Logic is not interested in numbers themselves. For example, we will not distinguish between concepts whose volume contains 5 elements and 7 elements. There are infinitely many natural numbers, and it is not our goal to distinguish infinitely many types of concepts. Therefore, we will consider numbers between which there is a clearly visible qualitative boundary. The first boundary is between zero and numbers greater than zero. In accordance with this, the concepts according to the number of volume elements are divided into empty And non-empty.
Emptyis a concept whose volume is an empty set, i.e. does not contain any one element.
Example. Perpetual motion machine, round square, mermaid, Pegasus- all these are various examples of empty concepts. Pay attention to the concepts " perpetual motion machine" And " round square" In the scope of both concepts there is not a single object, but how differently they do not exist. Round square you can’t even imagine (if you don’t believe it, try it!), but perpetual motion machine it is possible to conceive, but it is prohibited by the first law of thermodynamics; it does not exist in nature.
Non-emptyis a concept whose scope contains, at least one element.
In the set of non-empty concepts, one can draw another qualitative boundary between concepts whose scope contains exactly one element, and concepts whose scope contains more than one element. In accordance with this, we will distinguish between the concepts single And are common.
Singleis called a concept, the scope of which includes exactly one element.
Generalis a concept whose scope includes more than one element.
Example . « Moon», « first president of Russia», « first cosmonaut" - single concepts. " Earth satellite», « the president», « astronaut" - general concepts.
Thus, according to the number of volume elements, we have the following classification of concepts:
III. Types of concepts distinguished by the nature of volume elements.
A) Collective and divisive.
In practice, this is the most important distinction between the types of concepts, but the methods of action with concepts are directly related to the identification of these types. These types of concepts relate only to general concepts. Single (and, of course, empty) concepts can be neither dividing nor collective.
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:
collectiveis a concept whose volume elements themselves constitute sets of homogeneous objects.
Example . Collective concepts include: “ crowd", since the elements of the scope of the concept "crowd" are separate crowds, which, in turn, consist of homogeneous objects - people; " library" - since the elements of volume are both concepts separate libraries, which, in turn, consist of homogeneous objects - books; parliament, team, constellation, fleet and so on.
Dividingcalled concept, elements of volume which do not represent sets of homogeneous objects.
Example . Most concepts are divisive. Human, student, chair, crime– dividing concepts.
The main feature of the way of dealing with divisive and collective concepts is that they should be treated the same. The point of our discernment is to always be aware that O actually is element the scope of collective concepts, and what – dividing concepts. In the concept " library“The element of the concept’s scope is not books, but libraries. If they say that the library was flooded, this does not mean that every book perished in the water. An element of the scope of the concept " social class"are not individual people - bourgeois, peasants or workers, but large groups of people. And therefore, if they tell you 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. Just because a regiment was defeated does not mean that every soldier or officer was killed. You also need to be aware of what to count part of the volume such ponies. For example, part of the scope of the concept “ university"is this or that many universities, and not certain 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, 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 comply with the law” - this statement refers to everyone citizen, i.e. the concept of “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 looked at the word “abstraction” and found that it comes from a Latin word meaning “to distract.” What and from what are we distracting in the act of abstraction? The answer to this question is suggested by our ontology. There are objects in the world that have properties and between which there are relationships. In the act of abstraction, we abstract, separate a property from an object or a relationship from the objects to which they are inherent. Consideration of properties and relations in themselves, independently of the objects to which they belong or which they relate, is a characteristic feature of abstract thinking. Any thinking that pretends to generalize its conclusions is abstract. If we make some true judgments about properties or relations in themselves, independently of the objects to which they belong or which they relate, then we make true judgments about all these objects. Therefore, scientific thinking is always abstract.
This understanding of abstraction helps us understand what is meant by abstract and concrete concepts.
Abstractare called concepts, elements of volume which are properties or relations.
In other words, in these concepts it is not objects that are singled out and generalized, but their properties or relationship.
Example . « Justice», « white», « crime», « caution», « inherent», « paternity" and so on. - these are all abstract concepts.
Specificis a concept whose elements of scope 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 seen as objects(see Chapter 3, § 1), which gives us the opportunity to compose sets from them and consider them as elements of sets that make up the volumes of concepts. We remember that, in describing our logical ontology, we divided properties and relations, on the one hand, and objects, on the other. This division helps us think clearly about two different kinds of concepts: abstract and concrete.
Sometimes, based on specific concepts, abstract concepts associated with them are formed. For example, based on the concept " Human" we can form the concept " humanity", the volume element of which will be the complex property " being human" On the basis of such an operation, the famous ancient Greek philosopher Plato constructed such concepts as “ chairfulness», « equineity", which he calls ideas and which, in his opinion, serve as prototypes of things in the sensory world. According to Plato, sensible things are given to our senses, and such concepts as “ chairfulness», « equineity" and so on. - only to the vision of our mind 1.
The method of thinking by which abstract concepts are given an independent existence independent of objects is calledhypostatization.
Therefore, we can say that Plato hypostatized abstract concepts: “good,” “truth,” “good,” “beauty,” etc. Whether he did this correctly 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”, “brotherhood”, etc., are single concepts; since there is only one property of human actions “to be just”, one property of judgments “to be true”, one relationship between people “to be equal” or “to be a brother”. The concept of “justice” is always a single concept, regardless of whether just actions are performed or not, and how many of them are performed, since such a property still exists and, moreover, only one.
Some abstract concepts are still general. Let's consider the concept of “color”. The elements of the scope 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 volume 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, Sociologies are called values. This leads us to believe that the theory of abstract concepts can be used to define the concept of “value.”
To define value, we will try to find out the main features of this concept: 1) values are accepted/rejected consciously, 2) values speak about the properties or relationships of objects, 3) values declare objects that have the property specified in the value to be positively significant and not negatively significant ( in another interpretation also indifferent). This gives us the definition of value:
Value -is an abstract concept that divides the domain 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 is generalized and highlighted " be true" How does truth attach value to judgments that have this property (“true judgments”) positive meaning, and not those possessing this property (“false judgments”) – negative meaning.
