In the practice of thinking there are many definite and varied concepts. They are divided into types in accordance with two fundamental logical characteristics of any concept - content and volume.
Objective differences between the subject of thought are reflected in the differences between concepts, primarily in their content. In accordance with this feature, concepts are divided into the following most significant groups.
Concrete - concepts in which objects and phenomena themselves are reflected, having a relative independence of existence (a book, a pen).
Abstract - these are concepts in which the properties of objects or relations between objects are thought that do not exist independently without these objects (rigidity, electrical conductivity).
It must be borne in mind that if an abstract concept that reflects a property is used in relation to the objects themselves that have this property, then they become plural.
Those concepts that reflect the presence of any qualities, properties, and so on in objects of thought are called positive.
Negative concepts are concepts characterized by the absence of any qualities, properties, etc. in objects of thought. Negative particles (“not”) and negative prefixes (“without-” and “without-”) are used to express negative concepts. In addition to Russian, foreign negative prefixes can be used (“a-”, “anti-”, “des-”, “counter-”, etc.
Concepts are also divided into correlative and non-relative.
In correlative concepts, one subject of thought presupposes the existence of another and is impossible without it - it correlates with it (“parents” and “children”: one cannot be a son or daughter without parents).
In irrelevant concepts, an object is conceived that exists to a certain extent independently - separately from others: “nature”, “man”, etc.
Collective and non-collective concepts differ depending on how the thought relates to the objects they cover: with a group of objects as a whole or with each object of this group separately. One of the peculiarities of collective concepts is that they cannot be assigned to every object of the same class.
The peculiarity of non-collective concepts is that they refer not only to a group of objects as a whole, but also to each individual object of this group.
Empty concepts - they refer to really non-existing objects or phenomena ("mermaid", "goblin", "ideal gas").
Non-empty concepts refer to real objects ("city", "cosmic body").
Single concepts - the scope of the concept, constituting one subject ("Sun", "Russia").
General concepts - reflect in their volume a group of objects ("star", "planet").
The division of concepts into types according to their content and volume makes it possible to single out the largest and most widespread groups in a huge conceptual material, as well as more or less clearly imagine the features of these groups.
Singular concept- one that includes one element (the city of Saratov, Russia, etc.).
General concept- one that includes more than one element (student, soldier, criminal, etc.).
Concepts are divided into types according to: (1) quantitative characteristics of the scope of concepts; (2) type of generalized items; (3) the nature of the features on the basis of which objects are generalized and distinguished. For the most part, this classification refers to simple concepts (concepts whose contents are simple features).
According to the number of generalized objects, concepts are divided into concepts with empty (zero) volume and concepts with non-empty (non-zero) volume.
empty by volume a concept is called, in the scope of which there is not a single object from the universe of reasoning. The contents of such concepts are systems of attributes that do not belong to any object from the universe. Examples: (1) “perpetual motion machine”; (2) “a substance which is a metal and which is not electrically conductive”; (3) “a person who knows all European languages, but does not know Bulgarian, which is European”.
The emptiness of the above concepts is due to various circumstances. The first two are empty due to the inconsistency of their actual contents, i.e. due to the inconsistency of the contents within the existing knowledge. The content of the first is contradictory due to the law of conservation of energy. The content of the second is in the context with the knowledge “all metals are electrically conductive”.
The first two concepts have an empty actual volume. The logical volumes of these concepts are not empty. The content of the third of the above concepts is self-contradictory (logically contradictory). It has an empty logical volume.
The emergence of concepts, the logical content of which is contradictory, is associated with errors in cognition. Such mistakes are sometimes made in the formation of complex concepts, for example, in mathematics.
Concepts, the logical contents of which are consistent, but the factual ones are contradictory, arise in the following cases.
First. In science, concepts are formed not only about those objects whose existence has been established, but also about those whose existence is only assumed. In the formation of concepts of the latter type, the active nature of cognition is manifested. As a result of further research, it may turn out that nothing really corresponds to these concepts, and their actual content is contradictory. Such concepts are the concepts of caloric, world ether, living beings living on Mars. At the time of formation of such concepts, their actual content is not contradictory. It becomes such with the development of knowledge.
