It is the plan of this book to open discussions between a philosopher and his critics, benevolent or otherwise, for the purpose of creating an opportunity to clarify opinions and correct misinterpretations. I must confess that this program, welcome as it appears in many other cases, makes it somewhat difficult for me to contribute to the present volume. Bertrand Russell distinguishes himself from many other philosophers by the clarity with which he has always presented his ideas. An attempt to further clarification, therefore, seems to be out of place, and should be reserved for other sorts of philosophy. There are philosophies, indeed, which were so vaguely stated that every school of philosophy was able to give them interpretations corresponding to its own views. Many a philosopher derives his significance from the obscurity of his exposition rather than from the weight of his ideas, and I should like to believe that such ideas would have lost their persuasive power had they been formulated more precisely and coherently. Bertrand Russell is certainly not a philosopher of this sort. He constitutes a fortunate example showing that a philosopher may owe his success to clarity and cogency, to painstaking analysis and the renunciation of the mysterious language of oracles. It seems therefore scarcely necessary to provide for a second edition of his philosophy enlightened for the use of posterity by the criticism of opponents. What makes the present writer even more unsuitable for such a purpose is the fact that he does not even feel himself an opponent, that he agrees very much with most of the fundamental views of Bertrand Russell, to whom he is deeply grateful for the instruction and enlightenment which he has always gathered from Russell's books.

In order that this essay may serve the general purpose of the present volume I shall therefore try to follow another plan. I shall attempt to summarize the contributions which Russell has made to modern logic, hoping that Mr. Russell will correct my summary wherever it is incomplete or where the emphasis is on the wrong point. I hope, in addition, that Mr. Russell will answer some questions as to the genesis of his ideas and thus give us some valuable information concerning the history of modern logic. Finally I am optimistic enough to assume that at least on some points there will remain a diversity of opinion which may supply the reader with the most coveted fruit on the tree of philosophy: a philosophical dispute.

Let us have a short view of the situation within the history of logic at the time when Russell entered its field. Aristotelian logic, which for two thousand years had dominated Western thought, had finally been superseded by the symbolic logic constructed by such men as Boole, de Morgan, Peirce, Peano, Cantor, Frege, and Schröder. Boole's work, from which we may date the modern period of logic, was already fifty years old. But the new ideas had not yet acquired any significant publicity, they were more or less the private property of a group of mathematicians whose philosophical bias had led them astray into the realm of a mathematical logic. The leading philosophers, or let us better say the men who occupied the chairs of philosophy, had not taken much notice of it and did not believe that Aristotelian logic could ever be surpassed, or that a mathematical notation could improve logic. Russell, then at the age of twenty-eight, had read the writings of this group and attended a congress of logic In Paris in the year 1900, where he met Schröder, Peano, Couturat and others. A few years later he wrote his Principles of Mathematics, followed after some further years by the Principia Mathematica, written in cooperation with Whitehead. Why is it that from the appearance of these books we date the second phase of modern logic, the phase which united the various starting points and logistic calculi into one comprehensive system of symbolic logic?

There are several reasons which made Russell's work the beginning of this new phase. The first is given by a number of technical improvements over the symbolic systems of his predecessors. The second is that he combined the creation of a symbolic logic with the claim of including the whole of mathematics, an idea which, controversial as it has always been, has never ceased to excite the minds of mathematicians and logicians alike. The third is that Russell, uniting in his books a skilfully chosen notation with the brilliant style of a writer, drew the attention of philosophers of all camps to symbolic logic, which thus was made palatable for the first time.

I shall try to summarize the technical improvements which Russell has contributed to symbolic logic.

There is to be mentioned first Russell's introduction of the concept of propositional function. The idea of conceiving grammatical predicates as classes is of course much older and goes back to Aristotle, Boole's algebra of logic makes wide use of it. But Russell's concept of propositional function extends the concept of a class to that of a relation and thus combines the advantage of the mathematical analogy with a closer correspondence to conversational language. This close relation to conversational language constitutes one of the strong points in Russell's logic. It manifests itself also in his theory of descriptional functions. Using the iota-symbol introduced by Peano, Russell showed the way to the understanding of the definite article "the" and similar particles of speech, and developed Peano's notation into a general syntax of a high degree of perfection. It is surprising to what extent the understanding of the logical nature of language is facilitated by the use of Russell's concepts. In many a logic course I have given I have had the occasion to watch this effect of Russell's logic. Through its clarification of the structure of language, symbolic logic awakens logical abilities till then dormant in the minds of the students.

Next I must mention Russell's decision to use material implication. This sort of implication with its puzzles, it is true, has been known for a long time, Peirce,¹ who himself saw the advantages of this implication, quotes Sextus Empiricus as the first to have pointed out the nature of this relation, and justifies its use by showing that its queer consequences cannot lead to wrong results. But Russell was the first to discover that the whole system of logic can be consistently developed by the use of this sort of propositional operation. He saw that this is a point where the correspondence to meanings of conversational language must be abandoned, if a satisfying logic is to be constructed, and his logic thus was the first which is consciously extensional. He was not afraid to use propositions like "snow is black implies sugar is green," since he saw that the meaning of the word "implies" used here can be clearly defined and leads, unreasonable as it appears at first sight, to a reasonable logical calculus. He deliberately postponed the construction of concepts better fitting conversational usage, in the hope that this might be possible within the frame of an extensional logic, by the introduction of more complicated relations. His formal implication represents a stepping stone on this path, Russell saw that it corresponds much more closely to what is usually meant by an implication, although he frankly stated the limitations holding even for this generalized implication. This line of development has later been continued in Carnap's discovery, according to which a reasonable implication can be defined by the use of the metalanguage, a further continuation of this line of thought, which leads from tautological implications to a more general kind of implication corresponding to natural laws, has been given by the present author.²

It is the advantage of extensional operations that they permit us to define the notion of tautology. Although the formal definition of tautology on the basis of truth-tables seems to have been an idea of Wittgenstein's, I have no doubt that Russell has always clearly seen this fact and used it for the definition of logical formulae. The necessity expressed by logical formulae derives from the fact that they are true whatever be the truth-values of their constituents. This tautological character of logical propositions, on the other hand, represents the ground of their emptiness, and Russell has always emphasized that a logical formula states nothing. He saw, at the same time, that this result does not make logic superfluous. On the contrary, the use of logic within all forms of scientific thought is based on the fact that logic is empty. Were it not so, we would not be allowed to add logical formulae to empirical assumptions. Logical transformations exhibit the inherent meanings of such assumptions without secretly increasing their content. Moreover, although a tautology is empty, the statement that a certain formula is a tautology is not empty; and the discovery of new and intricate tautologies will always remain a challenge to the logician or mathematician. The history of mathematics, revealing more and more unexpected tautological relations, represents a proof of this contention.

