The whole process of concept formation consists only in the explanation and affirmation of the identity between individual representations, when definite connections are organized between them, and in these connections the entire given multitude of disparate facts of consciousness is brought to the unity of their mental expression. The affirmation of this unity without the affirmation of identity is absolutely impossible, because each fact, while preserving its real separateness, is necessarily determined in consciousness with its own features – the elements of thought – and by the force of this isolation must necessarily be thought separately. Therefore, the possibility of the mental unity of various facts arises only when in their content there are revealed signs that equally and in the same sense belong to each of them and, consequently, invariably enter into the content of thought about each of them. It is these very features that make it possible for the creative synthesis of thought to unite the diversity of the phenomena of consciousness in a single expression of thought common to them – in a conclusion from the correlation of identity established between the facts. Since this relation concerns only certain attributes, it goes without saying that the content of the general conclusion cannot be identical with the content of each of the original facts that substantiate it, because in each of these facts much more is given separately than how much is asserted about it in the conclusion. Consequently, the conclusion in this case is only a symbol under which thought stores the original facts of consciousness and from which it can again pass to them. This symbol is precisely the concept, an abbreviated expression for an infinite multitude of individual phenomena, each of which is much richer in its content than the composition of the concept, and therefore is expressed in it not by itself, but only through a group of homogeneous phenomena. Each individual thing in itself can be thought not in a concept, but only in a representation, because a representation is always singular and its content is not only equal to the content of the thing being known, but is also identical with it, i.e. it embraces in itself the sum total of all the judgments that can only be expressed about a given thing. A concept, by its very nature, is always abstract, and therefore it contains only the sum of such judgments as can define each given thing only as a part of a homogeneous whole, the scope of which is expressed by the scope of the concept.

If every concept is nothing but a certain sum of general judgments about a certain group of things, then it is clear that the whole process of concept formation is nothing but a process of inductive reasoning. According to the generally accepted definition, induction is an inference from the particular to the general. But this definition of induction does not in fact express the nature of this thought process, because it does not in the least correspond to its real content. It is impossible to infer from particular objects to general ones, because there are no general objects in the world. It is impossible to conclude from particular signs to general ones, because there are no and cannot be any grounds for such a conclusion. In the process of inductive reasoning, the operation of thought is carried out with such a attribute or with such attributes as are known as the attributes of one or several particular phenomena, and which, however, are affirmed as attributes of the whole range of phenomena to which the known ones belong. Consequently, one and the same attribute and at the same time is here thought of as particular and affirmed as general, precisely because the application of a given attribute is extended by thought to the full scope of the phenomena in which it is noticed, and it is precisely in this extension of its application that the whole essence of inductive inference consists. When, for example, we pronounce that all men are reasonable, we are of course asserting a proposition which can be justified by us only in the most insignificant part. For in comparison with the full extent of the human race, we know only the smallest number of men, and yet we affirm the attribute which is invariably observed in every unit of the quantity known to us as the attribute of every unit of the total number of men who have lived, live, and will live. On what basis is this extension made in the application of a feature known to us? If we answer this question, then this very answer will quite clearly define the basic condition of the very possibility of the inductive process of thought. And it is not very difficult to answer it, if only we understand the basis on which we link individual phenomena into complex groups and groups into classes, i.e., if we understand the law of concept formation.

