b) basic statements regarded as ground of all certain subsequent statements. Formal logic classifies those certain statements into six classes:
(1) Primitive statements - the truth of which the mind admits immediately such that the apprehension of the terms is sufficient for judging their truth, e.g., contradictories cannot both be true or [xof] that the part is smaller than the whole.
(2) Basic empirical statements - the truth of which we admit by sense-experience; these come to us either by outer sense, e.g. this fire is hot, or by inner sense, e.g., we are aware of our mental states.
(3) Universal empirical statements - the truth of which is admitted by the mind through repetitive sense perception, such as fire is hot, metal extends by heat.
(4) Testimonial statements - the truth of which we believe upon the testimony of others whose utterances we believe true, e.g., such as our belief in the existence of places unobserved by us.
(5) Intuitive statements - the truth of which is believed in virtue of strong evidence that dispels any doubt, such as our belief that the moon derives its light from the sun.
(6) Innate statements - these are similar to primitive statements except that the former needs a medium approved by the mind such that whenever an innate statement is present, the mind understands it by the aid of something also. E.g., 2 is half 4 because 4 is divided into two equal numbers, and this means its half.
Any premise derived from any of these classes of statements is also certain. Those classes are considered the basis of certain knowledge, and the premises derived from them form the body of knowledge.
Any derivation in this structure takes its ground from the correspondence between our belief in originally certain statements and our belief in their derivatives. Such structure of knowledge is called, in Aristotelian terms, 'demonstrative knowledge', and the inference herein used is called 'proof'.
Principles of other forms of inference
Principles of inference in formal logic are not confined to those of demonstration or not confined to the those of demonstration or proof, but there are also other principles inference such as probable commonsensical, acceptable, authoritative, illusive and ambiguous statements. There are classes of statements out of which one can establish uncertain inference. Let us make such classes of statement clear.
(1) Probable statements: those which admit either truth or falsehood, e.g., this person has no job therefore he is wicked.
(2) Commonsensical statements are those which derive their truth merely from familiarity and general acceptance, e.g., justice is good while injustice is bad, doing harm to animals is vicious.
(3) Acceptable statements are those which are admitted as true either among all people, or among a specified group.
(4) Authoritative statements are those admitted by tradition such as those come to us from holy books or sages.
(5) Illusive statements are false ones but which may be object of belief by way of sensual evidence, e.g., every entity is in space.
(6) Ambiguous statements are false ones but may be confused with Certain statements.
Now, all inference depending on certain statements is called demonstration, but when inference depends on commonsensical and acceptable statements it is called dialectic; and when inference is arrived at from probable and authoritative statements it is called rhetorical, and when it uses false statements it is called fallacy. Thus demonstration is the only inference that is certain and always true. If we examine the principles of inference, referred to above, we shall find that most of them are not really principles but derivatives.
For example generally acceptable statements, considered by formal logic [???] principles of inference, may be regarded as a starting point in a discussion between two persons; but they are not real principles of thought. Further, authoritative statements are also derivatives because regarding a statement as trustworthy on the basis of divinity of otherwise means deriving it from other statements based on divinity. And probable statements usually used by formal logic are really derived from other statements which are probable not certain. For example, in the inference 'this [piece of iron] extends by heat because it is metal and all metal extends by heat', 'this extends by heat' is certain though derived statement, and 'all metal extends by heat' is empirical and included under the six classes of certain statements, already given.
On the other hand, in the inference 'this person is rude because he has no job and nine of every jobless ten are rude', this is [???] rude' has 9/10 probability, and nine of every [... ???] rude' also empirical[???]. Now, difference between the two examples is that the former includes certain premises, while the latter does not. Finally illusive statements are in fact inductive, though the generalisation is false. We may now conclude that the six classes of certain statements are the ultimate principles of knowledge, and all other statements are derived from them; if these are logically derived they are also certain, but if they are mistakenly derived they become false or illusive.
In what follows, we shall discuss this theory of the sources of knowledge, adduced by formal logic. We shall ask the following questions. Is it valid to consider universally empirical, intuitive, testimonial and basic empirical statements as primitive? What are the limits of human knowledge if our interpretation of inductive inference is accepted? Is there is any a priori knowledge? Can knowledge have a beginning? And finally can primitive knowledge be necessarily certain?
Universal empirical statements
We have shown that universal empirical statements, for formal logicians, are among the classes of basic statements, though they are logically preceded in order of time by empirical statements. For we usually begin with such statements as 'this piece of iron extends by heat', and proceed to 'all iron extends by heat'. But formal logic in its classification of propositions does not consider universal empirical statements as inferred from basic empirical statements. For the former have more than the total of the latter by virtue of the process of generalisation.
Thus, when formal logic classifies statements into primary[/primitive???] and secondary, and includes universal statements among secondary ones, it regards them as derived from an important primary statement, namely, relative chance cannot prevail. Accordingly, on observing the uniform relation between the extension of iron and heat, we may infer that heat causes extension. For if this occurred by chance, we would have not observed the uniform relation. The basic statement would be 'relative chance cannot permanently recur' and such statement as 'all iron extends by heat' as inferred. Thus formal logic gives two different claims, namely, universal statements are basic, and they are inferred from the statement denying chance. And we have already argued that the latter claim, is not basic and independent of experience but it is derived from experience. This does not mean to deny that such statement could be a ground of empirical statements in latter stages of empirical thinking. That is, if we can empirically verify the statement 'relative chance cannot permanently recur', we may deduce from it other empirical statements.
But if we take empirical statements as a whole, we cannot take such statement as ground of them all. Thus formal logic in this is defective. Again, it is false to agree with formal logicians in claiming that empirical statements are primitive not derived from other inductive statements.
To make our criticism clear, we may first distinguish between two concepts of the relation between an empirical statement such as all iron extend by heat and particular statements such as this piece of iron extends by heat.
Any particular statement of this kind expresses only one particular case of a general statement, thus this latter contains more than what is conveyed by particular statements. But we may also regard a particular statement involving the whole content completely. Thus general statements are derivative in this sense. Accordingly, derivative empirical statements are three classes. First, particular statements which constitute general statement inductively. Second, the postulates required for inductive inference in its deductive phase, since these postulates are the ground for confirming any statement of the first class. And we have already seen that these postulates satisfy the a priori probability of causality on rationalistic lines. The third class contains the postulates required for probability theory in general, for determining degrees of credulity.
We may remark that inference from empirical statements is probable not certain. Hence any empirical statement is derived so for[???] as certainty is concerned, whereas certainty involved in empirical statements is not logically derived from other statements, but it is a result of multitudes of probabilities.
