Compound statements and logical expressions. Complex statements

23.09.2019

Negation, conjunction, disjunction.

Our reasoning is composed of statements. For example, in the inference “Some birds fly; therefore, some of the flying birds are two different statements.

A statement is a more complex formation than a name. When decomposing statements into simpler parts, we always get certain names. Let's say the saying "The sun is a star" includes the names "Sun" and "Star" as its parts.

Utterance is a grammatically correct sentence, taken together with the meaning (content) expressed by it and which is true or false.

The concept of an utterance is one of the initial, key concepts of logic. As such, it does not admit of a precise definition that is equally applicable in its various sections. It is clear that any utterance describes a certain situation, affirms or denies something about it, and is true or false.

A statement is considered true if the description given by it corresponds to a real situation, and false if it does not correspond to it. “Truth” and “false” are called the truth values of the statement.

From individual statements, you can build new statements in different ways. So, from the statements "The wind is blowing" and "It is raining", you can form more complex statements "The wind is blowing and it is raining", "Either the wind is blowing or it is raining", "If it is raining, the wind is blowing," etc. Expressions "And", "either, or", "if, then", etc., which serve to form complex statements, are called logical connectives.

A statement is called simple if it does not include other statements as its parts.

A statement is complex if it is obtained using logical connectives from other, simpler statements.

The part of logic that describes the logical connections of statements that do not depend on the structure of simple statements is called the general theory of deduction.

Negation is a logical connective with the help of which a new statement is obtained from a given statement, such that if the original statement is true, its negation is false, and vice versa. A negative statement consists of an initial statement and a negation, usually expressed by the words “not”, “it is not true that”. A negative statement is thus a complex statement: it includes as its part a statement different from it. For example, the negation of the statement "10 is an even number" is the statement "10 is not an even number" (or: "It is not true that 10 is an even number").

As a result of combining two statements using the word "and", we get a complex statement called conjunction. Statements connected in this way are called conjunction terms. For example, if the statements “Today is hot” and “Yesterday was cold” are combined in this way, the conjunction “Today is hot and yesterday it was cold”.

A conjunction is true only if both statements included in it are true; if at least one of its members is false, then the whole conjunction is false.

The definition of a conjunction, like the definition of other logical connectives that serve to form complex statements, is based on the following two assumptions:

each statement (both simple and complex) has one and only one of two truth values: it is either true or false;

the truth value of a complex statement depends only on the truth values of the statements included in it and the method of their logical connection with each other.

These assumptions seem simple. Having accepted them, it is necessary, however, to discard the idea that, along with true and false statements, there can also be statements that are indefinite in terms of their truth value (such as, say, "In five years it will rain and thunder at this time." etc.). It is also necessary to reject the fact that the truth value of a complex utterance also depends on the "connection in meaning" of the combined utterances.

In ordinary language, two statements are connected by the conjunction "and" when they are related to each other in content, or meaning. The nature of this connection is not entirely clear, but it is clear that we would not consider the conjunction “He wore a coat and I went to university” as an expression that has meaning and can be true or false. Although the statements "2 is a prime number" and "Moscow is a big city" are true, we are not inclined to consider their conjunction "2 is a prime number and Moscow is a big city" to be true either, since the statements that make it up are not related in meaning.

Simplifying the meaning of conjunction and other logical connectives and rejecting the vague concept of “connection of statements by meaning,” logic makes the meaning of these connectives both broader and clearer.

Combining two statements using the word "or", we get a disjunction of these statements. The statements that form the disjunction are called the members of the disjunction.

The word "or" in everyday language has two different meanings. Sometimes it means "one or the other, or both," and sometimes "one or the other, but not both." The statement “This season I want to go to The Queen of Spades or Aida allows for the possibility of two visits to the opera. In the statement, “He studies at Moscow or Leningrad University,” it is implied that the person mentioned is studying at only one of these universities.

The first sense "or" is called non-exclusive. Taken in this sense, a disjunction of two statements means only that at least one of these statements is true, regardless of whether they are both true or not. Taken in the second, exclusive sense, a disjunction of two statements asserts that one of them is true and the other is false.

The symbol V will denote a disjunction in a non-exclusive sense; for a disjunction in an exclusive sense, the symbol V will be used. Tables for two types of disjunction show that a non-exclusive disjunction is true when at least one of the statements included in it is true, and false only when both of its members are false; exclusive disjunction is true when only one of its terms is true, and it is false when both of its terms are true or both are false.

In logic and mathematics, the word "or" is always used in a non-exclusive meaning.