In every process of inductive inference there are two moments: the first moment of analysis, when the very thing that is thought of in the concept of the group is affirmed, and the second moment of synthesis, when a given element of the concept is affirmed as a common feature of all the phenomena of the group, not only present, but also possible. Consequently, the first point concerns the content of the group, the second – its volume. But it is not difficult to see that this second moment is wholly determined by the first moment, and therefore can always be returned to it. For a concept says nothing about its scope, it only says that the phenomenon defined by it has and must have certain attributes. Consequently, in this indeterminacy of the scope of phenomena, their universality is implicitly given, because in every concept it is not a question of any present phenomenon, but of phenomena in general which can be determined by this concept. Consequently, when we assert about the phenomena of a group the very thing that we think in the concept of it, we make our assertion independently of the composition of the group, and it is precisely this circumstance that justifies the moment of expansion of a given attribute in its application to an indefinite number of homogeneous phenomena, which is essential in the inductive process. If we think of a rational being in the concept of man, then the attribute of rationality refers not only to those people whose number forms the actual composition of the group united in the concept limited for each person, but to all people who can be defined by this concept. How many people have existed, are, and will exist in the world, we do not know, and cannot know, but we know for certain that, no matter how many of them there are, they must all have had and will have the property of rationality, because we think of this property in the concept of man, and, consequently, only under the condition of his existence can they be determined by us through this concept. If, for example, the human race ever existed without possessing the property of rationality, then before acquiring this property it would not be the human race for us, and if the human race ever lost the property of rationality, then from the time of this loss it would not again be the human race for us, because in the concept of man we also think of the sign of rationality.

Thus every judgment of inductive inference is possible only in relation to those attributes which we think of in a given concept, or, passing to the objective ground, any extension of the composition of a given group is possible only in relation to those phenomena which are homogeneous with its present composition. The full scope of each concept, or the composition of each group, is an indefinite number, of which only one part is known; but since this part is a part of a homogeneous whole, the whole of this whole is thought of in the concept through it, even though most of its elements have never appeared to the consciousness of man. From the psychological point of view, therefore, every judgment of inductive inference is only a simple formation and exposition of a given concept, but from the logical point of view it is an inference from a part to a whole, or a representation of a part as a whole. It is by virtue of this conception that all our general judgments are formed, and in them all our knowledge is composed. When, for example, a physicist says that the specific gravity of platinum is 21.5, he expresses this judgment not only in relation to the piece of platinum on which he has experimented, but in relation to all pieces of it, no matter how many of them exist in the world. It is clear that he considers all platinum to be homogeneous, and that is why he expands his judgment of the part to the full volume of the whole. When a commoner asserts that fire burns, he expresses such a judgment as he considers valid regarding all the phenomena of fire at all times and in every place. It is clear that he considers all the phenomena of fire to be homogeneous, and it is on this basis that he extends the judgment he has obtained from several experiments to the whole sum of possible experiments, whenever and by whomsoever they may have been made. Consequently, both the field of everyday knowledge and the field of scientific thought are composed of conclusions from the part to the whole, or of representations of the part as a whole.

3.

The process of systematization of concepts as a logical process of deductive reasoning.

In the process of concept formation, thought operates not with integral representations, but with separate elements in their content, so that the real connection of these elements in the concrete of a living representation is destroyed by thought, and they appear in consciousness not as a represented image of a thing, but as a conceivable sum of its attributes. Of this sum, those attributes which are identical in all representations of a certain kind, precisely by virtue of this identity, can be freely transferred from one representation to another, and indifferently express each of them in an abbreviation. These very features form by their connection the symbol of all representations of a certain kind – the objective concept. The same attributes by which the individuality of a thing is determined in consciousness, with the destruction of the living idea of it, remain unconnected with the other attributes and with each other, and therefore each of them takes the position of the primary dates of consciousness, so that their mental processing can be carried out only according to the law of the formation of these dates. They can be connected with each other only by the law of similarity, quite independently of the ideas from the living connection of which they fell out in the formation of the objective concept, and therefore by their connection they form the common symbol of all impressions of a certain kind, the abstract concept. For the formation of such a concept, it is completely immaterial in which exactly the represented connection of attributes is actually realized, for example, the attribute of whiteness. The whiteness of snow, the whiteness of paper, and the whiteness of every other object are expressed by thought in one and the same concept, which is thus an abbreviated expression for a whole multitude of impressions of similar content. Consequently, the type and meaning of the formation of these and other concepts, objective and abstract, is the same, and the only difference between them lies in the fact that some concepts serve as an abbreviated expression of ideas, while others serve as an abbreviated expression of impressions. It is absolutely impossible to draw any other boundary between them, because the world of objective concepts in its final development inevitably passes into the world of abstract concepts, and the world of abstract concepts, in the course of their further generalization, tends to pass into the world of objective concepts.

