A concept consists in the unification of several, or at least two, given representations, not in some new representation, but in one general position of thought for them, i.e., in a position which is equally valid in relation to each of the ideas united. Consequently, the content of concepts cannot include such elements from the united representations that characterize these representations in their particulars and therefore cannot be transferred from one of them to another. And from this circumstance it is self-evident that every concept is formed in a complex process of generalization, the first moment of which is always the unification of identical elements, the second moment is the subtraction of various elements, and the result is the establishment of a general formula of individual representations, or a general formula of the being of the things represented, equally determined in the sum of certain attributes, namely, in this definition connected by thought in the unity of the concept.

This identity of the elements of two different phenomena makes it possible to link them into a conceivable group under one common definition of thought. The group grows in proportion as we add new phenomena to it, and with it the content of the concept in which it is defined necessarily changes. The fact is that some attributes that characterize a few particular phenomena may not be found in other similar phenomena, and therefore cannot be included in the content of a definition common to the entire group of known phenomena. Consequently, such attributes must necessarily be excluded, and therefore the process of concept formation takes place by reducing the elements in the content of representations, and this decrease proceeds until the content of the concept becomes so stable that the series of new phenomena no longer require its reduction. With the attainment of such stability, each new phenomenon increases only the composition of the group, and does not produce any changes in the existing content of its mental expression, so that each member of the group is always and equally determined by this content. With the achievement of such results, the process of merging the views ends.

2.

Concept formation as a process of inductive inference.

In the organization of the world of impressions, the activity of thought is determined by the law of similarity, while in the organization of the world of representations it is carried out according to the law of identity. This circumstance is psychologically completely understandable. In the initial period of mental development, there are not things for a person – complex concepts of thought, but only phenomena as elementary states of consciousness. The latter, precisely because they are phenomena, are no longer the same today as they were yesterday, and the processes they initiate are no longer the same today as they were yesterday, but both are only repetitions, a new phenomenon of the previous, although experienced in the same way as the previous one, but in no way connected with it either by unity in being or by unity in the act of consciousness. This proposition changes essentially only with the emergence of the world of ideas, because with this emergence the things of knowledge are realized, and in these things the idea of permanence is revealed, i.e., the consciousness of the continuous possibility of given impressions, and it is precisely this consciousness that determines the unity in the position of impressions of different times, or their identity.

Discard only this possibility, assume, for example, that your acquaintance has died, and then even his absolute double, if possible, you will consider not for himself, but only for his double, although the former image will invariably arise in your consciousness at each meeting with his double. Consequently, the process of identifying a present image with a previously existing one, or the process of recognition, does not take place directly, but only through the medium of a special mental process, when it is thought that goes from the image evoked in consciousness, as previously received, to its object, and from the object again to the same image as present. Consequently, the process of identification of ideas is possible only in thought and through the operations of thought; so that separately and in themselves no representations can be recognized as identical, because the double position of one and the same representation can only be an empty phrase, and not at all an expression of thought. For when I say, for example, that A is A, I do not express any more than I have a consciousness of A, and I do not express any judgment about it; For the above expression is only a judgment in its form, and in its essence it is only a very strange attempt to express by the form of thought the decisive absence of thought. If, on the other hand, it is impossible to posit the same idea twice, because such a proposition would be utter nonsense, then, for the same reason, it is certainly impossible to be directly conscious of the identity of the idea with itself. Consequently, the consciousness of identity develops in some other direction, and it actually unfolds in an entirely different direction, and in its development necessarily passes through three successive moments. First of all, it is expressed as the identity of the representation and the thing represented, and then as the identity of the present idea with the past idea of the same thing, and finally as the identity of the thing with itself with the difference of ideas about it at different times. In this last moment, it becomes the basic law of thought in the process of unification of ideas and concepts.

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.