Example. " beauty" is an abstract concept, the scope of which contains the property " be beautiful" Accordingly, the value “beauty” gives a positive value to objects that have this property, and a negative value to those that do not have it 1 .
These examples show how the concept theory is used to give a clear and distinct interpretation of one of the most important concepts in the humanities.
Concepts are divided into types according to: the nature of the features on the basis of which objects are generalized and distinguished; quantitative characteristics of the scope of concepts; the type of generalized objects, that is, the nature of the elements of the scope of the concept.
According to the nature of the features included in the content, concepts are divided into positive and negative, relative and non-relative.
1. Concepts are divided into positive and negative depending on whether their content consists of properties inherent in the object or properties absent from it. Concepts whose content consists of properties inherent in an object are called positive. The concept xP(x) is positive if the feature P(x), that is, the specific difference, expresses the presence of some property or relationship in objects x. Concepts whose content indicates the absence of certain properties in an object are called negative. The concept xP(x) is negative if the feature P(x), that is, specific difference, indicates the absence of any property or relationship in objects x.
2. Concepts are divided into non-relative and correlative, depending on whether objects that exist separately or in relation to other objects are thought of in them. Concepts that reflect objects that exist separately and are thought of outside their relationship to other objects are called non-relative. The concept xP(x) is irrelevant if the feature P(x), that is, the specific difference, represents an attributive property. These are the concepts of “student”, “state”, “crime scene”, etc.
The concept xP(x) is relative if the feature P(x), that is, the species difference, represents a relational property. Correlative concepts contain signs indicating the relationship of one concept to another concept. For example: “parent” (in relation to the concept “children”) or “children” (in relation to the concept “parents”), “boss” (“subordinate”),
According to the number of generalized objects, that is, according to the number of elements of volume, concepts are divided into concepts with empty (zero) volume and concepts with non-empty (non-zero) volume.
The concept xP(x) is called empty in scope, 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) “a perpetual motion machine, (2) “a substance that is metal and is not electrically conductive,” (3) “a person who knows all European languages, but does not know the Bulgarian language, which is European.”
The emptiness of the above concepts is due to various circumstances. The first two are empty due to the contradictory nature of their actual contents, i.e. due to inconsistency of contents within the framework of 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 of the knowledge “all metals are electrically conductive.” The content of the third of the previously mentioned concepts is self-contradictory.
Among concepts with non-empty volume, single and general concepts are distinguished. Concepts are divided into individual and general, depending on whether they represent one element or many elements. The concept xP(x) is singular if its scope contains one element from the universe of reasoning (for example, “Moscow”, “F.M. Dostoevsky”, “Russian Federation”). The concept xP(x) is general if its scope contains more than one element from the universe of reasoning (for example, “capital”, “writer”, “federation”).
General concepts can be registering and non-registering. The general concept xP(x) is called registering, in which the set of elements conceivable in it can be taken into account and registered (at least in principle). For example, “a participant in the Great Patriotic War of 1941-1945,” “planet of the solar system.” Registering concepts have a finite scope.
A general concept that refers to an indefinite number of elements is called non-registering. The general concept xP(x) is non-registering if the number of elements conceivable in its volume cannot be counted (registered). Thus, in the concepts of “person”, “investigator”, “decree”, the multitude of elements conceivable in them cannot be taken into account: all people, investigators, decrees of the past, present and future are conceived in them. Non-registering concepts have an infinite scope.
According to the type of objects being generalized, that is, according to the nature of the elements of volume, concepts are divided into abstract and concrete, collective and non-collective.
Concepts are divided into concrete and abstract depending on what they reflect: an object (a class of objects) or its attribute (the relationship between objects). A concept is concrete if it generalizes the very objects that exist in the universe of reasoning. A concept is abstract if it generalizes individual aspects, properties, relationships of objects existing in the universe of reasoning.
Concepts are divided into collective and non-collective. A concept is collective if each element of its volume is a collection of homogeneous objects, conceived as a whole. A concept is non-collective if each element of its scope is a separate object.
Most likely, few people think about the fact that they think and reason using concepts. Concepts are like air: we don’t notice them, but at the same time we cannot think without them. Every child naturally learns to think with their help at the age of seven or eight, moving from operating with concrete objects to operating with ideas. However, this does not mean that everyone knows how to use them correctly, and without this skill the path to logical reasoning is closed. That's why in this lesson, we'll tell you what concepts are, what types of concepts there are, how different concepts relate to each other, and how to handle them correctly.
What is a concept?
What is a concept? It seems intuitively clear. Perhaps many will say: a concept is the same as a word or term. However, this definition is incorrect. Concepts are expressed in words and terms, but are not identical to them. Let us recall that in the last lesson we said that all the words of our language are signs that have two characteristics: meaning and meaning. Usually we use language intuitively, without thinking about meaning and meaning. We simply call some objects apples, others pears, and others oranges. Often we choose a particular word based on the context, that is, the boundaries of its use are blurred. Meanwhile, there are often situations when such intuitive use of words is unacceptable or leads to unpleasant consequences. Imagine, for example, that your whole family is going on vacation abroad. You apply for a visa together, and for this you need your spouse to take a salary certificate from work. You tell him: “Don’t forget to take the necessary paper.” In the evening he brings you a pack of beautiful A4 paper. In this situation, each of you understood the word “paper” in your own way, and this became the cause of mutual misunderstanding. In many areas (legislation, legal proceedings, job and technical instructions, science, etc.) such ambiguity should be eliminated. Concepts are designed to combat it.
From the point of view of logic, to understand a word means to be able to indicate exactly what objects it denotes, that is, to be able to establish in relation to any object whether it can be called by a given word or not. How to achieve this? Through concept formation.
Concept is a logical mental operation that, based on certain characteristics, selects objects from a set and combines them into one class.