Second. Concepts are formed in science, the content of which from the very moment of their formation is contradictory in the context of all existing knowledge. The objects generalized in these terms do not exist in reality. Examples of such concepts: “ideal gas”, “absolutely black body”. Concepts of this kind are necessary in the construction of theories. Within the framework of these theories (within the framework of the universe of reasoning), their contents are not contradictory.
Among concepts with non-empty volume allocate single And are common. The volume of a single concept contains one element, and the volume of a general concept contains more than one element. The general ones are divided into universal And non-universal. The scope of a universal concept is the entire universe, while the scope of a non-universal concept is not the entire universe.
According to the type of generalized objects, concepts are divided into collective And non-collective, and also on specific And abstract.
The elements of the volumes of collective concepts are sets of homogeneous objects, conceivable as a whole, i.e. like aggregates. Examples of collective concepts: “people”, “student group”. In these concepts peoples and groups are respectively generalized. These concepts are general. Collective concepts can be singular. Example: "Russian people".
The elements of non-collective concepts are separate objects. Examples: “planet solar system”, “Moscow State University”.
specific are called concepts in which the objects themselves that exist in the universe of reasoning are generalized. abstract - those in which certain aspects, properties, relations of objects that exist in the universe of reasoning are generalized. If the universe of reasoning is a set of bodies, then “hardness” is an example of an abstract concept.
By the nature of the features on the basis of which objects are generalized and distinguished, concepts are divided into positive And negative, and also on relative And irrelevant.
Let us characterize simple positive and negative concepts.
How to apply this division to complex concepts? We cannot give a general method.
An example of a complex positive concept: “a person who knows English, German and French". An example of a complex negative concept: "a person who knows English and does not know German or French."
As in the previous case, we first give a description of simple relative and non-relative concepts.
Relative is the concept, the content of which is the presence or absence of the relationship of selected objects to some other objects. Examples: "mother", "father".
Concepts, in one of which objects are distinguished on the basis of their relationship to other objects, and in the other - on the basis of their relationship to the first, are called correlative. Example: “cause”, “effect”.
In irrelevant concepts, objects are distinguished on the basis of the presence or absence of the characteristics of the objects themselves, which do not indicate the relationship of objects to other objects.
A complex concept is relative if, among the conjunction of features that make up its content, there are simple features that represent the presence or absence of relations. An example of a complex relative concept: “a person who has higher education and not knowing the Russian language.
The division into types of concepts about objects considered in this section can also be extended to concepts about systems of objects.
A) Types of concepts by volume.
When distinguishing the types of concepts, it is necessary to take into account their various features. The most important grounds for dividing concepts are: (1) the type of their scope, (2) the type of elements included in their scope, (3) the type of features on the basis of which the generalization is made.
By the nature of the volume concepts are divided into empty And non-empty.
– empty a concept is considered, in the scope of which there is not a single element (for example, "the person who is now the president of the USSR")
– non-empty a concept is considered, in the scope of which there is at least one element (for example, "a number that is even").
Non-empty concepts, in turn, are divided into single And are common.
– single a concept is considered, in the scope of which there is exactly one element (for example, "a number that is prime and even").
– General a concept is considered, the scope of which consists of more than one element (for example, "a person who is a student of a university").
General concepts are also divided into universal And non-universal.
– universal a concept is considered, the volume of which coincides with the universe (for example, "a square in which all sides are equal").
– non-universal a concept is considered, the volume of which is less than the universe (for example, "a quadrilateral in which all sides are equal")
C) Types of concepts according to the type of volume elements.
According to the type of volume elements, concepts are divided into
A) specific And abstract
– specific a concept is considered, the elements of the volume of which are objects or sets of objects (for example, "a person who knows how to play the violin")
– abstract a concept is considered, the elements of the volume of which are properties or relations (for example, “a state of passion caused by an emergency”).
b) collective And non-collective
– Collective a concept is considered, the volume elements of which are sets (for example, “a herd of deer grazing on the edge of a forest”).
– non-collective a concept is considered, the elements of which are separate objects, properties or relationships (for example, “fear experienced before visiting a dentist”).
Exercise 3. Determine the type of the following concepts according to the type of elements included in their scope.
a) a device designed to receive television programs (TV)
b) many books kept together and available for public use (public library)
c) a set of stable, socially significant properties of a person, manifested in his behavior (personality)
d) love that broke out suddenly at the first meeting (love at first sight)
C) Types of concepts by content.
