Noun. Section ii. logical naming theory

3.1. General logical characteristic of the name

The essential characteristic of a person is abstract logical linguistic thinking. It is based on the ability of a person, being distracted from specific objects and phenomena, to turn to their essence. At the same time, both real objects and phenomena (“house”, “morning”), and their properties (“purity”, “harmony”) are denoted by names in the language. Consequently, the name is the basic logical and semiotic unit, the elementary form, and the process of thinking is the process of operating with names and establishing special connections between them. The name denotes any object of thought in terms of its distinctive features.. IN

In a language, a name is expressed using words and phrases that are most often used in a sentence as the subject or nominal part of the predicate. Outside the verbal form, the name does not exist, but the name and the word are not identical: the same name in different languages ​​has a different language form, and many words have several meanings.

3.2. The scope and content of the name

IN logic, any name has scope and content. The content of the name represents its semantic meaning, that is, the totality of those features of objects and their classes that it designates.

The volume of a name is represented by a set of its bearers or designates, which can be both material objects and only conceivable ones.

The scope and content of the name, characterizing it from different angles, are closely related. The study of this relationship made it possible to identify a special pattern, which found expression in the law of the inverse relationship between the content and volume of the name: increasing the content of the name, we reduce its volume, and vice versa. The content of the name increases due to the inclusion of new features in it. For example, the name "student". Its volume includes all students of higher educational institutions of all forms of education (daytime, part-time, evening, distance learning, etc.). Adding to it a new sign - "correspondence student", we enriched the content of the name "student", but reduced its volume, excluding students of all other forms of education from it. The logical operation, during which we go from a name with a large volume to a name with a smaller volume, is called name scope limitation. The limits are

names with a minimum volume (single, most often proper).

The reverse operation in logic is called generalization of the scope of the name. It represents a transition from a name with a smaller volume to a name with a larger volume due to the exclusion of certain features from its content. For example, the name "textbook on logic". Excluding the sign from its content, we get a name with a large volume - "textbook", but with less content. At the same time, names with the widest possible scope are the limit of generalization - categories denoting extremely wide and abstract phenomena, processes and connections (“space”, “good”, “matter”, etc.).

3.3. Kinds of names

The type of a name depends both on the number of its designata and on the features it denotes. Names are divided into single, common and empty (null). In terms of content - concrete and abstract, positive and negative, relative and irrelevant, collective and non-collective.

A single name is a name that has one designatum (“first cosmonaut”, “Constitution of the Republic of Belarus”, “Immanuel Kant”). As a rule, proper names also belong to singular ones. Names that have two or more designata are called common (“student”, “law”, “constitution”). Names that do not have designata are called empty (null). Such names have a semantic meaning, but are devoid of subject matter. These include names from the sphere of human fantasy, fairy tales, myths (“mermaid”, “Snake Gorynych”, “unicorn”), scientific concepts as a result of extreme abstraction (“ideal gas”, “absolutely black body”) and names, in the content which signs are thought that contradict the nature of the designated objects (“triangular square”, “ice sun”).

Names are divided into abstract and concrete depending on what they mean. If the name denotes real objects and their classes, it is specific ("student", "house", "centaur", "thunderstorm"). Names denoting individual properties of objects, the relationship between them, are called abstract ("purity", "love", "courage").

Names are divided into positive and negative depending on whether they fix the presence of some sign in the designated object or its absence. A name is called positive, indicating the presence of a certain sign in the object (“believer”, “order”).

On the contrary, a name indicating the absence of a feature in an object is called negative (“asymmetry”, “inadequacy”). As a rule, negative names are formed with the help of negative particles (not-, without-, a-). If a name without a negative prefix is ​​not used for various reasons (language development, change in lexical norms), then it is positive (“hatred”, “dissonance”).

Irrelevant are names denoting objects in themselves, regardless of the relationship and connections of these objects with others (“man”, “house”). Relative names are names denoting objects that do not exist independently, but only as members of some kind of relationship (“good - evil", "day - night").

A name denoting a set of objects conceivable as a single whole is called a collective name (“constellation”, “service”). Moreover, the name of integrity does not coincide with the names of the objects that make it up. Thus, the designate of the name "constellation" is the Ursa Major Constellation and other constellations, and not the stars and celestial bodies. non-collective names are called that designate objects and their classes and are conceivable not as independent entities, but existing separately (“planet”, “window”).

Defining the types of the name by volume and content, we give it a complete logical description: planet - general, specific, positive, irrelevant, non-collective. A complete logical description allows you to clarify the scope and content of the name, more correctly use its verbal expression in the text, discussions, etc.

3.4. Relationships between names by scope

The entire set of names can be divided into comparable and incomparable. Comparable are names that have at least one common feature (“student” and “athlete”). Incomparable ones do not have common features, therefore, it is impossible to compare them. In logical relations there can be only comparable names. Comparable names, in turn, are divided into compatible and incompatible. Compatible names include names whose scopes completely or partially match, and incompatible names include names whose scopes do not match either completely or partially. Relationships between names have a graphic representation on Euler circles.

Types of compatibility:

1. Identity (equivolume).

A - student, B - university student.

whose volumes are exactly the same. At the same time, they have coinciding designations, since they denote the same subject, but their content may be different. The relationship between names of equal volume is shown in Fig.1.

2. Crossing

A - student, B - chess player, C - chess student.

IN in relation to the intersection, there are names whose volumes partially coincide. At the same time, as a result of the intersection of the scopes of names, a new class is formed, formed by designata common to the intersecting names. On fig. 2 shows the intersection relationship.

3. Submission

A - student, B - student.

In relation to subordination, there are names, the scope of one of which is completely included in the scope of the other, but not exhausting it. This relationship is shown in Fig. 3.

Types of incompatibility: 1. Subordination

A - university, B - BNTU, C - BSU.

IN there are two or more species of the same genus in a subordination relationship. In relation to the generic name, they are in a relationship of subordination, and among themselves - subordination, i.e. their volumes do not overlap. The relationship of subordination is shown in fig. 4.

2. Opposite

A - white, B - black, C - color.

IN In the relation of opposition (contrararity), there are names, one of which has some features, and the other excludes them, replacing them with opposite ones. This relationship is shown in Fig. 5.

From the point of view of logical grammar, the mechanism of human thinking is simple. There are diverse names (concepts) denoting individual objects or their sets. With the help of logical connectives (“there is”, “everything ... is ...”, “some ... are not ...”, etc.), statements are formed from concepts, from which, in turn, are composed all our reasoning. Those arguments in which some statements are taken as initial statements and a new statement is derived from them are called inferences.

Name (concept), statement and conclusion are the three central categories of logic. In the following, each of them will be considered in detail.

Names seem to be the simplest of all expressions in the language. They are everywhere, in ordinary life everything is named. Getting into a completely unfamiliar place, a person immediately gives him a name: "an unfamiliar place." Faced with what no one has yet observed, he first of all calls it: "what has not been observed before." Even a thing that does not have a name turns out to be the owner of a name - "a thing without a name."

Names are a necessary means of knowledge and communication. Denoting objects and their aggregates, they connect the language with the real world. They are so natural that they once seemed to belong to the things themselves, just as color, heaviness and other properties are inherent in them. Primitive people viewed their names as something concrete, real, and often sacred. Psychologist L. Levy-Bruhl, who proposed at the beginning of the 20th century. the concept of primitive thinking, considered such an attitude to names an important factor confirming the mystical nature of such thinking. In particular, he pointed out that the Indian considers his name not as a mere label, but as a separate part of his personality, such as eyes or teeth. He believes that he will suffer just as much from the malicious use of his name as from a wound inflicted on some part of his body. This belief is found among various tribes from the Atlantic to the Pacific.