Thus, three components are involved in the formation of a concept: a word or phrase (sign), a set of objects that it denotes (meaning), and some idea or distinctive feature that connects the word with the objects falling under it (meaning). It is this distinctive feature that acts as the heart of the concept, because it connects the word and objects. An example is the concept of a square. “Square” is a term, a distinctive feature is “a regular quadrilateral in which all angles and sides are equal,” objects are a set of geometric shapes that have this feature. What does the concept of a square do? From the entire set of geometric shapes, it singles out a certain group of shapes, because they have a set of some special characteristics.
It is important not to confuse the concept and the word by which it is designated. Sometimes different concepts can be associated with one word, depending on what is taken as a distinctive feature. For example, the following concepts can be associated with the word “man”: “a social being”, “a being with intelligence”, “a being capable of creating tools”, “a being with articulate speech”, etc. However, it must be taken into account that, for the sake of brevity, people most often simply talk about the concept of a square or the concept of a person, without specifying what specific distinguishing feature forms the basis for identifying this concept. This often leads to disagreements and so-called disputes over words. Therefore, before entering into an argument, it is useful to clarify exactly what concept your interlocutor puts into this or that word.
Types of concepts
Each concept has two characteristics: content and volume. Contents of the concept- this is the set of distinctive features on the basis of which objects are distinguished from the universe and generalized into one group. Scope of concept- this is the totality of all objects that have distinctive features. It is important to note that the scope of a concept is always specified relative to a certain universe of consideration, that is, a set of objects that, in principle, may have certain distinctive features. The universe of consideration can be people, living beings, numbers, chemical compounds, household appliances, science, food products, etc. Thus, the concept of “elephants” is given in the universe of living beings, the concept of “physics” - in the universe of sciences, the concept of “even numbers” - in the universe of numbers, the concept of “cheese” - in the universe of food products.
Depending on volume concepts are divided into empty and non-empty. The volume of empty concepts does not contain a single element. The scope of non-empty concepts contains at least one element. If there is only one element, then we are talking about a single concept (the author of “War and Peace”), if there are many of them, then we are talking about general concepts (“French kings”). If the scope of a concept coincides with the universe of consideration, then we speak of universal concepts (“numbers”, “people”)
Let's talk in more detail about empty concepts. We don't always notice it, but people use empty concepts quite often. This may happen unconsciously, but sometimes they try to mislead us with their help. We already encountered one example of an empty concept in the last lesson: “the current king of France.” In the entire universe of people there is not a single person who has the distinction of being the current king of France. It should be noted that in this case the concept turned out to be empty due to historical circumstances. If history had gone differently, this concept might not be empty. Another example of an empty concept is “perpetual motion machine”. Here the emptiness is not due to historical reasons, but to the laws of nature. As for scientific concepts, it is unknown for many of them whether they are empty or not. A good illustration of this is the concept of the “Higgs boson”, the non-emptiness of which was confirmed only recently with the discovery of a new particle that satisfies the distinctive features of this concept. A concept can also be empty due to the laws of logic. These are so-called self-contradictory concepts, for example, “round square”.
Depending on the types of generalized objects concepts are divided into collective and non-collective, abstract and concrete. Collective concepts include concepts about sets of objects or people. Such concepts usually contain the following terms: “set”, “class”, “collection”, “group”, “flock”, etc. Examples of collective concepts: “factory workers”, “rock band”, “constellation”. Non-collective concepts refer to single objects: “computer”, “tree”, “star”.
Concepts are considered concrete if the elements of their scope are individuals or collections of individuals. It is important to note that individuals here are understood not as people, but as individual objects, even if these objects are abstract entities. Therefore, an example of a specific concept could be “Solar system”, “natural numbers”. Abstract concepts include concepts whose volume elements are properties, subject-functional characteristics, relationships, for example: “beauty”, “hardness”.
By content type concepts are divided into positive and negative, relative and non-relative. Negative concepts contain a logical negation sign, positive concepts, accordingly, do not contain it. All the examples of concepts we gave were positive. An example of a negative concept: “odd numbers.” Relative concepts take so-called relational properties, that is, properties formed from some relation, as a distinctive feature of the objects falling under it. An example of a relative concept would be man as “a being capable of producing tools.” Among the relative concepts, we can distinguish pairs of interrelated concepts that presuppose each other: “teacher” and “student”, “seller” and “buyer”. Concepts about objects whose distinctive feature is not a relational property are called non-relative, for example: “citrus fruits”.
This entire rather complex typology of concepts is needed so that we can easily perform operations on concepts and determine the relationships they have to each other.
Relationships between concepts
Concepts are not isolated from each other; on the contrary, they are in many connections with other concepts. The ability to identify these connections is very important, since it allows us to identify when our interlocutor or the author of the text is mistaken in the use of concepts or even consciously manipulates them. Examples of such manipulation include the use of concepts whose volumes are not equal as interchangeable, an imperceptible transition to a concept with a smaller volume to facilitate the proof of one’s position, etc.
Before finding out the relationship between two concepts, it is necessary to determine whether they are comparable at all or not. Roughly speaking, the concept of “dogs” and the concept of “natural numbers” cannot be in any relation, because they refer to different universes of consideration: in the first case, animals, and in the second, numbers. Although if, for example, our universe of consideration is the things that people are interested in, then these two concepts become comparable, since people are interested in both. Thus, before comparing concepts, you need to make sure that, figuratively speaking, they have the same denominator - they refer to the same universe.
Logicians divide relations between concepts into fundamental and derivative. Fundamental relations are primary; with the help of their various combinations, all other relations can be defined. There are three fundamental relationships: compatibility, inclusion and exhaustion.
Concepts compatible, if the intersection of their volumes is non-empty. Accordingly, if the intersection of their volumes is empty, then the concepts are incompatible.
Concept A turns on into the concept B if every element of volume A is also an element of volume B.
Concepts are in relation exhaustion, if and only if each object from the universe of consideration is an element of the scope of either the first or second concept.