Intuitive statements are similar to universal empirical ones. An example of the former is 'the moon differs in shape according to its distance from the sun'; we intuitively know that the moon derives its light from the sun, in the same way that we know that heat is the cause of the extension of iron, owing to observing the concomitance between heat and extension. Formal logic considers intuitive statements as primary, but it considers them statement which is the ground of empirical statements, namely, that relative chance does not permanently recur. For unless the moon derives its light from the sun, the difference in the distance between them would not have been connected with the various forms of the moon.
Now, we take it that intuitive statements are inferred from particular statements constituting their general form. But intuitive statements are not certain. Certainty adduced to these statements is merely a degree of credulity. That is, we cannot confirm it by means of prior statements, but we cannot at the same time obtain such certainty except as an outcome of probabilities. Thus certainty attributed to empirical and intuitive statements presupposes prior statements, though not deduced from them.
This is the third class of certain statements for formal logic, for our belief in the persons or events we are told to exist is primary. This means that formal logic postulates that a great number of people cannot give lies, and this [is] similar to the postulate 'chance cannot permanently occur'. Thus giving lies cannot always occur.
Suppose a number of people have met in a ceremony, and asked each other who was the lecturer, and suppose all answers referred to one and the same person, therefore we say that the answer expresses a testimonial statement. Our belief in such statement is really based on induction not on reason. Testimonial statements are really inductive and based on inductive premises. Those statements are concerned with the second form of inductive inference. We have previously shown that induction has two forms, the first is concerned with proving that a causes [b???] though we know nothing of the essence of both. The second form of induction is concerned with the existence of [a???] and its being simultaneous with b, knowing that a causes b, but we doubt the existence of a. This form involves the question whether the cause of b is c or d. Testimonial statements deal with such sort of induction. For example, if a group of persons agreed on the name of the lecturer, here the latter is a and the various answers of persons are b. The alternative for a is to suppose that all persons have given a lie for some reasons. This enables us to form an indefinite knowledge containing probabilities about such reasons. These will be eight if we have 3 persons. We may have the probability that only one person has a personal interest in lying, or the probability that two have interest in lying, or the three, or else that such interest is absent in all.
Each probability involves three suppositions, thus the sum of supposition in this knowledge is eight assuming that we have three persons. Seven of those suppositions imply that at least one person has no interest in the lie, and the eighth, implying that all have personal interest in lying, is indifferent as to the truth or falsehood of the statement.
If the value of having the personal interest in the news given by each person is 1/2, then having three persons, the value would be 7.5/8 = 15/16 included in the indefinite knowledge of a; and if we have four persons the value rises to (15+ 1/2)/16 = 31/32, until we reach the value of a very small fraction in case of denying the statement expressed by the answer given. Then begins the second step of inductive inference where the small fraction is neglected and is transformed into certainty. For the necessary condition of the second step of induction is fulfilled, namely, the neglect of the small fraction of probability value contrary to fact does not rule out one of the equal values.
This condition is made clear as follows.
(1) knowledge which embraces all possible cases of supposing personal interests in giving news in the source of probability values on a certain matter, and such values supersede the value of the contrary probability. (2???) It is observed in this connexion that the non-occurrence of an event is not included in such indefinite knowledge, it is rather necessary in this knowledge, because it is the case which involves the supposition of the personal interest concerned. The non-occurrence of an event does not apply except in this case. We have already shown in considering the second phase of inductive inference that knowledge in such phase affords superseding one of its items.
Probability values of items are unequal within the knowledge embracing possible cases of assuming personal interests. This means that such knowledge affords superseding the probability of one of its items, without superseding other equal values. The reason why the values of items are unequal is that the value of the case assuming personal interest in informing news is smaller than that of any other probability, because the probability of recurring chances uniformly is smaller than other probabilities. If you try to throw a piece of coin ten times, the probability that it appears on its head or its tail all the times is less than any other; likewise, in testimonial statements, the case of there being personal interests in giving news about an event is less probable than any other case.
(3) We have explained this by introducing another indefinite knowledge in which this case has less value than other cases. The persons concerned have different circumstances and their difference are far more numerous than their agreements. And supposing the agreement among all testimonies in those circumstances resulting in the personal motive for the news, such supposition means that it is items of agreement which determine the judgment of all testimonies; and this makes the probability of the uniform recurrence of chance less effective than the other probability. In consequence, the indefinite knowledge involving possible cases of supposing the personal motive for giving certain news does not include equal terms of probability value, because the value of their being a personal motive of information is the intrusion of another indefinite knowledge. Thus, indefinite knowledge may possibly supersede the probability value of such personal motive, without leading to the ruling out of one of its equal values. Accordingly, we can distinguish testimonies agreeing on a certain matter form those which disagree. When there is complete agreement on some fact, the belief that at least one person gives us the true news is more trustworthy than the case in which each person of a group gives different information. Testimonial statements are then inductive inference deals with any inductive statement, in two stages, namely, the calculus of probability and the grouping of probability values toward one direction.
Testimonial statements and a priori probability
These statements give rise to a problem concerned with a priori probability, to which we may turn. Although indefinite knowledge deluding the probabilities of truth and falsity issues the grouping large values cannot determine the ultimate value of testimonial statement. But we may here consider the a priori probability of this statement derived from prior knowledge, in order to determine the ultimate value by multiplying one knowledge in another.
For example, suppose we have a piece of paper on which are written words containing a hundred letters, but we know nothing more about such words. We have then a great number of a priori probabilities because there are 28 probabilities in each of the 100 letters, thus the sum of possible probabilities is the product of 28 in itself hundred times. And this is a fabulous number constituting an indefinite knowledge, let us call it 'a priori indefinite knowledge". If hundred men inform us of a definite form of those various forms of words and that each man in his information is moved with a personal interest with the probability 1/2, then we get an indefinite knowledge of the possible forms of the being or absence of personal interests, such knowledge may by called a posteriori indefinite knowledge'. The number of such forms is 2 X 2 hundred times.
For each man has in his information two equal probabilities, namely, that he may or may not have personal motive, and by multiplying the two probabilities, for each man we obtain a great number of possible forms. All these forms, except one, involve that at least one of the hundred men has no personal motive, and means that the testimonial statement is true. But this exceptional form is indifferent. When we compare the probability value depending on the knowledge expressing the testimonial statement with the value depending on the a priori knowledge denying that statement, we find that the latter value is larger than the former. For the favourable value depends on the grouping of the values of the items of the a posteriori knowledge, with the exception of half value of one item, and it is the truth of the testimonial statement. And the unfavourable value depends on the grouping of the values of the items of a priori knowledge. The number of the items of this latter knowledge is much greater than those of a posteriori knowledge, because the items or the a priori knowledge are equal to the multiplication of the 28 letters in themselves hundred times, while the items of a posteriori knowledge are equal to the multiplication of 2 in itself hundred times.
