• Water supply
• Pumps
• Water purification
• Wells
• Filters

Symbol designation logic. Logical operations and their properties. Poland and Germany

LOGIC AND LANGUAGE. SEMANTIC CATEGORIES OF LANGUAGE

Modern formal logic is called symbolic because it uses a special language to analyze the structure and laws of thinking.

The necessary connection between thinking and language, in which language acts as the material shell of thoughts, means that identifying logical structures is possible only by analyzing linguistic expressions. Just as the kernel of a nut can only be reached by opening its shell, so logical forms can only be revealed by analyzing language.

Logical language is based on certain premises. On the one hand, these are philosophical ontological assumptions. Ontology - from the Greek ontos - being and logos - doctrine, which means the doctrine of being. Ontological assumptions are expressed in a certain picture of the world, in knowledge about the structure of the world, its properties and patterns. On the other hand, since the logical theory of thinking is based on the analysis of the properties of linguistic thinking, the logical theory includes certain premises about language and its structure.

The main building material for constructing a language is the signs used in it. Sign- this is any (sensually perceived (visually, auditorily or otherwise) object that acts as a representative of another object. Among the various signs, we distinguish two types: signs-images and signs-symbols.

Signs-images have a certain resemblance to the designated objects. Examples of such signs: copies of documents; photographs; some road signs depicting children, pedestrians and other objects.

Signs-symbols have no resemblance to the designated objects. For example: musical notes; Morse code characters; letters in the alphabets of national languages.

The set of original signs of a language constitutes it alphabet.

A comprehensive study of language is carried out by the general theory of sign systems - semiotics, which analyzes language in three aspects: syntactic, semantic and pragmatic.

Syntax is a section of semiotics that studies the structure of language: methods of formation, transformation and connections between signs. Semantics deals with the problem of interpretation, i.e. analysis of the relationship between signs and designated objects. Pragmatics analyzes the communicative function of language - emotional, psychological, aesthetic, economic and other relations of the native speaker to the language itself.

Let us briefly consider the composition and structure of this language.

Designed for the logical analysis of reasoning, the language of predicate logic structurally reflects and closely follows the semantic characteristics of natural language. The main semantic category of the language of predicate logic is the concept of name.

Name- this is a linguistic expression that has a certain meaning in the form of a separate word or phrase, denoting or naming some extra-linguistic object. A name as a linguistic category thus has two obligatory characteristics or meanings: subject meaning and semantic meaning.

Subject meaning (denotation) of a name- this is one or many objects that are designated by this name. For example, the denotation of the name “house” in Russian will be the whole variety of structures that are designated by this name: wooden, brick, stone; single-story and multi-story, etc.

The semantic meaning (meaning, or concept) of a name is information about objects, i.e. their inherent properties, with the help of which many objects are distinguished. In the above example, the meaning of the word “house” will be the following characteristics of any house: 1) this structure (building), 2) built by man, 3) intended for housing. The name denotes, i.e. denotes objects only through meaning, and not directly. A linguistic expression that has no meaning cannot be a name, since it is not meaningful, and therefore not objectified, i.e. has no denotation.

The types of names in the language of predicate logic, determined by the specifics of naming objects and representing its main semantic categories, are the names of: 1) objects, 2) attributes and 3) sentences. Item names denote single objects, phenomena, events or sets of them. The object of research in this case can be both material (airplane, lightning, pine) and ideal (will, legal capacity, dream) objects.

Names are distinguished by composition simple, which do not include other names (state), and complex, including other names (Earth satellite). According to denotation, names are single And are common. A singular name denotes one object and can be represented in language by a proper name (Aristotle) ​​or given descriptively (the largest river in Europe). A common name denotes a set consisting of more than one object; in language it can be represented by a common noun (law) or given descriptively (big wooden house).

The names of attributes - qualities, properties or relationships - are called predicopores. In a sentence, they usually serve as a predicate (for example, “to be blue,” “to run,” “to give,” “to love,” etc.). The number of names of objects to which the predicator refers is called its locality. Predicators that express properties inherent in individual objects are called one-place (for example, “the sky is blue”). Predicators that express relationships between two or more objects are called multi-place. For example, the predicator “to love” refers to doubles (“Mary loves Peter”), and the predicator “to give” refers to triples (“The father gives a book to his son”).

Sentences are names for expressions of language in which something is affirmed or denied. According to their logical meaning, they express truth or falsehood.

The alphabet of the language of predicate logic includes the following types of signs (symbols):

1) a , b, With,...- symbols for single (proper or descriptive) names of objects; they are called subject constants, or constants;

2) X,y,z, ... - symbols of common names of objects that take on meanings in one area or another; they are called subject variables;

3) P 1, O 1, K 1,... - symbols for predicates, the indices over which express their locality; they are called predicate variables;

4) R,q, r, ... - symbols for statements, which are called expressive or propositional variables (from the Latin proposito - “statement”);

5) ,  - symbols for quantitative characteristics of statements; they are called quantifiers: - general quantifier; it symbolizes expressions - everything, everyone, everyone, always, etc.;  - existence quantifier; it symbolizes expressions - some, sometimes, happens, occurs, exists, etc.;

6) logical connectives:

– conjunction (conjunction “and”); – disjunction (conjunction “or”);

-> - implication (conjunction “if..., then,..”);

 - equivalence, or double implication (the conjunction “if and only if..., then...");

 - negation (“it is not true that...”).

Technical language symbols: (,) - left and right brackets.

This alphabet does not include other characters. Acceptable, i.e. Expressions that make sense in the language of predicate logic are called well-formed formulas - PPF. The concept of PPF is introduced by the following definitions:

1. Every propositional variable – p, q, r... is a PPF.

2. Any predicate variable taken with a sequence of subject variables or contact, the number of which corresponds to its location, is a PPF: A 1 (x), A 2 (x, y), A 3 (x, y, z), A n (x, y, … , n) , Where A 1 , A 2 , A 3 , A n– metalanguage signs for predicators.

3. For any formula with objective variables, in which any of the variables is associated with a quantifier, the expressions  xA(x) And  xA(x) there will also be PPFs.

4. If A and B are formulas (A and B are metalanguage signs for expressing formula schemes), then the expressions:

are also formulas.

5. Any other expressions other than those provided for in clauses 1–4 are not PPF of this language. Using the given logical language, a formatted logical system called predicate calculus is built. Elements of the language of predicate logic will be used in the subsequent presentation to analyze individual fragments of natural language.

Concept as a form of thinking. Concept formation.

Word and concept. Education of concepts

Language and thinking are inextricably linked in our cognition. The word is a powerful tool for analyzing the world; it takes us beyond sensory experience and allows us to penetrate into the realm of the rational.

The meaning of a word is a function of identifying individual features in an object, generalizing them and introducing the object into a certain system of categories.

The main stages of concept formation.

First, we identify individual features of the subject we are interested in (perform an analysis).

Then we consider the selected features separately (abstraction operation).

The next operation is comparison; it involves identifying common features and discarding specific ones.

At the synthesis stage, we combine common features into a single whole, into a mental image of an object.

And finally, with the help of cognitive generalization based on the selected features, we think of the entire set of objects that have this feature.

Explanation of the concepts used

Analysis- mental division of objects into their component parts, mental identification of features in them.

Abstraction– mental selection of some features of an object and distraction from others; Often the task is to highlight essential features and abstract from unimportant, secondary ones.

Comparison– mental establishment of the similarity or difference of objects according to essential or insignificant characteristics.

Synthesis- mental combination into a single whole of parts of an object or its characteristics obtained in the process of analysis and comparison.

Cognitive generalization- mental unification of individual objects in a certain concept.

The concept is inextricably linked with the linguistic unit – the word. Concepts are expressed and fixed in words and phrases, for example, “right”, “law”, “complicity”, etc. Words are the material, linguistic basis of concepts, without which it is impossible to form them or operate with them.