By combining these fundamental relationships, fifteen derived relationships between concepts can be defined. We will only talk about those that operate with non-empty and non-universal concepts. There are only six of them.
This is a relationship in which the volumes of two concepts completely coincide.
With equal volume, the concepts A and B live in the same circle. An example is the pair of concepts: “triangle with equal sides” and “triangle with equal angles.” Both of these concepts denote the same set of objects.
It occurs when the scope of one concept is completely included in the scope of another concept.
Circle B is completely located in circle A, and at the same time circle A is larger than B in volume, that is, A includes objects that are not included in B. An illustration of subordination is the relationship between the concepts “citrus fruits” (A) and “oranges” ( IN).
This is a relationship in which the scopes of concepts intersect, but do not completely coincide.
An example of intersection is the relationship between the concepts of “women” and “leaders”. There are people who have both the first and second characteristics.
This is a relationship when two concepts intersect and at the same time exhaust the entire universe of consideration.
I specifically depicted concepts A and B in different colors so that it would be clear that the circle in the center is not a separate concept, but the result of their intersection. The complementarity relation exists, for example, between the concepts “temperature above 0°C” and “temperature below 30°C”. The volumes of these concepts intersect, and at the same time the volume of their addition is equal to the volume of the universe of consideration.
This is a relationship in which the volumes of concepts do not intersect and exhaust the entire universe.
If, for example, the universe of consideration is people, then A can be the concept “employed”, and B can be “unemployed”. Every person can be either employed or unemployed, but not both of them and not something third.
It arises when the scopes of concepts do not intersect, but at the same time do not exhaust the entire universe of consideration.
I’ll say right away that I don’t know what motivated those who called this relationship subordination. In my opinion, it is more about independence from each other. Apparently, what is meant is that both concepts are in a relationship of subordination to some third concept - in this case, the entire universe of consideration. Let us assume that the universe of consideration is animals. Then concept A is “lizards”, concept B is “cats”. Both lizards and cats are animals. The scopes of these concepts do not overlap. At the same time, the scope of the universal concept “animals” contains many elements that do not fall under A and B.
The law of the inverse relationship between the content and volume of a concept
At the very beginning, we said that a concept has two characteristics: content and volume. Accordingly, when we determine the relationship between concepts, not only their volumetric characteristics matter, but also their content. In particular, logicians have discovered that there is a so-called inverse relation law between the volume and content of concepts. The essence of this law is as follows: if the first concept is narrower in scope than the second concept, then the first concept is richer in content than the second. By and large, this law operates when we are faced with a relationship of subordination between concepts. Suppose the first concept is “flowers”, the second concept is “daisies”. The concept of “daisies” is narrower in scope than the concept of “flowers,” that is, it includes fewer elements. But it is richer in content. This means that we can extract more information from the concept “daisies” than from the concept “flowers.” If a certain object falls under the concept of “daisy,” then we automatically know that it will also fall under the concept of “flowers,” but a conclusion in the opposite direction cannot be made. If a certain object is an element of the concept “flowers,” this does not mean at all that it will also be an element of the concept “daisy.” It could well be peony, rose, lavender, etc.
Operations on concepts
The main goal of operations on concepts is the formation of a new concept, with its own volume and content, from existing other or more concepts. The basic operations performed on concepts are called Boolean operations. They received this name in honor of the English mathematician and logician J. Boole, who developed a kind of logical mathematics. True, the operations performed on concepts are similar to the operations that we learned to perform with numbers in elementary school. These include: intersection, union, subtraction, symmetric difference, addition.
Conception is an operation during which two or more concepts are taken and, as it were, superimposed on each other. As a result, at the intersection of their volumes, a new concept is formed, the elements of which will be those objects that simultaneously possess the distinctive features of all intersected concepts. To visualize this, let's look at the pictures:
The result of the intersection is a shaded area. For example, if we take the concept of “police officers” and the concept of “corrupt officials” and perform an intersection operation on them, then the shaded area will contain only those people who are both police officers and corrupt officials. This is how we formed a new concept of “corrupt police officers.” As you can see, the intersection operation is based on the intersection relation. This means that if two concepts are in an intersection relationship, then we can easily form a new concept with their help.
An association concepts is similar to addition: we take several concepts, combine their volumes and thereby form a new concept, the elements of which will be those objects that have at least one of the distinctive features of the combined concepts.
To illustrate, we can take the concepts of “smokers” and “people who drink alcohol” and, by combining, form the concept of “people who smoke or drink alcohol.” In this case, the concept will include not only those people who both smoke and drink, but all those who have at least one of these bad habits. Therefore, we shaded both circles.
Subtraction concepts are again very similar to mathematical subtraction. When subtracting, two or more concepts are taken and the volumes of the remaining ones are subtracted from the volume of one. Thus, a new concept is formed, the elements of which will be objects that have a distinctive feature of the first concept, but do not have the distinctive features of those concepts that were subtracted from it.
Let's assume that concept A is “people with diabetes” and concept B is “people who are overweight.” If we subtract concept B from concept A, we get the new concept “people who have diabetes but are not overweight.” It is shown as a shaded area.
This is an operation, in a sense, the opposite of intersection. It is also necessary to take two or more concepts and superimpose them on each other, but the new concept formed as a result of this superposition will contain only those elements that have no more than one distinctive feature of the original concepts.
The shaded area shows this new concept. Items falling under this concept must have attribute A or B, but not both. Let A be the concept of “doctor”, B - “man”. Then we get the following concept: “to be a doctor, but not to be a man, or to be a man, but not to be a doctor.”
This is an operation during which a concept is taken, and then its volume is subtracted, as it were, from the entire universe of consideration. This creates a new concept, the elements of which will be only those objects that do not have the distinctive feature of the initially taken concept.
The new concept A’ is an addition to the concept A. If the universe of our consideration is animals, the concept A is “mammals,” then A’ is “animals that are not mammals.” The complement operation should not be confused with the complementarity relation.
In addition to Boolean operations, a whole series of operations can be performed on concepts: restriction, generalization, division.