And this means that the probability value of the testimonial statement cannot be large enough, thus inductive inference in the way stated hitherto cannot explain testimonies.
Solution of the Problem
This problem can be solved with an application of the third additive postulate (the dominance postulate) instead of the postulate of inverse probability, because the probability value favouring a testimonial statement dominates the value inconsistent with it. For the a priori knowledge is concerned with something universal, i.e. one of the possible construction of the hundred letters. We know that the actual form of letters on the paper is that for which there is no personal motive, and this is the content of the a priori knowledge. Now, if we look at any value involving that at least one of the 100 information has no personal motive, such value is inconsistent with the truth of any other combination of words contained in the a priori knowledge. This proves that the value favouring the testimonial statement dominates the value contrary to it, and thus the faintness of the a priori probability value of testimonials cannot hinder inductive inference.
But the faintness of the a priori probability of testimonial statements cannot be an obstacle to induction if this faintness arises out of various alternatives to testimonial statements, such as we have seen in the last example, that the actual combination of words of which there is a complete consent as one of the great number of possible combinations. In such a case the probability value derived from the a posteriori knowledge favouring the testimonial statement, dominates the value derived from a priori knowledge denying the statement.
On the other hand, if the faintness of a priori probability of a statement depends, not on the multitude of alternatives, but on probability calculus in the stage of giving a reason for this testimonial statement, the faint value will have a positive role hindering inductive inference. For example, suppose an Arab write on a piece of paper hundred letters, and informs many persons that he has written hundred letters in Chinese. Then we notice that a priori probability of writing hundred letter in Chinese is very small, the cause writing of hundred Chinese letters depends on knowing Chinese which is not familiar among Arabs.
Suppose that in every ten million Arabs, there is our knowing Chinese; this means that the probability of knowing that someone knows Chinese among that number of men is one - ten millionth, and that there are ten million probabilities constituting an indefinite knowledge. The largest value in this knowledge denies that x knows Chinese; in consequence, there arises a large negative probability value of x's knowing Chinese. In such a case we obtain three kinds of indefinite knowledge: (a) the knowledge that the writer writes either Chinese or Arabic, (b) the knowledge that there are people saying that he wrote Chinese letters, that the items of such indefinite knowledge is the product of 2 in itself as times as the number of the people giving testimony, provided that the probability of there being or not being a personal motive is 1/2; (c) the indefinite knowledge that the person writing Chinese letters is one of the ten million people, that it has ten million items one of which involves knowing Chinese while others involve ignorance of Chinese.
If we take notice of the value that x wrote Chinese letters on the ground of the first knowledge, we see that it is 1/2, provided we have only two languages. But if we look at the value within the second knowledge, we find it very large, because most of the values here deny any personal motive by testimonies. Again, the probability value within the third knowledge is found very small, because most of the values here deny that x knows Chinese, and this means that the value depending on the first knowledge mediates two inverse attractions.
We have already stated that the large probability value, affirming testimonial statements derived from the second indefinite knowledge, dominates the value denying that statement derived from the first indefinite knowledge, we similarly claim that the large value denying testimonial statements and derived from the third indefinite knowledge dominates the value affirming them and derived from the first knowledge.
In order to confirm such dominance, we say that the first indefinite knowledge is concerned with a restricted universal, namely, that the author wrote a language known to him. The large value denying the testified statement and derived from the third knowledge denies that the author knows Chinese, thus it denies the fact of Chinese script. In consequence, the probability value affirming the Chinese script and derived from the first knowledge is dominated by the probability value denying that the writer knows Chinese which is derived from the third indefinite knowledge. And the value denying such knowledge is dominated by the value derived from the second indefinite knowledge; the former value assures that at least one testimony is not based on a personal motive. Therefore appears the positive role played by the a priori probability.
But if we do not know yet that x knows Arabic, only we know that x knows either Arabic or Chinese, and that the probability of his knowing Chinese is one ten millionth according to the third knowledge, then it is impossible to explain the dominance of the value, derived from the third knowledge, on the value derived from the first knowledge on the basis of third additional postulate. For in such a case both values give rise to the denial of the restricted universal belonging to the other knowledge, namely, the writing of a language which the writer knows.
Whereas the restricted universal belonging to the third knowledge is that the writer knows the language written on a paper. The value derived from the third knowledge, denying the knowledge of Chinese script is inconsistent with the universal belonging to the first knowledge.
This position can be attacked with the help of the fourth additional postulate which says that real, not artificial, restriction produces dominance. This letter postulate states that the probability value determined by indefinite knowledge of causes, dominates the value determined by knowledge of effects. The case with which we are now concerned is one to which the fourth postulate applies, because the third indefinite knowledge is that of causes and the first knowledge is that of effects.
Belief in rational agent
We usually believe that other men, whom we know, have minds and thought. When we read a book consistently written for example, we believe that its author is a rational being, and deny the probability that he is irrational or lunatic and that such book is produced by mere chance.
It may be claimed by someone, who thinks on Aristotelian lines, that inferring that such author has a mind is inference from effect to cause. Indeed, the book is an effect produced by some author, but such book does not logically prove that the author is a rational thinking being. It may be so, but it may be also that the author is a lunatic having some random ideas which constitute the book. In both cases the principle of causality is at work. Inductive inference is a basis of the first probability but not the second. For the second supposition involves many particular suppositions according to the number of the contents of the book. In such a supposition, there is no connection or consistency among the successive contents of the book; and this means that this second supposition cannot explain the rational production of the book.
On the other hand, the first supposition involves that ideas expressed in the book are connected and systematically related to each other. Suppose the word boiling occurred in the book hundred times, defined, explained and exemplified in the relevant way. This explains that the author has understood that word, and that he is a thinking being. In consequence, two sorts of indefinite knowledge arise. First, the knowledge which includes the probabilities required of the first supposition, suppose we have three ideas a, b, and c; here we have eight probabilities as to their truth and falsity. It may happen that (a) only or (b) only or (c) only is true, or all are true.
Such indefinite knowledge denies the first supposition with a great probability. For all its items, except the one in which all ideas are true, deny the first supposition. The exceptional case will be indifferent, because if a, b and c are all true, they may be so as a result of rational process or of chance.
