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.

The process of systematization of concepts consists in the assertion of identity between separate concepts and groups of concepts in one or more respects, in which precisely the various objects of concepts can be expressed by one and the same general judgment. This transfer of judgment from one object to another, or from one concept to another, constitutes a process of deductive reasoning, the sole significance of which is precisely that it organizes certain connections of representations or concepts by establishing or denying the relation of identity between them. It is true that such an understanding of deduction is far from being justified by certain forms of syllogistic thinking, but in judging these forms it must be borne in mind that they are expressions of a very complex formation of the deductive process of thought, and therefore in themselves they still require an explanation of their possibility. When, for example, the conclusion is drawn that all men are mortal, and that Caius is a man and therefore mortal, then thought obviously operates in this construction with propositions which could appear only in the preliminary processes of other inferences and with other elements. In order to affirm the thesis about the mortality of people, it is necessary to affirm the identity in the individual representations of individual people, i.e. it is necessary to form a concept of man; and in order to affirm the humanity of Caius, it is necessary to affirm the identity in the content of the idea of him and in the content of the concept of man. Only by the force of this identity can the subject of the first premise be thought of in the form of an indefinite number of individual representations, each of which can be combined with the representation of Kai in the general concept of man. Consequently, for all these ideas, whatever their number, the concept of man will be common, i.e. in its content it will be completely identical with the content of the subject of the first premise and, as identical, can always replace it. Consequently, instead of the proposition that all men are mortal, an identical proposition can always be posed: man is mortal, and in the conception of man all men are thought not with the exception of Caius, but together with him. Consequently, the conclusion is not something new in comparison with its foundation, but only a simple translation of the general proposition into a particular one. The new does not actually consist in the conclusion, but only in the products of identification that substantiate this conclusion, in the formation of a general concept of man and in the subsuming of a single idea under this general concept, while the conclusion is a simple exposition of the concept in relation to the idea reduced to it, and consequently not as separate from it and alien to it, but as one which is already contained in it and through it is expressed in all those judgments. the sum of which forms the content of this concept.

In view of the fact that the process of identification expresses the whole essence of deductive inference, it is not difficult to find this process even in constructions in which the act of identification is directly negated and the conclusion is made only by the force of this negation. Let us assume, for example, the following construction: true Christians live and act according to the spirit of the faith of Christ; some people who consider themselves Christians do not live and act according to the spirit of the faith of Christ; Consequently, some people who consider themselves Christians are not true Christians. The process of this inference obviously consists in establishing the relation of the predicates of the two premisses, but the real basis of this inference lies not in this relation, but behind it. From the proposition that one magnitude is not equal to another, it does not in the least follow that it is not equal to a third, unless it is established or assumed beforehand that the second and third magnitudes are equal to each other. Consequently, the negation of the predicate of the first premise in its application to the subject of the second premise does not in the least extend this negation to the subject of the first premise in relation to the subject of the second premise, unless it is established or assumed beforehand that the subject of the first premise fully covers its predicate, and vice versa, that the predicate of this premise fully covers its subject. i.e., unless it is established or assumed in advance that the subject and the predicate of the first premise are identical concepts. In this case, the negation of the predicate eo ipso is also the negation of the subject of the first premise in its relation to the subject of the second premise. Consequently, the essential character of deductive reasoning appears here no less sharply than in the first example, i.e., the determination of the correlation of two concepts through the medium of a third under the point of view of the principle of identity. If, however, this is the whole essence of deduction, then it goes without saying that every deductive inference, both in content and in its formal consistency, must be determined not by the scope of the concepts and not by their quality, but solely by the possibility or impossibility of the said correlation.

It is true that logical science still indicates the fundamental rule of deductive reasoning in the well-known scholastic formula, "e mere negativis et e mere particularibus nihil sequitur," but this indication is in fact only a great misunderstanding. A conclusion can be drawn both from particular premises and from negative premises, as long as there are conditions for the possibility of affirming or denying the identity between these concepts. To clarify this point, we will take an example from the Logic of Prof. Vladislavlev: "No science should borrow its material from second sources; fairy tales cannot be called science; no conclusion is drawn from this." But we think that the learned author, who is absolutely convinced of the justice of his "fundamental rule" (e mere negativis...), did not try to draw any conclusion from this, and yet this conclusion follows, and moreover in the most natural way: "Fairy tales can also take their material from second sources." For by denying the identity between the subject of the first proposition and the subject of the second proposition, I thereby also deny the predicate of the first proposition in its relation to the subject of the second proposition, i.e., by denying the scientific character of fairy tales, I can no longer transfer to them the obligatory attribute of scientific thinking, I cannot say of them that they should not take their material from the second sources. It is, of course, true that the negation of one predicate does not provide any basis for the direct affirmation of its opposite predicate, because such a statement requires a positive reason; but it is also true that this negation determines the possibility of an opposite affirmation. A conclusion can be drawn precisely from this foundation and within its limits, i.e. a conclusion can only be made in the form of a simple expression of this possibility. The only question is whether such a conclusion will always be consistent and what exactly determines its consistency? That in a whole mass of cases such a conclusion will be untenable is beyond doubt, and that its inconsistency will be determined by the real impossibility of combining the negation of the predicate of the first premise with the negation of the subject in their relation to the subject of the second premise. But if it is certain that in one mass of cases the conclusion will be untenable, while in another mass of cases it will be valid, then it is self-evident that there must be some grounds or conditions for its validity, i.e., there must be such grounds or conditions by the force of which the logical possibility of transferring the negation from the subject to the predicate of the first premise passes into necessity. And this necessity coincides with the real necessity of such a translation. For example, let's assume the following construction: a birch cannot grow without roots, an oak is not a birch, therefore, an oak can grow without roots. The investigation is clearly ridiculous, because it clearly contradicts reality. And it is not difficult to guess the reasons for this absurdity, if only we compare this construction with the construction given from the Logic of Prof. Vladislavlev. We shall then observe that there are two fundamental differences in the relations between the elements of the two constructions. The first concerns the relationship between the subjects of both premisses, the second the relationship between the subject and the predicate of the first premise. Science and the fairy tale are thought of as opposites, and therefore the relation aimed at uniting the second premise with the first is necessarily the negation of the predicate of the first premise on the part of the subject of the second. In the second construction, however, the subjects of both premises (birch and oak) are formed from homogeneous concepts, and therefore there can be no opposition between them, and consequently no mutual negation. From this it is self-evident that the second premise in this construction can be directed not at negation, but only at distinguishing its subject from the subject of the first premise. Once this distinction has been established, identity is, of course, denied, but since concepts are homogeneous, it cannot be denied unconditionally, but necessarily under certain conditions, i.e., in some definite relations. Yet these conditions or relations are not expressed at all in the second premise, and therefore the conclusion from this is quite impossible. It may well be that the difference between the subjects of the two premisses is expressed by the predicate of the first premise, but it is very possible that it is also expressed by some other premise, which are completely unindicated in either premise. In this case it is expressed by other attributes, and not by a predicate, and therefore the expression of the subject's negation of the second premise of the subject of the first premise in this very predicate turns out to be absurd.