All sciences that study language are engaged in the study of names as one of the basic concepts of both natural and artificial languages. And above all, logic, for which names are one of the main categories.

In different scientific disciplines, the “name” refers to different, and sometimes incompatible things. Logic has gone to great lengths to clarify what a name is and what principles govern the operation of naming or designation. Perhaps nowhere are names treated so comprehensively, deeply and consistently as in logical studies.

Name - language expression denoting a separate object or set (class, set) similar items.

For example, the name "Cicero" denotes an individual - the Roman philosopher Cicero, who wrote, in particular, the treatise "On Duties"; the name "government" - the highest collegial executive body of the state; the word "black" can be considered as designating a class of black objects; the word "further" - as a designation of a certain relationship between objects, etc.

The ability to designate, or name, something, that is, to refer us to some objects or their combinations, is a specific feature of the name. In logic, the word "subject" is understood as broadly as anything that can be named. Strictly speaking, being a name and designating are one and the same thing.

The word "Plato", denoting an individual, is also a name. Another name is "Aristotle's teacher", but the same person is meant. The name "teacher" designates a class of people, each of whom is engaged in teaching. The word “suspect” is any person detained on suspicion of committing a crime, as well as a person to whom a measure of restraint has been applied before a charge is brought, and the word “large” is any object that is large, etc.

Thus, the word "name" refers not only to language expressions that name individual objects ("Cicero", "Plato"), but also to expressions denoting groups of objects ("teacher", "suspect"), as well as the properties of objects ( “black”, “big”) and even sometimes relationships between objects (“higher”, “more”).

Logic greatly expands the usual use of the word "name". This is due to many reasons, and above all, her desire for the ultimate generality of her reasoning. However, it is not difficult to get used to the wide use of this word and not to confuse names in general with their special case - proper names.

This use of names is far from definite and consistent. Logic seeks to bring order into this procedure, to establish the principles to which the latter must obey.

In logic, names are subject to two main requirements in particular.

The principle of unambiguity: a name denotes only one item, item class, or property. However, both in everyday language and in legal language, this principle is often violated due to the ambiguity of words and expressions. We should strive to ensure that, at least within one context or one discourse, our words and expressions refer to the same objects. Otherwise, logical errors are inevitable.

The principle of objectivity: every sentence must speak of those objects which are designated by its expressions. For example, in the sentence “The loan agreement is different from the storage agreement”, we are not talking about the names “loan agreement” and “storage agreement”, but about the mutual relations of these different types of agreements. The principle of objectivity seems obvious, but when we start talking about the linguistic expressions themselves, a confusion of the name with the objects it denotes can occur.

1. SUBJECT AND LANGUAGE OF LOGIC

Logics is a science that explores the structure of thinking, reveals the patterns underlying it.

Thinking is inextricably linked with language. The content of thinking only through language becomes a reality. The structure and method of using language gives us knowledge about the forms and laws of thinking.

In logical analysis, language is considered as a sign system.

Sign is a material object used to refer to any other object. Logic explores signs-symbols , constituting the majority of natural language words. Their connection with the designated objects is established either by agreement or spontaneously during the formation of the language.

Signs-symbols have substantive and semantic meaning. subject matter possesses the object that is represented (or denoted) by the sign; semantic meaning - the characteristic of the object expressed by the sign. An example of a semantic meaning is a sign that carries information about this object. The subject matter is often called simply value, and the meaning is meaning. For example, the value of the sign "a number that is prime and even" is the number 2; that is what this phrase refers to. The meaning of this sign is the information that it contains about the number 2, namely, the complex sign of the number "to be prime and even."

The science of signs is called semiotics. This science is divided into three sections - syntax, semantics And pragmatics which is connected with the existence of three aspects of language.

Syntactic aspect makes up a variety of relations between signs and includes the rules for the formation of some signs from others, the rules for changing signs (declension, conjugation), etc.

Semantic aspect constitutes a set of relations of signs to the objects they represent, i.e., the meaning and meaning of signs.

Pragmatic aspect includes the relationship of a person to signs, as well as the relationship between people in the process of sign communication.

In the logical analysis of the language, pragmatic characteristics are abstracted.

There are natural and artificial languages. Natural (national) languages emerged as a means of communication between people; their formation and development is a long historical process and occurs mainly spontaneously. Constructed languages consciously created by a person to solve certain problems. One such language is the formalized language of logic. It is characterized by accuracy, brevity, strict rules for the formation of complex expressions from elementary ones and the transformation of some expressions into others.

Logic explores the form of thoughts, abstracting from the specific content. logical form - it is a way of connecting meaningful parts of thought. Meaningful parts of thought - names and sayings nia , which are fixed using the variables A, B, C, D, etc.

Meaningful specification of variables is called the values ​​of these variables. To link variables, we use lo logical constants , which retain their value in any reasoning. The words “and”, “or”, “if, then”, “it is not true that”, “all”, “some”, etc. act as logical constants. form (see section "Statement"). Names and statements are the main semantic (logical) categories.

So, to reveal the logical form (structure) of a thought means to formalize it. Thus, the statements: “All graduates have a higher education”, “All rectangles are quadrangles”, “All metals are conductors of electricity” - have the same construction scheme: “All S is P”. Consider more complex examples: “If all students of our course study logic, and I am a student of our course, then I study logic”, “If all metals are simple substances, and lithium is a metal, then it is a simple substance.” These arguments are built according to the scheme: "If A and B, then C." The selected schemes are logical forms.

The correct connection of thoughts is determined by the laws of logic, which warn against errors in reasoning, regardless of the specific content.

logical law ~ it is a logical form that generates a true statement with any substitution of their values ​​instead of variables.

Reasoning, the form of which is a logical law, is called correct. Correctness is distinguished from the truth of thinking. Thought is true, if it is true. You can reason correctly, but start from false data, which will lead to a false conclusion. So, from the false statement "All alloys are simple substances" the statement "Some simple substances are alloys" is derived, which is also false.

Correctness with true input leads to true results. This property of thinking was noticed in ancient times. Logic as a separate science developed in the 4th century. BC. Its founder is the ancient Greek philosopher Aristotle, who formulated the basic laws of logic and developed the doctrine of syllogistic reasoning .

The teachings of Aristotle were further developed in the Middle Ages and in modern times. An important addition to this teaching was theory of induction , developed by the English philosopher F. Bacon in the 16th-17th centuries. and systematized by the English logician in the 19th century.

The deductive logic of Aristotle and the inductive logic of Bacon-Mill were the main directions in the development of logic until the middle of the 19th century. The logic founded by Aristotle is called formal or traditional logic .

In the second half of the XIX century. formed symbolic or mathematical logic . It arose as a result of the application of mathematical methods to the solution of logical problems. The idea of ​​using computational methods in any science belongs to the German thinker Leibniz (XVII - XVIII centuries); in fact, it was embodied in the works of J. Boole, W. Jevons, G. Frege, B. Russell and other scientists who created the main sections of mathematical logic, which became the most important branch of formal logic. Mathematical logic has found wide application in technology, where, thanks to information-logical machines, complex calculations are carried out, automatic devices are controlled, etc.

Today, the development of formal logic goes in the direction of development non-classical logics (logic of assessments, questions, temporal, inductive, etc.), creating their general theory and expanding the scope of formal logic.

Modern logic includes two relatively independent sciences: formal logic and dialectical logic. Formal logics studies the forms of thinking, reveals the structure common to thoughts that are different in content. dialectic logics explores the basic laws of the process of cognition, its emergence, change and development.

