Human Knowledge and Probability

By: Martyred Ayatullah Muhammad Baqir As-Sadr

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 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.[18]
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 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
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.

Testimonial statements
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 though we know nothing of the essence of both. The second form of induction is concerned with the existence of 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. 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, 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.

Is There A priori Knowledge?
We have already referred to the main difference between rationalism and empiricism concerning the source and ground of knowledge. Philosophers through the ages have been divided into two classes[/groups] on this problem, those who believe that human knowledge involves an a priori element independently of experience, and those who believe that experience is the only source from which all sorts of knowledge spring, and that no a priori element is involved, even our knowledge of logic and mathematics. According to the former class (rationalists), man is believed to have some a priori ideas regarded as basis of our knowledge and stimulating our experience and explaining it.
But as the empiricists maintain that we have nothing a priori, that all knowledge derives from experience, through which everything is explained. In consequence, starting points of knowledge, for empiricism, are particular ideas supplied by experience, and all that this gives us is particular. But this situation involves that any general statement has more than what is particular, thus we are not justified in the certainty of such statements. Conversely, rationalism affords an explanation of this certainty on the ground of a priori ideas assumed.
Now, we must have before us a criterion by means of which we can compare and evaluate rationalism and empiricism. This criterion may be reached by pointing the minimal degree of belief in the truth of both formal and empirical statements. And any theory of knowledge disclaiming such criterion is doomed to failure, while it is approved when is consistent with such criterion. Now what is the minimal degree of credulity in formal and empirical statements?

Empirical Statements
This class of statement is given by rationalists a high degree of credulity reaching sometimes the degree of certainty while empiricism denies certainty to these statements since they depend on induction, but they acquire a higher degree of probability. Thus both theories of knowledge agree in regarding empirical statements as highly probable, and their probability increases by the increase of more instances. The question now arises, which of the two theories gives more satisfactory explanation than the other. We have already maintained that such degree of belief rests on the application of probability theory to induction. This theory has its own postulates some of which are mathematical in character. It is necessary then to regard such postulates as a priori statements independent of induction. This, we notice, is more consistent with rationalism than empiricism. For empiricism has to maintain, within its principles, that probability postulates are derived from experience; thus it has no basis for exceeding the values of probabilities. In short, empiricism cannot explain the minimal degree of belief in the truth of empirical statements.

Formal Statements
By these we mean in this context mathematical and logical statements. These have always made a problem for empiricists, in order to explain their certainty and the way they are distinguished from empirical statements. It is commonplace that a mathematical or logical statement is certain; thus if it is claimed that all knowledge derives from experience and induction, it follows that '2+2 = 4' or 'a straight line is the shortest distance between any two points' are inductive. If so, these statements are the same as empirical statements. Hence, empiricists have to choose either to ascribe certainty to formal statements only, or to make formal and empirical statements on the same footing. Either alternatives is a dilemma for them. For if they hold formal statements to be certain, then these cannot be inductive, it follows that we have to admit that they are a priori. And if formal statements derive from inexperience and not a priori, how can we explain their certainty?
The differences between empirical and formal statements are as follows. (1) Formal statements are so absolutely certain that they cannot conceivably be doubted, while empirical statements are not. Statements such as '1 +1 = 2' 'a triangle has three angles', or 'two is half of four' are very different from statements such as 'magnets attract iron', 'metals extend by heat', or 'men are mortal'.
The former cannot conceivably be doubted however sure we are about them. If we imagine someone we trust saying that there is water which does not boil when heated or that some metal does not extend by heat, we may possibly doubt general empirical statements. Whereas we cannot conceive denying such logical truth as 'two is half of four', even if the greatest number of men told us that two is not half four.
(2) More instances do not make mathematical statements more certain, while they do confirm empirical statements. When we supply more examples or new experiences about the expansion of metal by heat, we are more justified in claiming the truth of the statement. But if we observe only once that a magnet attracts iron, we have not established the truth of the statements unless we provide more and more and more instances. The case with formal statements is different, because when I can add five books to five others and realize that the sum in ten, then I judge that every two fives equal ten, whatever kinds of things I add, and the judgment is always true without giving more instances. In other words, once we hear or read a mathematical or logical statements and understand its meaning, we are sure of its absolute truth and certainty without the least of doubt; whereas the more we are supplied with instances that confirm an empirical statement, the more its truth is vindicated.
(3) Though general empirical statements are not confined to our actual observations and experiments, they concern our physical world and do not transcend it. For example, when we say that water boils at a certain degree by heat, we transcend our actual observation but not our empirical world. But if we can conceive another world in which water boils at a different degree by heat, then we are not justified in making the general statement that water boils at that degree in that conceivable world. Conversely, mathematical and logical statements [admit] of different consideration. The statement 2+2=4 is always true in any world we may conceive, and we cannot conceive a world in which a double two equal five; and this means that formal statements transcend the real in our sensible world.
Such differences between formal and empirical statements caused empiricism a dilemma in the way formal statements are to be explained, since to be consistent it has to explain then within its experience alone. Empiricism had to give formal statements a purely empirical explanation for some time, and thus made both kinds of statements on the same footing in that both are probably not certainly true. For empiricists, the statement 1+1=2 was probable and involved all the logical inadequacies ascribed to empirical generalisation. But this position proved empiricism to be on the wrong track and gave rationalism utmost credit, since the latter could explain the certainty of formal statements in terms of a priori knowledge and probability of empirical statements in terms of experience.