This is an operation that represents, as it were, a narrowing of a concept. To limit the concept A means to move to the concept B, such that its scope will be strictly included in the scope of the concept A. Moreover, this transition from A to B represents a transition from a generic concept to a specific one.
As can be seen from the picture, as a result of the restriction, the circle representing the volume of the concept becomes smaller. We restrict concept A to concept B, and then concept B to concept C. We can assume that concept A is “fish”. We can limit it to the concept B - “sharks”. The scope of concept A is broader, since fish are different, they include many species - not just sharks. In this case, the scope of concept B is completely included in the scope of concept A, because all sharks are fish. The concept of “sharks” can be limited to the concept C - “white sharks”. Again, the concept of “white sharks” is fully included in the concept of “sharks”, but is smaller in scope. The limit of limitation of a concept is a single concept. In our drawing it would represent a point in the center that can no longer be narrowed.
The operation of limiting concepts is often accompanied by errors. Most often, they are due to the fact that the limitation of concepts is confused with the division of objects, that is, a concept is limited not on the basis of generic characteristics, but on the basis of those parts into which the elements of their volumes are divided. For example, let’s take the concept of “cars”. Based on generic characteristics, we can limit it to the concepts of “cars with manual transmission” or “electric cars”. And this is the right limitation. However, a car consists of many components: headlights, wheels, steering wheel, windshield wipers, engine, etc. Therefore, you can come across this option: the concept A - “cars” is limited to the concept B - “wheels”. Although wheels are part of a car, this limitation is incorrect. There is an easy way to avoid this mistake. Given the correct restriction of concept A to concept B, the statement “All B is A” must be true: “All sharks are fish,” “All electric cars are cars.” If we apply this formula to cars and wheels, it turns out: “All wheels are cars.” The statement is incorrect, which means that the restriction operation was carried out incorrectly.
This is the inverse operation of a constraint. This time we are not narrowing, but expanding the concept. To generalize concept B means to move to concept A, so that the scope of concept B will be strictly included in the scope of concept A. Here a transition is made from a specific concept to a generic one.
We generalize the concept C, represented by the smallest circle, to the concept B, which in turn we can further generalize to the concept A, and C is completely included in B, and B is completely included in A. Let C be the concept “gold”, then we can generalize it to the concept B - “metals”, and the concept B - to the concept A - “chemical elements”. The limit of generalization is a universal concept, that is, a concept whose scope coincides with the universe of consideration. In our example, the concept of “chemical elements” can be considered as universal.
The operation of generalizing concepts can be subject to the same error as restriction: often people generalize concepts based not on generic characteristics, but on their constituent parts. In particular, the concept of “wings” is generalized to the concept of “birds,” which is incorrect. The way to check is the same: see if the statement “All B is A” is correct. Obviously, the statement “All wings are birds” is incorrect.
Division- this is an operation consisting of taking a concept, highlighting some characteristic and, based on varying this characteristic, the original concept is divided into several parts, resulting in a set of new concepts. The original concept is called a divisible concept. Those concepts that are formed after division are members of division. The characteristic on the basis of which division is carried out - the basis of division.
The entire circle is the volume of the concept of the divisible concept A. B, C, D and E are members of division, that is, concepts formed as a result of dividing concept A. For illustration, assume that concept A is “months”. The basis of division is belonging to the season. Then the newly formed concepts B, C, D and E are “winter months”, “spring months”, “summer months” and “autumn months”. Obviously, as a result of division, a different number of concepts can be obtained: everything depends on the concept being divided and the basis of division.
For the division to be correct, the following conditions must be met:
- Division must be carried out using only one base. If we use our example with the concept of months, then I cannot divide it into the following sub-concepts: “winter months”, “spring months”, “summer months”, “autumn months” and “my favorite months”. In this division, two characteristics are used: belonging to the season and my attitude to a specific month. This is called confused division. Also, if you use more than one division base, you can make a so-called division leap, which consists in the fact that some division members are species of A, and others are its subspecies. For example, the initial concept is “wine”, the basis of division is color. As a result of correct division, we should get three new concepts: “white wine”, “rosé wine” and “red wine”. But if a leap is made in the division, then you can come to the following result: “white wine”, “rosé wine”, “cabernet”, “shiraz”, “merlot”, “pinot noir”. In this case, two bases were combined: color and variety, and the members of the division simultaneously included species (white, rosé) and subspecies (cabernet, shiraz, etc.).
- Division members B, C, etc. must represent species in relation to the generic concept A. This is the same condition that we encountered in limiting and generalizing. It is impossible to divide the concept of “car” into the concepts of “wheels”, “engine”, “steering wheel”, etc. Again, you need to ask yourself whether the statement “All B is A”, “All C is A” is true, and so on for all members of the division. If you are still interested in the wheels and the engine, then you need to replace the concept being divided with “parts of the car”, then the division will become correct.
- The volumes of the division terms do not intersect, that is, none of the elements can simultaneously fall into B and C or into B and E, etc.
- Division terms cannot be empty concepts. Suppose that the original concept A is “kings currently reigning.” The basis of the division is belonging to countries. So, among the members of the division there cannot be the concepts “currently ruling French kings” or “currently ruling German kings,” since these are empty concepts.
- If we perform a union operation on all division terms B, C, D, E, then we must obtain the volume of the divisible concept A.
There are two types of division: dichotomous division and division by modification of the base. The word “dichotomous” is literally translated from Greek as “dividing into two.” When it is implemented, the original concept is divided into only two new concepts. Any basis of division, that is, a sign, is selected, and depending on the presence or absence of this sign, all volume elements are divided into two parts. Let the divisible concept be the concept of “people”; let the division be based on the presence of higher education. In this case, our initial concept will be divided into two: “people with higher education” and “people without higher education.” Another example: let’s take the concept of “dog”, the basis of division is thoroughbred. As a result of the dichotomous division we get the concepts: “pedigreed dogs”, “mongrel dogs”.