I should like to add here a remark concerning Russell's distinction between inference and implication. Although at the time he wrote the Principia the present distinction between levels of language (with which we have to deal later) was not yet known, Russell clearly saw that inference and implication are of a different logical nature. Whereas implication is an operation connecting propositions and leading to a new proposition, inference represents a procedure, performed on propositions. Russell emphasized that inferences cannot be stated in a formula, a result which may appear somewhat paradoxical, since he symbolized it in the traditional schema

We know today that the correct formulation is to say that the schema belongs to the metalanguage; that the formalization of inference can be given, not in statements of the object language, but only in the metalanguage. This formulation given at a later stage was anticipated by Russell's original distinction of formalizable and non-formalizable parts of logic.

I have mentioned here only a few prominent points among Russell's technical contributions to symbolic logic, since an extensive historical study is not included in the program of my paper. There remains to be given an analysis of Russell's views on the foundations of logic. But we cannot go into this problem without having first outlined his theory of the relation between logic and mathematics.

What Russell claims to have shown is the identity of logic and mathematics, or, more precisely, that mathematics is a part of logic. The proof of this thesis is given in two steps. On the first he gives a definition of the positive integers, or natural numbers, showing that they are expressible in terms of purely logical notions including the operators "all" and "there is." On the second he shows, in correspondence with theories developed by other mathematicians, that the whole of mathematics is reducible to the notion of natural number.

The enormous significance of this theory is evident. If it is true, the whole of mathematics is reducible to logical statements containing only the simplest logical concepts; although the translation of a complicated mathematical formula into such simple notions cannot actually be carried through because of the limitations of man's technical abilities, the statement that such a translation can be carried through in principle represents a logical insight of an amazing depth. The unification of mathematics and logic so constructed can be compared to the unification of physics and chemistry attained in Bohr's theory of the atom, a result which also can be stated only in principle, since the actual translation of a chemical reaction into quantum processes involving only protons, electrons, and so on, cannot be carried through. Here, as in the case of Russell's logical theory of mathematics, it is the fact that there is such an ultimate unity which has excited the admiration of scientists and philosophers alike.

I shall not here go into the discussion of the second step. The reducibility of mathematics to the theory of natural numbers is regarded as possible by the majority of mathematicians. The interesting version given to this theory by Russell, according to which the irrational numbers are to be conceived as classes of rational numbers, is a continuation of a principle which in its full import was introduced by him within his analysis of the first step; and we shall discuss this principle of abstraction in that connection.

Let us, therefore, enter directly into an examination of the first step. Russell's definition of number is based on the discovery, anticipated in Cantor's theory of sets, that the notion of "equal number" is prior to that of number. Using Cantor's concept of similarity of classes, Russell defines two classes as having the same number if it is possible to establish a one-to-one coordination between the elements of these classes. Thus when we start from the class constituted by the men Brown, Jones and Robinson, the class constituted by Miller, Smith and Clark will have the same number because we can establish a one-to-one correspondence between the elements of these classes. Now the class of all classes which have the same number as the class constituted by Brown, Jones and Robinson is considered by Russell as constituting the number 3. A number is therefore a class of classes.

It may appear strange that a number, which seems to be a very simple logical element, is to be interpreted by so complicated a notion as a class of classes, or a totality of totalities, of physical things; a concept which includes so many classes of unknown objects. But Russell shows that this definition provides us with all the properties required for a number. When we say that there are 3 chairs in this room, all we wish to say is that there is a relation of one-to-one correspondence between the class of these chairs and certain other classes, such as Brown, Jones and Robinson; a relation which can be expressed, for instance, in the fact that, if Brown, Jones and Robinson sit down on these chairs, there will be no chair left, and each of the men will have his chair. It is this property of the class of the chairs which we express by saying that this class has the number 3; and since having a property is translatable into being a member of a certain class, we can state this property also by saying that the class of the chairs is a member of the class of classes which by the above definition is called the number 3.

This definition of number represents an illustrative application of the principle of abstraction, which has been made by Russell one of the cornerstones of logic. To define a property by abstraction has usually been interpreted as a rule singling out the common property of the objects concerned. Russell saw that the rule: "consider the property which such and such objects have in common," is in this form of a questionable meaning. He replaces it by the rule: "consider the class constituted by all objects having a certain given relation to each other," i.e., he defines the common property in extension rather than in intension. Once more we see here the principle of extensionality at work. Russell shows that it is unnecessary to go beyond the extensional definition. All that can be said about the common property can be replaced by the statement that something is a member of this class. Thus in order to say what "green" means we shall point to a green object and define: a thing is green if it has the same color as this thing. The meaning of the word "green" therefore is definable by the statement: something is green if it is a member of the class of things which have the same color as this thing. We see that the principle of abstraction expressed in this sort of definition represents an application of Occam's razor, it would be an "unnecessary multiplication of entities," if we were to distinguish the meaning of the word "green" from the membership in the class defined. In the same sense Russell's definition of number constitutes a standard example of an application of Occam's razor.

Since a definition by abstraction refers to physical objects as determining the property under consideration, the definition of number above given is an ostensive definition. For example, in order to define the number 3 we point to some objects such as Brown, Jones and Robinson and say it is the number of this class which we call '3'. Russell has however given another definition of number which is not of the ostensive type, and we must now analyze the nature of this definition.