Logical Positivism
Empiricism has not changed its position in its empirical explanation of the truth of formal statement until the appearance of logical positivist movement in the present century[19]. Logical positivism admits the difference in nature between mathematical and logical statements on the one hand and empirical ones on the other. Such movement classifies mathematical statements into two classes, those of pure mathematics such as 1+1=2, and those of applied mathematics such as Euclidean postulates, e.g., any two straight lines intersect in only one point. The former are in essential, logical statements, and all logical and purely mathematical statements are necessary and certain, because they are tautologies.
The statement 2+2=4 does not give us information about anything empirical, but it is analytic. We may make clear this logical positivistic distinction between analytic and synthetic statements in some detail.
Synthetic or informative statements give us information about the world, in other words, the predicate in statements of this type is not included in the very meaning of the subject. 'Mortal' in the statement 'man is mortal', or Plato's teacher' in 'Socrates is Plato's teacher' is not part of the meaning of man or Socrates; so these statements give us new knowledge about men and Socrates. But analytic or tautological statements are those whose the predicate is part of the concept of the subject; the statement here gives us no new empirical knowledge but only analyses its subject. The bachelor is unmarried' is an example of analytic statements, because 'unmarried' is part of the meaning of ' bachelor'.
Now, logical positivists have tried to consider statements of pure mathematics and logic as analytic and explain their absolute certainty by means of their uninformative function. '1+1 = 2' is, for them, trying and sterile, because 2 is a sign identical to the signs (1 + 1), and then say that the two signs are identical. But statements of applied mathematics, e.g., postulates of Euclidean geometry, give us new information and knowledge. For positivists, 'straight line is the shortest distance between any two given points' is not analytic because shortness and distance are not part of the meaning of a straight line.
These statements are not considered by them necessary and a priori. "It was said about Euclidean geometry or any other deductive system that it deduces its theorems from certain axioms, these require no proof because they are self evident and necessarily true, although self-evidence is relative to our past knowledge .... But you may logically doubt the truth of that past knowledge thus the so called axiom is no longer self-evident. Euclidean system was supposed for centuries to be based on self-evident axioms, being indubitable ... But such supposition is now mistaken. The appearance of Non-Euclidean geometries made possible other geometries based on axioms different from Euclid's thus we reach different theorems"[20].