Formal and dialectical logic develop in close interaction, which is manifested in the practice of scientific and theoretical thinking, which uses both the formal logical apparatus and the tools developed by dialectical logic in the process of cognition.

So, the study of logic allows you to master the forms, laws and methods of correct thinking, which guarantee the competent transformation of statements, a clear formulation of definitions, confidence in argumentation, etc.

2.1. The main characteristics of the name. Types of names.

Name is the basic semiotic unit. Therefore, any mental operation is the establishment of a special type of relationship between names. In natural language, the name can be expressed by the word ("student") or the phrase ("student - excellent student of the 1st year of the PSF"). Names perform the function of replacing objects in the process of thought. In logic, an object is understood to be everything that thought can be directed to, i.e. an object is not only a real-life object (“book”), but also an abstract quality, relation or type of connection (“beauty”, “equality”, "symmetry"). In reality, each object has a large set of different properties and features, but not all of them are equivalent; some fix what is stable in an object, something that reflects its essence, distinguishes it from other objects similar to it.

A name is a word or phrase that expresses an idea about an object in terms of its essential and distinctive features.

The totality of homogeneous objects, i.e., objects that are similar in their essential features, is called a class.

In logic, any name consists of two structural components - scope and content.

name scope - a set of objects that have a specific feature for this class.

There is an inverse relationship between volume and content: the greater the content of a name, the smaller its volume, and vice versa.

Let's illustrate this with a specific example. Let's take three comparable names - A ("settlement"), B ("city"), C ("capital"). Obviously, in this example, the volume of the name A will be maximum, because it includes the volumes of the names B and C, without completely exhausting it (there are many settlements that are not cities - villages, farms, agglomerations, etc.). With regard to the content of the name, the regularity is different: in this case, the content of the name C will be the maximum, because it includes all the signs related to A and B, as well as specific only to C (concentration of the central legislative, executive and judicial authorities, the presence of the main communication nodes , main educational and cultural centers, etc.).

The names with the maximum content are the names corresponding to single, unique objects (“the capital of Belarus”), and the names with the maximum volume are the extremely general, fundamental names that reflect the most significant, regular connections and relationships (these include the categories: “matter”, “ consciousness", "space", "time", "movement" and many others).

The logical operation of moving from a name with a larger volume to a name with a smaller volume is called name scope limitation , and the reverse operation, i.e. the transition from a name with a smaller volume to a name with a larger volume, - a generalization of the scope of the name.

Despite the fact that there are an almost infinite number of names, there are few logical relationships between them. Names are divided into two large groups : comparable and incomparable. Comparable names have at least one common feature, incomparable names do not have such features. Therefore, only comparable names can be found in different types of logical relations, which, in turn, are divided into two subgroups: compatible and incompatible.

Compatibility relationship types

1. Identity (equivolume).

Rice. 2.1: A -; V - founder of Moscow State University

The graphic notation symbolizes that the volume of the name A (denoted by @) is identical to the volume of the name B. The content characteristics of the names A and B can be either identical or different.

2. Intersection.

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Rice. 2.3: A - lawyer; B is a lawyer

The scope of the name B is fully included in the scope of the name A, but does not exhaust it.

Types of incompatibility relationships

I. Subordination.

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Fig.2.5: A - courage; B - cowardice

A and B occupy extreme places in the series of relations, without exhausting the scope of the name C. The name B not only denies the content of the name A, but also replaces its signs with the opposite (“old - young”, “white - black”).

3. Controversy

A - white; non-A - non-white

The names A and A (not-A) are in a contradiction relationship if the name A denies the features of A without replacing them with any other features. The name A is positive, A is its corresponding negative name.

Name classification

By volume names are divided into single, common and empty.

Single - these are names, the volume of which is equal to one, i.e., in reality, a given name corresponds to the only possible object. For example: "Belarus", "the deepest lake in the world."

Are common - these are names whose volume is equal to or greater than two, i.e., objects corresponding to these names are not unique. Such names are the vast majority. For example: "student", "atom", "constellation", "geometric figure".

empty are names whose volume is zero. They do not have a corresponding subject interpretation in reality, but they have a certain content. This type includes names from the sphere of human fantasy (“centaur”, “goblin”), as well as names formed with violations of logic (“round square”). Empty names can perform the function of a model representation (“absolutely black body”, “ideal gas”). They are formed by emphasizing one distinguishing feature of the subject while completely abstracting from others, even inextricably linked with it, qualities and relationships. The selection of an essential feature (in this context) allows this type of names to perform a reference function in scientific knowledge.

1) concrete and abstract;

2) positive and negative;

3) relative and irrelative;

4) collective and non-collective.

specific a name is called that denotes a material or ideal class of objects (“electron”, “number”, “book”, “earthquake”).

abstract a name is called that denotes individual features, qualities or properties of the object of thought, taken separately from the object itself (“symmetry”, “inequality”, “rigidity”, “whiteness”).

positive a name is called that fixes the presence of a certain quality or attitude in the subject of thought (“neatness”, “literacy”).

negative a name is called that fixes the absence of a certain quality or relationship in the subject of thought. In Russian, as a rule, these names are formed using the negative particle "not" (sloppy, asymmetrical). If the name without the particle “not” is not used, it is positive (carelessness, bad weather).

Relative a name is called that reflects such an object of thought, which always implies the presence of another name that is paired with it (day - night, plus - minus, numerator - denominator).

Whatever a name is called in the event that the object designated by it does not imply another object correlated with it (drawing, house).

Collective a name is called in which a group of homogeneous objects is conceived as a single whole (flock, collective, ensemble).

non-collective a name is called, which suggests the possibility of its application with respect to each element of the class (flower, building).

2.2. Boolean operations with names

Name definition is a logical operation that reveals the content of a name by pointing to its essential features.

In the definition structure, a defined name (what is defined) and a defining name (what it is defined with) are distinguished.

Definitions are divided into explicit and implicit.

IN explicit definitions there is a clear indication of the essential features inherent in this name.

Implicit definitions define all kinds of relationships in which the name being defined can be related to other names.

The main varieties of explicit definition are definition through genus and species difference and genetic definition.

Definition through genus and specific difference- this is a definition, the essence of which is to indicate the nearest generic name and a specific feature that distinguishes the desired name from the class of similar (generic) names. For example: "Barometer is a meteorological instrument (generic name) designed to measure atmospheric pressure (species feature)."

Genetic definition- definition through an indication of the method of formation of the subject (its genesis). For example: "A ball is a geometric body formed by the rotation of a circle around one of the diameters."

Implicit definitions include description, characterization, comparison, indication of the relation of an object to its opposite, contextual definition, etc.

In order for a definition to be logically correct, it must obey certain rules:

1 . The definition must be proportionate, i.e., the defined and defining names must be of equal length to each other. For example: "A cylinder is a geometric body formed by the rotation of a rectangle around one side."

If this rule is not observed, two main logical errors are possible:

1) too broad definition:"A nation is a stable historical community of people"; an error occurs if the defining name is larger than the defined one; in this case not

a specifying feature is indicated that distinguishes the nation from other stable historical communities, i.e. the requirement of proportionality is violated;

2) too narrow definition:"Theft - secret theft of public property"; an error occurs if the defining name is less than the defined one; in this case, the scopes of the defining and the defined names are in relation to subordination, therefore, the requirement of proportionality is also violated.

2. The definition must not contain a circle. A circle in a definition arises if the name being defined is defined through the defining name, and the latter, in turn, through the name being defined. For example: "Quantity is a characteristic of an object from its quantitative side."

3. The definition should be as simple as possible. form of all possible: "Law is an essential connection between objects."