The second indefinite knowledge includes the probabilities required of the second supposition, and since the latter is more complex than the first supposition, its items are much more than this. This second knowledge denies the second supposition with a greater probability value than the value given by the first knowledge to deny the first supposition. But the two negative values are incoherent because one of the suppositions in fact occurs. Thus we must determine the total value by means of the multiplication rules and here we get a third indefinite knowledge which embraces all possible probabilities. In this last knowledge the negative value of the second supposition will be very large. And this application of induction belongs to the first of the cases of the second forms of induction.
Inductive proof of God's existence
Instead of the example of the book, we may now suppose as object of induction, a group of physical phenomena. We may use inductive inference to conclude that such phenomena have a wise Maker. When we consider the conceivable hypotheses relevant to explain a group of phenomena, such as those of which the physiological composition of a particular man consists, we might have before us the following hypotheses:
(1) explaining those phenomena by virtue of a wise Creator,
(2) or by mere chance,
(3) or by virtue of an unwise maker having non-purposive actions
(4) or by means of non-purposive causal relations produced by matter.
What we hope to show is to verify the first hypothesis and refute the other ones. To accomplish this and, we offer the following points.
1. We must know to begin with low[???] to determine the value of the a priori probability of the hypothesis in question, that is, what is the probability value of there being a wise Creator having the required consciousness and knowledge for when we obtain an a posteriori indefinite knowledge increasing this probability inductively, we can compare the value of a priori probability and that of a posteriori probability, and by multiplication we come to the required value.
We need to suppose certain opinions to defend the hypothesis that the physiological composition of Socrates for instance is due to a wise Maker. Any of these opinions is regarded as elements in the hypothesis in question, and its value may be determined a priori by 1/2. For the being or non-being of such element is involved in the second additional postulate, this we obtain an indefinite knowledge having two members, the value of each of which is half, and this value is not refuted by introducing causes or effects. Now, if the value of each element of hypothesis is 1/2, then the value of all the elements is 1/2 multiplied in the number of elements. This value is included in an indefinite knowledge, different from the first, let it be called knowledge. Thus, we get an idea about evaluating the a priori probability of the hypothesis in question. But it is difficult to determine its value, because we do not know the number of elements of the hypothesis thus we cannot know the number of elements included in knowledge 1.
2. Suppose for the moment, that we confine ourselves to Socrates' physiological constitution within two hypotheses only, namely, that it is due to a wise Maker or to absolute chance. Now, we want to get an indefinite knowledge determining the value of a posteriori probability of the first hypothesis, let that knowledge be knowledge 1.
This is formulated thus: if there were no wise Being creating Socrates, the non-existence of Socrates would have been probable, or Socrates would have been existed in any other way consistent with the way he in fact is. All probabilities of the consequent, except the last, refutes the antecedent, thus we deny this latter, that is, affirm the first hypothesis.
3. In order to determine the total value of the probability of the first hypothesis, we have to multiply the number of items of knowledge 1 in those of knowledge 1, and subtract the improbable cases. But here we have before us the problem, that we do not know yet all the items of sorts of knowledge.
4. In consequence, we have to offer a rule which enables us to get the value of the probability of any items in that knowledge, the items of which we do not know. But since we do not know this, the value of the fraction cannot be determined. However, we can get the approximate value if we follow the following points.
First, if we have two sorts of indefinite knowledge the elements of which we do not know, and if the probability of the number of elements in one knowledge is equal to that in the other, then the number of elements in each is equal to that in the other. That is the actual value of one element in the one knowledge is equal to the actual value of one element in the other, and the value of the element belonging to a knowledge, the number of the elements of which we do not know, may be determined in the following way.
We assume that n2 is the probability value that the items of the indefinite knowledge are two, that n3 is the value that the number of items are three, and so on. We also assume that x2 is the value of the one item supposing n2 and x3 is the value of the one item supposing n3, and so on. Thus we determine the value of this element thus: n2x2+ n3x3+ n4x4 + .....
When we clearly face two sorts of indefinite knowledge in the way aforementioned, the process determining the value of an item in each knowledge will be similar to that which determines the value of a item in the other knowledge. Therefore their values are equal. This means that the value of the denial of a determined item in one knowledge is at the same time the same value of the denial of a determined item in the other.
Secondly, whenever we have two sorts of indefinite knowledge (let us say a, b), [and] the number of their members is unknown except that a is larger than b, and whenever we have two other sorts of indefinite knowledge (c, d), but we know only that c is larger in number than d, here we have four indefinite knowledge the number of their members we know only that a is larger than b and c larger than d. In such a situation, the actual value of a equals that of c, and that of b is equal to that of d. This means that the value of a member of a is less than the value of a member of d, still less than a member in b which we already know to be less than a. On the other hand, the value of the denial of one member in a is larger than the value of denying another member in d. For all probabilities assuming that the members of a are not less than the members of c, show that the members of (a) are larger than those in (d), since there is a chance that members of (a) may be more than those of (d) while there is no contrary chance that members of (d) are more than those of (a).
Thirdly, assuming that we have four kinds of indefinite knowledge a, b, c, and d; that we do not know the number of items in each, but we only know that items in (a) are larger than those in (b), that those in (c) are more than those in (d), that we also know that the ratio of increase in the former is more than the latter- in such a case (a) would be more in the number of items than (c), in the sense that the value of the one item in (a) is less than that in (c), and that the inverse value of the one item in (a) is larger than that in (c). For all not exceed (b) entail that (a) is larger than (c). Whereas the probabilities implying that (d) exceeds (b) do not entail that (c) is larger than (a). Thus there are probability values denoting that (a) is larger than (c), but there is none denoting the contrary.
Fourthly, if we keep (a), (b), and (d), and know that (a) has more members than (b), but know nothing about (d), and do not assume (c), then (a) has more members than (d), because all probabilities implying that (d) does not exceed (b) entail that (a) has more members than (d). But the probabilities implying that (d) exceeds (b) do not entail the converse. In consequence, the value of a member in (a) is less than that of a member in (d), and the value of denying a member in (a) is larger than that in (d) All these statement form a rule for the relative determination of the value of a member belonging to acknowledge the members of which we do not know.
Fifthly, in view of what has been said, we may suppose that the number of members of knowledge 1 and that of knowledge 1[???] is identical. That is, knowledge which includes all the elements of the hypothesis of a Supreme Being, is equal in its value to the knowledge which includes all the elements of the hypothesis of chance. For we have no idea of the number of elements in each. It follows that knowledge 1 provides a favourable value to refuting the first hypothesis, and that knowledge 1 provides favourable value to such hypothesis. And if we assume the two hypotheses to be equal then any multiplication would also give equal values.