Each word not only denotes an object, but also produces much deeper work. It identifies a feature that is essential for this object and analyzes this object.

Signs- this is what objects are similar to each other or different from each other; properties or relations are signs.

Signs can be significant or insignificant.

Essential feature- this is, firstly, a feature inherent in all objects of a given class, and secondly, a feature without which we cannot imagine this object. The second characteristic of an essential feature reflects the relativity of the philosophical concept of essence. The essence of a thing is a reflection of the depth of our knowledge of this thing at a given time stage. For example, the ancient Greeks identified certain primary elements as the beginning of all things: water, fire, air and earth. But these primary elements themselves could be defined through combinations of basic qualities: wet, dry, hot, cold. With this approach, water was defined as wet and cold, this was understood as its essence. And for a modern schoolchild, the essence of water will be expressed by the formula H 2 O.

Insignificant signs- these are signs that are not decisive in relation to the qualitative specificity of objects generalized in the concept.

Signs can be distinctive or non-distinctive.

Distinctive features of a class of any objects are features inherent only to objects of this class.

Non-distinctive features are features that belong not only to these objects.

The concept is the main form of thinking through which we identify certain classes of things and distinguish them from each other. The concept appears, firstly, as a result of abstraction and comparison, i.e. mental identification and separation of essential properties of things from non-essential ones, and, secondly, as a generalization of these essential properties into a single concept.

Let's give a more concise definition: A concept is a thought that identifies objects of a certain set and distinguishes this set according to its essential and distinctive characteristics..

In language, concepts are designated names. Proper names (“Moscow”, “Pushkin”) correspond to specific objects, while general names (“capital”, “man”) correspond to entire sets of objects. We can say that a concept is the meaning of a name.

A word not only denotes a thing, but also generalizes things and assigns them to a certain category. For example, “a crime is a socially dangerous act provided for by criminal law.”

In the concept " crime“we identify two significant and (together) distinctive features: (1) to be a socially dangerous act; (2) be provided for by criminal law.

The word not only denotes an object, but also performs the function of analyzing the object, conveying experience formed in the process of historical development of generations of people. So, “samovar” means an object that cooks itself; “telephone” denotes an object that transmits sound at a distance, “TV” denotes an object that makes it possible to see at a distance.

Names and concepts are the initial, elementary means of understanding, forming and expressing thoughts. Concepts are made up, and reasoning is made from judgments: explanations, doubts, objections, evidence and any other ways of “unfolding” thoughts. That is why concepts act as the meaning of a name, something that should be understandable during adequate communication, when transmitting scientific or business information.

Such characteristics of understanding as clarity of perception, accuracy in expression and understanding of meaning, clear awareness of all relationships between concepts depend on:

Degree of systematization of knowledge;

The degree of certainty of knowledge and its expression;

The degree of development of formulations, completeness of disclosure of details.

In live communication, success largely depends on the ability to “grasp” the essence of the conversation as a whole, on the ability to clarify details through analysis and identification of details, on the ability to anticipate possible interpretations and interpretations of concepts. In conflict situations, certainty and consistency of presentation are often sacrificed, allowing vagueness and streamlined formulations in order to achieve agreement.

If we mean systematized (scientific) knowledge, then the canon is the following: it is necessary to clearly, clearly and in detail provide information about the objects generalized in the concept. For this purpose, logic distinguishes between the volume and content of concepts.

The concept is the main form of thinking through which we identify certain classes of things and distinguish them from each other. The concept appears, firstly, as a result of abstraction and comparison, i.e. mental identification and separation of essential properties of things from non-essential ones, and, secondly, as a generalization of these essential properties in a single concept. To form a concept, it is necessary to identify the essential features of an object, using a number of logical techniques for this purpose: comparison, analysis, synthesis, abstraction, generalization. These techniques are widely used in cognition. They play an important role in the formation of concepts based on identifying essential features: - to form a concept about an object, you need to compare this object with other objects, find signs of similarity and difference. A logical technique that establishes the similarity or difference of objects is called comparison. - identification of features is associated with the mental division of an object into its constituent parts, sides, elements. The mental dissection of an object into parts is called analysis. -selection using feature analysis allows you to distinguish essential features from unimportant ones and to distract yourself and abstract from the latter. Mentally highlighting the features of one object and distracting from other features is called abstraction. -elements, sides, features of an object, identified through analysis, must be combined into a single whole. This is achieved using a technique opposite to analysis - synthesis, which is a mental connection of parts of an object dissected by analysis.

Content concepts constitute all its elements, which can be distinguished as separate concepts. Volume a concept is all other concepts for which it serves as a sign, their main part. The first can be designated by the symbol A, then the second will look like Aa, Av, Ac, Ad... If our symbol A (content), for example, means the concept of “state”, then the other symbols (Aa, Av, Ac...) (volume) will be mean “slave state”, “feudal state”, “bourgeois state”, “totalitarian state”, “democratic state”, etc. It is easy to notice that A acts as a subordinate (generic), and Aa, Av, As... are subordinate concepts.

From what has been said, it follows that if the presence of the scope of concept A is recognized, then this means that the presence of concepts must be recognized, for each of which it is part of the content. Their absence means the absence of the concept A itself, since no concepts there is no volume. This can be understood in such a way that each concept always corresponds to a real object. If we are dealing with concepts about fantastic creatures (“centaur”, “faun”, “naiad”, etc.), then they also have scope in a logical sense, although we do not know the real objects for them.

What is the relationship between the volume and content of a concept? From the above reasoning, we can conclude that if the content of concept A is in the content of concept B, then B is in the scope of concept A. And vice versa, if concept B is contained in the scope of concept A, then the latter is part of the content of the former. The content and scope of the concept are thus in an inverse relationship .

The law of the inverse relationship between the volume and content of concepts is valid only for those concepts, one of which is generic (subordinate) and the other specific (subordinate). Let's explain this with an example.

Let’s take the generic concept of “cosmic body”; here we understand everything that is common to all cosmic bodies. The specific concept will be “stellar systems”, by which we mean a class of cosmic bodies that have special distinctive features (these systems exist as clusters of stars, where they interact with each other, arise and “die”, “black holes” appear in their place and etc.). Let's now relate these concepts. The generic concept “cosmic body” “absorbs” (or includes) the specific concept “stellar systems,” but our generic concept also includes other specific concepts, for example, “planetary systems,” “planets,” etc. Including the most general characteristics characteristic of all cosmic bodies, the generic concept tends to narrow its scope, but at the same time, in terms of content, it includes many concepts that reveal specific characteristics, and thus our generic concept tends to expand. The specific concept “stellar systems” represents a richer content (includes more features), but it turns out to be narrower in scope, because it is “absorbed” by the generic concept “cosmic body”.

This law reflects the objective fact that the number of general characteristics of an object and the number of objects possessing these characteristics are in an inverse relationship (young specialist - engineer - Ivanov A.P.).

In the practice of thinking, it is necessary to distinguish the relationship of genus and species from the relationship of part and whole. The whole consists of its parts, and the genus in the logical sense consists of species as its parts only when the genus and species are considered from the side of their volumes. Taken from the content side, they are in the opposite ratio, i.e. a genus is part of a species. For example, “a rose is a plant.” A part that is not a species cannot be said to be a whole. For example, hair is a part of the human body, but it cannot be said that hair is the human body. In addition, there are concepts that are only genera in relation to other concepts, but cannot be species. Such concepts are called categories. They have the largest volume and the smallest content compared to other concepts. Let's take the philosophical concepts of “time”, “space”, “movement”, “quantity”, “quality”, “property”, “relationship”, etc. They will differ from the categories of particular sciences, which cannot be species in relation to the concepts of the same sciences, but are species in relation to philosophical categories. Examples of categories in special sciences: “living organism” - in biology, “elementary particle” - in particle physics, “figure” - in geometry, “atom” - in chemistry, etc.