Hence the question naturally arises: under what conditions can the negation of the identity between the two subjects be expressed through the negation of the predicate of one of the premisses in its relation to the subject of the other premise? The answer to this question is indicated by the second fundamental difference in the constructions we have cited. When it is said that science should not take its material from second sources, in this case an essential demand is made of science, i.e., a demand that is thought of in the very concept of science. Consequently, the subject and the predicate in this position are analytically connected, i.e. they are thought of and considered as identical expressions in the relation in which they are connected. Therefore, any negation of the subject of such a proposition is at the same time a negation of its connection with the predicate, not in relation to itself, but in relation to this connection with that element of the second premise by which the subject of the first premise is denied. That is, the predicate of the first premise cannot be connected with the subject of the second premise in the same way as it is connected with the subject of the first premise, because this subject is negated by the subject of the second premise, while the predicate of the first premise is identical with its subject. The second formation is completely unsuitable for such a case. The concept of the impossibility of growth without the intermediary of roots, while forming a predicate in this judgment about the birch, does not, however, in the least express the subject of this judgment in its particularity, does not in the least express what actually makes the birch a birch, and not some other kind of plant. As a consequence, the subject and the predicate of a given judgment cannot be thought of and considered as identical expressions, and therefore the negation of the subject of this judgment on the part of the subject of another judgment can in no case be transferred to a given connection between it and the predicate. In other words, the subjects of both premisses may not be identical, but the relation of the predicate of one of the premisses to both subjects may be the same, because the predicate itself is not identical with either subject. Consequently, it is possible to transfer the negation of the subject of the first premise to its connection with the predicate in relation to this relation to the subject of the second premise only on the condition that the predicate is the essential predicate of its subject, when it is the subject itself to its particularity. In this case, the negation of the subject consists precisely in the fact that its essential predicate is denied, i.e., the conclusion affirms the impossibility of connecting this predicate with the subject of the second premise by the same connection as it is connected with the subject of the first premise.

Consequently, the conclusion from these judgments can only be an expression of this negation: a person (not-not) can be mistaken, i.e. the conclusion affirms only the impossibility of identity in connection with both subjects.

Then the compared concepts would be identified in the conclusion. Suppose, for example, that a student who is beginning to study philosophy knows the general proposition that all preachers of pessimism do not recognize the good of life, but who exactly preached pessimism is not told to him. He studied the history of philosophy and found that Schopenhauer did not recognize the good of life. From this he draws the direct conclusion that Schopenhauer was a preacher of pessimism.

Obviously, this conclusion is determined by the identity of the connection of the predicate of the first premise with both subjects, and is substantiated by the identity of the content of this predicate with the content of its subject. The same predicate which is thought of in the first proposition is also affirmed in relation to the subject of the second premise. But this predicate is an identical expression of its subject: the preachers of pessimism are the very people who do not recognize the good of life, and vice versa – it is precisely those people who preach pessimism who do not recognize the good of life. Consequently, the predicate in this case expresses the only attribute of its subject and, consequently, it expresses in itself the entire content of its concept. Therefore, the connection of this predicate with the subject of the second premise eo ipso is the connection of this subject with the concept whose entire content is expressed by the predicate. The predicate, of course, is not identical with the subject of the second premise, but by the force of the very fact of its connection with this subject, it determines it in the relation in which it is identical with the subject of the first premise. Hence, the idea of Schopenhauer is not only the idea of a man who did not recognize the good of life. It also contains many other attributes which are not expressed by the predicate, but in the process of inference these attributes play no role at all, because the subject of the judgment cannot be thought of in any other respect than that which is determined by its predicate. In this case, therefore, the idea of Schopenhauer is only the idea of him as a man who did not recognize the good of life, and consequently in this respect the idea of Schopenhauer is identical with the concept of a preacher of pessimism, i.e., identical with the concept of the subject of the first premise.