But Socrates is not the only human being, but there might be Smith for example who owns a set of phenomena to be explained in terms of each of these two hypotheses; thus we obtain knowledge 2 and knowledge 2. We may construct another indefinite knowledge having more members than knowledge 1 and knowledge 2, which is a product of the members of both, let this new sort of knowledge be knowledge 3. Now, if this has more members than the others, then the value of the probability of absolute chance is much less than the value of the probability of absolute chance belonging to Socrates or Smith alone. And since knowledge 3 has more members than knowledge 2 and knowledge 1, it is also larger than knowledge 1 and knowledge 2, because knowledge 3 represents (a), knowledge 1 and knowledge 2 represent (b), and knowledge 1 and knowledge 2 represent (d). As we obtain the indefinite knowledge 3, we can obtain knowledge 3, which determines the probability value of a Supreme Maker of Socrates and Smith. But this knowledge has no more members than those in knowledge 1 or knowledge 2, because the elements of the hypothesis of a Maker of Socrates are the same as those of a Maker of Smith.
Thus we have before us six sorts of indefinite knowledge: knowledge, knowledge 2, knowledge 3, knowledge 1, knowledge 2, knowledge 3[???]. We do not know, we only know that the members of knowledge 3 exceed those of knowledge 1 or knowledge 2, and that the ratio of excess in the former is larger than the latter. Thus, we may argue that knowledge 3 has more may argue that knowledge 3 has more members than knowledge 3, because knowledge 3 represents (a), knowledge 3 represents (c), knowledge 1 and knowledge 2 represent (b), and knowledge 1 and knowledge 2 represent (d). But we have already argued that (a) has more members than (c) and this means that the value of a member in knowledge 3 is less than that of a member in knowledge 3 and that the value of denying a member in knowledge 3 is larger than the value of denying a member in knowledge 3.
Since the value of denying a member in knowledge 3 is larger than the value of denying a member in knowledge 3, then the value of refuting the second hypothesis is larger than the value of refuting the first hypothesis. And when knowledge 3 and knowledge 3 are multiplied and a third indefinite knowledge is obtained to determine our values, then the value of refuting the second hypothesis will be much larger than the value of refuting the first hypothesis. Thus the number of factors refuting the first and second hypothesis is constant in the third indefinite knowledge.
And since we know that knowledge 3 has more members then knowledge 3[2???], the value of the probability of the first hypothesis derived from the third knowledge is necessarily much larger than the value of the probability of the second hypothesis derived from this knowledge. Therefore, the probability of the first hypothesis increases in value.
Sixthly, likewise, we can explain the developing value of the first hypothesis in opposition to the third hypothesis supposing phenomena to be result of irrational being. If we supply additional phenomena we get a new knowledge, and consequently a new knowledge 3. Here, we find that the value of the probability of there being an irrational entity, producing all phenomena, will be very faint, whereas the value of the probability of the Wise Being hypothesis will not be so. For supposing an irrational being producing all phenomena implies new numerous elements not involved in the first hypothesis.
There is a final hypothesis stating that Socrates' physiological constitution is explained by some causal relations among such constitution and other phenomena.
We have to re-formulate this hypothesis in order to falsify it. For if we add Smith [???] for example, Socrates, we do not obtain more elements in the hypothesis because causal relations are connotational dealing with essence. If we assume that the matter of which Socrates is composed entails his physiological constitution, this means that there is a similar relation between Smith's matter and constitution.
Now, in order to get a new hypothesis we have to imagine a different kind of matter for each. Thus, we can construct on indefinite knowledge 3 which these later, except one, refute the fourth hypothesis, knowledge 3 does not include such number. Therefore, we may speak of the physiological constitution of sexual reproductive system in the male, and a different constitution in the female.
But, though they are different, they have something in common which could be explained only by supposing a Supreme Being.
Basic Empirical Statements
We have already mentioned the six-fold classification of statements from the Aristotelian point of view, and have now considered five of them. Basic empirical statements are the last class now to be considered. For formal logic, this class is first step to acquire human knowledge.
Basic empirical statements are divided into two kinds, what belongs to outer sense and what belongs to inner sense. "The sun is now rising' is an example of the former, 'I feel pain' is one of the latter. Basic empirical statements belonging to inner sense is doubtless basic because the appeal to inner private sense is the only test for its truth. Whereas statements belonging to outer sense involve the existence of an external world, and thus may be doubted. There are two formulae of such empirical statements, which justify us in doubting their certainty.
First, in our perception of lightning, for example, our direct awareness of it does not itself enable us to claim that there is something external to us called lightning; perception itself does not enable us to distinguish subjective states from objective reality. My perception of lightning is a basic empirical fact but the being of lightning is not. Secondly, even if we could distinguish the subjective from the objective elements in perception, perception of an object is not the objective fact itself but a subjective event in our brain or mind. It may be said that such subjective event is causally related to an external object, but the latter is not itself revealed in perceptual situation itself. Both formulae denote one thing, namely, that objectivity or external reality is not an immediate given to sense, thus this reality is still to be argued for. Therefore, idealism denies the belief in the existence of external objects on the ground that our empirical knowledge does not justify this belief. However, we have noticed that Aristotelians claimed that the objectivity of the event perceived is involved in basic empirical knowledge.
The supposition of objective sensible reality is not without justification as idealism claims, not is it indubitable basic knowledge as Aristotelians have argued. Such supposition is got by inductive inference. For belief in objective reality is based on the grouping of probability value in a definite direction, and such grouping of values is transformed into certainty if certain conditions, hitherto stated, are fulfilled. In what follows, we give some of the inductive ways by which we arrive at basic empirical statements. We shall consider the two formulations of our doubts in objectivity separately.
Inductive ways concerning the first formulation
(1) Suppose I am in a situation in which I perceive lightning and under [???], and do not know yet whether they are merely subjective states of the mind or also refer to a physical fact. I have a doubt not in the perceptual fact, but in interpreting it, whether it is caused by me or has an external cause. Now, both hypotheses are equally probable. This means that the value of the proposition 'the occurrence of lightning is an objective fact5 is equal in its probability to the proposition that my perception is a subjective event. Both probabilities are represented as items in an indefinite knowledge which we may determine the a priori probability that the event in question is objective.
(2) Before considering whether the event is subjective or objective, we must maintain the principle of causality inductively, and this we have given in detail in the previous chapter.
(3) Very often and through uniform observation, we see certain [???lings] succeeding each, such as light and sunrise, thunder and lightning, boiling and heat etc. Such permanent concomitance may be taken as ground of causal relations between any items.