In formal logic, the categories “thing”, “property” and “relation” are widely used. Therefore, there is a need to consider their content. Objects denoted by the category “thing” differ from objects denoted by the category “property” and the category “relation”. Objects designated as “things” have a relative independence of existence, manifested in the fact that each thing (stone, apple, moon, river, elementary particle, etc.) has special spatial boundaries and is different from another thing. The properties of things, for example, color, hardness, smell, etc., do not have independent spatial boundaries; they are “attached” to things. The same can be said about relationships. For example, the relationships “more - less”, “dark - light”, “good - evil”, etc. do not exist outside of things or people - their carriers.

Each thing is a collection of properties, and they are located within the same spatial boundaries in which the thing itself exists, no matter how these boundaries change in connection with the movement of the thing, i.e. its change. In any case, properties do not exist separately from the thing that bears them. A thing can lose some property, as well as individual relationships with other things, but at the same time it remains itself. Both the properties of things and their relationships are manifested in the relationships of things to each other, or in the relationships of one part of a thing to another.

Operations of restriction and generalization of concepts

Operations on concepts- these are logical actions as a result of which new concepts are formed. Since the volume of concepts is considered as a class with which these operations are carried out, the latter are called operations with classes; as a result of their operations (operations on concepts) they acquire new classes. Let us consider the following operations on concepts: a ) composition, b) multiplication, c) negation, d) generalization and limitation understand.

A The operation of adding concepts is to combine two or more classes into one class

Thus, the operation of adding the concepts \"conviction\" and \"acquittal\" consists of combining the class of convictions with the class of acquittals into one class or into one concept \"about binival sentence \"letter A, and the concept\" acquittal sentence \" - the letter B, then the result of this operation can be displayed graphically as follows (see Fig. 7) The shaded surface is the class of the sentence. 7.

Using the operation of addition, you can combine classes (concepts) that are in the same relationships with each other: identity, subordination, intersection, subordination, contradiction. For example, when combining the concepts “witnesses” (A) and “relatives” (B ), which are in relation to the intersection, we will receive a new class (Fig. 8), which will include not only witnesses who are not relatives and, and relatives who are not witnesses, but also relatives-witnesses When drawing up the concepts of \"agreement\" (A) and "agreement" (B), between which there are relations of subordination, will receive a new class (shaded surface in Fig. 9), which will include not only transactions that are not contracts, but also agreements.

When adding concepts together, the conjunction \"or\" is often used. It is used not in a dividing, but in a connecting-dividing sense. This should be kept in mind when interpreting legal norms.

The scope of the concepts \"A or B\", obtained as a result of the addition operation, and the union of classes correspond to the concepts A and B. Therefore, the expression \"A or B\", for example \"students or athletes\", means that this new class includes not only students who are not athletes, and athletes who are not students, but also students who are also athletes.

B The operation of multiplying concepts consists of searching for such objects (elements) included simultaneously in the class of both concepts. For example, the operation of multiplying the concepts “witness” (A) and “relative” (B) consists of searching for such elements among the class of witnesses and such elements among the class of relatives who are simultaneously included in both classes, that is, such people who are both witnesses and relatives.

Graphically, the result of this operation can be reflected as follows (see Fig. 10) The shaded part of the surface means the desired class of objects, that is, those people who are both witnesses and relatives

The multiplication operation can be carried out with concepts that are in different relationships with each other. For example, if we need to carry out the multiplication operation of the concepts “crime” (A) and “official crime” (B), that interruptions occur in relation to subordination, then we We highlight the following elements of subordination, which are simultaneously included in both of these classes, that is, we find such crimes in general that are at the same time.

Graphically, the result of the operation of multiplying these concepts will be the following display (see Fig. 11) The shaded surface denotes the class of those elements (crimes) that are simultaneously included in concept A (\"crime\") and in concept B (\"office crime).

When multiplying concepts whose volume does not coincide, we get a zero concept. For example, we need to perform a multiplication operation on the concepts “pretend” and “carelessly.” Since the volume of these concepts does not have common elements, the multiplicity obtained as a result of the multiplication operation The action is both intentional and careless and will be class zero.

The multiplication operation is expressed mainly with the help of the union \"and\" (\"student and athlete\", \"law and state law\", \"bribe and negligence\"), which is used in the connective tissue sense

B The operation of negating the concept A consists in the formation of a new concept - not-A, the volume of which, combined with the volume of the concept A, constitutes the logical class of the sphere of objects about which we are discussing

For example, the scope of our discussion is legal agreements. Denying the concepts of "purchase and sale" (A), we get the concept of "not purchase and sale" (non-A) By adding the concepts of "purchase and sale" and "not purchase and sale\", we will receive a legal class.

Graphically, the result of this operation can be represented as follows (see Fig. 12) Here the square is the sphere of objects that we are discussing (in this case, legal agreements) The circle of the concept (A) \"purchase and sale\" The shaded part of the square is the concept (not -A) \"not buying and selling\" The concept of non-A, denies the concept of A, has a certain scope. So, the scope of the concept \"not buying and selling\" (not-A) will not include everything, an object of reality, for example a tree , house, person, etc., but only those elements of the class of legal transactions that are not purchase and sale, are not included in the scope of concept A. And since each object or phenomenon of the material world can be considered by us as part of various classes of objects, the scope of a specific concept is not -And depend on the volume of the sphere of objects about which we are talking.

For example, if the scope of objects that we are thinking about is the class of crimes in general, then the scope of the concept “not theft” (not-A), obtained by negating the concept “theft” (A), will include all crimes, but namely: all state crimes, all crimes against property, with the exception of theft, crimes against life, health, freedom and dignity of the individual, etc. If the scope of the subjects are crimes against the personal property of citizens, then the scope of the concept "not theft" (not-A), formed by negating the concept of "theft" (A), will no longer include all crimes provided for by the code, except for theft, but only crimes against the personal property of citizens, which is not theft, that is, robbery, robbery, fraud , blackmail, etc. The concepts (A and not-A), obtained through the operation of negation, are in a relationship of contradiction with each other

Generalization and limitation of concepts

In the practice of thinking, we often have to move from one concept to another. Thus, we can move from the concept of "negligence" to the concept of "malfeasance," from the concept of "malfeasance crime," from the latter to the concept " acts\" and, conversely, from the concept of "action" to the concept of "crime", from it to the concept of "official crime".

The logical operation by which a transition occurs from a concept with a smaller scope to a concept with a larger scope is called generalization. To generalize a concept means to move from species to genus

A logical action, during which a transition occurs from a concept with a larger volume to a concept with a smaller volume, is called limitation. When limiting, we move from genus to species

For example, when we move from the concept of "agreement" to the concept of "transaction", and from it to the concept of "civil legal relations", and then to the concept of "legal relations", we generalize the concept of "agreement" we move on to the concept of "insurance", and from it to the concept of "property insurance", then we limit the concepts (see Figure 13).

The process of generalization and limitation of concepts is not endless

The limit of generalization is categories. Categories are concepts with an extremely wide scope. Categories have no gender, therefore they cannot be generalized. For example, categories such as “matter”, “consciousness”, “movement”, “essence”, \"phenomenon\", \"quantity\", \"quality\", etc.,

The limit of the limitation is a single concept. Thus, the limitation of the concept \"theft\" will be \"theft committed by Petrov\"

Generalization and limitation can be both correct and incorrect. For these operations to be correct, it is necessary to move from species to genus when generalizing, and from genus to species when limiting. If, when generalizing, we move to a concept that is genus relative to the original concept, then the generalization rate will be incorrect. It is impossible, for example, by generalizing the concept of “theft,” to move on to the concept of “robbery,” since robbery is not a genus for theft.

When limiting, errors occur when the concept one arrives at is not a species relative to the concept that is being limited. If, for example, limiting the concept “state”, we move on to the concept “family”, then such a restriction will be incorrect

Generalization and limitation of concepts allows us to clarify the content and scope of concepts, establish relationships between them, which is very important for cognition

Types of concepts

Concepts are usually divided into the following types: 1) singular and general, 2) collective and non-collective, 3) concrete and abstract, 4) positive and negative, 5) irrespective and correlative.