According to the type of signs, concepts are divided into
A) positive And negative
– positive a concept is considered in which objects are generalized on the basis of the attribute they possess (for example, “a book taken from a library”).
– negative a concept is considered in which objects are generalized on the basis of a sign that they do not possess (for example, “a person, Not fluent in Japanese).
b) relative And irrelevant
– Relative is considered a concept in which objects are generalized on the basis of their relationship to other objects. For example, the concept of a wife is relative - “a woman who is married to some man” - since its attribute distinguishes women not by their own qualities, but through attitude to some men, that is, as one of the parties of a married couple.
– Whatever is considered a concept in which objects are generalized on the basis of their own properties. For example, the concept of a ballerina is “a woman doing ballet”, about a beauty is “a woman with a beautiful appearance”, etc. Here, women stand out based on their own characteristics.
Note that a relative concept can always be matched with another, correlative , that is, to implement conversion . For the above concept of a wife, the concept of a husband is correlative: "a man who is married to some woman." For the concept of the parent, the concept of the child will be correlative, for the concept of the cause, the concept of the effect, and so on.
Exercise 4. Determine the type of the following concepts according to the type of features, on the basis of which the generalization is made. For relative concepts, select correlative ones.
a) a number that has no divisors other than itself and one (a prime number)
b) a feudal lord who is personally dependent on some other feudal lord (vassal)
c) a girl who is the daughter of a woman's husband, but is not her own daughter (stepdaughter)
d) a philosopher who was the teacher of Alexander the Great (Aristotle)
Implement complete logical analysis of the concept means to determine its universe (genus), scope and content, as well as to establish to which species it belongs according to all the above grounds for division.
According to the nature of the features of the content, the following types of concepts are distinguished:
1. Positive and negative concepts. Positive - these are those concepts in the main content of which there are only positive signs. They reflect the presence of any qualities, properties, etc. in objects. For example: “a crime is a socially dangerous act provided for by the criminal code”. Such concepts are called negative, in the main content of which there is at least one negative sign. They are characterized by the absence of any qualities, properties, etc. in objects. For example, the concept of "autocracy", in the content of which there is a sign of "absence of truly representative institutions", is negative.
2. Absolute and relative concepts. Absolute concepts are those in the main content of which there are only signs-properties (“a square is a rectangular, equilateral quadrilateral”). Relative - concepts in the main content of which there is at least one attribute-relationship ("debtor", "creditor", "brother").
According to the number of volume elements, concepts are divided into empty and non-empty. Concepts are called empty, the volume of which is an empty set, i.e. does not contain any element. These include: concepts that have a fantastic (mythological) character (“centaur”, “mermaid”); concepts that were put forward as scientific or technical concepts, but in the course of the development of science and technology, their inconsistency (“perpetuum mobile”); concepts of idealized objects that play an auxiliary role in the sciences (“ideal gas”, “absolutely black body”, “ ideal state»); concepts of the really non-existent, but possible (“aliens”, “non-terrestrial civilization”). Non-empty are concepts whose volume contains at least one element (“city”, “cosmic body”). The division of concepts into empty and non-empty is to a certain extent relative, primarily because of the mobility of the boundaries between the existing and the non-existent. What does not exist under one set of conditions can become existent under another, and vice versa.
According to the nature of the elements of the volume, the concepts are divided into the following types:
1. Correlative and non-relative concepts. In correlative terms, one object presupposes the existence of another and is impossible without it (“parents”, “children”, “teacher”, “student”, etc.). An object that exists to a certain extent independently, “separately” from others (“nature”, “plant”, “animal”, “man”, etc.) is conceived in irrelevant concepts.
2. Collective and non-collective (separating) concepts. Collective - these are concepts, the volume elements of which themselves constitute a set of homogeneous objects (for example, "crowd", "library"). One of the features of collective concepts is that they cannot be attributed to every subject of a given class: one book is not yet a library, one person is not a crowd. Separating concepts are those whose volume elements do not represent sets of homogeneous objects. There are a majority of such concepts (for example, "tree", "man", "student", "chair", "logic"). The peculiarity of dividing concepts is that they refer not only to the group of objects as a whole, but also to each individual object of this group. For example, a “tree” is both the whole set of trees in general, and each specific tree separately - birch, pine, oak, etc.