(4) It often happens that we perceive the effect (b) without perceiving its cause (a) which is already known by induction to be prior. We may see light without seeing the sun, hear thunderstorm without seeing lightning. In such situations, we have a cause which would be (a) known inductively to be its cause, or any other unknown event, (c) for instance. This case is represented in another indefinite knowledge called 'second a priori indefinite knowledge'. In the latter, the objectivity of (a) is involved since its existence is assumed though unperceived.
(5) Added to the first and second a priori indefinite knowledge, here is 'a posteriori indefinite knowledge which determines that (c) is not cause of (b) because (c) as not cause is only a probability not certainty, and being probable there be an indefinite knowledge, does not only deny that c is cause, but affirms something - since (a) occurs in the vicinity of (b) and (a) is known inductively to be cause of (b), thus we get the affirmation that (a) is the cause of (b); and a fortiori (a) as objective. We now conclude that the probability that (c) is not a cause is inconsistent with the probability that the event (b) is merely a subjective mental state.
(6) When we compare the improbability of (c) as cause with the second a priori knowledge, we notice that the former dominates the value determined by the latter. For such improbability denies any essential relation between (b) and (c). And this involves the exceeding probability that (a) is the cause of (b).
(7) When we compare the improbability of (c) as cause with the first a priori knowledge, we find that such improbability supersedes one of the hypotheses of that knowledge namely, the subjectivity of (a).
(8) The outcome is that we obtain probability values increasing the objectivity of the event in question-such values deny that (c) is a cause of (b) and determine the degree of increase of objectivity if we multiply the items of a posteriori knowledge and the first a priori knowledge. And the more cases we have the larger the value we get for the objectivity of events.
Inductive ways concerning the second formulation(9)In the previous paragraphs, we have applied inductive inference to basic empirical statements within the first formulation of the possibility of doubt in the truth of those statements, wherein we arrived at the objectivity of the events (a) and (b). Now, we consider the second formulation of the possibility of doubt in those basic statements. This formulation is that the event (b) is a felt subjective state of the mind, and we have no indefinite knowledge as to whether it is really subjective or has it objective reference. In what follows we give the inductive steps by which we establish those basic empirical statements.
First, when we perceive the event (b) without perceiving the event (a), for example when we hear thunder without seeing lightning, we have an indefinite knowledge that (b), being surely subjective, is either caused by an objective fact (a)[x(b)] or caused by another subjective event (c).
Secondly, if our perception of (b) is caused by another subjective event (c), this latter in turn requires a hypothesis to explain it, namely, that (c) has a cause (d). Or we may suppose that the subjective event is caused by an objective fact (a) [x(b)]. Now, even if the latter requires another fact[xpact] as causing it, it is still valid to suppose that the subjective event (b) is caused by an objective fact. Such concomitance between the subjective [xobjective] (b) and the objective (a) is constant; thus the inductive argument that the objective fact (a) is cause of the subjective [xobjective] fact (b). Whereas the hypothesis that the subjective event (b) is caused by another subjective event (c) is not constant, hence, the increase of the probability that our subjective events require objective reference outside our minds.
Thirdly, there is [xhere] a point implicitly assumed [here], namely, all events are either caused by objective events or by subjective events. Thus if we regard (b) being subjective, as caused by an objective fact (a) [x(b)], we are considering the first hypothesis stating the regular concomitance between subjective [xobjective] (b) and objective (a). But there is nothing to justify this point - nothing prevents supposing that (b) subjectively given is not preceded by (a) also caused by the objective fact (a) [x(b)].
Fourthly, it is possible to increase the probability of objectivity if we assume the objectivity of all events or the subjectivity of them all, thus the value of absolute objectivity exceeds the value of absolute subjectivity.
However, this is insufficient to increase the probability of objectivity in a single event. To overcome this difficulty, we may try the following formulation. When we have in our experience subjective [???] and (b), we obtain an indefinite knowledge the (b) is caused either by a subjective (c) [x(a)] or objective (a) [x(b)]; and when we perceive the subjective event (b) without the subjective event (c) [x(a)], we obtain another indefinite knowledge that the subjective (b) is caused either by an objective (a) [x(b)] or another subjective event (c). This means that the objective (b???) causing the subjective (b) is common in that two kinds of knowledge, and this increases the probability of causality between the objective (b???) and the subjective (b) much more than the causality between the subjective (b) and the subjective (c). Therefore, the probability of the objective explanation of the subjective event (b) exceeds that of explaining it subjectively in relation to (c). For, the former implies there being a causal relation between the objective (???) and the subjective (b). Whereas the explanation of (b) in relation to (c) implies a causal relation between the subjective (b) and the subjective (c). And since the first hypothesis is more probable than the second, then the probability of objectivity exceeds that of subjectivity.
(10) We may confirm the previous point with an argument from constancy, namely, when we abstain from perceiving a certain situation, and return to it, we perceive the same. This we make clear in the following points.
First, if we suppose that the perception of an object is purely subjective state of mind, then the probability that the perceived object recurs is very faint. For, the subjective situation ceases to exist after it has been recognised, and we may perceive a different object in the next moment. Thus we may claim that when the object is purely subjective it can never recur exactly as before.
Secondly, if we claim that involves a real object external to us, this is more probable on account of similar recurrence of the same object.
Thirdly, when we perceive the previous object again, and realise it to be almost the same, we may argue that this object of perception is objectively real, on the assumption that if it is not so it could not have been the same. And since the consequent is false, the antecedent is also false; thus the objectivity of perceived objects.
However, such objectivity depends on the probability value of the sameness of the object in the two successive instants, and that it is greater than the difference between them. For if both values are equal in force, then the truth of the matter between subjectivity and objectivity is indifferent. Now, if we suppose that the object perceived is a subjective state, then it is caused by something also subjective; and in the next experience, very probably I will perceive something different, thus the probability of subjectivity is very faint. On the other hand, if we suppose the object of perception to be objective in character, this involves that there is something in common among the object on successive intervals, and the permanence of such common characters is a probability but higher in value.
It may be argued that in supposing the objective character of the object of perception we may have more than two probabilities. Suppose we are looking at a friend, John for example; then when we see all his body in the normal way on two intervals in the same manner, we can say that the object of seeing is objective. If we start assuming that the object is a subjective state, then we may see John with one arm or three arms, or other probabilities. But in this case we naturally say for example, that the lost arm is broken, or the third arm is unusual scene. And this indicates that to be objective, an object must remain the same on successive situations.