This logical definition of number applied to the number 1 is written in the form

This definition states that a class F has the number 1 if the class has a member so that if anything is the member of the class it is identical with this member. Similarly we can state that a class F has the number 3 by the following formula

This is equally a logical definition since it does not refer to three physical objects in an ostensive way. It is true that the definition itself contains three symbols, namely existential operators, which thus represent a class of three physical objects in extension. But the definition does not refer to these objects, since it does not speak about the signs occurring in it. It would be different if, for instance, we were to write the word "green" always in green ink and then to say: green is the color of this sign. Such a definition refers to a property of a symbol occurring in it and is therefore ostensive.

In order to see clearly what is achieved in Russell's logical definition of number let us consider his definition of the number 1. Here the meaning of the term "one" is reduced to the meaning of some other terms including the term "there is a thing having the property F." The meaning of this latter sentence must be known when we wish to understand the definition. It is a primitive term in Russell's sense. Now it is clear that this term practically contains the meaning of "one." For Instance, we must know that the sentence "there is an apple in the basket" is true even when there is only one apple in the basket. We could define the existential operator in such a way that an existence statement is true only when there are at least two objects of the kind considered. That this is not the ordinary meaning of the phrase "there is," is something we have to know when we apply existential operators. Therefore the meaning of the term "at least one" is antecedent to Russell's definition of the number 1. This does not make this definition circular, since, as the definition shows, the meaning of the number 1 is given by a rather complicated combination of primitive terms, among which the primitive "at least one" is only one constituent.

Let us now consider the relation of Russell's definition to Peano's axiomatic definition of natural numbers.

Peano in his famous five axioms has stated the formal properties of natural numbers. These axioms contain the two undefined concepts "the first number," and "successor." Peano then defines by the use of his axioms what a natural number is.³ His definition is a recursive definition, therefore we can paraphrase it by saying that something is a natural number if it is derivable from the two fundamental concepts in compliance with the rules stated in the axioms. It is well known that Peano's definition admits of a wider interpretation than that given by the natural numbers. The even numbers, e.g., satisfy Peano's axioms if the interpretation of the term successor is suitably chosen. The series defined by Peano has therefore been given the more general name of a progression. This result shows that, as in the case of all axiomatic definitions, or implicit definitions, we must distinguish between the formal system and its interpretation.

This may be illustrated by the example of geometry. An axiomatic construction of Euclidean geometry, such as that given by Hilbert, though fully listing all internal properties of the fundamental notions, must be supplemented by coördinative definitions of these notions when the formal system is to be applied to reality. Thus physical geometry is derived from Hilbert's system by the use of coördinative definitions, according to which straight lines are interpreted as light rays, points as small parts of matter, congruence as a relation expressed in the behavior of solid bodies, etc. This interpretation is not a consequence of the formal system; and there are many other admissible interpretations. But these other interpretations do not furnish what we call physical geometry.

Similarly it is only one of the interpretations of Peano's system which represents the series of natural numbers. It is here that Russell's definition comes In: this definition furnishes an interpretation of Peano's system. Russell's definition can be used to define the first number (it may be advisable to use here the number one, and not the number zero used by Peano, in order to make the definition clearer), and in addition to define the successor relation. All the rest is then done by Peano's axioms, these axioms will lead to the consequence that all numbers are classes of classes in Russell's sense. The system so obtained may be called the Peano system in Russell's interpretation. It is this system which we use in all applications.

The necessity of combining Peano's definition with his own has been recognized by Russell in his discussion of the principle of mathematical induction. This principle, also called the principle of recurrence, is used in the famous inference from n to n + 1, applied in many mathematical proofs. When it is shown that the number 1 has a certain property, for instance that of satisfying a certain equation; and when it is shown in addition that if a number n has this property the number n + 1 must have this property also, then we regard it as proved that all numbers have this property. How do we know the validity of this inference? We can actually perform this inference only for a certain number of steps, and we cannot run through an infinite number of such steps and therefore cannot extend the inference in this way to all numbers. Poincare therefore regarded the principle of mathematical induction as a synthetic principle a priori. It was Frege and, independently, Russell who recognized that a very simple solution to this problem can be found: the principle must be considered as constituting a part of the definition of natural numbers. It thus distinguishes this series from other series which do not have this property, and represents a specific feature which less strictly is expressed by saying that every element of the series can be reached in a finite number of steps, although the number of all elements is infinite.

When this conception is to be utilized for Russell's definition of number we must notice that this is possible only because of a certain peculiarity of recursive definitions. The Peano system contains the three fundamental notions "first number," "successor," and "natural number." But only the first two of these are undefined; the third is defined in terms of the two others. Therefore, only these first two fundamental notions require coördinative definitions, the interpretation of the third notion then is determined by the given two coördinative definitions in combination with the formal system. In other words, the totality of physical objects belonging to the system is defined in a recursive way in terms of the interpretations of the first two fundamental notions.

To make this clear let us consider a similar example limited to a finite series. Let us assume that there is a certain male fly with pink wings, and that there is a law according to which the male descendants of such a fly will also in general have pink wings. The first fly may be called the Adam fly. We now define the term "color family derived from the Adam fly" as follows: 1) the Adam fly belongs to this family, 2) if any fly belongs to the family then each one of its male offspring belongs to it which has the same color of wings as its male parent. These two definitions are sufficient to determine a totality of flies; it is this totality to which we give the above name. The family will presumably be finite, because at a certain stage there will be no male offspring with pink wings or no offspring at all. It is not necessary, however, to give any direct definition of this totality, i.e., a definition which allows us to determine whether a given individual fly belongs to this totality without examining its relations to the Adam fly. In the same sense the totality of natural numbers is defined if the first number and the successor relation are defined, as soon as the limitation of the totality through the inductive axiom has been added.