The second type of division is division by modification of the base. As a result, we can get more than two new concepts. Here, any subject-functional characteristic of the elements of the scope of the original concept is chosen as the basis. In our example with months, this characteristic was belonging to the season. If our divisible concept is “people,” then we can take eye color, hair color, nationality, etc. as the basis for division. If the concept being divided is “poems,” then the basis for the division may be their genre. To illustrate, let’s take the concept of “playing cards”, and use the suit as the basis for the division:
The division operation underlies the compilation of classifications and typologies. Classification is carried out by sequentially dividing a concept into its types, types into subspecies, etc. Classification, first of all, is important in scientific knowledge. It can act as both a result of studying a certain subject area (Carl Linnaeus’s general classification of plants and animals) and a driver of research (Mendeleev’s periodic table of chemical elements). In addition, classifications are very important in learning: people perceive information much easier if it is organized into categories. Often, without even noticing it, we use classifications in everyday life: ranking employees in the office, organizing clothes in a closet, distributing goods into departments in a store - these are just a few examples.
Correctly done classification is like an upside-down tree (in my opinion, more like an upside-down bush). The top of the classification - the original divisible concept - is called the root. The lines radiating from it are like branches. They lead to division members, from which, in turn, branches also diverge to new concepts. Each concept in the classification is called a taxon. Taxa are grouped into tiers. On the first tier is the root of the classification A. On the second tier are the taxa B 1 -B n, formed using the first division operation. On the third tier are taxa C 1 -C n, formed as a result of the second division operation, etc. Each tier can contain any number of taxa.
When constructing classifications, both types of division are used: dichotomous and by modification of the base. Moreover, they can coexist even in the same classification. The fact is that within the classification, each individual division operation can be performed according to its own basis. Let's give an example. Let’s take the concept of “writers” as the root of the classification, the basis of the division - whether the writer was Russian or not. Accordingly, we make a dichotomous division, as a result of which we obtain two new concepts at the second level: “Russian writers” and “foreign writers”. Then we can divide the concept of “Russian writers” according to the modification of the basis. As a basis, let’s take the characteristic: “in what century did the writer live?” We get new concepts: “Russian writers of the 11th century,” “Russian writers of the 12th century,” and so on up to “Russian writers of the 21st century.” As for the concept of “foreign writers,” it can also be divided according to the modification of the basis, but take the nationality of the writers as the basis. Thus, we get: “Spanish writers”, “French writers”, “German writers”, etc.
The sign [...] indicates missing division terms. Further, each taxon can be divided according to some other characteristic. The main thing in each individual division is to follow the rules listed above.
It should be noted that drawing up classifications is not as simple a task as it might seem at first glance. Situations are not uncommon when it is difficult or impossible to determine which taxon a particular item should be classified as. In our example with writers, in particular, cases are possible when a writer was born and began to create in one century, and died in another, like Chekhov. Where should he be classified - among the writers of the 19th century or the 20th century? Sometimes there are objects that, in principle, do not fit anywhere. Then a separate taxon is created for them or they are placed in the so-called “settlement tank”. It can be designated by the words “everything else,” and the objects located in it are not connected by anything other than the fact that they cannot be defined anywhere.
Exercises
Chinese Encyclopedia
Borges in one of his works cites an excerpt from a mysterious Chinese encyclopedia. This “divine repository of beneficial knowledge” says that “animals are divided into: a) those belonging to the Emperor, b) embalmed, c) tamed, d) suckling pigs, e) sirens, f) fairy tales, g) stray dogs, h) included in real classification, i) raging, as if in madness, j) innumerable, k) painted with a very thin brush of camel hair, m) and others, p) having just broken a jug, o) from afar seeming like flies" (Borges H.L. Analytical the language of John Wilkins // Works in 3 volumes, Vol. 2. Riga: Polaris, 1997, p. 85).
Try to imagine this classification of animals as a tree. Do you think it was done correctly? If yes, then prove that none of the rules of division are violated. If not, then explain exactly what rules were violated. How could this classification be corrected?
Meat is not food
Cat. Please forgive me for my indiscretion. This is what I've been wanting to ask you for a long time...
Cat. How can you eat thorns?
Donkey. And what?
Cat. There are, however, edible stems in the grass. And the thorns... so dry!
Donkey. Nothing. I love it spicy.
Cat. What about meat?
Donkey. What - meat?
Cat. Have you tried eating it?
Donkey. Meat is not food. Meat is luggage. They put him in the cart, you fool. (E. Schwartz, “Dragon”)
Define the relationships between the concepts of “food”, “sharp objects”, “spicy food”, “thorns”, “meat” and “luggage”. Depict these relationships using graphical diagrams. Remember that concepts can only be compared if they belong to the same universe of consideration.
Conversation between husband and wife
Husband: Honey, you're wrong.
Wife: Oh, I'm wrong. So I'm lying. I'm lying, which means I'm a bad person, that is, a non-human. Are you saying that I'm an animal? Mom, he called me a beast!
Determine whether the transition between the concepts “a person who is wrong”, “a liar”, “a bad person”, “a non-human”, “animal”, “a brute” was made correctly. Justify your position. What operations on concepts were used during this transition? What are the relationships between these concepts? Depict them using graphic diagrams.
Test your knowledge
If you want to test your knowledge on the topic of this lesson, you can take a short test consisting of several questions. For each question, only 1 option can be correct. After you select one of the options, the system automatically moves on to the next question. The points you receive are affected by the correctness of your answers and the time spent on completion. Please note that the questions are different each time and the options are mixed.
A) Collective and divisive.
In practice, this is the most important distinction between the types of concepts, but the methods of action with concepts are directly related to the identification of these types. These types of concepts relate only to general concepts. Single (and, of course, empty) concepts can be neither dividing nor collective.
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 a concept whose elements of scope themselves constitute sets of homogeneous objects.