1. Concepts are divided into single and general V depending on whether one element or many elements are thought of in them. A concept in which one element is thought of is called single (for example, “Moscow”, “L.N. Tolstoy”, “Russian Federation”). The concept in which many elements are thought of is called general (for example, “capital”, “writer”, “federation”).

General concepts can be registering and non-registering. Registrants are called concepts in which the multitude of elements conceivable in it can be taken into account and registered (at least in principle). For example, “participant of the Great Patriotic War of 1941-1945,” “relatives of the victim Shilov,” “planet of the solar system.” Registering concepts have a finite scope.

A general concept relating to an indefinite number of elements is called non-registering. 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.

2. Concepts are divided into collective and non-collective. Concepts in which the characteristics of a certain set of elements that make up a single whole are thought of are called collective. For example, “team”, “regiment”, “constellation”. These concepts reflect many elements (team members, soldiers and regiment commanders, stars), but this multitude is thought of as a single whole.

The concept in which the attributes relating to each of its elements are thought is called non-collective. Such, for example, are the concepts of “star”, “regiment commander”, “state”.

3. 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).

The concept in which an object or a set of objects is thought of as something independently existing is called specific; the concept in which the attribute of an object or the relationship between objects is thought of is called abstract. Thus, the concepts “book”, “witness”, “state” are specific; the concepts of “whiteness”, “courage”, “responsibility” are abstract.

The difference between concrete and abstract concepts is based on the difference between an object, which is thought of as a whole, and a property of an object, abstracted from the latter and not existing separately from it. Abstract concepts are formed as a result of distraction, abstraction of a certain feature of an object.

4. 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. Concepts whose content indicates the absence of certain properties in an object are called negative. Thus, the concepts “literate”, “order”, “believer” are positive; the concepts of “illiterate”, “disorder”, “non-believer” are negative.

5. Concepts are divided into non-relative and correlative in depending on whether objects are thought of as existing separately or in relation to other objects.

Concepts that reflect objects that exist separately and are thought of outside their relationship to other objects are called irrelevant. These are the concepts of “student”, “state”, “crime scene”, etc. Correlative concepts contain signs indicating the relationship of one concept to another concept. For example: “parents” (in relation to the concept of “children”) or “children” (in relation to the concept of “parents”), “boss” (“subordinate”), “receiving a bribe” (“giving a bribe”). The concepts “part”, “reason”, “brother”, “neighbor”, etc. are also correlative. These concepts reflect objects, the existence of one of which is not conceivable outside of its relationship to the other.

Relationships between concepts

Depending on the content and volume, all concepts are divided into specific types. For clarity, let's present them in the form of a diagram, and then consider each type in more detail. Single concepts in which one object is conceived are called (for example, “the great Russian writer Alexander Nikolaevich Ostrovsky”, “United Nations”, “the capital of Russia” and others).

General is a concept in which many objects are thought of (for example, “capital”, “state”, “lawyer”, “economist” and others). General concepts can be registering and non-registering. Registrants concepts are called in which a multitude of objects conceivable in them are submitted to accounting and registration (for example, “participant of the Great Patriotic War”, “people's deputy of Russia” and others). Non-registering is a general concept that refers to an indefinite number of objects (for example, “man”, “philosopher”, “scientist” and others). Non-registering concepts have an infinite scope.

Zero(empty) are concepts whose volumes represent classes of really non-existent objects and the existence of which is in principle impossible: “perpetual motion machine”, “mermaid”, “goblin”, etc.). One should distinguish from the zero ones concepts that reflect objects that do not really exist at the present time, but existed in the past or whose existence is possible in the future: “ancient Greek philosopher”, “thermonuclear power plant”. Such concepts are not null.

Specific- these are concepts in which an object or a set of objects is thought of as something independently existing: “academy”, “student”, “romance”, “house”, “A. Blok’s poem “The Twelve”, etc.

Abstract- these are concepts in which it is not the object itself that is thought of, but one of the attributes of the object, taken separately from the object itself: “courage”, “conscientiousness”, “courage”, “blueness”, “identity”, etc.

Relative- these are concepts in which objects are conceived, the existence of one of which presupposes the existence of the other: “parents” - “children”, “teacher” - “student”, “boss” - “subordinate”, “plaintiff” - “defendant” and etc.

Irrelevant- these are concepts in which objects are conceived that exist independently, regardless of another object: “farmer”, “rule”, “village”, “man”, etc.

Positive- these are concepts whose content consists of properties inherent in the subject: “principle”, “noble act”, “living within one’s means”, “successful student”, etc.

Negative are concepts whose content indicates the absence of certain properties in an object (for example, “an ugly act”, “an unpainted house”, “an unmown meadow”, etc.). In the Russian language, negative concepts are usually expressed by words with negative prefixes “not” or “without” (“demon”): “illiterate”, “unbeliever”, “lawlessness”, “disorder”, etc. In words of foreign origin - most often in the words with a negative prefix “a”: “agnosticism”, “immoral”, etc.

Collective are concepts in which a group of homogeneous objects is thought of as a single whole: “forest”, “constellation”, “grove”, “student construction team”, etc. The content of a collective concept cannot be attributed to each individual element included in the scope of this concept.

Non-collective- these are concepts whose content can be attributed to each subject of a given class, which is covered by the concept: “tree”, “star”, “student”, etc.

Determining which of these types a specific concept belongs to means giving it a logical characterization. For example, the concept of “inattention” is general, non-collective, abstract, negative, irrespective. The logical characterization of concepts helps clarify their content and scope, develops skills for more accurate use of concepts in the process of reasoning.

Logical relationships between concepts

Since all objects of the world are in interaction and interdependence, then the concepts that reflect the objects of the world are also in certain relationships. Specific types of relationships are established depending on the content and scope of the concepts that are being compared.

If concepts do not have common features and are far from each other in their content, then they are called incomparable (for example, “symphonic music” and “solar eclipse”, “air space” and “library”). Comparable concepts are those that have common characteristics (for example, “language” and “foreign language”, “economist” and “bank employee”). Comparable concepts are divided by scope into compatible and incompatible.

Compatible are concepts whose scopes coincide completely or partially. Incompatible are concepts whose volumes do not coincide in any element.

Relationships between concepts are usually illustrated using circular diagrams (Euler circles), where each circle denotes the volume of the concept, and each point represents an object included in its volume. Circular diagrams allow you to visualize the relationships between various concepts and better understand and assimilate these relationships.

In identity relations there are concepts that differ in their content, but whose volumes coincide. In such concepts, one object or a class of homogeneous objects is conceived. However, the content of such concepts is different, since each of them reflects only a certain aspect (attribute) of a given object or class of homogeneous objects. For example, “the author of the story “The Man in the Case” and “the author of the story “Kashtanka”

In relation to intersection, there are concepts whose volumes partially coincide. The content of these concepts is different. For example, the overlapping concepts are “student” and “philatelist” (A and B): not all students are philatelists, and not all philatelists are students. In the combined (shaded) part of the circles are those students who are philatelists.

In relation to subordination there are concepts, the scope of one of which is completely included in the scope of the other, constituting its part. In this relationship, for example, are the concepts of “hero” (A) and “theater hero” (B). The scope of the first concept is wider than the scope of the second concept: in addition to the theatrical hero, there are other types: literary, artistic, television, cinematic and others. The concept of "theatrical hero" is fully included in the scope of the concept of "hero".

When illustrating the relationships between incompatible concepts, there is a need to introduce a broader concept that would include the volumes of incompatible concepts.

In relation to subordination there are two or more non-overlapping concepts belonging to a common generic concept. Subordinate concepts (B and C) are species of the same genus (A), they have a common generic characteristic, but the specific characteristics are different. For example, “offense of office” (A), “bribe” (B), “embezzlement” (C).