The interpretation of Peano's axioms given by Russell's definition of number is of a peculiar kind. It does not refer Peano's undefined notions to empirical terms, as is done by the physical interpretation of geometry. When we use Russell's logical definition of the number 1 we do not introduce any new notions into the Peano system. All the notions used in the above logical definition of the number 1 are equally used in Peano's formal system. Thus the statement that each element of the progression has one and only one successor, when formalized, is written in the same way as the definition of the number 1, by the use of an existential operator followed by a qualification in terms of an all-operator and an identity sign. Russell's interpretation of the Peano system must therefore be called a logical interpretation, to be distinguished from an empirical interpretation.

For this reason it is possible to regard Russell's definition of number, not as an interpretation but as a supplementation of Peano's system. We can simply write Russell's definition of the number 1 as a sixth axiom, to be added to Peano's five axioms. Similarly the definition of the successor relation can be expressed in a purely logical way and then added as a seventh axiom. The Peano system thus is made complete and loses its character as a system of implicit definitions, since the terms "first number" and "successor" are no longer undefined. When used in the first five axioms they stand only as abbreviations for what is said in the sixth and seventh axiom. It then is even possible to prove Peano's axioms, with the exception of the axiom of Infinity. The latter axiom seems to be a condition which we must write as an implicans before the whole of mathematics, thus conceiving mathematics ultimately as a system of implications.

I think, therefore, that Russell is right when he says a logical definition of natural number can be given. He is also right when he insists that the meaning of the number 1 used in mathematics is expressed by his definition, and that the mathematical number 1 is not completely defined when it is conceived as a term defined implicitly in Peano's five axioms. This is clear also from the fact that Peano's five axioms use the complete meaning of "one," in Russell's sense, within the statement that each element of the series has one and only one successor. We saw that the formalization of this expression requires all the means used in Russell's definition of "one." Using this result we can say that, in the formalist interpretation, the Russell numbers are implicity contained; they occur in such notations as "the first successor," "the second successor," "the twelfth successor," etc. What the formalist uses, when he applies Peano's axioms to arithmetic, are not the undefined elements, but these successor numbers. All that Russell says, then, is that the latter numbers should be used for the interpretation of the undefined elements of the system. To refuse this would simply represent a tactics of evasion.

I should like to add a remark concerning the application of arithmetical concepts to physics. The formalists are inclined to regard this application as being of the same type as the application of geometry, namely as based on coördinative definitions of the empirical kind. The first to express this conception was Helmholtz.⁴ He explains that, e.g., the concept of addition can be realized by various physical operations, we then must check whether the operation used actually has the properties required for an addition. Thus, if we empty a bag of apples into a basket containing apples also, this operation has the character of an arithmetical addition. On the other hand, mixing hydrogen molecules and oxygen molecules at a rather high temperature does not have the character of an addition since these molecules will disintegrate into atoms and then combine to water molecules in such a way that the number of water molecules is not the sum of the numbers of hydrogen molecules and oxygen molecules. This conception seems to contradict the logical interpretation of arithmetic, according to which no empirical coördinative definition concerning the arithmetical fundamentals is necessary.

I think this contradiction can be eliminated as follows. We frequently do apply arithmetic in Helmholtz's sense by the use of coördinative definitions of the empirical type. But there is, besides, the purely logical application in Russell's sense. The latter is given only when the arithmetical operations are not connected with any physical change of the objects concerned. Thus adding five apples to seven apples in Russell's sense means that as long as a class of five apples and a separate class of seven apples exist, these two classes can be simultaneously conceived as one class of twelve apples. Russell's conception does not say whether this additive relation holds when the joined class is the result of a physical process to which the original classes are submitted, e.g., by putting the apples into the same bag. A statement that in the latter case we also can speak of arithmetical addition is of the Helmholtz-kind. In this case we leave the sign of arithmetical addition logically uninterpreted, and interpret it, instead, by means of a coördinative definition of the empirical type. The logical definition of number operations can be conceived as the limiting case of empirical definitions holding when no physical changes are involved. It does not depend on physical assumptions, because its application is empty, like all statements of deductive logic, and leads merely to logical transformations of statements. It is true that the practical value of arithmetic derives from its frequent use in combination with coördinative definitions of the empirical kind. But it is also true that such definitions would be useless if we had no purely logical definition of number: we state, in such cases, that the addition, which has been defined empirically, leads to the same result as an addition which is logically defined. If numbers were not used in the meaning given by the logical definition, arithmetic could not be applied to physical operations. It is the historical significance of Russell's logic to have pointed out this fact.

I must turn now to a discussion of Russell's theory of types. After having discovered the antinomy of the class of classes which do not include themselves, Russell saw that too liberal a use of functions of functions, or classes of classes, leads into contradictions. He therefore introduced a rule narrowing down such use. This is the rule of types.

It is the basic idea of this theory that the division of linguistic expressions into true and false is not sufficient, that a third category must be introduced which includes meaningless expressions. It seems to me that this is one of the deepest and soundest discoveries of modern logic. It represents the insight that a set of syntactical rules - Carnap now calls them formation rules - must be explicitly stated in order to make language a workable system, and that it is a leading directive for the establishment of such rules that the resulting language be free from contradictions. We need not ask whether or not a certain expression is "really" meaningful, it is a sufficient condition for absence of meaning when a certain sort of expression leads to contradictions. It is from this viewpoint that I have always regarded the theory of types. This theory is an instrument to make language consistent. This is its justification, and there can be no better one.

In the further development of the theory of types Russell introduced a second form, to the simple theory of types he later added the ramified theory of types. The simple theory of types states that a function is of a higher type than its argument, it follows that classes which contain themselves cannot be defined. This simple rule has found the consent of most logicians and appears at present so natural to the younger generation of logicians that it has acquired an almost self-evident character. Such is the fate of all great discoveries, artificial and sophisticated as they may appear at the time when they are first pronounced, after a while nobody can imagine why they had not been recognized from the very beginning. "Truth is allotted only a short period of triumph between the two infinitely long intervals in which it is condemned as paradox or belittled as trivial," says Schopenhauer.