Criticism
This positivistic view of mathematical statements may be criticized on the following lines. First, if we agree that all statements of pure mathematics are analytic and tautologies, this does not solve the problem which empiricism faced, namely, explaining mathematical statements empirically, because it has still to explain necessity and certainty in those analytic statements. Take the typical analytical statement 'A is A'; its certainty is due to the principle of non-contradiction. This states that you cannot ascribe a predicate and its negation at the same time to a given subject. Since this principle is the ground of certainty in analytical statements, then how can we explain certainty and necessity of this principle itself? It cannot be said that the principle is itself analytic because impossibility is not involved in the being of a predicate and its negation. If we consider the principle of non-contradiction synthetic or informative, we are again required to explain its necessity. For to say that the principle is synthetic is to deny the distinction between the principle and empirical statements. On the other hand, if we admit that the principle is a priori not empirical we are rejecting the general grounds of empiricism. In other words, is the statement 'A is A', being certain and analytic, identical with 'it is necessary or not necessary A is A' or 'A is in fact A'? If the former then the principle is synthetic because necessity or impossibility are not involved in the concept of A; if the latter then the principle is not necessary.
Secondly, we may say that statements of applied mathematics are not absolutely necessary but have restricted necessity. For instance, axioms of Euclidean geometry are not absolutely true to any space but only true to space as plain surface; thus this geometry involves an empirical element. In consequence, it is possible that there may be other geometries different from Euclid's. But this does not deny the necessity of Euclidean axioms provided that space is a plain surface. Thus Euclidean axioms are hypothetical statements, the antecedent of which is that space is plain surface. Such statements are unempirical and thus as necessarily true as those of pure mathematics. But the former differ from the latter in that they are not analytic because the consequent is not implied in the antecedent. For that the angles of a triangle are equal to two right ones is not part of the meaning of space as plain surface. They are necessary synthetic statements.
Therefore we have to reject empiricism owing to its failure to explain the necessity of formal statements, in favour of rationalism in this respect. Further, the empiricistic dictum that sense experience is the sole source of human knowledge is not itself a logical truth, not is it itself derived from experience. It remains that this dictum is obtained a priori; if true, then empiricism admits a priori knowledge; and if it is empirical and a priori then it is probable. This implies that rationalism is probably true for empiricism.
Empiricism and Meaning of Statements
Empiricism does not only maintain that experience is the source of all knowledge, but maintains also that experience is the ground on which the meaning of statements is based. Empirically unverified statements, for empiricism, are logically meaningless, neither true nor false. That is one of Logical positivism's principal contentions.
Let us discuss this point. We have before us three theories in this context. First, we have the theory which maintains that any word having no empirical reference is without meaning, "when I am told", an eminent logical positivist writes, "that you do not understand a certain statement, this means that you cannot verify it in order to know whether it is true or false. If you tell me that there is a ponsh in this box, I understand nothing because you cannot have an image of a ponsh when you look into the box". That is to say the word 'ponsh' has no meaning because it applies to nothing in experience.
This situation depends on the view that sense is the sole source of forming concepts. If we have a statement every word in which has empirical import, it has then a meaning denoted by the possibility of forming images or concepts of each word. The following statements: 'John is a living creature', 'John is not Peter', there are bodies' are meaningful because each term in them has empirical application. Thus the statement 'there can be life without body' has a meaning because we can form a complex concept of its terms, though such concept is not in fact to be found in experience.