4. The definition must be public, i.e. when constructing it, words of common vocabulary should be used. If this requirement is not observed, an error occurs in determining the unknown through the unknown. For example: "Holism is an idealistic philosophy of integrity, close in its ideas to the theory of emergent evolution."

5. The definition, if possible, should not be denied telny. This requirement is related With the fact that with such a definition there is no identification of essential features of the subject.
A negative definition is limited to only indicating the absence of features that do not belong to the name, therefore the cognitive, and even more so, the prognostic function of such a definition is very insignificant.

Division- this is a logical operation, by means of which the volume of a divisible name (genus) is divided into a number of subsets (types) taking into account the chosen basis (criterion) of division.

Basic division rules:

1. The division must be proportionate, i.e. the volume of the dividend name must be equal to the sum of the volumes of the division members. For example: "Electric current is divided into direct and alternating."

If this rule is not followed, two errors are possible:

1) incomplete division; an error occurs if the scopes of the division members do not exhaust the scope of the dividend. For example: “Arithmetic operations are divided into addition, subtraction, multiplication, division, exponentiation” (no “root extraction” is indicated);

2) division with an excess term, for example: "Forests are divided into deciduous, coniferous, mixed"; here “mixed” is an extra member, since the scope of the name “forest” is exhausted by the scope of the names “leaf
vein forest" and "coniferous forest".

2. The division must be based on one basis, i.e., two or more signs cannot be used as the basis for division: "The sciences are divided into humanitarian, natural and technical."

Members of the division must be mutually exclusive, i.e., their volumes should not be in relation to the intersection: "Students are divided into full-time students, evening students and correspondence students."

3. The division must be continuous; if this rule is violated, a “jump in division” error occurs. For example: "Crimes are divided into intentional, negligent and theft." IN
in this case, theft is a kind of intentional crime and, therefore, cannot act as an independent member of the division.

Division as a logical operation must be distinguished from division into parts: if the division members always have the attribute of a divisible name (genus), then the parts do not have the attribute of the whole. For example, let's compare: “The year is divided into 12 months”, - division into parts, because the month does not have the sign of the year; “Angles are divided into sharp, straight and obtuse”, - the division of volume, since the members of the division have a generic feature (they are varieties of the angle).

Exercises

1. Give a logical description of the following names: student; negligence; North Pole; Noah's ark; night; flock; quantity; square; a person who lived 300 years; abstraction; denominator.

2. Draw graphically the relationship between the following names:

1). Stone house, three-story house, one-story house, unfinished house.

2). Plant, ornamental plant, medicinal plant, wormwood.

3) Fire, lightning, natural disaster, natural phenomenon.

4) "Satellite of the planet, natural satellite, satellite of the Earth, Jupiter, satellite of Jupiter, Moon.

5). Fire, cause of fire, atomic bomb explosion, arson.

6). Heroism, cowardice.

7). Congruent, incongruent.

8). Geometric figure, rhombus, trapezium, square.

9). Hour, second, minute.

10). University, institute, faculty.

eleven). Scientist, Doctor of Sciences, Doctor of Historical Sciences, Nobel Prize Laureate.

3. Choose names that match the given schemes.

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4. Perform a restriction operation on the following names Keywords: periodical, textbook, constellation, organism, crime, metal, capital, elementary particle.

5. Perform a generalization operation: autumn, electron, athlete, Buddhism, table, rose.

6. Is the restriction of the following names correct: 1). Building - room; building - gazebo.

2). Settlement - the capital - the center of the capital - the center of the modern capital.

3). Geometric figure - triangle - isosceles triangle.

4). Hour - minute - second.

5). Tree - deciduous tree - birch - crown.

7. To these names, select subordinates and subordinatesnames: school, textbook, transformer, form of government, lake, comedy, elementary particle.

8. Find out in which of the given examples there is a division of names by volume, and in which - the division of an object into parts: metal - tin; geometric figure - rhombus; fraction - numerator; a number is a rational number, an apartment is a room; book - chapter; building - facade; arithmetic operation - extracting the root; the triangle is the hypotenuse.

9. Check the correctness of the division; in the wrong divisiondetermine which rules are violated:

1). Sciences are divided into natural, technical and humanitarian.

2). Languages ​​are divided into natural, artificial and folk.

3). The state can be feudal, capitalist and socialist.

4). Sentences are accusatory, acquittal and unjust.

5). Republics are divided into presidential, parliamentary and unitary.

6). Rays are divided into ultraviolet, visible and infrared.

7). The art forms are fiction, music, sculpture, architecture, and portraiture.

8). Houses are divided into single-storey, multi-storey and equipped with an elevator.

9). Animals are divided into vertebrates, invertebrates and herbivores.

10. Perform the operation of dividing the following names:

science, government, media, logic, engineer, arithmetic operation, human.

11. Specify the type of definition, the defined and defining name, in the latter - the generic name and specific difference:

1). Logic is a philosophical science about the forms in which human thinking proceeds, and about the laws to which it obeys.

2). The science of the general laws governing the processes of control and transmission of information in complex machines, living organisms and society is called cybernetics.

3). A word used in a figurative sense is called a metaphor.

4). A regular rectangle is a polygon that has all

sides are congruent and all angles are equal.

5). Acids are complex substances formed from acid residues and hydrogen atoms that can be replaced by metal atoms or exchanged for them.

6). Corrosion of metals is a redox process that occurs as a result of the oxidation of metal atoms and their transition into ions.

12. Find out the correctness of the following definitions:

1). A cylinder is a geometric body formed by the rotation of a rectangle around one of its sides.

2). Value is that which can be reduced or increased.

3). A fraction whose numerator is less than the denominator is called

correct.

4). Robbery - theft of state property, committed openly.

5). A student is a student.

6). A fraudster is a person who engages in fraud.

7). Stubbornness is the vice of a weak mind.

8). Barometer is a meteorological instrument.

Relationship" href="/text/category/vzaimootnoshenie/" rel="bookmark">terms (S and P) are in a relationship.

Attributive statements often use existential and general quantifiers.

Attributive statements are divided according to quality and quantity.

By quality, they are divided into affirmative and negative. IN affirmative indicates the belonging (presence) of the sign, conceivable in the predicate, to the subject of the statement: "S is P". For example: "Plato is an idealist philosopher." IN negative indicates that the predicate does not belong to its subject: "S is not P".

According to the number of statements are divided into single, private and general. This refers to the totality (number, quantity) of individual items that make up the name of the subject class.

IN single In utterances, the subject consists of one object.

Private statements are of the form: "Some S are (are not) P".

IN general In utterances, the subject embraces all objects. Such statements have the form: "All S is (is not) P".

Statements are classified by quality and quantity. There are 4 classes of statements:

1) general affirmative (A) - general in quantity and affirmative in quality (“All S is P”);

2) private affirmative (J) - private in quantity and affirmative in quality ("Some S are R");

3) common negative (E) - general in quantity and negative in quality (“Not a single S is P”);

4) private negative (ABOUT)- private in quantity and negative in quality ("Some S are not P").

In each class of statements, the ratio of the volumes of S and P (terms) is different. In logic, the problem of the ratio of volumes S and P is called term distribution problem. A term is distributed if it is completely included in the scope of another term or completely excluded from it.

In class A |AllSis R| the subject is fully distributed in the predicate, and the predicate is not distributed.

https://pandia.ru/text/78/045/images/image017_55.gif" width="146 height=2" height="2"> In class O [SomeSis not R| the subject is not distributed, but the predicate is distributed.