3. Concrete and abstract concepts. Concrete are concepts whose volume elements are objects and phenomena that have a relative independence of existence (“chair”, “shadow”, “music”, “crime”). Abstract - these are concepts in which the properties of objects or relations between objects are thought that do not exist independently, without these objects: "justice" (for example, societies), "whiteness" (for example, paper), "caution" (for example, a person).
A) Collective and separating.
In practice, this is the most important distinction between types of concepts, but because the methods of action with concepts are directly connected with the selection of these types. These concepts only apply to general concepts. Singular (and, of course, empty) concepts can be neither divisive nor collective.
The elements of the scope of a concept can be of two types: 1) they can be single objects, 2) they themselves can be sets of objects. In connection with this division, two types of concepts are distinguished:
A collective concept is one whose volume elements themselves constitute sets of homogeneous objects.
Example . Collective terms include: crowd”, since the elements of the scope of the concept of “crowd” are individual crowds, which, in turn, consist of homogeneous objects - people; " library» - since the volume elements of this concept are separate libraries, which, in turn, consist of homogeneous objects - books; parliament, team, constellation, fleet and so on.
A separating concept is a concept whose volume elements do not represent sets of homogeneous objects.
Example . Most concepts are separative. Human, student, chair, crime- separation concepts.
The main feature of the way of dealing with divisive and collective concepts is that they should be handled equally. The meaning of our distinction is to always be aware that O actually is element volume of collective concepts, and what - dividing concepts. In the concept " library The element of the scope of the concept is not books, but libraries. If they say that the library was flooded, this does not mean that every book perished in the water. The volume element of the concept " public class"are not individual people - bourgeois, peasants or workers, but large groups of people. And so if you are told that something is in the interests of such and such a class, this does not mean that it is in the interests of every worker, bourgeois, peasant. From the fact that the regiment was defeated, it does not follow that every soldier or officer was killed. You also need to be aware of how part of the volume such concepts. For example, part of the scope of the concept " university"is this or that many universities, and not those or other faculties of a given university. Here we should remember the earlier distinction between the relation of genus and species and the relation of part and whole.
However, the difficulties with the phenomenon of "collectivity" do not end there. The fact is that many concepts can be used both in a divisive and in a collective sense. "The citizens of our state support the idea of private property" does not mean that every citizen of the state supports this idea. According to the author of this statement, the citizens of our state generally support this idea. Here the concept of "citizens of our state" is used in a collective sense. “Citizens of our state are obliged to abide by the law” - this statement refers to everyone citizen, i.e. the term "citizens" is used here in a divisive sense.
b) abstract and concrete.
This division of concepts into types is most important philosophically. We have already considered the word "abstraction" and established that it comes from a Latin word meaning "distract". What and from what do we abstract in the act of abstraction? Our ontology suggests the answer to this question. There are objects in the world that have properties and between which there are relationships. In an act of abstraction, we abstract, separate a property from an object, or a relation from the objects to which they are inherent. The consideration of properties and relations in themselves, independently of the things to which they belong or to which they relate, is feature abstract thinking. Any thinking that claims to be general in its conclusions is abstract. If we make certain correct judgments about properties or relations in themselves, independently of the objects to which they belong or which they relate, then we make correct judgments for all those objects. Therefore, scientific thinking is always abstract.
This understanding of abstraction helps us understand what is meant by abstract and concrete concepts.
Concepts are called abstract if their volume elements are properties or relations.
In other words, in these concepts, not objects are singled out and generalized, but their properties or relationship.
Example . « Justice», « white», « crime», « caution», « inherence», « paternity" and so on. are all abstract concepts.
A concept is called concrete, the volume elements of which are objects.
Example . « Chair», « table», « crime», « shadow», « music”- all these are specific memories.
In abstract concepts, properties and relations do not turn into objects. They are regarded as objects(see Chapter 3, § 1), which makes it possible for us to compose sets from them and consider them as elements of sets that make up the volumes of concepts. We remember that when describing our logical ontology, we have separated properties and relations, on the one hand, and objects, on the other. This division helps us to think clearly different kind concepts: abstract and concrete.
Sometimes, based on specific concepts, they form abstract concepts associated with them. For example, based on the concept Human"can form the concept of" humanity", whose volume element will be the complex property " being human". On the basis of such an operation, the famous ancient Greek philosopher Plato constructed such concepts as " chairness», « horsepower”, which he calls ideas and which, in his opinion, serve as prototypes of things in the sensible world. According to Plato, sensible things are given to our senses, and such concepts as " chairness», « horsepower" and so on. - only to the sight of our mind.