Example . Collective concepts include: “ crowd", since the elements of the scope of the concept "crowd" are separate crowds, which, in turn, consist of homogeneous objects - people; " library" - since the elements of volume are both concepts separate libraries, which, in turn, consist of homogeneous objects - books; parliament, team, constellation, fleet and so on.
A concept whose volume elements do not represent sets of homogeneous objects is called separative.
Example . Most concepts are divisive. Human, student, chair, crime– dividing concepts.
The main feature of the way of dealing with divisive and collective concepts is that they should be treated the same. The point of our discernment is to always be aware that O actually is element the scope of collective concepts, and what – dividing concepts. In the concept " library“The element of the concept’s scope is not books, but libraries. If they say that the library was flooded, this does not mean that every book perished in the water. An element of the scope of the concept " social class"are not individual people - bourgeois, peasants or workers, but large groups of people. And therefore, if they tell you 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. Just because a regiment was defeated does not mean that every soldier or officer was killed. You also need to be aware of what to count part of the volume such ponies. For example, part of the scope of the concept “ university"is this or that many universities, and not certain 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, 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 comply with the law” - this statement refers to everyone citizen, i.e. the concept of “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 looked at the word “abstraction” and found that it comes from a Latin word meaning “to distract.” What and from what are we distracting in the act of abstraction? The answer to this question is suggested by our ontology. There are objects in the world that have properties and between which there are relationships. In the act of abstraction, we abstract, separate a property from an object or a relationship from the objects to which they are inherent. Consideration of properties and relations in themselves, independently of the objects to which they belong or which they relate, is a characteristic feature of abstract thinking. Any thinking that pretends to generalize its conclusions is abstract. If we make some true judgments about properties or relations in themselves, independently of the objects to which they belong or which they relate, then we make true judgments about all these objects. Therefore, scientific thinking is always abstract.
This understanding of abstraction helps us understand what is meant by abstract and concrete concepts.
Abstract are concepts whose elements of scope are properties or relationships.
In other words, in these concepts it is not objects that are singled out and generalized, but their properties or relationship.
Example . « Justice», « white», « crime», « caution», « inherent», « paternity" and so on. - these are all abstract concepts.
A concept whose elements of scope are objects is called concrete.
Example . « Chair», « table», « crime», « shadow», « music" - all these are specific memories.
In abstract concepts, properties and relations do not turn into objects. They are seen as objects(see Chapter 3, § 1), which gives us the opportunity to compose sets from them and consider them as elements of sets that make up the volumes of concepts. We remember that, in describing our logical ontology, we divided properties and relations, on the one hand, and objects, on the other. This division helps us think clearly about two different kinds of concepts: abstract and concrete.
Sometimes, based on specific concepts, abstract concepts associated with them are formed. For example, based on the concept " Human" we can form the concept " humanity", the volume element of which will be the complex property " being human" On the basis of such an operation, the famous ancient Greek philosopher Plato constructed such concepts as “ chairfulness», « equineity", which he calls ideas and which, in his opinion, serve as prototypes of things in the sensory world. According to Plato, sensible things are given to our senses, and such concepts as “ chairfulness», « equineity" and so on. - only to the vision of our mind.
The method of thinking by 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 this correctly 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”, “brotherhood”, etc., are single concepts; since there is only one property of human actions “to be just”, one property of judgments “to be true”, one relationship between people “to be equal” or “to be a brother”. The concept of “justice” is always a single concept, regardless of whether just actions are performed or not, and how many of them are performed, since such a property still exists and, moreover, only one.
Some abstract concepts are still general. Let's consider the concept of “color”. The elements of the scope 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 volume 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, Sociologies are called values. This leads us to believe that the theory of abstract concepts can be used to define the concept of “value.”
To define value, we will try to find out the main features of this concept: 1) values are accepted/rejected consciously, 2) values speak about the properties or relationships of objects, 3) values declare objects that have the property specified in the value to be positively significant and not negatively significant ( in another interpretation also indifferent). This gives us the definition of value:
Value - is an abstract concept that divides the domain 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 is generalized and highlighted " be true" How does truth attach value to judgments that have this property (“true judgments”) positive meaning, and not those possessing this property (“false judgments”) – negative meaning.
Example. " beauty" is an abstract concept, the scope of which contains the property " be beautiful" Accordingly, the value “beauty” gives a positive value to objects that have this property, and a negative value to those that do not.
These examples show how the concept theory is used to give a clear and distinct interpretation of one of the most important concepts in the humanities.
§ 2. Relations between concepts
Av: Hello, friends! Think about the following problem: who are more in the world - fathers, sons or men?
SS: Of course, men.
Av: And then?
SS: Well, probably fathers, and then sons. Although it’s not very clear with sons and fathers.
Art. Wait, we already know how to depict the volumes of concepts using Euler circles. (Goes to the board and draws the following picture:
It will turn out like this! Great, you took it and drew your thoughts!
Ss: Are you sure this is correct?
St: You said so yourself.
Ss: I said something... But did I say it correctly?
Av: Yes, that's a very good question. Let's get a look. (Refers to the drawing of the slow-witted student). Let's consider some object that is included in the scope of the concept “father”, but is not included in the scope of the concept “son”, as shown in your picture. (He approaches the board and puts a dot in the “fathers” circle as follows:
What happens? You have fathers who are not sons. This is good?
St: No, this cannot be.
SS: Yes, but the same can be said about the concepts of “son” and “man”. We found out that not every man is a son.
Av: We'll have to look into this matter.
Our consideration of the scope of concepts and sets shows that the same object can be an element of the scope of different concepts. Thus, Ivan Petrovich Sidorov can simultaneously be an element of the scope of the concepts “person”, “student”, “man”, “athlete”, “voter”, etc. This simple fact already shows that these concepts enter into certain relationships with each other, since they have a 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.
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 because their contents have a common attribute of “being human.”
Almost all concepts are comparable. Even God's gift And fried eggs in our logical ontology they are objects, and therefore have a common attribute in their content. Please note that this definition is not about the main content, but about everyone content of the concept. Therefore, almost every pair of concepts can have a common feature.