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statement

3.2. Relationships between simple propositions in terms of truth

Questions on the course "LOGIC" (test)

1. The subject of logic as a science. The logical form (structure) of thought.

2. Formalization as a means of identifying the logical form.

3. General characteristics and language of propositional logic.

4. Mistakes in thinking. Their classification.

5. The concept of "statement". Types of statements. The language of propositional logic.

6. Types of complex statements. The meaning of logical unions.

7. Tabular method for determining logical laws.

8. Elementary laws of logic: the law of identity, the law of contradiction, the law of the excluded middle.

9. Logical characteristic of the name: volume and content.

11. The scope of the name. Types of names by volume.

12. Relations between names. Euler circles as a means of analyzing the relationships between the volumes of names.

13. Restriction and generalization of the name.

14. Division as a logical operation. division structure.

15. Types of division (logical, analytical).

16. Rules of logical division. Division errors.

17. Definition as a logical operation. Definition structure.

18. Rules for determining and possible errors in case of their violation.

19. Structure and types of attributive statements.

20. Distribution of terms in attributive statements.

21. Relations between attributive statements. logical square.

22. Direct syllogistic conclusions: transformation (overversion), conversion (conversion), opposition to the predicate (partial counterposition).

23. The basic rule of direct syllogistic conclusions.

24. The structure of a simple categorical syllogism.

25. General rules of simple categorical syllogism.

27. Enthymeme. The procedure for restoring the enthymeme to a complete syllogism.

28. Argumentation: its structure, types and rules.

29. Errors in argumentation.

30. Logical requirements for the creation of a scientific text.

Compiled by: Associate Professor of the Department

Philosophy of Culture, Ph.D., Associate Professor Malaya N.V.

LOGICS. The subject of logic as a science

Logic diagram- this is its side that does not depend on the specific content, but serves to connect, organize and transform its elements.

Types of logic circuits. The reasoning is correct, risky and absurd.

logical law- a scheme that, for any content, takes only true values, and the corresponding reasoning is correct.

Executable scheme - a logical scheme that, with some substitutions, is transformed into true expressions, and with others, into false expressions, and the reasoning corresponding to it is risky.

Controversial scheme - a logical scheme, which, with any substitution, is converted into false expressions, and the reasoning corresponding to it - absurd.

The ratio of correctness and truth

A thought is true if it corresponds to reality. Correctness characterizes thought from the point of view of the internal connection between its elements. Being correct with true inputs always leads to true results.

Cognitive errors in reasoning

Cognitive errors associated with misconceptions about the actual state of affairs are called meaningful .

Errors associated with violations of correct thinking are called formal , or logical . They are divided into paralogisms and sophisms.

Paralogism is an unintentional logical fallacy. Sophism is a deliberate violation of the requirements of logic, a method of intellectual fraud associated with an attempt to pass off a lie as the truth, or vice versa.

SECTION 1. STATEMENTS

General characteristics of propositional logic

Saying - a linguistic expression about which only one of two things can be said: it is true or false.

Statements (as well as the construction schemes corresponding to them) are divided into simple and complex ones. A complex statement can be broken down into simple ones. A simple statement is not divided into simpler ones. When constructing schemes, lowercase letters of the Latin alphabet are usually used as variables for simple statements: p,q,r,s,…; for any (sometimes it doesn’t matter to us whether this statement is simple or complex) - capital letters of this alphabet: A, B, C, D, ...

The utterance schema accepts boolean- "true" or "false".

The logical meaning of a complex statement scheme in modern logic is made dependent (is a function) on the logical values ​​of simple schemes.

Definitions of the most important schemas of propositional logic

Compound statements and their corresponding schemes are formed with the help of special expressions, which are called functors(negation, conjunction, disjunction (weak and strong), implication, equivalence). It is customary to call a complex scheme the name of the functor with which it is formed, i.e. if, for example, a schema is formed with the help of a conjunction, then the schema itself is called a conjunction.

Denial A is a schema denoted by ØA (read: "not-A", "it is not true that A"), which evaluates to "true" if and only if A evaluates to "false". This definition can be expressed using the following table (truth table), where "u" means "true" and "l" means "false":

Table 1

Conjunction A and B are circuits, denoted by AÙB, that evaluate to true if and only if true evaluates to A, so and B(see 3rd column of Table 2). Expression AÙ B reads: " A And B».

table 2

Disjunction is weak A And B is a schema, denoted by AÚB, that evaluates to "true" if and only if at least one of A and B evaluates to "true"(cm . 4th column of the table. 2). Expression AÚ B reads: " A or B».

The disjunction is strong A And B - scheme, denoted by the expression A Ú B, which is true if and only if only one of A and B is true(cm . column 5th table. 2). Expression AÚ B reads: "either A, or B».

implication A and B are circuits, denoted by A®B, that evaluate to false if and only if A evaluates to true and B evaluates to false(see 6th column of Table 2). Expression A® B reads: "If A, That B».

Equivalence A and B is a circuit, denoted by the expression A"B, which evaluates to true if and only if the logical values ​​of A and B are the same(see the 7th column of Table 2). Expression A« B reads: " A if and only if B».

The propositional logic alphabet includes the symbols:

1.p, q, r, s, … are symbols that denote variables for simple statements; A, B, C, D... - symbols that designate variables for any statements;

2.Ù, Ú, Ú, ®, «, Ø - symbols for logical unions;

3.(,) - brackets as indicators of logical actions.

There are no other symbols in propositional logic.

A meaningful expression in the language of propositional logic is defined as follows:

1. Every variable is a meaningful expression;

2.If A- meaningful expression ØA, A Ù B, A Ú B, A Ú B, A®B, A«B - also meaningful expressions;

3. There are no other meaningful expressions in propositional logic.

Laws of Propositional Logic

Truth tables can be used to identify forms that are logical laws. A schema that generates only true compound statements is LOGICAL LAW.

The simplest laws of propositional logic are those that can be expressed with one variable- the law of the excluded middle, the law of contradiction, the law of identity, the law of removal of double negation, the introduction of double negation, etc.

Law of the excluded middle- scheme AÚØ A – two statements negating each other are not together false, one of the possibilities is fulfilled: if one of these statements is false, then its negation is true, and something else is excluded.

Law of contradiction- scheme Ø( AÙ Ø A) - two statements that negate each other are not both true, one of them is false.

Law of Identity- scheme A« A – every statement is equivalent (identical) to itself, hence, in correct reasoning it agrees with itself.

Double Negative Removal Law– scheme ØØ A® A- the negation of some proposition twice forms its assertion.

Law of introduction of double negation- scheme A® ØØ A- the assertion of some proposition forms its double negation. The validity of the considered laws with one variable is easily checked in a tabular way (see Table 5).

Table 5

SECTION 2. NAMES

The main characteristics of the name

Name- a language expression denoting a single object or a set, a collection of objects.

A set (collection, class) of objects denoted by a name is called volume name. Individual items included in the scope of the name are called elements name size. Name scope subclasses are called volume parts .

The signs that make up the content of the name can be generic, specific and individual. If we single out a narrower class of objects within a fairly wide class of objects, then the features that distinguish a wider class will be considered generic , and the features that distinguish a narrower class are specific . Individual features are those that uniquely distinguish a given single object.

main content a name can be called that minimal part of its content, from which, in the theory to which the name belongs, all the rest of the content of the name is logically deduced (which in this case is called derivative ). The totality of the main and derivative contents of the name is its complete content.

NAME TYPES

If the volume of the name includes only one object, then such a name is called single.

common name is a name that contains more than one element. A class that is the scope of a common name is called value this name.

A special kind of common names are universal names, or universes . They fix all classes of objects, all elements studied in one or another field of knowledge. Names belonging to the same universe are called related .

Null (empty) names in the most general form are defined as names, the scope of which does not contain a single element. A class that does not contain any element is called null or empty.