The method of thinking, with the help of which abstract concepts are given an independent existence, independent of objects, is called hypostatization.
Therefore, we can say that Plato hypostatized abstract concepts: “good”, “truth”, “good”, “beauty”, etc. Whether he did it right or not is no longer a matter of logic, this question is considered by philosophers.
Most abstract concepts, such as the concepts of "justice", "truth", "equality", "brothers", etc., are single concepts; since there is only one property of human actions "to be just", one property of judgments "to be true", one relation between people "to be equal" or "to be brother". The concept of “justice” is always a single concept, regardless of whether fair actions are performed or not, and whether there are many of them, since such a property still exists and, moreover, only one.
Some abstract concepts are still general. Consider the concept of "color". The scope elements of this concept are the following properties: yellow, blue, red, etc., i.e. some simple properties of objects. Consequently, a concept can be abstract, but at the same time general, since its scope contains more than one element.
The examples of abstract concepts that we considered above show that among abstract concepts there are such concepts as "justice", "truth", "beauty", "goodness", "equality", etc. Such concepts in philosophy, psychology, sociology are called values. This leads us to think that the theory of abstract concepts can be used to define the concept of "value".
To give a definition of value, let's try to find out the main features of this concept: 1) values are accepted / rejected consciously, 2) values talk about the properties or relations of objects, 3) values declare objects that have a property indicated in the value of positively significant, and not having negatively significant ( in another interpretation also indifferent). This is where the definition of value comes from:
Value - this is an abstract concept that divides the area of objects to which it applies into two classes - positively significant and negatively significant objects.
Example. " True" is an abstract concept in which the property of judgments " be true". How does truth attach value to judgments that have this property (“true judgments”) positive meaning, and not possessing this property (“false judgments”) - negative meaning.
Example. " beauty” is an abstract concept, the volume of which contains the property “ be beautiful". Accordingly, the value of "beauty" gives objects that have this property a positive value, and those that do not have it - a negative value.
These examples show how the theory of the concept is applied in order to give a clear and distinct interpretation of one of the most important concepts of humanitarian knowledge.
§ 2. Relations between concepts
A: Hello friends! Think about the following problem: who is more in the world - fathers, sons or men?
SS: Of course, men.
Av: And then?
SS: Well, probably, fathers, and then sons. Although it is not very clear with sons and fathers.
Art. Wait, we already know how to depict the volume of concepts using Euler circles. (Goes to the blackboard and draws the following picture:
Get it like this! Great, they took and drew thoughts!
SS: Are you sure this is correct?
ST: You said so yourself.
SS: I said… But did I say it right?
Av: Yes, that's a very good question. Let's get a look. (Referring to the drawing of the slow-witted Student). Consider some object that is included in the scope of the concept of "father", but is not included in the scope of the concept of "son", as shown in your picture. (Goes to the blackboard and puts a dot in the circle "fathers" as follows:
What happens? You have fathers who are not sons. This is good?
ST: No, it can't be.
SS: Yes, but the same can be said about the concepts of "son" and "man". It turned out that not every man is a son.
A: We'll have to sort this out.
Our consideration of the volumes of concepts and sets shows that one and the same object can be an element of the volume of different concepts. So, Ivan Petrovich Sidorov can simultaneously be an element of the volumes of the concepts "person", "student", "man", "sportsman", "voter", etc. This simple fact alone shows that these concepts enter into certain relations with each other, since they have common element. But a priori it can be assumed that those concepts that do not have common elements also enter into certain relations - after all, this is already a certain relation in itself.
Consider an arbitrary pair of concepts A And B.
The concepts A and B are called comparable if the contents of these concepts have at least one common feature.
Example. Concepts " man" And " woman” are comparable, since their contents have a common feature of “being human”.
Almost all concepts are comparable. Even God's gift And fried eggs in our logical ontology are objects, and therefore have a common feature in their content. Please note that this definition is not about the main content, but about everyone the content of the concept. Therefore, almost every pair of concepts can find a common feature.
The concepts A and B will be called incomparable if the contents of these concepts do not contain a single common feature.
We will not deal with incomparable concepts, so we will not consider them in detail.