In relation to opposition (contrary) there are concepts that are species of the same genus, and moreover, one of them contains some characteristics, while the other not only denies these characteristics, but also replaces them with others that exclude (i.e., opposite) signs). For example, “democratic state” and “totalitarian state” (A and B), “us” and “alien”, “bravery” and “cowardice”, etc. Words that express opposite concepts are antonyms. The volumes of opposite concepts constitute in their sum only a part of the volume of the generic concept common to them.

In relation to the contradiction, there are two concepts that are species of the same genus, and at the same time, one concept indicates some characteristics, while the other denies these characteristics, excludes them, without replacing them with any other characteristics. For example, “knowing philosophy” and “ignoring philosophy”, “friend” and “enemy”, etc. The volumes of two contradictory concepts constitute the entire volume of the genus of which they are species. Thus, understanding the logical structure of a concept, revealing their types and relationships between comparable concepts makes it possible to move on to the consideration of logical actions, or operations, on concepts.

Definition of concepts and types of definitions. Techniques similar to definition.

DEFINITION OF THE CONCEPT AS A LOGICAL OPERATION

Definition is a logical operation that reveals the content of a concept.

Types of definitions:

1) nominal is a definition by which a new term (name) is introduced instead of describing an object. The purpose of this definition is to form a new term. For example, the discrepancy between a person’s subjective ideas and the objective state of affairs is called delusion. In this case, we introduced a new term - delusion - instead of describing the process;

2) real- this is a definition that reveals the essential characteristics of an object. For example, logic is a philosophical science about the laws and forms of human thinking, considered as a means of understanding the surrounding reality.

Since the definition of a concept consists in establishing its essential features, the rules of definition must obviously contain instructions for methods with the help of which the essential, and not other, features of the concept being defined can be found.

In many cases, listing all such signs is too lengthy. There is another way, which consists in indicating, firstly, the closest genus to which the given concept being defined belongs. Secondly, a special feature is indicated by which this concept differs as a species from all other species of the specified genus. This characteristic is called “specific difference,” and the method of definition itself is called determination “through the nearest genus and through specific difference.”

Definition through the nearest genus and species-forming difference is applied wherever previous research has revealed that the concept being defined is the concept of an object belonging to one of the species of a certain genus. These are many concepts of mathematical, physical and other sciences. For example, logic can be defined as the philosophical science of the laws and forms of human thinking, considered as a means of understanding the surrounding reality. This is a definition through genus and species difference.

Definition through the closest genus and species-forming difference presupposes that the concept being defined is the concept of an object that:

1) has already arisen and exists;

2) is bound by a certain relation of belonging to another class of objects, which includes it in itself in the same way as a genus includes a species.

At the same time, the method of origin of the object is not noted in the definition itself.

Techniques similar to definition: description, comparison, characterization, distinction.

Description- listing, as a rule, the external characteristics of an object. It plays an important role in activities. Thus, when making any decision, it is necessary to strive for the most complete description of all the consequences to which this action will lead.

Characteristic- this is an indication of the distinctive, characteristic features and characteristics of a single object.

Comparison is a technique that is used to figuratively characterize an object.

With the help of discrimination, signs are established that distinguish one object from other objects similar to it. For example, in the practice of an investigator, so-called “special signs” are often encountered.

Stefan Zweig of the appearance of Honore Balzac, the appearance of his father and other people, description of landscapes, trees, birds, etc.), in historical literature (description of the Battle of Kulikovo, description of the appearance of military leaders, monarchs and other personalities); Special technical literature provides a description of the appearance of machines, including computers, and a description of the designs of various objects (for example, locks, electric refrigerators, electric heating devices, etc.).

When searching for criminals, a description of their appearance and, first of all, special features is given so that people can identify them and report their location.

Characteristic gives a listing of only some internal, essential properties of a person, phenomenon, object, and not its appearance, as is done with the help of a description.

Sometimes a characteristic is given by indicating one characteristic. K. Marx called Aristotle “the greatest thinker of antiquity,” and Lunacharsky characterized Klim Samgin (from the novel by M. Gorky) as “a microscopic individuality on the big heels of self-conceit.” K. D. Ushinsky wrote: “Laziness is a person’s aversion from effort.”

The Guinness Book of Records (1988) gives the following characteristics: “Sergey Bubka (USSR). The first pole vaulter to overcome the six-meter mark"; "Sir Edmund Hillary (New Zealand). His outstanding achievement is that he was the first to conquer Everest"; “The most expensive painting. "Sunflowers", one of a series of 7 paintings by Vincent van Gogh, was sold at Christie's on March 30, 1987 in London for £22,500,000. Art."

Characteristics of literary heroes are given by listing their business qualities, moral, socio-political views, as well as corresponding actions, character traits and temperament, and the goals that they set for themselves. The characteristics of these characters allow us to clearly and accurately notice the typical features of a particular collective image.

For example, Aristotle gave this description of the ideal person. “The ideal person experiences joy in doing good to others; but he is ashamed to accept benefits from others. Elevated natures do good, lower natures accept it.”

J.-J. Rousseau believed that you can make a person kinder by changing his needs. Developing this idea, K. D. Ushinsky also gives the characteristics of a strong and weak creature: “He whose strength exceeds his needs, be it an insect or a worm, is a strong creature; the one whose needs exceed his strength, be it an elephant, a lion, be it a winner, a hero, be it a god, is a weak creature.” And further: “...a feeling of kindness appears when our strength exceeds the demands of our aspirations.”

Dale Carnegie gives this description in combination with comparisons. “One of the most tragic aspects of human nature, as far as I know, is our tendency to postpone the fulfillment of our aspirations until the future. We all dream of some magical garden full of roses that can be seen somewhere over the horizon - instead of enjoying the roses that grow under our window today. Why are we such fools - such terrifying fools? “How strangely we spend this little period of time called our lives,” wrote Stephen Leacock. - The child says: “When I become a young man.” But what does this mean? The young man says: “When I become an adult.” And finally, as an adult, he says: “When I get married.” Finally, he gets married, but little changes from this. He starts thinking, “When can I retire.” And then, when he reaches retirement age, he looks back on the path he has taken in life; it’s as if a cold wind is blowing in his face, and the cruel truth is revealed to him about how much he has missed in life, how everything is irrevocably gone. We understand too late that the meaning of life lies in life itself, in the rhythm of every day and hour.”

A combination of description and characterization is often used. It is used in the study of chemistry, biology, geography, history and other sciences. For example, “Oil is an oily liquid, lighter than water, dark in color, with a pungent odor. The main property of oil is flammability. When burned, oil produces more heat than coal. Oil lies deep in the earth." This technique is often used in fiction.

Explanation by Example is used when it is easier to give an example or examples illustrating a given concept than to give its strict definition through genus and specific difference.

The concept of “desert fauna” is explained by listing the species of its inhabitants: camel, goitered gazelle, turtle, monitor lizard, kulan, etc.

The concept of “mineral resource” is explained by listing the types (examples): oil, coal, metals, etc. Explanation through example is used in both secondary and primary schools.

A variation of this technique is ostensive definitions that are often used when teaching a foreign language, when an object (or a picture of it) is named and shown. The same thing is sometimes done when explaining incomprehensible words in the native language.

Another technique that replaces the definition of concepts is comparison. Comparison is used both at the level of scientific knowledge and at the level of artistic reflection of reality. V. A. Sukhomlinsky used a comparison of a child’s brain with a rose flower: “We, teachers, are dealing with the most gentle, the most subtle, the most sensitive that is in nature - the child’s brain. When you think about a child’s brain, you imagine a delicate rose flower with a drop of dew trembling on it. What care and tenderness are needed so that, when you pick a flower, you don’t drop a drop. We need the same caution every minute: after all, we are touching the subtlest and most delicate in nature - the thinking matter of a growing organism.”