There are also names descriptive And own . Descriptive names designate objects by indicating their respective attributes. Proper names denote objects by direct correlation with them, due to the fact that certain traditions and naming norms have developed in the culture of the human community.

It is important to distinguish between collective and non-collective names. non-collective such a name is called, each element of the volume of which is something single, integral. Collective such a name is called, each element of which is a collection, a collection, an association of some objects.

Allocate positive and negative names. It is based on the fact that objects can be characterized both by the presence and the absence of certain properties in objects. positive the name is considered, in the content of which the properties inherent in the objects are indicated. negative a name is considered, in the content of which properties are indicated that are absent from objects.

Finally, we indicate the division of names into clear and fuzzy. If the name is such that, with respect to any object, it is possible to accurately, unambiguously decide whether this object is included or not included in the scope of the given name, then this name is called clear (precise, definite) in scope (eg rational number, subsistence economy, criminal liability). Otherwise, the name is considered fuzzy (vague, vague, fuzzy, inaccurate) in scope (e.g. expensive goods, young man, good looks).

RELATIONSHIP COMPATIBILITY

Names are considered compatible if their volumes at least partially coincide, i.e. these volumes have common elements.

Types of compatible names:

1) Equivalent (equivalent) names are considered, the volumes of which completely coincide (Fig. 1). With the relation of equivocality of names A And B each item with a name A, can be denoted by the name B, and vice versa.

2) The names are in relation subordination , if the volume of one is completely included in the volume of the other, but does not coincide with it. In this case, the inclusive name is called subordinate, or generic, and the included name is called subordinate, or species. If the name A obeys the name B(Fig. 2), then all signs B inherent in the content of the name A, and each item denoted by the name A, can be named B(but not vice versa).

3) Intersecting (crossing) are such names, the volumes of which only partially enter into each other. At the same time, some objects denoted by the name A, can be named b, and vice versa. If the names A And B are in relation to the intersection (Fig. 3), then the objects included simultaneously in the volumes of names A And B, that is, located at the intersection of these volumes, have the same features.

Relationships between related names.

Relationship of incompatibility

In the case of incompatibility in the content of one of the names, signs are indicated that exclude the signs of the content of the other name.

Types of incompatible names:

1) Conflicting two incompatible names are called, the specific content of one of which (i.e., the totality of its specific features) is a negation of the specific content of the other. Such names completely exhaust the volume of the third, subordinating their name (Fig. 4).

2) Outside such incompatible names are called, the volumes of which in total make up a part of the volume of some subordinate (generic) name. Because the A And B, being external, are simultaneously subordinated WITH, since they are also called subordinate relatively WITH(Fig. 5).

3) Opposite names are called, the contents of which express some extreme characteristics in a certain ordered series of gradually changing properties (Fig. 6).

Generalization and restriction as operations on names

Volume Generalization A- a logical operation, as a result of which a name with a volume is formed B containing the volume A. In other words, generalize the name A means to form such a different name B(genus), which would subordinate the name A(view). The limit of generalization in each specific case is a certain universal name.

Limitation- a logical operation inverse to generalization. It consists in finding a name with a volume B, which is contained in the volume A. Limit Volume A means to find such another name B(kind), which would be in a relationship of subordination to A(genus). The limit of the restriction is the names, the volumes of which are equal to one subject (single names).

A special kind of constraint is type allocation, or typing . A type is a name to which homogeneous objects correspond to one degree or another. If some objects make up the scope of the name A and among them there are those that unconditionally (that is, with a degree equal to 1) belong to the volume B, and others have this property to some (less than 1) degree, then a name with a scope B represents a type.

Attachment to volume A new objects that are identical with the old ones in some way is called a logical operation extensions volume A.

The operation is the reverse of the expansion, i.e. removal from the volume A objects that are identical with the remaining ones in some way is called localization name scope A.

Logical operations with the scope of names should not be confused with mental transitions from a part to a whole and vice versa, from a whole to a part. The specificity of the latter is most clearly revealed when they are compared with the operations of generalization and restriction.

DIVISION OPERATION

division logical is a logical operation by which the scope of the name (genus) is distributed among classes (species).

Analytic division - this is an operation associated with the mental isolation of its parts as a whole. These operations should not be mixed.

The division can be classical or non-classical. At classical division, both genus and species - names with a clear volume, with non-classical they are fuzzy, vague names, or types.

Classical logical division consists in finding for the name A such names A 1 , A 2 , ..., A n ( n is a finite number) that:

a) each of the volumes A 1 , A 2 , ... , A n is in relation to subordination to volume A);

b) sum of volumes A 1 , A 2 , ... , A n is equal to the volume A;

c) each pair of volumes A 1 , A 2 , ... , A n is related by the relation of incompatibility. At the same time, the name A called shared name , A A 1 , A 2 , ... , A n- division members .

It is possible that the basis of the division is a sign that is inherent only in a part of objects of a certain class. In this case, objects are divided into those that have this feature, and those that do not. This division is called dichotomous(Greek dicho - into two parts, tome - section). In contrast, division according to a feature that all objects of a genus possess and which varies in species is called polytomic Greek polis - a lot).

The difference between division and dismemberment is based on the different nature of the relationship "whole - part" and "genus - species".

DIVISION RULES

1. adequacy rule.The division must be proportionate. This means that in case of division each of the volumes A 1 , A 2 , ... , A n must be a type of volume A, and the sum A 1 , A 2 , ... , A n must exhaust the entire volume A;in case of dismemberment mental connection of parts should be equal to the whole. Deviation from this rule leads to errors, the most famous of which are: " division with extra members"when some of the volumes (parts) A 1, A2, ... , A n is not a species A(not part of a whole) A); "incomplete division", when not all types (parts) of the divisible genus (whole) are named, and the sum of the volumes of the division members is less than the volume of the divisible name.

2. Rule of demarcation. Members of division (dismemberment) must exclude each other, i.e. their volumes should not have common elements in the case of classical division, and the parts should not overlap each other in the case of dismemberment.

3. Rule of uniqueness of base. Division must be based on the same base.. When this rule is followed, the objects included in the scope of the divisible name are endowed with one single attribute - the one that acts as the basis for division. Deviation from this rule leads to an error, which is called mixing bases.

Instead of the term "division", the term "classification" is sometimes used as a synonym. Narrow classification (it is in this sense that we will use this term in the future) is a multi-stage, branched division, such that each of the members obtained during this operation becomes the subject of further division.

According to the classical and non-classical division, one should distinguish between classical and non-classical classification. The latter is called typology .

So far, a simple and unambiguous term has not been assigned to a multi-stage and branched dismemberment. This operation can be called hierarchization .

Classification and hierarchization obey all the rules of division. In addition, they have their own special rules.

1. sequence rule . In the case of classification, one should proceed from the genus to the nearest species, and in the case of hierarchization, from the whole to its parts of the same level, without skipping them. If this rule is violated, the permissible error is “ leap in classification (hierarchization) ».

2. Reason materiality rule . Classification (hierarchization) should be carried out according to essential features. The criterion for the significance of a feature is the ability of the object possessing it to serve as a means of solving the task.

A special case of division is periodization .Its feature is, firstly, an indication of the development of the displayed object in time. Secondly, the members of the division (periods) differ in their measure as a unity of the qualitative and quantitative characteristics of the object.

DEFINITION, OR DEFINITION (GENERAL CHARACTERISTICS)

In logic, one distinguishes first of all two different senses of the term "definition". First, under definition is an operation that allows you to select a certain object among other objects, uniquely distinguish it from them. This is achieved by pointing to a sign that is inherent in this, and only this, subject. Such a feature is called distinctive (specific). What do we do, for example, if we want to select squares from a class of rectangles? We point to a feature inherent in squares and not inherent in other rectangles, – to the equality of their sides.

