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?

    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.

    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.

    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 [xTato'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].

    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 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.

    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 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 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/n_2. 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].

    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.

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