In science, comparison allows us to find out the similarities and differences of compared objects. The biology textbook makes the following comparisons: “The body of a jellyfish is gelatinous, similar to an umbrella”; “Kidneys are small paired organs shaped like beans”; “The pea flower resembles a sitting moth”; “The ovaries of rosehip pistils are hidden in an overgrown receptacle, similar to a glass.” In all the above comparisons, the common feature (the basis of comparison) is the form.

Comparison at the level of artistic reflection of reality allows us to notice what is common, similar in two objects, and to express this similarity in a vivid form, figuratively. M. Gorky uses the following comparison: “Rudeness is the same ugliness as a hump.”

Artistic comparisons often include words: “as”, “as if”, “as if”, etc.

V. Nabokov in the story “Spring in Fialta” uses the following interesting comparisons: “... the Christmas trees silently traded their bluish pies”; “... someone, escaping, falling, crunching, laughing out loud, climbed onto a snowdrift, ran, gasped at the snowdrift, amputated his felt boot”; “... as if a woman’s love was spring water containing healing salts, which she willingly gave to everyone from her ladle, just remind.”

Arthur Conan Doyle uses three techniques in one sentence to replace a definition (he gives a description, a characteristic and a number of comparisons). “As soon as I close my eyes now, Marie stands in front of me: her cheeks are dark, like the petals of a muscat rose; the look of brown eyes is gentle and at the same time bold; hair black as pitch awakens excitement in the blood and begs for poetry; and the figure is like a young birch tree in the wind.”

Discrimination there is a technique that allows you to establish the difference between a given object and similar objects. For example, “Hysteria is not a disease, but a character: the main feature of this character is self-suggestibility” (P. Dubois).

Determination rules. Errors in definitions.

Compliance with these rules is mandatory to avoid committing logical errors. These rules are as follows:

1. The definition must be proportionate , i.e. the scope of the defined concept must coincide with the scope of the defining one, they must be equivalent concepts. This proportionality is easily verified by rearranging the positions of the members of the definitive judgment. Let's give examples. “The science of the laws and forms of correct thinking is logic.” If you rearrange this logical equation, you can find an identity, as in the first case. It’s a different matter when we resort to the following examples: “A young man with a diploma is a specialist.” If we rearrange the defined and the defining, we can see that the concept of “specialist” is broader than the concept of “young man with a diploma.” This means that in this case this rule is violated.

2. No circles should be allowed in the definition , i.e. when the determinant itself is explained through the defined concept. Violation of this rule leads to a logical error - tautologies . Here are some examples of tautology: “A criminal is a person who has committed a crime”; “Comparative analogy” (from the newspaper “Telegorod”, No. 21, 2003). Here it is clear that the defining concept repeats what is said in the defined, without revealing its meaning. To avoid this mistake, you need to remember that the defined and defining concepts are equal in volume, but not identical in content, they represent independent concepts.

3. The definition should not be only negative . After all, the purpose of the definition is to answer the question: what is the given object reflected in the concept. To do this, it is necessary to identify and list in an affirmative form its essential features. The negative definition marks only missing features, i.e. indicates what the item is not. However, a negative element in the defining concept is sometimes necessary; it more clearly highlights the subject of our thought. For example, the concept of “invisible world” does not give a positive idea of ​​this world, but emphasizes the object itself, which is reflected in the concept.

4. The definition must be concise, precise and clear .

A definition that is too verbose goes beyond its intended purpose and threatens to become a mere description. In the definition, it is necessary to avoid ambiguous, vague terms that can be interpreted in different ways. A vague definition leads to a lack of understanding of the subject, to vague ideas and confusion.

The accuracy of the definition presupposes its unambiguity throughout the entire argument (speech before an audience, written text, process and conclusion). This is required by the logical law of identity. In practice, it often becomes necessary to change the definition, but a special clause must be made. The clarity of a definition depends on its brevity and precision.

Division of concepts and its types

Rules for dividing concepts in logic

1. The division must be proportionate.

The task of division is to list all types of the concept being divided. Therefore, the volume of division terms must be equal in their sum to the volume of the concept being divided. If, for example, when dividing crimes depending on the nature and degree of public danger, crimes of minor gravity, moderate gravity and serious crimes are distinguished, then the rule of proportionality of the division will be violated, since one more member of the division is not indicated: especially serious crimes.

This division is called incomplete.

The rule of proportionality will also be violated if extra division terms are indicated, i.e. concepts that are not species of a given genus. Such an error will occur if, for example, when dividing the concept of “criminal punishment”, in addition to all types of punishment, a warning is indicated, which is not included in the list of penalties in criminal law, but is a type of administrative penalty.

This division is called divisionwith extra members.

PROPERTIES OF LOGICAL OPERATIONS

1. Designations

1.1. Notation for logical connectives (operations):

a) negation(inversion, logical NOT) is denoted by ¬ (for example, ¬A);

b) conjunction(logical multiplication, logical AND) is denoted by /\
(for example, A /\ B) or & (for example, A & B);

c) disjunction(logical addition, logical OR) is denoted by \/
(for example, A \/ B);

d) following(implication) is denoted by → (for example, A → B);

e) identity denoted by ≡ (for example, A ≡ B). The expression A ≡ B is true if and only if the values ​​of A and B are the same (either they are both true, or they are both false);

f) symbol 1 is used to denote truth (true statement); symbol 0 – to indicate a lie (false statement).

1.2. Two Boolean expressions containing variables are called equivalent (equivalent) if the values ​​of these expressions coincide for any values ​​of the variables. Thus, the expressions A → B and (¬A) \/ B are equivalent, but A /\ B and A \/ B are not (the meanings of the expressions are different, for example, when A = 1, B = 0).

1.3. Priorities of logical operations: inversion (negation), conjunction (logical multiplication), disjunction (logical addition), implication (following), identity. Thus, ¬A \/ B \/ C \/ D means the same as

((¬A) \/ B) \/ (C \/ D).

It is possible to write A \/ B \/ C instead of (A \/ B) \/ C. The same applies to the conjunction: it is possible to write A /\ B /\ C instead of (A /\ B) /\ C.

2. Properties

The list below is NOT meant to be complete, but we hope it is representative enough.

2.1. General properties

1. For a set of n there are exactly logical variables 2 n different meanings. Truth table for logical expression from n variables contains n+1 column and 2 n lines.

2.2.Disjunction

1. If at least one of the subexpressions to which the disjunction is applied is true on some set of values ​​of the variables, then the entire disjunction is true for this set of values.
2. If all expressions from a certain list are true on a certain set of variable values, then the disjunction of these expressions is also true.
3. If all expressions from a certain list are false on a certain set of variable values, then the disjunction of these expressions is also false.
4. The meaning of a disjunction does not depend on the writing order of the subexpressions to which it is applied.

2.3. Conjunction

1. If at least one of the subexpressions to which the conjunction is applied is false on some set of variable values, then the entire conjunction is false for this set of values.
2. If all expressions from a certain list are true on a certain set of variable values, then the conjunction of these expressions is also true.
3. If all expressions from a certain list are false on a certain set of variable values, then the conjunction of these expressions is also false.
4. The meaning of a conjunction does not depend on the writing order of the subexpressions to which it is applied.

2.4. Simple disjunctions and conjunctions

Let us call (for convenience) the conjunction simple, if the subexpressions to which the conjunction is applied are distinct variables or their negations. Similarly, the disjunction is called simple, if the subexpressions to which the disjunction is applied are distinct variables or their negations.

1. A simple conjunction evaluates to 1 (true) on exactly one set of variable values.
2. A simple disjunction evaluates to 0 (false) on exactly one set of variable values.

2.5. Implication

1. Implication A →B is equivalent to disjunction (¬ A) \/ B. This disjunction can also be written as follows: ¬ A\/B.
2. Implication A →B takes the value 0 (false) only if A=1 And B=0. If A=0, then the implication A →B true for any value B.