The ramified theory of types, on the contrary, has met with much aversion on the side of the logicians. According to this theory every type must be subdivided into functions of different orders so that each order can contain only lower orders as their argument. Russell saw that this restriction excludes too great a part of mathematics. To save this group of mathematical theorems he introduced the axiom of reducibility, according to which to every function of a higher order there exists a corresponding function of the first order which is extensionally equivalent to it. Russell himself seems not to have been too much pleased with this axiom, although he sometimes defended it as being of the same sort as Zermelo's axiom of choice.

Meanwhile a more convenient solution of the difficulties was given by the line of thought which was attached to Ramsey's classification of the paradoxes into logical and semantical ones and which has been continued by Carnap and Tarski. Logical paradoxes are those in which only functions are involved, in semantical paradoxes, on the other hand, we are concerned with the use of names of functions alongside of the functions themselves. A paradox of this sort is the statement of the Cretan who says that all Cretans lie. For the purpose of logical analysis, this historically famous paradox is better simplified to the form "this statement is false," where the word "this" refers to the sentence in which it occurs. It was only this sort of paradox which made the introduction of the ramified theory of types necessary, for the paradoxes of the logical sort the simple theory of types is sufficient. Now it has been shown that the semantical paradoxes can be ruled out if in addition to the theory of types a theory of levels of language is introduced. According to this theory the object language must be distinguished from the metalanguage, a distinction carried on to the introduction of a meta-metalanguage, and so on. Disregarding some exceptions, it is in general considered as meaningless if a linguistic expression refers to the language in which it is contained. This extension of the theory of types to a theory of levels of language was anticipated by Russell himself who, in his Introduction to Wittgenstein's Tractatus, referring to the problem of generality, wrote:⁵

It seems to me, therefore, that the theories of Carnap and Tarski about the distinction of levels of language represent merely a continuation of ideas originating from Russell himself, a continuation which perhaps also includes ideas derived from Frege and Hilbert. Russell has recently published a statement⁶ expressing on the whole his consent to Carnap's exposition of this linguistic solution of the semantical paradoxes. Thus it seems that this is at least a point on which a general consensus of opinion is attainable.

I should like now to discuss a question as to the foundation of logic; a question which has often occurred to me when I was studying Russell's logic.

Russell has emphasized that logic is not purely formal, that it contains some primitive terms whose meaning must be understood before we can enter into formal operations. He has listed these primitive terms, among them propositional operations and the existential operator, or, instead of the latter, the all-operator. Equally, some of the axioms of logic must be understood as necessarily true; then other formulae can be formally derived from them. Later analysis has shown that it is possible to eliminate all material thinking from the object language and to define logical necessity as a formal property of formulae, namely the property of being true for all truth-values of their constituent propositions. But the results of this formalization should not be overestimated, since in performing it we cannot dispense with material thinking in the metalanguage. Russell is therefore justified in that primitive notions and propositions will remain necessary at least on one level of language. Although they can also be eliminated from the metalanguage, they will reappear in the meta-metalanguage and so on. For instance, in the construction of truth-tables, which belong to the metalanguage, we take it for granted that for two elementary propositions only the four combinations "true-true," "true-false," "false-true," and "false-false" are possible. This means an application in the metalanguage of the same sort of tautologies as are formally proved within the object language.

It appears indeed inevitable that the directive of self-evidence has to be followed. In saying so I do not intend to introduce a sort of apriorism into logic. When we use a logical statement as self-evident we do not combine with this use the claim that the statement will always appear evident. If tomorrow we discover that we were mistaken we shall be ready to correct our statement, and shall follow new evidence, once more without the claim of eternal validity. It seems to me that in this sense the concept of posit, which I have introduced within the analysis of inductive inference, applies also to deductive logic. It is true, if we make an inductive posit we can very well imagine that it be false; whereas when we state a logical tautology we cannot imagine that the formula be false. But we can imagine that tomorrow we shall say that it is false. The procedure of positing is here in the metalanguage. The formula "p ∨ not-p" is a tautology; but that it is a tautology is not a tautology, but an empirical statement concerning a group of signs given to the sense of sight.⁷ The statement about the tautological character has therefore only the reliability of empirical statements, and can only be posited.

I should believe that this conception corresponds to Russell's views, and I should like to know whether he considers it as a satisfactory solution of the problem of self-evidence. The revision of opinion which we reserve with every statement of a self-evident character can be of two sorts. First, we must always envisage the possibility of an error in the sense of a slip of the controls of our thinking. Of this sort are errors made in the addition of numbers, or in the committing of logical fallacies. A second sort of error is of a deeper nature. It consists in not seeing that our statement is true only within certain limitations, that it depends on certain assumptions which we do not explicitly state, but which if once stated can be abandoned. We shall thus arrive at generalizations within which our former statement appears as a special case. It is self-evident for this special case, but outside this usage it is simply false.

An example of this sort seems to me to be given in the teritum non datur. This principle has long been considered as one of the pillars of logic; and it is used so not only in traditional logic, but equally in Russell's logic. But modern developments have shown that the principle can be abandoned. That either "p" or "non-p" is true, holds only for a two-valued logic; but if we use, instead, a three-valued logic, the principle is false. An unqualified validity must be replaced by a qualified validity, the tertum non datur is valid only with respect to certain assumptions about the nature of propositions.

Propositions can be classified in various ways. It is customary to divide them into the two categories of true and false propositions, if this is done the tertium non datur is self-evident. But that propositions must be divided into these two categories can by no means be asserted. The necessity of the tertium non datur is therefore of a relative nature, the formula is necessary with respect to a dichotomy of propositions. This division of propositions, on the other hand, has the character of a convention. It therefore can never be proved false, but it can equally be replaced by another convention, e.g., a trichotomy of propositions. Which sort of classification is to be used will depend on the purposes for which the classification is made. When the dichotomy leads to a system of knowledge satisfying the exigencies of human behavior it will be considered as reasonable, this is the ground on which we use a two-valued logic in conversational language and equally in the language of classical science. But it may happen that for certain purposes a dichotomy will appear unreasonable and that a classification of propositions into three categories will seem preferable. We then shall not hesitate to use a three-valued logic and thus abandon the tertium non datur.