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 different ways (through “and”, “or”, etc.). Therefore, difficulties arise when considering the relationship of concepts in content. To avoid inaccuracies, one could limit ourselves 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, we must keep in mind that in this case the comparability and incomparability of concepts will depend on how we formulate the main content of the concepts.
More accurate is the theory of relations of concepts by volume.
Let's consider a couple of comparable concepts A And B.
Concepts A and B will be called compatible if the scopes of these concepts have at least one common element
Example. Concepts " football player" And " genius"are compatible because there are brilliant football players, for example, Eduard Streltsov or Pele.
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 suggested in the proverb “confused God’s gift with scrambled eggs,” are incompatible, i.e., no object called “God’s gift” is at the same time an object called “scrambled egg.” In short, this proverb says that no scrambled egg is God's gift and vice versa.
If we denote the scope of a concept with the same symbol as the concept itself, then the condition for the compatibility of 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
Let us imagine possible cases of compatibility of two concepts A And B. First, it may be that the scope of concepts A And B match up. Secondly, it may be that the scope of the concept B is entirely included in the volume 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 scopes of concepts have a common part, but there are such elements of the scope of the concept B, which are not elements of the scope of the concept A and vice versa.
Let's look at these three cases in more detail.
Concepts A and B will be called equivalent if the scopes of these concepts consist of the same elements.
It is convenient to illustrate the relationships between volume concepts with Euler circles. In this case, you will get the following picture:
Example. The following concepts 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 notice that the type of a concept is subordinate to this concept itself, and any concept is subject to its genus.
Using Euler circles, we depict this relationship 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- all these are pairs of concepts, of which the first is subordinate to the second.
Concepts A and B are in a crossing relation 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.
Using Euler circles, the crossing relationship can be depicted 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 - all these are pairs of intersecting concepts.
How to establish in what relation compatible concepts are located? To do this, we need to ask our concepts A And B two questions:
1. Are all A's B's?
2. Are all B's A's?
If we're on o ba We answer the question "Yes", then we get the relation equivalence.
If we're on first we answer the question "Yes", and on second- "No", then the concept A obeys concept B.
If we're on first we answer the question "No", and on second- "Yes", then the concept B obeys concept A.
If we're on both We answer the question “ No", then we get the relation crossing,
Example . Let's consider the concepts " son" And " man" Moreover, by man we mean a male person. Let's ask our questions.
Are all sons men?? - 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 received a relation of subordination, and the concept of “father” is subordinate to the concept of “son”.
This gives us a 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 relations, we get the following diagram.
According to the type of volume elements, concepts are divided into:
A) specific And abstract
– Specific is considered a concept whose elements of scope are objects or sets of objects (for example, “a person who knows how to play golf”)
– Abstract is considered a concept whose elements of scope are properties or relationships (for example, “a state of affect caused by an emergency”).
b) collective And non-collective
– Collective is considered a concept whose elements of scope are sets (for example, “a crowd of people gathered for a rally”).
– Non-collective is considered a concept whose elements of scope are individual objects, properties or relationships (for example, “excitement experienced before an exam”).
End of work -
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Let us consider these relations between judgments using a logical square. In relation to subordination there are judgments of forms A and I, as well as judgments E and O. For example, the general affirmative su
Types of attributive judgments
1. Establish a correspondence between the types of attributive judgments and the formulas that express them. (1) All S are P (A) S a P (2) No S is P (B) S e P (3) Some S
Oslash;A
Thus, the correct conclusions are from the statement of the antecedent (A) to the statement of the consequent (B) and from the negation of the consequent (&
At work. Therefore, he was not in class."
Conditional disjunctive (lemmatic) inferences. These inferences contain several implicatures and one disjunctive premise. The disjunctive premise separates certain options
Direct inferences
Direct inferences are those in which the conclusion is drawn from one premise. Despite the triviality, in the practice of argumentation such conclusions are given a very important role.
P – ~S ~P – S ~P – ~S
opposition contrast opposition of subject to predicate to subject and predicate Each of them can be reduced to a combination inverted
As already noted, a syllogism is a conclusion made from more than one premises. In this broad sense, syllogisms are, for example, inferences
S P S P S P S P
Figure I Figure II Figure III Figure IV The mode of a syllogism is a type of figure determined by the type of premises and conclusion included in it. Sokra
Enthymemes and polysyllogisms
An enthymeme (from the Latin “enthyme” - “in the mind”) is a shortened syllogism in which one of the premises or conclusion is missing. In the practice of argumentation
Parcel rules
1. According to the general rules of a syllogism, if one of its premises is negative, then the conclusion must be 1. particular 2. general 3. affirmative
Simple categorical syllogism
1. Establish a correspondence between the terms and their role in the structure of the syllogism 1. middle term (A) term present in both premises 2. larger term (B) predicate of the conclusion
A1, …, An ú B
Such conclusions are called inductive (from the Latin “inductio” - “guidance”), or plausible. Among the plausible conclusions
Statistical induction
Statistical induction is a generalized induction that establishes the relative frequency of possessing property P for an arbitrary
Argumentation and proof
Argumentation is a complete or partial substantiation of the truth of a judgment with the help of other judgments. It is assumed that in the correct argumentation
Refutation and criticism
Activities that are opposite in purpose and content to argumentation are refutation and criticism. A refutation is a complete substantiation of the falsity of the thesis, and
Basic rules of argumentation
In the process of argumentation and criticism, two types of errors can be made: intentional and unintentional. Deliberate errors are called sophistry, and persons who commit such errors are called
Dictionary of logical terms
1. Abduction (from the Latin abducere - reduction) is a form of inference in which a new judgment (hypothesis) is selected from the initial judgments (describing the properties of some phenomena).
Software and Internet resources
1. http://www.logic.ru/Russian/: LOGIC OF RUSSIA 2.http://www.logic.ru/Russian/LogStud/index.html: Electronic journal “Logical Research”. 3. http://www.iph.ras.ru:8100/~logi
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