Conjunction or logical multiplication (in set theory, this is intersection)

A conjunction is a complex logical expression that is true if and only if both simple expressions are true. This situation is possible only in a single case; in all other cases the conjunction is false.

Notation: &, $\wedge$, $\cdot$.

Truth table for conjunction

Picture 1.

Properties of conjunction:

1. If at least one of the subexpressions of a conjunction is false on some set of variable values, then the entire conjunction will be false for this set of values.
2. If all expressions of a conjunction are true on some set of variable values, then the entire conjunction will also be true.
3. The meaning of the entire conjunction of a complex expression does not depend on the order in which the subexpressions to which it is applied are written (like multiplication in mathematics).

Disjunction or logical addition (in set theory this is union)

A disjunction is a complex logical expression that is almost always true, except when all expressions are false.

Notation: +, $\vee$.

Truth table for disjunction

Figure 2.

Properties of disjunction:

1. If at least one of the subexpressions of the disjunction is true on a certain set of variable values, then the entire disjunction takes on a true value for this set of subexpressions.
2. If all expressions from some list of disjunctions are false on some set of variable values, then the entire disjunction of these expressions is also false.
3. The meaning of the entire disjunction does not depend on the order in which the subexpressions are written (as in mathematics - addition).

Negation, logical negation or inversion (in set theory this is negation)

Negation means that the particle NOT or the word FALSE is added to the original logical expression, WHAT and as a result we get that if the original expression is true, then the negation of the original will be false and vice versa, if the original expression is false, then its negation will be true.

Notation: not $A$, $\bar(A)$, $¬A$.

Truth table for inversion

Figure 3.

Properties of negation:

The “double negation” of $¬¬A$ is a consequence of the proposition $A$, that is, it is a tautology in formal logic and is equal to the value itself in Boolean logic.

Implication or logical consequence

An implication is a complex logical expression that is true in all cases except when truth follows falsehood. That is, this logical operation connects two simple logical expressions, of which the first is a condition ($A$), and the second ($A$) is a consequence of the condition ($A$).

Notation: $\to$, $\Rightarrow$.

Truth table for implication

Figure 4.

Properties of implication:

1. $A \to B = ¬A \vee B$.
2. The implication $A \to B$ is false if $A=1$ and $B=0$.
3. If $A=0$, then the implication $A \to B$ is true for any value of $B$ (true may follow from false).

Equivalence or logical equivalence

Equivalence is a complex logical expression that is true for equal values ​​of the variables $A$ and $B$.

Notation: $\leftrightarrow$, $\Leftrightarrow$, $\equiv$.

Truth table for equivalence

Figure 5.

Equivalence properties:

1. The equivalence is true on equal sets of values ​​of the variables $A$ and $B$.
2. CNF $A \equiv B = (\bar(A) \vee B) \cdot (A \cdot \bar(B))$
3. DNF $A \equiv B = \bar(A) \cdot \bar(B) \vee A \cdot B$

Strict disjunction or addition modulo 2 (in set theory, this is the union of two sets without their intersection)

A strict disjunction is true if the values ​​of the arguments are not equal.

For electronics, this means that the implementation of circuits is possible using one standard element (though this is an expensive element).

The order of logical operations in a complex logical expression

1. Inversion(negation);
2. Conjunction (logical multiplication);
3. Disjunction and strict disjunction (logical addition);
4. Implication (consequence);
5. Equivalence (identity).

To change the specified order of logical operations, you must use parentheses.

General properties

For a set of $n$ boolean variables, there are exactly $2^n$ distinct values. The truth table for a logical expression of $n$ variables contains $n+1$ columns and $2^n$ rows.

In what follows, no special logical symbols are used. Taking into account, however, that the reader may also have to read books in which such symbolism is used, we will give as an example the basic, most frequently used logical symbols.

For more than two thousand years, traditional logic has used ordinary language to describe thinking. Only in the 19th century. The idea gradually became established that for the purposes of logic a special artificial language was needed, built according to strictly formulated rules. This language is not meant for communication. It should serve only one task - identifying the logical connections of our thoughts, but this task must be solved with utmost efficiency.

The principles of constructing an artificial logical language are well developed in modern logic. Its creation had approximately the same significance in the field of thinking for the technique of logical inference as the transition from manual labor to mechanized labor had in the field of production.

A language specially created for the purposes of logic is called formalized. Words in ordinary language are replaced by individual letters and various special characters. A formalized language is a “thoroughly symbolic” language in which there is not a single word of ordinary language. In a formalized language, meaningful expressions are replaced by letters, and logical symbols

(logical constants) symbols with strictly defined meanings are used.

In the logical literature, various notation systems are used, so two or more symbol options are given below.

Signs used to indicate negation; read: “not”, “it’s not true”;

Signs to denote a logical connective called conjunction; read: “and”;

A sign to denote a logical connective called a non-exclusive disjunction; reads: “or”;

A sign to denote a strict, or exclusive, disjunction; reads: “either, or”;

Signs to indicate implication; read: “if, then”;

Signs to indicate the equivalence of statements; read: “if and only if”;

General quantifier; reads: “for everyone”, “everyone”;

Existence quantifier; read “exists”, “there is at least one”;

L, N, - signs to denote the modal operator of necessity; read: “it is necessary that”;

M is a sign to denote the modal operator of possibility; reads: “it’s possible that.”

Along with those listed, various systems of logic also use other specific symbols, and each time it is explained what exactly this or that symbol means and how it is read.

As in the language of mathematics, parentheses are used as punctuation marks in artificial languages ​​of logic.

Let’s take, for example, some meaningful statements and present next to them their notation in the language of logic:

A) “He who thinks clearly speaks clearly” -; the letter A denotes the statement “The person thinks clearly”, B - the statement “The person speaks clearly”, - the connective “if, then”;

B) “He is an educated person and it is not true that he is not familiar with Shakespeare’s sonnets” -; A - the statement “He is an educated person”, B - “He is not familiar with Shakespeare’s sonnets”, - the connective “and”,

C) “If light has a wave nature, then when it is represented as a stream of particles (corpuscles), an error is made” -

; A - “Light has a wave nature”, B - “Light is represented as a stream of particles”, C - “Error is allowed”;

D) “If you were in Paris, then you saw the Louvre or saw the Eiffel Tower” - “You were in Paris”, B - “You saw the Louvre”, C - “You saw the Eiffel Tower”;

4. Logical symbolism

D) “If a substance is heated, it will melt or evaporate, but it can also explode” - (A ^ (B v C v D)); A - “The substance heats up”, B - “The substance melts”, C - “The substance evaporates”, D - “The substance explodes”.

Let us give another simple example of the transition from an artificial language of logic to ordinary language. Let variable A represent the statement “Darwin’s theory is scientific”, B - “Darwin’s theory can be confirmed by experimental data”, C - “Darwin’s theory can be refuted by experimental data”. What meaningful statements are expressed by formulas:

A) A ^ (B ^ C);

B) (V l ~ C) ^ ~ A;

B) (~ V l ~ C) ^ ~ A?

The answer to this question is, respectively, three statements:

A) If Darwin's theory is scientific, then if it can be confirmed by experimental data, it can also be refuted by them;

B) If Darwin's theory can be confirmed by experimental data, but cannot be refuted by it, it is not scientific;

C) If Darwin's theory cannot be confirmed by experimental data and cannot be refuted by it, it is not scientific.

We have already talked about signs. Now let's look at this issue in more detail. Sign- this is a material object that acts in the process of cognition or communication as a representative of an object.

The following three types of signs can be distinguished: (1) index signs; (2) symbols-images; (3) signs-symbols.

Index signs are connected with the objects they represent materially, for example, as effects with causes. Thus, smoke over a forest indicates the presence of fire there, an increased temperature of a person indicates a disease, a change in the color of a person’s nails indicates a disease of the internal organs, and a change in the height of the mercury column indicates a change in atmospheric pressure.