So far, we have been talking about the content of concepts. The content is a complex feature in which many simple features can be found, connected in various ways (through “and”, “or”, etc.). Therefore, with the consideration of the relationship of concepts in terms of content, difficulties arise. To avoid inaccuracies, one could restrict oneself to the basic content of concepts, as defined in § 2 of this chapter. To do this, it is necessary to replace the word "content" in the definitions with the word "main content". However, it must be borne in mind that in this case the comparability and incomparability of concepts will depend on how we formulate the main content of concepts.
More accurate is the theory of relations of concepts by volume.
Consider a couple of comparable concepts A And B.
The concepts A and B are called compatible if the volumes of these concepts have at least one common element
Example. Concepts " soccer player" And " genius"are compatible, because there are brilliant football players, for example, Eduard Streltsov or Pele.
The concepts A and B will be called incompatible if there is not a single common element in the scope of these concepts.
Example. Concepts " God's gift" And " fried eggs”, as the saying “confuses God’s gift with scrambled eggs” suggests, are incompatible, that is, no object named “God’s gift” is at the same time an object named “fried eggs”. In short, this proverb says that no scrambled eggs are a gift from God and vice versa.
If we designate the scope of a concept with the same symbol as the concept itself, then the compatibility condition for two concepts can be written as follows:
A IN Æ,
and the incompatibility condition is:
A B=Æ .
In contrast to the comparability-incomparability of concepts, we will be interested in both types of compatibility and types of incompatibility of concepts.
Types of compatibility
Imagine possible cases of compatibility of two concepts A And B. First, it may be that the volumes of concepts A And B match up. Secondly, it may be that the scope of the concept B entirely included in A, but at the same time there are such elements A, which are not elements of the scope of the concept B. Thirdly, it may be that the volumes of concepts have a common part, but there are such elements of the volume of the concept B, which are not elements of the scope of the concept A and vice versa.
Let's consider these three cases in more detail.
The concepts A and B are called equivalent if the volumes of these concepts consist of the same elements.
It is convenient to illustrate relations between concepts in terms of volume with Euler circles. In this case, the following figure will be obtained:
Example. The following terms are equivalent: A) Moon And ( B) natural satellite of the earth; (A) square And ( B) equilateral rectangle; (A) daughter And ( B) woman; (A) son And ( B) man; (A) son And ( B) grandson.
The concept B is subordinate to the concept A if the volume B is a proper subset of the volume A.
It is easy to see that the type of a concept is subject to this concept itself, and any concept is subject to its kind.
With the help of Euler circles, this relation can be represented as follows:
Example : The following concepts are in relation to subordination: ( B) student And ( A) Human; (B) Human And ( A) animal; (B) historian And ( A) humanitarian; (B) mother And ( A) daughter are all pairs of concepts, of which the first is subordinate to the second.
Concepts A and B are in the relation of crossing if they are compatible and there are elements of the scope of concept A that are not elements of the scope of concept B, and elements of the scope of concept B that are not elements of the scope of concept A.
With the help of Euler circles, the crossing relation can be represented as follows:
Example. ( A) a student and (B) an athlete, (A) a woman and (B) a beautiful person, (A) a monarchy and (B) a democratic state are all pairs of overlapping concepts.
How to establish in what relation are compatible concepts? To do this, we need to set our concepts A And B two questions:
1. Are all A's B?
2. Are all Bs A?
If we are on ba we answer the question "Yes", then we get the ratio equivalence.
If we are on first we answer the question "Yes", and on second- "No", then the concept A obeys notion B.
If we are on first we answer the question "No", and on second- "Yes", then the concept B obeys notion A.
If we are on both we answer the question " No", then we get the ratio crossing,
Example . Consider the concepts son" And " man". And by a man we mean a male person. Let's ask our questions.
Are all sons male? - Yes.
Are all men sons? - Yes.
Therefore, we have obtained an equivalence relation.
Example . Now consider the relationship between the concepts of "son" and "father".
Is every son a father? - No.
Is every father a son? - Yes.
We have obtained a relation of subordination, and the concept of "father" is subordinate to the concept of "son".
This gives us the solution to the problem given in our characters' dialogue at the beginning of this paragraph. Graphically, this solution can be represented as follows:
If we summarize our consideration of the types of compatibility relationships, we get the following diagram.
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