The second theory to explain the relation of meaning to experience states that experience makes a difference as the truth or falsehood of the statement concerned. The statement 'there can be life without body' is meaningless on this theory because the complex involved in the statement cannot be empirically tested because experience is indifferent as to disembodied life: it is not found in experience nor does experience deny it.
The third theory does not merely state that each word in a statement must have an empirical import to have a meaning, or that experience must make a difference as to the truth or falsehood of the statement. The theory states also that the statement in question must be capable of being verified. That is, unverified statements are meaningless, thus its meaning is constituted by its being verified empirically.
In consequence, a number of statements considered meaningful, if the above account is correct, are rendered by positivists meaningless. For example, 'rain has fallen in places not seen by us, has meaning on our account because its terms have empirical import and because our notion of experience makes difference as to the truth or falsity of the statement. But the statement is meaningless on the third account because it is not possible to be verified empirically, because any rain to the seen would not verify the statement. To this third account we now turn to comment.
We cannot accept the positivistic identification between the meaning and method of verifying a statement for following reasons. First, such identification is contradiction in terms, because to say of a statement that it is subject of verification or falsification is to say that it may be true or false, and a fortiori that it has meaning. And this involves that the meaning of a statement is not derived from its verifiability, but the latter presupposes its meaningfulness.
Secondly, there can be statements which are not only meaningful but we also believe in their truth, and yet they are empirically unverifiable; for instance, 'however wide human experience is, there can be things in nature that cannot be subject to our experience', or there can be rain falling not seen by anybody'. These statements and the like are intelligible and true although they cannot be empirically tested.
Thirdly, we may like to know what is meant by experience by virtue of which verification is possible. Is it meant to be my private experience or anyone else? If it is meant to be my own experience, this means that the statement which expresses a fact beyond my own experience has no meaning, e.g., 'there were men who lived before I was born'; but this certainly is meaningful. Further, if by experience is meant that of other people, this is inconsistent with positivistic principles because experiences of other minds lie beyond my own, but they are known to me inductively. Thus, any such statement is meaningful. For example, the belief in causality as involving necessary relation between cause and effect has meaning though it is not immediately verified by me but it is inductively verified.
Fourthly, we may ask again, whether the criterion of the meaning of a statement is its actual verification or its verifiability. If we assumed the former, then what cannot be actually verified is meaningless, for logical positivism, even if the statement is concerned with nature. The statement 'the other side of the moon is full of hills and valleys' is not actually verified because this other side is not seen by anyone on the earth and so no one is able to verify its truth. However, it is false to say with the positivists that such statement is meaningless.
Science often provides propositions to be examined even before we possess the crucial experiment which testifies their truth. And scientific activity in testing hypotheses would be frivolous if scientific hypotheses were meaningless.
On the other hand, suppose we assume that the positivists actually claim that the meaning of a statement is its verifiability in principle, in the cases in which actual verification is empirically impossible but still logically possible. We must now examine this claim, how do we know that a statement is verifiable? If we do know this in a way different from sense experience, then positivists admit a sort of knowledge independent of experience which is inconsistent with its principles. And if they identify verifiability with actual verification, they consider many statements meaningless though they are concerned with nature and are intelligible.
In fact we need understand a criterion of the meaning of a statement before we test its truth or falsity. Truth or falsity presupposes one image comprising the concepts of the terms and the relations among them in a statement. If we can grasp such complex image, we can get its meaning.