Secondly, the definition is called the logical an operation that makes it possible to reveal, clarify or form the meaning of some linguistic expressions with the help of other linguistic expressions. So, if a person does not know what the word "vershok" means, they explain to him that an vershok – this is an ancient measure of length, equal to 4.4 cm. Since a person knows in advance what "an ancient measure of length equal to 4.4 cm" is, the meaning of the word "vershok" becomes clear and understandable for him.

A definition that gives a distinctive characteristic of an object is called real. A definition that reveals, clarifies or forms the meaning of some linguistic expressions with the help of others is called nominal.

The method of establishing the meaning of a linguistic expression by its direct correlation with the designated object or its image is called ostensive definition.

IN definition structure there are three parts:

1) the name being defined or the expression containing it (denoted by the sign Dfd – short for lat. definitionum);

2) an expression that reveals, clarifies or forms the meaning of the name being defined (indicated by the sign Dfn - abbreviation from Latin definiens);

3) a definitive connective that relates Dfd and Dfn by their meaning (denoted by º).

Formally, the structure of the definition is represented by the expression: Dfd º Dfn.

RULES OF DETERMINATION

1. Proportionality rule. Dfd and Dfn must be of equal volume.

Deviation from the proportionality rule leads to errors:

1) "too broad definition" - the volume Dfn is greater than the volume Dfd;

2) "too narrow definition" - the volume Dfn is less than the volume Dfd;

3) "simultaneously too broad and too narrow definition" - the volumes Dfd and Dfn are in relation to the intersection.

4) definition via empty name- Dfd and Dfn are incompatible.

2. No vicious circle rule. It is forbidden to define Dfd through Dfn, which, in turn, is defined through Dfd. The resulting violation is called " vicious circle in definition". A special case of the "vicious circle" is tautology – repetition of Dfd and Dfn (albeit in a different verbal form) without establishing the meaning of Dfd.

3. Unambiguity rule. Each Dfn must correspond exactly to one single Dfd, and vice versa. This rule eliminates the phenomena of synonymy and homonymy, prohibits the use of metaphors, artistic images.

4. Simplicity rule. Dfn should be expressed by a descriptive name that characterizes the defined objects only by their main features. Otherwise, the definition will be redundant. In the classical definitions, this rule is fulfilled under the condition that: such that no other name subordinate to the genus and subordinate to Dfd has been previously defined; b) there are no expressions in Dfn that are related to following (subordination).

5. The rule of competence. Dfn can only include expressions whose values ​​are already accepted or previously defined. Deviation from this rule is called "defining the unknown through the unknown".

abbreviated syllogisms

For intellectual speech activity, expressions with missing but implied parts are used. These expressions include enthymemes (from the Greek. en time - in the mind), - abbreviated syllogisms in which one of the premises or conclusion is omitted.

The methodology for restoring and evaluating an enthymeme for its consistency consists of the following steps:

1. The enthymeme is written in the standard form: the available premises are placed above the line, the conclusion - below it.

2. In accordance with the accepted classification, the type of this conclusion is established (it can be a categorical syllogism, a conditional syllogism, etc.).

3. In accordance with the definitions of premises and conclusion, it is established which of the parts of the conclusion is implied.

4. Using the definitions and rules specific to this class of inferences, the missing part of the inference is restored.

5. An analysis is made of the links between the premises and the conclusion for compliance with logical rules. Violation of at least one of the rules indicates the presence of a formal error in the enthymeme.

6. The restored parcel is analyzed for compliance with the actual state of affairs. Its falsity means the presence of a meaningful error in the enthymeme.

SECTION 4. ARGUMENTATION

BELIEVABLE ARGUMENTATION

The most important property DEDUCTIVE ARGUMENTATION ( correct conclusions) - the presence succession relations between premises and conclusions, as a result of which the truth of the premises guarantees the truth of the conclusions. Plausible argumentation (plausible conclusions) is characterized by the absence of this relation, and the truth of the premises does not guarantee, but does not exclude the truth of the conclusions, makes it possible, plausible. The most important varieties of plausible argumentation are analogy, induction and abduction.

ANALOGY

Analogy is a type of argumentation characterized by the transfer of a feature inherent in one object to another, similar to the first, object.

The analogy is built as follows:

S 1 is P 1 , P 2 , P 3 , ... , P n-1 , P n.

S 2 is P 1 , P 2 , P 3 , ... , P n-1

S 2 is P n

Here S 1 and S 2 - the names of the matched items, P 1 , P 2 , P 3 , ... , P n-1 - names of features common to items S 1 and S 2 , P n- the name of the attribute belonging to the subject S 1 and transferred to the subject S 2. The breaking line indicates the plausibility of the conclusion.

An object whose attribute is transferred to another object is calledmodel; an object to which the attribute of another object is transferred is calledprototype. Along with the term "prototype", the terms "original", "sample", etc. are also used.

A textbook example of inference by analogy is the argument about the possibility of life on Mars. Proponents of this hypothesis pay attention to the fact that between the Earth ( S 1) and Mars ( S 2) a lot in common: these are two nearby planets of the solar system ( P 1), here and there there is water ( P 2), atmosphere ( P 3), the surfaces of these planets have approximately the same temperature ( P 4) etc. But there is life on earth P n). Therefore, it is quite plausible that there is life on Mars ( P n).

Argumentation by analogy is widely used in a wide variety of areas of human activity - in science, art, everyday life. In particular, the mental schemes developed in the course of the centuries-old practice of mankind, we transfer to reasoning with the most diverse content. The solution of any problem is due to the fact that methods and means are used that have justified themselves in solving other problems. The origin of many mysterious natural phenomena finds its explanation by analogy with those objects, the essence of which is already known. Fables, fairy tales, parables, proverbs, sayings have prototypes in everyday life. Thanks to the analogy, scope is opened for creative imagination, the exit of human thought into such areas where ties with the real world can be broken is carried out. In a number of cases, analogy underlies intuitive cognitive processes.

INDUCTION

Special consideration deserves a kind of argumentation - induction. In the history of logic and the methodology of science, it was usually opposed to deduction and along with it, in contrast to other logical forms of argumentation, it became widely known.

Induction(from lat. inductio - guidance) - a form of justification in which the conclusion (thesis) is obtained by summarizing the information contained in the premises (arguments).

In the simplest case, namely, when the premise and conclusion are attributive statements, the scheme of inductive inference takes the following form:

S 1 is P

S 2 is P

S n There is P

S 1, S 2,...… S n essence S

All S essence P

Example:

Copper is a good conductor of electricity.

Aluminum is a good conductor of electricity.

Iron is a good conductor of electricity.

Lead is a good conductor of electricity.

Gold is a good conductor of electricity.

Copper, aluminum, iron, lead, gold - metals

All metals are good conductors of electricity.

The plausible nature of this argument is indicated by the fact that from the conclusion, which has the form "All S essence P", and sending the form " S 1 , S 2 , ..., S n essence P” follows each of the other premises: “ S 1 is P», « S 2 is P" etc. But this is a special kind of reduction: here the conclusion generalizes single facts belonging to the same class of objects.

There are cases when a generalizing conclusion (thesis) is adopted on the basis of statements covering all individual cases of a feature belonging to objects of a certain class. This kind of induction is calledcomplete. When, for example, a teacher, after making a roll call of his students and making sure that each of them is present at the lesson, notices with satisfaction that all his students have come to the lesson, then he argues in accordance with the principle of complete induction. Otherwise, induction is calledincomplete.