The reasons determining the usefulness of a classification of propositions will depend on the purposes for which the classification is used and on the means by which it is carried through. To speak of "Truth in itself" and "Falsehood in itself," existing as Platonic ideas, constitutes a method which has no relation to actual procedures of knowledge. We cannot use this sort of truth-value. The notion of truth used in actual knowledge is so defined that it is related to what actually can be done. We have methods to find out the truth, and if no such methods existed it would be no use to speak of true propositions. This does not mean that we are always able to apply these methods; there may be technical limitations to them. But we require that in principle such methods should be given; otherwise the notion of truth would be a castle in the air.

These considerations show that when we speak of truth in ordinary language we actually mean verifiability, i.e., the possibility of verification. Russell has objected that, by a restriction of language to verifiable statements, many statements which we usually regard as meaningful, would be cancelled from the domain of meaning. I do not think that a theory replacing truth by verifiability must lead to these consequences. If the notion of verifiability is defined in a sufficiently wide sense, it will include all sorts of statements which Russell would like to consider as true or false, such as his example "it snowed on Manhattan Island on the first of January in the year 1 A.D."⁸ This aim can be reached if the term "possibility," applied within the expression "verifiability," is suitably defined. It certainly would narrow down meanings too much if we should require that a sentence be true only when it is actually verified. In the latter case it is known to be true; but "true" is defined in the wider meaning that a sentence is true if it can be shown to satisfy certain conditions, called verification. Similarly, a sentence will be called false if it can be shown that the sentence does not satisfy these conditions.

Are we then allowed to say: for every sentence it is possible to show that it is either true or false? I do not think that a logician can have the courage to assert such a far-reaching statement, if he does not have a proof for it.

Arguments of this sort have first been used by Brouwer in his famous criticism of mathematical methods. His three-valued logic is somewhat complicated because of its application to mathematics. Mathematics is a completely deductive science; its truth is determined by logical methods alone and does not refer to observation. The only way to determine whether a mathematical formula is true, is by deriving it from the axioms of mathematics, whose truth may be regarded as shown by self-evidence. When a mathematical formula of a syntactically correct form is given, is it possible, in principle, either to derive this formula or its negation from the axioms? Brouwer has raised this question; he regards it as unanswerable and therefore insists on a division of mathematical statements into the three categories of true, false, and indeterminate. If we could give an affirmative answer to that question, Brouwer's trichotomy would be dispensable. But we all know that thus far such a proof has not been given. Gödel's theorem has shown that if we submit mathematical demonstrability to certain limitations there certainly are "undecidable" formulae. But Gödel also shows that the truth or falsehood of these formulae can be found out by methods using the metalanguage. Thus the controversy is still open.

Russell answers considerations of this sort by distinguishing truth from verifiability;⁹ he thinks that independently of whether we are able to find the truth we should assert the principle that a sentence is, or is not, true. I do not see what this principle can mean other than a convention. If we are not given any methods to find out a truth, all we can do is to say that we wish to retain the formula "p ∨ non-p" for all sorts of statements. But if this convention is established for a purely deductive science such as mathematics, there will arise the question of consistency. If it were possible to show that the postulate of the tertium non datur will never lead to contradictions, its establishment would represent a permissible convention. But Hilbert and his collaborators, in spite of ingenious advances in this direction, have so far not been able to give the proof.

For empirical sciences the situation is different. Here the methods of verification are widely dependent on conventions, at least when physical objects which are not directly observable are concerned. It is therefore possible to combine the postulate of the tertium non datur with the principle of verifiability, if suitable conventions as to the method of verification are introduced. But if this is done, another problem may arise which represents, for empirical languages, the correlate of the problem of consistency existing for a deductive science: this is the problem whether the use of a two-valued language is compatible with certain other fundamental principles usually maintained for empirical sciences.

A case of this sort has turned up in recent developments in physics, namely in quantum mechanics. We are facing here the question whether we shall introduce rules determining the values of unobserved entities, and thus introduce a two-valued logic in the sphere of the quanta. Now the results of quantum mechanics can be so interpreted that when we insist upon constructing the language of physics in a two-valued manner it will be impossible to satisfy the postulate of causality, even when an extension of causal connections to probability connections is admitted. The violations of the principle of causality are of another kind; they consist in the appearance of an action at a distance. On the other hand, it can be shown that causal anomalies disappear when the statements of quantum mechanics are incorporated into a three-valued logic. Between true and false statements we then shall have indeterminate statements; and the methods by which the truth-values of statements are derived from empirical observations are so constructed that they will classify any quantum mechanical statement in one of the three categories.¹⁰

This situation resembles very much the development of the problem of physical geometry. After it had been shown that in addition to Euclidean geometry several other geometrical systems can be constructed, the question as to which geometry holds for the physical world could be answered only on the basis of a convention. It could be shown, furthermore, that some of these conventions, if used for the description of the physical world, will lead to causal anomalies. Thus Einstein's theory of general relativity leads to the result that a use of Euclidean geometry for the description of the physical universe leads to causal anomalies. This is the reason that Euclidean geometry has been abandoned and replaced by a Riemannian geometry.¹¹ Similarly, we must distinguish between various logical systems and make the question of application dependent on the sort of physical system so obtained.

I do not see why this conception of the tertium non datur should lead to difficulties. I do not quite understand Russell's¹² insistence upon the law of excluded middle, in particular I am not clear whether he considers the law as a priori or has other reasons for insisting upon this law. I should be glad to get Professor Russell's opinion on this point.

One argument may be stated in favor of the superiority of the teritum non datur. The multi-valued logics introduced by Brouwer, Post, Lucasiewicz and Tarski, including the three-valued logic of quantum mechanics, are so constructed that the metalanguage coordinated to them is two-valued. Thus we can say in the metalanguage of a three-valued logic of this type, "a proposition is either true or it is not true," in the ordinary meaning of the word not. That the category "not true" divides into the two categories "indeterminate" and "false," makes for the metalanguage no more difficulties than, for our ordinary two-valued logic, a division say of a country's armed forces into the three categories of army, navy and air force. It is this use of a two-valued metalanguage which makes the multi-valued logical system very simple and easy to handle. I do not think, however, that it is necessary always to use a two-valued metalanguage. Elsewhere I have given an example¹³ of a multi-valued logic applied within an infinite series of metalanguages. It is true that the metalanguage in which this theorem is stated, and which is not contained in the denumerable infinity of the metalanguages to which the theorem refers, is two-valued. But it should be possible to define a method by which each two-valued language on every level can be translated into a multivalued language; this method would then be applicable also to the language in which it is stated.