Decomposition of some statement into simple, further indecomposable parts gives two types of expressions, called proper and improper symbols. The peculiarity of their own symbols is that they have some kind of content, even taken by themselves. These include names (denoting some volumes), unresolved (referring to a certain area of objects), statements (describing some situations and being true or false). Improper symbols do not have an independent content, but in combination with one or more of their own symbols form complex expressions that already have an independent content. Improper symbols include, in particular, logical connectives used to form complex statements from simple ones: "... and ...", "... or ...", "either ... or ..." , "If ..., then ...", "... then and only if ...", "neither ..., nor ...", "not ..., but ... "," ..., but not ... "," it is not true that ... ", etc. The word itself, say" or ", does not mean any object. But in combination with two proper denoting symbols, this word gives a new denoting symbol: from two statements “Letter received” and “Telegram sent” - a new statement “Letter received or telegram sent”.

The central task of logic is to separate the correct schemes of reasoning from the wrong ones and to systematize the former. Logical correctness is determined by the logical form. To identify it, you need to abstract from the meaningful parts of the reasoning (proper symbols) and focus on improper symbols that represent this form in its purest form. Hence the interest of formal logic in such words that usually do not attract attention, such as “and”, “or”, “if, then,” etc.

Saying - a grammatically correct sentence, taken together with the meaning (content) expressed by it and which is true or false.

The concept of an utterance is one of the initial, key concepts of modern logic. As such, it does not admit of a precise definition that is equally applicable in its various sections.

A statement is considered true if the description given by it corresponds to a real situation, and false if it does not correspond to it. "Truth" and "falsehood" are called "truth values of statements."

From individual statements, you can build new statements in different ways. For example, from the statements “The wind is blowing” and “It is raining”, you can form more complex statements “The wind is blowing and it is raining”, “Either the wind is blowing or it is raining”, “If it is raining, then the wind is blowing”, etc.

The saying is called simple, if it does not include other statements as parts of it.

The saying is called complicated, if it is obtained using logical connectives from other simpler statements.

Let's consider the most important ways to build complex statements.

Negative statement consists of an initial statement and a negation, usually expressed by the words "not", "it is not true that." A negative statement is thus a complex statement: it includes as its part a statement different from it. For example, the negation of the statement "10 is an even number" is the statement "10 is not an even number" (or: "It is not true that 10 is an even number").

Let us denote statements by letters A, B, C,... The full meaning of the concept of denial of a statement is given by the condition: if the statement A is true, its negation is false, and if A false, its denial is true. For example, since the statement “1 is a positive integer” is true, its negation “1 is not a positive integer” is false, and since “1 is a prime number” is false, its negation “1 is not a prime number” is true.

The combination of two statements using the word "and" gives a complex statement called conjunction. Statements put together in this way are called "conjunction terms."

For example, if the statements “Today is hot” and “Yesterday was cold” are combined in this way, the conjunction “Today is hot and yesterday it was cold”.

A conjunction is true only if both statements included in it are true; if at least one of its members is false, then the whole conjunction is false.

In ordinary language, two statements are connected by the conjunction "and" when they are related to each other in content or meaning. The nature of this connection is not entirely clear, but it is clear that we would not consider the conjunction “He wore a coat and I went to university” as an expression that has meaning and can be true or false. Although the statements "2 is a prime number" and "Moscow is a big city" are true, we are not inclined to consider their conjunction "2 is a prime number and Moscow is a big city" to be true either, since the statements that make them are not related in meaning. Simplifying the meaning of conjunction and other logical connectives and refusing for this from the vague concept of "connection of statements by meaning", logic makes the meaning of these connectives both broader and more definite.

The combination of two statements using the word "or" gives disjunction these statements. The statements that form a disjunction are called "members of the disjunction."

The word "or" in everyday language has two different meanings. Sometimes it means "one or the other, or both," and sometimes "one or the other, but not both." For example, the statement “This season I want to go to The Queen of Spades or Aida allows for the possibility of two visits to the honra. In the statement, “He studies at Moscow or Yaroslavl University,” it is implied that the person mentioned is studying at only one of these universities.

The first meaning of "or" is called non-exclusive. Taken in this sense, a disjunction of two statements means that at least one of these statements is true, regardless of whether they are both true or not. Taken in the second, excluding or in the strict sense, a disjunction of two statements asserts that one of the statements is true and the other is false.

A non-exclusive disjunction is true when at least one of the statements included in it is true, and false only when both of its terms are false.

An exclusive disjunction is true when only one of its terms is true, and it is false when both of its terms are true or both are false.