Our knowledge of the external world is inductive
In view of what has passed, we are entitled to claim that our belief in the external world depends on induction, because 'the external world' means that we can entertain statements which involves a reality outside our perception of it. And we have just argued that belief in the objective reality of perceptual statements is inferred inductively, that is, our knowledge of the world is accumulation of various beliefs in the objective reality of empirical statements. Thus the inductions confirming the objectivity of those statements assure us of objective reality. Consequently, we may face idealism which denies any justification for believing in the physical world, because this belief is inductively justified. Further, the common sense view of the world, being a reaction of idealism, is also answered, that is, this view which maintains that our knowledge of objective reality is so primary and immediate that it needs no inference. This commonsense view is answered by saying that at least some empirical statements are true owing to their probability values, and this explains their obviousness.
Belief in the conditions of perception is inductive
Our belief in the objective reference of empirical statements is the belief that when we get a sensible image of any object, and there exists certain conditions for its objectivity, then such object has objective reference.
But we possess further the belief that when we are confronted with an external object, we obtain a corresponding image, provided that certain conditions are fulfilled, and this other belief is inductively acquired. By conditions here we refer to the normal position the perceiver, the absence of a dim curtain, the normal quantity of light and the like. When these conditions are fulfilled uniformly or regularly, we inductively conclude that such relation between objective reality and those conditions has not randomly occurred, but that it is causal. If it happens that we do not get a certain image we infer that the object corresponding to it does not exist. For the absence of effect always denotes the absence of cause. When I believe that I am sitting in my study alone with nobody else, I assert a statement in the inductive way hitherto explained.
As it is an inductive matter to consider a physical object as cause of its image in my mind under certain conditions, it is likewise inductive to explain the occurrence of our ideas under these conditions. When I notice that a visual image in my experience gradually disappears, while another visual image of another object appears which seems nearer to me than the first object, we inductively conclude that whenever we have a visual percept of an object which seems nearer to us than another object, we lose sight of the latter, under certain conditions. Thus our belief that we cannot see a person's hand put behind his back is inductive, that is, inferred from the fact that the hand is not seen under certain conditions.
Resemblance between percepts and realities
We usually believe in the resemblance, of a certain degree, between the sensible image of what we perceive and the object perceived. This belief is inductively acquired and not immediately given, because in our perception of the external world we have no immediate knowledge of physical objects, but we know the latter by the mediation of sensible images or percepts. When we see a square piece of wood, for instance, we are seeing an image, in our brain, having the property of squareness, and that it is an effect of the piece of wood really there. And we believe that this property noticed in our perception is also ascribed to the physical object. True, this point is usually owned by the commonsense view of the world but it is approved with lesser degrees by those who disclose more subjective factors in perception. However, it seems to be a minimum degree of resemblance between the percept and the object perceived. When we usually see round objects such as apples or oranges we do not usually ascribe to them squareness, though we have a priori basis of claiming that any physical object must cause a percept having the properties that it has; there is no self contradiction in saying there is a round object there causing in my mind a square-like shape. [this can be taken as a refutation to Kant's theory, which considers external world to get moulded into the constructions of space and time in mind, making the external objectivity a subjective (and distortion of the objectivity) in mind, refer to details in "Our Philosophy", reader's note]
Therefore, our belief in such resemblance is gained inductively, that is, what we see round in shape is really round. Suppose someone assumes that such roundish percept corresponds to a really square physical object. Here we have two alternatives: either we do not see the part of an object which faces us or see an object which is not really there. When we see a round sheet of paper and suppose it to be really square, then if this square is that kind of shape which can be drawn inside the round shape that we see, this implies that the spatial area of the image is bigger than the real sheet. If, conversely, this square is larger than what can be drawn inside the round shape we see, this implies that we do not see part of the square. Therefore, if I see all the parts and sides of the object then the percept resembles the real original.
When we have an a priori indefinite knowledge that the sheet of paper must have a shape, we know also that it actually has the specific shape that we see, if our perception has objective reference. And when we see the paper as round, we notice that the conditions being the objectivity of our perception, if it is regular, as the round shape is the specification of the a priori knowledge of the shape.
For, provided that our perception involves objective reference, the paper cannot be square. But if the condition is not fulfilled, that is, if our perception has no objective reference, then the paper probably has any other shape. So we conclude that the object of our indefinite knowledge is restricted to a hypothetical statement; and the restriction is that the paper is round if our perception is objective. But the latter fails, that is, the paper may have any shape.
Thus, any probability value affirming the antecedent affirms that the paper is round. We reach such value by induction. We notice the concomitance of seeing many sheets of paper with a certain shape, so we arrive at a greater value of the objectivity of our perception.
Beliefs in resemblances of particulars
We believe that certain things have something in common, so we call them a's, and other group of things having something is common, let us call them b's. Such belief is acquired inductively. We have already argued that there are resemblances between the physical objects and our perceptual images of them. When this happens we say that such many physical objects resemble each other. Resemblances among percepts are immediately recognised by us, but resemblances among things are inferred from resemblances of the former.
But our belief in resemblance between things and perceptual images is not itself sufficient for knowing resemblance among kinds of things, because we need suppose that nothing has changed in our sensory system and mental activities. For perceptual images depend on two factors, namely, the existence of the external world, on the one hand, and physical, physiological and psychological conditions of perception on the other hand. Now, if our sensory system is the same, we obtain the same images uniformly as before; but if the internal conditions have [underently???]. that is, it is probable that we get two similar percepts when there are two different objects before us, or two different percepts when the physical object is the same on two successive intervals.
In order to prove resemblance between any two physical objects on the basis of our perception of them, we have to obtain a significant probability value opposing any change in our subjective system. We recognise such resemblance by induction, and this process is necessary condition of proving that a is cause of b, (cause being a relation between two meanings), when we notice concomitance between a's and b's. In order to reach this conclusion, we must discover all a's to belong to one kind, all b's to another. Then we are able to increase the probability of causality between these kinds of things.
We have so far explained four, out of the six, classes of statements, and concluded that all empirical, basic empirical, testimonial and intuitive statements are inductive depending on the accumulation of probabilities in a certain direction, according to two steps of inductive inference. We do not mean that certainty in these statements depends for every body on induction; belief in the objective reference of empirical statements depends for some people on confusing subjective and objective elements of perception. Again, belief in empirical statement may depend on purely psychological factors, or on expectation in terms of habit and conditioned reflex. What we mean is that our objective belief in those statements is based on induction.
Primitive and innate statements
These two classes of statements are considered by Aristotelians as certain, a priori and starting points of human knowledge; they are said to be apprehended independently of sense experience. Sense experience is introduced only, in relation to those statements, when we want to explain belief in them. Belief in these statements derives from conceiving their subjects and predicates, and experience supplies the mind with various images and meanings, these being the materials for conception. What can we say of this theory?