Our preference for a two-valued logic seems to be based on psychological reasons only. This logic is of a very simple nature, and we shall therefore prefer it to other conventions concerning a classification of propositions. On the other hand, a closer consideration shows that the two-valued logic which we use in all these cases is never strictly two-valued, but rather must be considered as resulting from probability logic by the method of division, which I have described elsewhere.¹⁴ Such a logic satisfies the usual rules only with exceptions. Thus if "p" is true and "q" is true, it may occasionally happen that "p and q" is false. These discrepancies can be eliminated when the two-valued logic is replaced by a probability logic; the logic of the metalanguage used will then be once more of the approximately two-valued type, but with a higher degree of approximation, i.e., with fewer exceptions. This process can be continued. The replacement of a two-valued logic by a multi-valued logic and the use of a two-valued logic on a higher level, therefore, seems to represent only a method of proceeding to a higher degree of approximation. But a strictly two-valued language is perhaps never used.

Russell's logic is a deductive logic. It never was intended to be anything else; and its value derives from the fact that it represents an analysis of the analytic, or demonstrative, components of thought. But Russell has frequently recognized that there are other components which have a synthetic character and which include inductive methods.

I think we should be grateful that a man, who has devoted so much of his work to deductive logic and who has given a new foundation to this science, which in its modern form will for ever be connected with his name, has never pretended that deductive operations can cover the whole of cognitive thought. Russell has repeatedly emphasized the need for inductive methods and recognized the peculiar difficulties of such methods. He thus makes it clear that he does not belong to the category of logicians who claim that the cognitive process can be completely interpreted in terms of deductive operations, and who deny the existence of an inductive logic. It is indeed hardly understandable how such utterances can be made, in view of the fact that knowledge includes predictions, and that no deductive bridge can lead from past experiences to future observations. A logic which does not include an analysis of inductive inference will always remain incomplete.

Now it is a perfectly sound method to restrict one's field of work to one sort of thought operations and to leave the analysis of another sort to others. And yet I should like to ask Professor Russell to tell us a little more about his personal opinions in this other field. His books occasionally contain very interesting remarks on induction. Thus we find in one of his writings a well-aimed caricature of a familiar misinterpretation of the inductive inference. The latter is regarded by some logicians as being of the form: p implies q, now we know q, therefore p. Russell¹⁵ illustrates this inference by the example: "If pigs have wings, some winged animals are good to eat, now some winged animals are good to eat, therefore pigs have wings." I should like to add that I do not regard this sort of inference as being improved if the conclusion is stated in the form: "p is probable." I do not think it is probable that pigs have wings. Actually, the calculus of probabilities knows no such inference, and it appears hardly understandable why some logicians try to impose upon scientific method the use of an inference which every mathematician would refuse to recognize. I do not see, either, that the logic of the inference is improved when it is named an inference by confirmation.

I think an analysis of the problem of induction must be attached to the form of inductive inference which has always stood in the foreground of traditional inductive theories: the inference of induction by enumeration. It can be shown that all inductive methods, including the so-called inference by confirmation, are ultimately reducible to this sort of inference, more precisely: it can be proved that what such methods contain in addition to inductive inferences by enumeration, belongs to deductive logic. This is shown by the axiomatic construction of the calculus of probabilities.¹⁶ I should like to believe that Russell agrees with this statement.

As to the analysis of induction by enumeration, the traditional discussion has been greatly influenced by the criticism of David Hume. I think Hume's proof that the conclusion of inductive inferences can never be proved to be true, is unquestionable. But I do not think that Hume's interpretation of induction as a habit is able to point a way out of the difficulties. Russell occasionally follows Hume by remarking that, in regarding inductive inference as a method of cognition, we turn causes of our belief into grounds of such belief.¹⁷ If this were the only answer which could be given to the problem of induction, we should frankly state that modern logic is unable to account for scientific method.

Now it seems to me that Hume's treatment of the problem of induction, apart from its healthy refutation of all sorts of rationalism, has seriously biased later philosophies of induction. Even the empiricist camp has not overcome Hume's tacit presupposition that what is claimed as knowledge must be proven as true. But as soon as this assumption is discarded, the difficulties for a justification of induction are eliminated. I do not wish to say that we can at least demonstrate the inductive conclusion to be probable. The analysis of the theory of probability shows that not even this proof can be given. But a way out of Hume's skepticism can be shown when knowledge is conceived, not as a system of propositions having a determinable truth value or probability value, but as a system of posits used as tools for predicting the future. The question of whether the inductive inference represents a good tool can then be answered in the affirmative by means of considerations which do not use inductive inferences and therefore are not circular.

It is only within the frame of a system of knowledge which as a whole is posited that we can coordinate probability values to individual propositions. Here, in fact, probability replaces truth in so far as no empirical sentence is known to be true, but can be determined only as more or less probable.

Russell has argued that such a usage of probabilities does not eliminate the notion of truth. He contends that even in a probability theory of knowledge every sentence should be regarded as true or false, and that what a degree of probability refers to is the degree to which a proposition is true.¹⁸ I do not think that this conception is necessary. The sentences "p" and "p is true" are equipolent, and therefore it is of course permissible to attach the degree of probability either to the one or to the other. But it appears an unnecessary complication to use the second version; it is simpler to say directly that "p" is probable. Such a semantical interpretation of probabilities can be consistently carried through.¹⁹

The question may be asked whether the concept of truth is completely redundant. Russell is inclined to say that it is not, and that if it is eliminated from one place it will reappear in another place of the system of knowledge. I think this leads back to what was discussed in section VI. There is no doubt that, if a probability logic is used for the object language, the notion of truth is dispensable for this language. What can be asked is only whether the concept will reappear in the metalanguage. Now it seems that what I said about the approximately two-valued character of this language holds equally whether the object language is conceived as a three-valued system or as following the rules of a probability logic. Actually, what is called truth in conversational language has never had more than a high degree of probability. The truth of statements made under oath, for instance, is certainly not more than a probability of high degree. It seems that truth is a concept which we use only in idealized logical systems, but which in all applications is replaced by a substitute sharing only to a certain extent the properties of truth.