On the other hand, abstract concepts can be gradually reduced to objective concepts. Thus, for example, the concept of whiteness can be consistently expressed in the content of the concepts of the color of the visual impression, the fact of consciousness, and the fact of the cosmic. The whole series of these concepts constitutes a successive transition to objective reality, and therefore the pure abstract concept, at first unconditionally indefinable, acquires a certain degree of definiteness in each successive concept, and finally reaches a complete mental definition in the highest objective symbol.

This translation of concepts from one to another is of particular importance from the epistemological point of view, because it expresses and explains to us the whole essence of our knowledge.

Consequently, all our knowledge is nothing but the development of concepts and the progressive classification of phenomena. Therefore, it is precisely with the formation of concepts that the first possibility of systematic thinking is realized, since the world of concepts, translated into the phenomena of reality, is a continuous classification of these phenomena. Objective reality itself is composed of individual things that exist side by side in an immense mass and follow one another and do not represent any system of facts. But thought divides them into genera and species, organizes them into subclasses and classes, and thus creates for itself a complete system of the world in the organized world of its concepts.

Every single thing in a concept is necessarily posited as homogeneous with many other things which are expressed by the same concept as the given thing. Therefore, individual things that are actually different in the concept are separated into a separate group, and this group is defined in the field of thought as a homogeneous whole, in relation to which all individual things are its components. By virtue of such a mental definition, the entire infinite multitude of individual things is organized into a special world of homogeneous groups within themselves, and the mental definition of each group expresses the concept of it. But since the whole process of this organization takes place only in thought and for thought, homogeneity and heterogeneity are obviously reduced only to the identity and difference of attributes, and consequently have a purely relative significance. If, for instance, the separate pieces of gold were not equally manifested in the same cases, they would be as heterogeneous to each other as copper and platinum are. And if the pieces of copper and platinum were constantly manifested in the same characters as they appear within themselves, they would be as homogeneous as each of these metals is homogeneous within itself. Therefore we call gold homogeneous only in so far as each particle of it has the same attributes as all its other particles. In the same way, we call gold and copper heterogeneous only in so far as the particles of these metals do not coincide with each other in the characteristics in which they always coincide within themselves. Consequently, the assertion of heterogeneity between them does not in the least exclude the possibility of their coincidence in some other characteristics than those in which they are heterogeneous. And if there should be such attributes which belong equally to the particles of gold and to the particles of copper, then with regard to these attributes gold and copper would be found to be homogeneous, and in the very relation in which they are homogeneous, the judgment of them can be expressed in a general manner. This circumstance determines the possibility of differences in concepts in relation to their scope.

Concepts can express a different number of groups of phenomena. The concept of gold, for example, has in mind only one group of phenomena, the concept of metal – several groups, the concept of an element – more than seventy.

Therefore, in relation to their content, concepts can be of different generality. The wider the process of unification of the heterogeneous, the more differences are subtracted and the fewer similarities in the content of different groups of phenomena are affirmed, so that the scope of the application of a concept always increases at the expense of a decrease in its content. This scope can be carried to its ultimate limit when the concept embraces the whole of being, but apart from this consciousness of being it does not contain a single element, when, consequently, it will be impossible to make a single statement about the object of the concept except the very fact of this impossibility, except that being is nothing, emptiness, non-being. But between this ultimate concept of being and those elementary concepts which serve as the first products of intellectual creativity lies an uninterrupted series of concepts of increasing complexity and generality, of breadth of application and poverty of content. Each of these concepts embraces a definite department of being, and each of the elements of their content expresses in a general way the whole department embraced by its concept. In this way, the process of systematization of concepts is accomplished and their organized world is created.