Signs-images are those signs that themselves carry some information about the objects they represent (terrain map, painting, drawing), since they are in a relationship of similarity with the designated objects.

Signs-symbols are not materially related and are not similar to the objects they represent.

Logic explores signs of the latter type.

Signs have, as already mentioned, objective and semantic meanings. Subject meaning is an object that is represented (or denoted) by a sign. Subject meaning is often called simply meaning.

Semantic meaning is a characteristic of an object expressed by a sign, of which the sign is a representative, i.e. information about this object. There are two types of information. Information of the first type is called the meaning of the sign, and information of the second type is called a visual image, or intuitive representation. Meaning is information expressed in language that allows one to distinguish objects that are the meaning of a sign from all other objects. Information of the second type is also called an idea. As has already been said, semantic meaning can include both meaning and idea. It may only be a meaning, or it may only be an idea.

Some signs have no meaning, that is, they represent objects that do not exist in the area of ​​reasoning (“perpetual motion”).

Among the signs-symbols, logical signs and non-logical signs are distinguished. Non-logical signs are also called descriptive signs.

Logical signs express the most general characteristics of things and phenomena, as well as thoughts. These include conjunctions “and”, “or”, “if..., then...”, negation “it is not true that” (“not”), words characterizing the number of objects about which something is affirmed or denied: “ all” (“none”), “some”, the connective “essence” (“is”), the word “therefore”, etc. Since all the listed expressions in everyday language are used in different senses, they are not yet signs. For them to be signs, they need to be given meaning. Once meaning is given to these expressions, they become signs and are called logical terms.

Example. The conjunction “and” can be used in different senses, including the following.

First. A union expresses the simultaneous existence of two situations. (It is raining and snowing.) In logic, in order to fix the meaning of a conjunction, a special language is used, called the language of symbols. In the language of symbols, the conjunction “and” in the indicated sense is denoted as follows: 8s.

Second. The sequential existence or occurrence of two situations is expressed. (Petrov went outside and (then) met a friend.) Designation: 8C

Third. A certain situation arises, the second situation arises later than the first, but continues to exist when the first has not yet ended. (Summer has come and the flowers have bloomed.) Designation:

Other logical terms are introduced below.

Descriptive terms. Signs-symbols are names. Name- is a word or phrase that denotes an object. The names were the symbols described above. As was said, signs, and therefore names, have semantic and (or) subject meanings. A name denoting a single object is called single. A name whose volume consists of more than one subject is called general Common names can be universal. Universal is called a general name, the scope of which is the entire universe of reasoning (the subject area about which the reasoning is being conducted). For example, “a person who knows some foreign languages ​​or does not know any foreign languages.” The universe of reasoning here is the set of (all) people. The scope of the name is the same set. The name “a person who knows some foreign languages” is not universal, since its scope does not coincide with the set of (all) people. The universe of reasoning is determined by the context in which the name is used.

There can be names with different meanings and the same volume (for example, “the largest city in England” and “the capital of England”), but there cannot be names with the same meaning but different volumes. Names in the scope of which there is not a single subject from the field of reasoning are called imaginary. Here you should pay attention to the fact that areas of reasoning (subject areas) may be different. The name “perpetual motion machine” is imaginary if the area of ​​​​discussion is material objects that actually exist, or those that can exist as material. A geometric point does not exist as a material object (in the real world there are no objects that have neither length, nor height, nor width), but it exists in the domain of geometric objects. In relation to the area of ​​geometric objects, the name “point” is not imaginary.

There are names that have their own meaning and names that do not have their own meaning. Names that have their own meaning are descriptive names like “the largest river in Europe.” The meaning of such names is determined by their structure and the meanings or meanings of the names that make up these descriptive names. If the names included in a complex name do not make sense, then a descriptive name can still make sense in this case. This meaning consists in indicating the relationship between the meanings of the constituent names, distinguished on the basis of ideas. Non-descriptive names like "Volga" do not have their own meaning. If they have any meaning, it is only a given one. Non-descriptive names are given meaning by means of descriptive names that are associated with them. Descriptive names, in turn, include non-descriptive names. They are also given meaning through descriptive language. Obviously, such a process cannot be infinite, that is, some non-descriptive names have meaning, but have no meaning, although they have ideas. These names designate objects, but do not carry information about them expressed in language that allows them to be distinguished from other objects. They are introduced on the basis of visual images or intuitive ideas, ideas. Names that have no meaning are often names with underdefined meanings. These names do not express concepts, but they are mistakenly called fuzzy concepts. These are the so-called “evaluative concepts”: “cruelty to animals”; “animal” (when addressing the issue of cruelty to animals).

The underdetermination of the meanings of names that have no meaning is due to the fact that visual images and intuitive ideas about the objects denoted by such names in many cases are different for different people, that is, they contain elements of subjectivity, which is presented in the following diagram.

The use of names is subject to certain requirements (principles). Let us formulate two of these principles.

First. Principle of objectivity: in sentences something must be affirmed or denied not about names, but about the meanings of names. For example, if we say that the Earth is a planet, then we are not talking about the word “Earth”, but about the Earth itself. Of course, sometimes you have to affirm or deny something about names. Then the so-called "quoted names". For example, the sentence ““Earth” is the name of the planet” does not speak about the celestial body “Earth”, but about the name of this celestial body. Sometimes in natural language there are cases where the name of a name is the original name itself. For example, in the sentence “The table consists of four letters,” the word “table” is the name of the word itself. This use of names is called autonomous. Autonomous use of names is unacceptable in scientific languages.

Comment. This principle is often violated when teaching children to read. Learning begins not with learning letters, but with learning the names of letters. If a child knows the names of letters, it is not necessary that he knows letters. For example, the name of the letter b is the expression bae. The names of the vowels are the letters themselves. After the child learns the names of consonants, he is taught to read syllables: bae And A read ba, ah And bae read ab etc. This method of teaching reading is extremely difficult. The best way is to teach the child not the names of the letters, but the letters.

Second principle- the principle of unambiguity. According to this principle, an expression used in business or scientific language as a name must be the name of only one object, if it is a single name, and if it is a general name, then the expression must be a name common to objects of the same class. This principle is not always observed by people with a low logical culture.

Another type of descriptive terms are signs of object functions, or subject functors. These signs express objective functions.

Function is called a correspondence by virtue of which objects (an object, a pair, a triple of objects, etc.) from a certain set, called the domain of definition of a function, are correlated with objects from another or the same set, called the values ​​of the function. Everyone knows mathematical (numerical) functions - addition of numbers, subtraction, multiplication, division. In logic, the understanding of a function is generalized.

Subject is a function whose values ​​are any objects. Examples of subject functions: weight, length of service, average monthly income, father, capital. Applying the functional sign “mass” to the singular name “Earth,” we obtain as a value the singular name “mass of the Earth,” denoting a certain quantity, i.e., an object. Thus, this function compares objects (material objects with mass) with other objects (mass values). The domain of definition of the function “work experience” is a set of people. The range of values ​​is a set of named numbers (many years of work). Applying this function to a person, for example, to Petrov, we get a named number, for example, 20 years. The domain of definition of the “father” function is a set of people. Applying this function, for example, to Socrates, we get a specific person as the value.

Some logical terms are also understood as functions. These are already functions of a different type - logical functions. For example, the logical term “it is not true that” (negation) is considered as a function that compares a true sentence with a false one, and a false one with a true one. Applying negation to the true sentence “There is life on Earth,” we get the false sentence “It is not true that there is life on Earth.” Applying negation to the false sentence “Moscow is a big village,” we obtain the true sentence “It is not true that Moscow is a big village.”

• Leonardo da Vinci writes: “...If you say that touching a certain surface with the very end of a pencil is the creation of a point, then this will be wrong; we will say that such a touch produces a surface surrounding its middle, and in this middle is the location of the point.” . See: Zhukov A. N. UnknownLeonardo: parables, allegories, facets. Rostov, 2007. P. 79.