This result, it seems to me, applies also to the problem of basic statements. I think it is an outstanding feature in Russell's philosophy that he attaches so much importance to the empirical nature of basic statements. He has emphasized the necessity of an observational basis of science in discussions with some authors who apparently attempted to discard the notion of observation entirely from the exposition of scientific method. Russell here has carried on the empiricist tradition in opposition to logical systems which, in spite of their claims to cover the method of empirical science, resemble only too much a modern form of rationalism. But, in spite of this agreement in general, I have to raise some objections to Russell's theory of basic statements as given in one of his recent publications.²⁰

It seems to me that Russell's attempt to reduce the content of immediate observations to sense-data springs from the desire to find a basis of knowledge which is absolutely certain. I do not quite understand whether he wishes to say that sense-data statements are absolutely certain, or whether they possess only the highest degree of certainty attainable. But it seems clear that he wishes to construct a system of basic statements of such a kind that no basic statement can ever be shaken by later observations.²¹ Now everybody will agree with Russell, I think, when he says that basic statements must be logically independent, i.e., that they must be so formulated that none of them can ever logically contradict another. But I cannot see how such an independence can be maintained when inductive methods are admitted.

Russell has argued²² that basic statements cannot be empty, because if such statements were empty their sum also would be empty, and no synthetic knowledge could be derived from them. This, I think, is a sound argument. But I should like to use it in reverse also: if by the use of inductive methods basic statements will lead to a prediction of future observations, then such observations conversely will also make the original statements more or less certain. Inductive methods always work both ways. The rule of Bayes represents an inference by which probabilities holding in one direction are transformed into inverse probabilities. If, therefore, the system of knowledge is construed as being derived from basic observations in terms of inductive methods, this will include the admission that the totality of observations can be used as a test for the validity of an individual observation.

If this is once recognized, it appears no longer necessary to look for basic statements different from statements of simple physical observations, i.e., to regard sense-data as the immediate object of observation. A set of less reliable statements will do, if such statements are of the observational type, i.e., if they are reports about concrete physical objects. If such basic statements have a sufficiently high initial weight, or in other words, if they are at least subjectively true in an approximate meaning of truth, they can be used for the construction of knowledge, and the probability of this knowledge can, on the whole, be even greater than the probability of any individual basic statement. That such possibilities are given by the use of probability methods may be illustrated by the fact that the average error of the mean of a set of observations is smaller than the error of the individual observations of this set. It is the advantage of the probability theory of knowledge that it frees us from the necessity of looking for a basis of data having absolute certainty.

I have tried to outline the major results of Russell's logic; and I have ventured to criticize Russell's views on certain points. But I think that my criticism concerns what, for Russell's logic, are only minor points. This logic is not of a type which needs to be afraid of critique.

My exposition would be incomplete without the addition of some words concerning the influence of Russell's logical work on the present generation of philosophers. Comparing the general level of philosophical writings at the time when Russell wrote the Principia with that of today, we find a remarkable change. Studies in mathematical logic, which forty years ago appeared only occasionally and were read only by a small group of experts, today occupy a great part of the space filled by philosophical publications. A school of younger logicians has grown whose work, to a great extent, has been stimulated by the study of Russell's books and who have tried to continue Russell's methods even beyond the goal for which they were created. The knowledge of Russell's symbolism is today a necessary condition to pass any academic examination in logic; the discussion of Russell's theory of mathematics and of his theory of types plays a prominent part in philosophical seminars, and Russell's methods have become the tools by which a younger generation tills the philosophical soil. The logic and epistemology of today is unthinkable without Russell's contributions; his work has been assimilated even by those who in part contradict his views and look for other solutions.

It would be too optimistic an interpretation of this situation if we were to believe that the use of mathematical logic and its methods will always indicate profundity. Some decades ago we hoped, and I think I can include Russell in this "we," that if mathematical logic should some day become a part of general philosophical education, the times of vague discussions and obscure philosophical systems would be over. We cannot help admitting that our belief was based on a fallacious inference. We see today that the knowledge of symbolic logic is no guarantee for precision of thought or seriousness of analysis.

This has been shown, in particular, by some criticisms of Russell's more recent writings. I do not say this with the intention of condemning a critique of Russell's views. But I do think that such criticism should bear the mark of the same seriousness which distinguishes Russell's thought. Who criticizes Russell should first try to understand the major issues which always stand behind Russell's conceptions. It does not make a good show when a critic, who learned his logic from Russell, indicates with friendly condescension between his lines that he regards Russell's recent writings as not quite up-to-date. The reader might be induced to discover that the use of a metalinguistic vocabulary is not a sufficient criterion for a more advanced state of logical analysis. What use is it to make minor distinctions, if these discriminations are irrelevant for the problems considered? There is such a thing as a fallacy of misplaced exactness; this may be mentioned to those who are inclined to strain out the gnat but to swallow the camel. A truly philosophical attitude will be shown in the ability to balance purpose and means, in the subordination of technical research to the general issue for which it is being undertaken.

It is this balancing of purpose and means which we can learn from Russell himself. The enormous technical work of the Principia was done by him in the pursuit of a major philosophical aim, the unification of logic and mathematics. Russell's work bears witness that logical analysis can become an instrument for the solution of major philosophical problems. Let us not forget that a display of logical symbolism is not in itself the aim of philosophy. There are philosophical problems still unsolved, let us try to use logical technique in order to solve them. Let us look at Bertrand Russell as a man who, by the precision of his methods and by the largeness of his mind, has opened an approach to a philosophy adequate to our time.