Primitive and innate statements, we claim, are inferred inductively. Let us explain. For Aristotelians, a primitive a priori statement is one the predicate of which is ascribed to its subject necessarily, that is, any subject of this kind implies a certain predicate. The statement 'the whole is larger than the part', 'or all rights angles are equal' are a priori, in the sense that the whole' implies being larger than any of its own part, or that if being right angle' is common among angles, this implies their being equal. But such implication relation between subject and predicate may be reached inductively, if we put forward two presuppositions: (a) the ascription of a predicate to a subject is a necessity for the latter, (b) such ascription is a consequence of an element different from the very notion of the subject.
Now, these presupposition are similar to those which we recognise when we observe (b) succeeding (a), and say either that (a) is cause of (b), or that (c) is cause of (b) while (a) and (c) are concomitant.
Now, if we suppose the implication relation between subject and [predicate??? precedent], we mean a relation between two concepts. But if we suppose that the ascription of predicate to subject is a result of a third element, then we have many probabilities before us in order to determine which thing is such element. In such a situation, inductive inference enables us to favour the first probability by the help of an indefinite knowledge which includes all forms of supposing what may be a cause of a certain predicate or property instead of triplication relation. For such indefinite knowledge gives us the accumulation of the probable values of that cause, thus affirming he implication relation between subject and predicate, save one value, i.e. that the only alternative to implication is involved in all cases when subject and predicate are correlated; and in this point, the exception is indifferent to all probabilities.
Therefore, we may affirm the implication relation inductively. And here we need not consider here the value of a priori probability of implication on the ground of indefinite knowledge prior to induction. For, the probability value of implication involved in the a posteriori knowledge dominates the a priori value of the probability of implication or its improbability.
There are two exceptions to inductive application of primitive statements, namely, the principle of non-contradiction and postulates of inductive inference itself.
First, the former states the impossibility of affirming two both contradictory statements. This cannot be proved by induction but must be presupposed a priori. For if we do not assume its truth beforehand, we cannot accumulate probable value in a certain problem, since such accumulation depends on the fact that any probability implies the negation of its contradictory.
Second, as to the postulates of inductive inference, any degree of approval of any statement is presupposed a priori. When we claim that it is possible to apply induction to all primitive statements, except the two principles, just referred to, we do not mean of course that such primitive statements are inductive not a priori, but we do mean that they can be theoretically explained in terms of inductive methods, though they may still he considered a priori.
Differences between primitive and inductive statements
One main difference between these two classes of statements is that more evidence would give inductive statement more truth, while more evidence adds nothing to the truth and clarity of a priori primitive statements. More inductive examples do not add to their truth because their truth is independent of experience, that is, it is a priori statement. The statement '1 + 1 = 2' does not become clearer or more true when we increase its relevant applications. Conversely, if we take the statement 'metals extend by heat', the more we collect instances of it, the more it is confirmed and its probability is established; thus it is inductive.
However, the difference is not easily made to distinguish a priori statements from empirical ones. For, there are empirical statements which become already settled by instances, so that new evidence does make no difference. For example, 'men who are beheaded die', or 'fire is hot' are inductive but they are so true and confirmed that they need no more confirmation. Thus, we do not consider such type of inductive statements in order to test their truth by looking for more evidence. In such cases we may not find the difference aforementioned relevant between a priori and empirical statements. Now, there is another difference between those types of statement, namely, if we have a feeling of the possibility of abandoning a statement, provided there is evidence against it, then this statement is empirical and inductive otherwise it is a priori and primitive. Suppose some trustworthy persons tell me that they have seen a beheaded [???] we probably believe them; nevertheless have a feeling of disbelieving them.
The third difference between a priori and empirical statements is that the latter cannot be always true in all possible worlds, however much we collect favourable evidence, and that they are true only in our sensible world. Whereas primitive a priori statements are absolutely true in our world and in any possible world. 'Tire is hot' is an empirical statement, for though it is obviously true in our world it is not necessarily true in any possible world, but we may intelligibly suppose a world in which there is cold fire. Whereas 'contradictory statements cannot be true together' is always true in any possible world we may imagine or conceive, because we cannot suppose a world where affirmation and negation or truth and falsity can be consistent with each other.
[Thus,]There are three differences between a priori or primitive and empirical or inductive statements, which are taken criteria for picking up primitive and innate statements.
Induction and mathematical statements
We have already shown that Aristotelian logic classifies statements into primary and secondary; the former is in turn classified into six certain types of statements. This logic regards any inferred statement from these formal statements out of which our knowledge can be established. As such formal statements can be inferred from primary ones, they can also be inferred from those inferred from these. Thus formal statements derive from primary certainly true ones either directly or indirectly. The only way to infer the formal statements and prove them is through syllogism, the conclusion of which is implied in its premises.
We have made clear the relation of those six certain classes of statements to induction, and argued that they are inductive. Now, what relation there is between induction and formal statements, such as 'the angles of a triangle are equal to two right ones'. To give an answer to this question, we must distinguish the statements from they way it is inferred and whether the conclusion is validly inferred from its premises. As to the statement itself, it is no doubt possible to reach at [xI] it by induction not by deduction from a priori premises. Instead of inferring the previous example about triangle from the fundamental postulates of Euclidean geometry, it may be inferred inductively, in the same way in which the six classes of certain statements, discussed above are considered.
That is, we begin with assuming two probabilities:
(a) that a triangular figure implies that its angles equal two right ones,
(b) that such equality is due to an external reason (c). When we take notice of a triangle and observe that in every case, drawing it is always connected with the its three angles equalling 180°, we obtain an indefinite knowledge including the probability and improbability of (c), and we get a high probability value for the implication hypothesis. Such value derived from that knowledge dominates the value of a priori probability.
Further, as to the way in which our statements is inferred, it may be said that it is deduced a priori from certain postulates, but either the deducer is feeling sure that he proceeds validly, or that we may test his work on inductive grounds. First, the deducer may feel sure that he is right in choosing relevant postulates and proceeding rightly from premises to conclusions, and feels that conclusions follow from premises and that he does not fall in any mistake. Such feeling has nothing to do with induction but it is a personal direct feeling involving alertness and belief in the truth of its object.
Second, we may examine such deductive process inductively, that is, we may observe the number of mistakes the deducer has committed and then we determine the probability value of his falling into mistakes, and such determination is inductive. For the value is known through observing our activity and is considered as inductive inference showing that such value is not random but expresses the proportion of mistakes relative to right steps made.