With complete induction, the conclusion (thesis) necessarily follows from the premises. Therefore, it is legitimate to consider it a deductive conclusion. (It is no coincidence that a complete induction is sometimes called an inductive syllogism.)

Incomplete induction is divided into simple and scientific. Forsimple inductiona purely formal approach is characteristic, when a generalization is made on the basis of the first available, and therefore, random facts. Therefore, there is a real danger of a false conclusion. So, contemplating the animal world, you can find the following similar facts:

In humans, the lower jaw is mobile.

The horse is the same.

The goose is the same.

The pike is the same.

The snake is the same.

These facts, based on the knowledge that a person, a horse, a goose, a pike, a snake are vertebrates, “lead” to the conclusion:

All vertebrates have a movable lower jaw.

However, the probability of the truth of this conclusion turns out to be equal to zero, because there are facts that contradict it. For example, in a crocodile, it is not the lower, but the upper jaw that is mobile.

scientific inductionrelies in its premises not on any, but on the essential features of the class of objects under consideration. The identification of such signs requires a purposeful selection of premises in accordance with the methods and criteria developed in science. Having entered the church and saw a large mass of praying people, it is easy to succumb to suggestion and draw a conclusion about the continuous religiosity of the population of this area. But such generalizations, at first glance, contradict the scientific approach. In order to investigate the degree of religiosity of the population in a certain area, a sociologist will do a lot of preparatory work: he will single out different groups of people, distributing them by occupation, education, age, place of residence, etc., establish quantitative relationships between them, carefully formulate and select questionnaire questions, will subject the received answers to statistical processing, etc. Thus, the premises of scientific induction are not just some random information, but experimental data with additional features that make it possible to reveal the essential in the subject being studied - some natural connection. It is clear that in the case of scientific induction the degree of probability of the conclusion is much higher than in the case of simple induction.

2.4 Types of names

Singular and common names. subject meaningssinglenames are separate objects ("Volga", "Socrates", "natural satellite of the earth", "the highest mountain in the world"), i.e. a single name denotes one thing.Generalthe name can be a sign of any object from a certain class of objects (it is common for objects of a certain class) and therefore the entire given class (“river”, “man”, “celestial body”) is considered the object value of a common name.

Descriptive and non-descriptive names. Both general and singular names are divided into descriptive (complex) and non-descriptive (simple). For example,simple(non-descriptive) are the names "Everest", "mountain", "river", "Volga".complex(descriptive)are the names "the largest river in Europe", "a flat, closed figure bounded by three sides."

Real and imaginary names. In relation to a given universe (reality, set), names are divided intovaliddenoting objects from the given universe, andimaginarydenoting objects that are not included in this universe.

Example. In relation to objective reality, the names “man”, “internal combustion engine” will be real, and the names “mermaid”, “perpetual motion machine” will be imaginary, since in objective reality neither mermaids nor perpetual motion machine exist.

Review questions

1. 
   What is a sign, the meaning of a sign and the objective meaning of a sign?
2. 
   Why is language a sign system?
3. 
   What are the types of signs?
4. 
   What is a semantic category? List the main semantic categories of language expressions.
5. 
   What are the types of names? Describe them.

Topic 3. Formalized logical languages

3.1 The language of predicate logic

Many sciences (mathematics, physics, chemistry, etc.) use special symbols in their languages ​​(+; ?; 2 2 ; H 2 O, etc.). The advantage of any symbolic language is that it is more concise and, above all, more precise than the natural language we speak in everyday life. Logic also has its own symbolic language, which was created specifically for accurate and clear reproduction of the structures of human thinking, and was called the language of predicate logic (from Latin proedicatum - said).
Source characters:
p, q, r, s, p 1 ... - propositional variables (symbols for designating entire declarative sentences);

a, b, c, d, a 1 ... - subject constants (symbols for denoting single names);

x, y, z, x 1 ... - subject variables (symbols for denoting common names);

P, Q, R, S, P 1 ... - predicate symbols (symbols for designating properties and relationships);

 () – logical negation (“not” or “it is not true that”);

 (&) – conjunction (“and”);

 - disjunction ("or");

 - strict disjunction (“either ... or ...”);

 (?) – implication (“if…, then…”);

 () – identity (equivalence) (“if and only when…”);

 – universality quantifier (“everything”, “each”);

 – existential quantifier (“some”, “exist”);

In addition, technical signs are used in the record: brackets and a comma.

The expressions of the predicate logic language are called formulas.

When translating propositions into the language of predicate logic, there is a difference between writing features-properties and features-relations.

The fact that the subject A owns property R, in the language of predicate logic we write R(A), but what the subject b owns property Q – Q(b). That some property R belongs to an arbitrary subject X from some area we have chosen, will be recorded R(X).

Example 1. The statement "This tree is tall" in the language of predicate logic is written as follows: R(A), Where A- "this is a tree"; R- "high".

Example 2. "Some trees are tall" in the language of predicate logic is written as  xp(X), Where X- "trees"; R- "high";  - the existential quantifier, indicating that the statement refers only to some elements of the set "trees".

That which is between two arbitrary objects X And at there is a relation R, will be recorded R(x,y).

Example 3. The statement “Every positive number is greater than any negative number” can be represented as a formula as follows:  XatR(X,at), Where X- "positive numbers"; at- "negative numbers"; R- attitude "to be more".

Example 4. "Five is greater than three" in the language of predicate logic is written R(a,b), Where A- "five"; b- "three"; R- to be more.

Example 5. "Moscow is located between St. Petersburg and Yekaterinburg." In this statement, there is a relationship between the three objects "Moscow", "Petersburg", "Yekaterinburg". The expression formula will be as follows: R(a,b,c), Where a- "Moscow"; b- "Petersburg"; c - "Ekaterinburg"; R- the relation "to be located between".

Example 6. The statement "If some body invades the Earth's atmosphere, then it flares up" in the language of predicate logic will be written as follows:

x(P(x,a)Q(x)),

Where R- the attitude "intrudes"; Q- "flashes"; A- "Earth's atmosphere"; X- "body".

Formulas R(A), R(X), R(X,at), R(a,b,c) etc. called predicates. A predicate must be distinguished from a predicator. Predicators (see Topic 2) are building blocks of predicates. The difference between them lies in the fact that if we are talking about characteristics (properties and relations, as well as characteristics of an object-functional type) without attributing them to certain objects, then they are called predictors. If we're talking about predicates, then we mean the characteristics of certain, given objects. Thus, unlike predicators, predicates are not just signs of properties or relations, but signs signs. For example, the word "white" as a sign of a property abstracted from objects is a predicate, and as a sign of an attribute of an object "sweater" ("white sweater") or "snow" ("white snow") - a predicate.

When writing statements in the language of predicate logic, one must keep in mind that in logic there is a concept relational properties. A relational property is formed from some relation and indicates the presence or absence of a relation of this subject to some others.

Example 7. The statement "Moscow is located between St. Petersburg and Yekaterinburg" can be written as R(A), Where A- "Moscow"; R– the relational property “to be located between St. Petersburg and Yekaterinburg”.

3.2 The language of propositional logic

Sometimes, in the process of logical analysis of a language, it is not necessary to take into account the structures of simple propositions. Then one can use a simpler version of the symbolic language of logic, the language of classical propositional logic, which uses only propositional variables and logical terms.

Example. “You will get a positive mark in logic if and only if you solve all the tasks offered to you and do not make noise in lectures.” Let us denote simple statements by means of propositional variables: p- "You will receive a positive assessment in logic"; q- "You will solve all the tasks offered to you"; r- “You will make noise at lectures” (we will introduce the negation into the formula with the appropriate sign). We get:

p q r.