Has knowledge Necessarily A Beginning?
If human knowledge is established such that certain items are derived from others either by deduction or induction, then it must have a beginning with certain premises un-derived in any way. For otherwise we fall in an infinite regress, and thus knowledge becomes impossible.

Reichenbach's Position
Reichenbach claims the possibility of knowledge without any beginning, and argues (a) that human knowledge is all probable, (b) that probable knowledge can be explained in terms of probability theory, and (c) that the theory of probability he adheres to is frequency theory, and that the proportion of the frequency of past events is constant and regular. In consequence, any probability involves a certain frequency, the proportion of which can be determined by means of other frequency probabilities, without beginning. Lord Russell illustrates Reichenbach's theory by the example of the chance that an English man of sixty will die within a year. "The first stage is straightforward: Having accepted the records as accurate, we divide the number of dead people within the last year by the total number. But we now remember that each item in the statistics may get some set of similar statistics which has been carefully scrutinized, and discover what percentage of mistakes it contained. Then we remember that those who thought they recognised a mistake may have been mistaken, and we set to work to get statistics of mistakes about mistakes. At some stage in this regress we must stop; wherever we stop, we must conventionally assign a "weight" which will presumably be either certainty or the probability which we guess would have resulted from carrying our regress one stage further"[21].

Russell's Objection
Russell objects to this point of Reichenbach by saying that this infinite regress makes the value of probability determined in the first stage of the regress almost zero. For we can say the probability that an (a) will be a (b) is m1/n1; at the level, we assign to this statement a probability m2 /n2, by making it one of some series of similar statements; at the third level, we assign a probability m3 /n3 to the statement that there is a probability m2 /n2 in favour of our first probability m1 /n1 and so we go on for ever. If this endless regress could be carried out, the ultimate probability in favour of the rightness of our initial estimate m1 /n1would be an infinite product: m2/n2. m3/n3 . m4/n4 ........ which may be expected to be zero. It would seem that in choosing the estimate which is most probable at the first level we are almost sure to be wrong[22].

Discussion
But Russell's objection may be retorted by saying that any estimation we impose on endless regress which may be mistaken admits of two alternatives: the mistake may arise when we realise that the proportion of mistakes in statistics is greater than that which we found in the list discovering mistakes in this statistics, or the mistake arises when we realise that the former proportion is lesser than the latter. For example, so we may suppose that the value of the probability of the death rate among Englishmen over sixty is 1/2, on the ground of the frequency of death rate in statistical records.
Now if we look back into these records and found that the rate of mistakes in such records is 1/10, this means that the value 1/2 has the chance that it may be mistaken with the probability value 1/10. Thus the possibility of mistake involves two equal probabilities, i.e., either that the first value is really over 1/2, or that the second is really less, not that the value is 1/2 x 1/10.
We believe that Reichenbach is mistaken in dispensing with the absolute beginning of knowledge by recourse to endless regress. For no knowledge is possible without real starting point. For instance, the probability which determines our knowledge that Englishmen over sixty die cannot be interpreted except in terms of probability theory with all the axioms and postulates connected with it. Thus, in applying such theory, we have to assume prior knowledge of those axioms, and these constitute our starting point. And those axioms cannot be applied, as we having already shown, except on the basis of indefinite knowledge. Therefore there can be probable knowledge without prior knowledge.
As to the beginnings of knowledge, we may assume two kinds of knowledge: one presupposed by the axioms of theory of probability, the other is that of the nature of sensible experience regardless of its contents. When we see clouds in the sky for example, then clouds make the object of our seeing, but our awareness of seeing is a primary knowledge and not inferred. Now, we may ask whether such primary knowledge is certain or not. It is not necessarily certain but may be probable.
Primary probable knowledge applies to two fields. First, it applies to sensible experience. Usually I am certain about what I experience, but it may happen that I am doubtful about what I see or hear when the object is dull or faint or distant in my perceptual field; in this case I get probable knowledge. The second field of primary probable knowledge is that of primary propositions in which the relation of subject to predicate is immediate without a middle term. Such propositions are the ground of all syllogistic inferences, and can themselves be reached by direct awareness. Such awareness may get the utmost degree of certainty, and may gain lesser degree of credibility. In consequence, since those propositions may have probability values we may increase their values by virtue of probability theory.
The object of our study in this book is two fold. First, we are concerned to show the logical foundations of inductive inference which embraces all scientific inferences based on observation and experiment. In this context we have offered a new explanation of human knowledge based on inductive inference. Secondly, we are interested to show certain conclusions connected with religious beliefs based on our study of induction. That is, the logical foundations of all scientific inferences based on observation and experiment are themselves the logical foundations on which a proof of the existence of God can be based. This proof is a version of the argument from design, and is inductive in its character.
Now, we have to choose the whole scientific knowledge or reject it, and then an inductive proof of the existence of God would be on the same footing as any scientific inference. Thus, we have found that science and religion are connected and consistent, having the same logical basis; and cannot be divorced. Such logical connection between the methods of science and the method of proving God's existence may be regarded as the ground of understanding the divine direction, in the Koran, the Holy Book of Muslims, to observe the workings of the natural world.
The Koran is encouraging people to scientific knowledge on empirical grounds. And in this sense, the argument from design is preferred in Koran to other proofs of the existence of God, being akin to sense and concreteness and far from abstractions and sheer speculations.
Notes:
[19]Hume claimed the a priori character of formal statements before Logical Positivists.
[20]Zaki Naguib Mahmoud. Positivistic Logic, P. 324, Cario. 1951
[21](1) Russell, Human Knowledge, P. 433.
[22]Ibid, p. 434