In logic and mathematics, the word "or" is almost always used *** in a non-exclusive meaning.

Conditional statement - a complex statement, usually formulated with the help of the link "if ..., then ..." and establishing that one event, state, etc. is, in one sense or another, a basis or condition for another.

For example: “If there is fire, then there is smoke”, “If the number is divisible by 9, it is divisible by 3”, etc.

A conditional statement is composed of two simpler statements. The one to which the word "if" is prefixed is called basis, or antecedent(previous), the statement that comes after the word "that" is called consequence, or consequent(subsequent).

In asserting a conditional statement, we first of all mean that it cannot be so that what is said in its foundation took place, and what is said in the corollary was absent. In other words, it cannot happen that the antecedent is true and the consequent is false.

In terms of a conditional statement, the concepts of a sufficient and necessary condition are usually defined: an antecedent (reason) is a sufficient condition for a consequent (consequence), and a consequent is a necessary condition for an antecedent. For example, the truth of the conditional statement “If the choice is rational, then the best available alternative is chosen” means that rationality is a sufficient reason for choosing the best available opportunity and that the choice of such an opportunity is a necessary condition for its rationality.

A typical function of a conditional statement is to justify one statement by reference to another statement. For example, the fact that silver is electrically conductive can be justified by referring to the fact that it is a metal: "If silver is a metal, it is electrically conductive."

The connection between the justifying and justified (grounds and consequences) expressed by a conditional statement is difficult to characterize in general view, and only sometimes the nature is relatively clear. This connection can be, firstly, the connection of logical consequence that takes place between the premises and the conclusion of the correct inference ("If all living multicellular creatures are mortal, and the medusa is such a creature, then it is mortal"); secondly, by the law of nature ("If a body is subjected to friction, it will begin to heat up"); thirdly, by causality (“If the Moon is in the node of its orbit on a new moon, a solar eclipse occurs”); fourthly, a social pattern, rule, tradition, etc. (“If the society changes, the person also changes”, “If the advice is reasonable, it must be followed”).

With the connection expressed by a conditional statement, the conviction is usually combined that the consequence with a certain necessity "follows" from the foundation and that there is some general law, having managed to formulate which, we could logically deduce the consequence from the foundation.

For example, the conditional statement “If bismuth is a metal is plastic”, as it were, presupposes the general law “None of metals are plastic”, which makes the consequent of this statement a logical consequence of its antecedent.

Both in ordinary language and in the language of science, a conditional statement, in addition to the function of justification, can also perform a number of other tasks: to formulate a condition that is not associated with any implied general law or rule (“If I want, I will cut my cloak”); to fix any sequence (“If last summer was dry, then this year it was rainy”); express disbelief in a peculiar form ("If you solve this problem, I will prove the great Fermat's theorem"); opposition ("If an elderberry grows in the garden, then an uncle lives in Kiev"), etc. The multiplicity and heterogeneity of the functions of the conditional statement significantly complicates its analysis.

The use of a conditional statement is associated with certain psychological factors. Thus, we usually formulate such a statement only if we do not know with certainty whether its antecedent and consequent are true or not. Otherwise, its use seems unnatural ("If cotton wool is metal, it is not an electric wire").

The conditional statement finds very wide application in all areas of reasoning. In logic, it is represented, as a rule, by means of implicative statement, or implications. At the same time, logic clarifies, systematizes and simplifies the use of "if ... then ...", frees it from the influence of psychological factors.

Logic is abstracted, in particular, from the fact that the connection of the basis and the effect, which is characteristic of a conditional statement, depending on the context, can be expressed using ns only "if ... then ...", but also other linguistic means. For example, "Since water is liquid, it transfers pressure in all directions evenly", "Although plasticine is not a metal, it is plastic", "If wood were metal, it would be electrically conductive", etc. These and similar statements are presented in the language of logic by means of implication, although the use of "if ... then ..." in them would not be entirely natural.

In asserting an implication, we assert that it cannot happen that its foundation takes place, and the effect is absent. In other words, the implication is false only if the reason is true and the effect is false.

This definition assumes, like the previous definitions of connectives, that every statement is either true or false and that the truth value of a complex statement depends only on the truth values of its constituent statements and on the way they are connected.

An implication is true when both its basis and its effect are true or false; it is true if its foundation is false and the effect is true. Only in the fourth case, when the foundation is true and the effect is false, is the implication false.

The implication does not imply that the statements A and V somehow related to each other in content. If true V saying “if A, then V" is true regardless of whether A true or false and it is connected in meaning with V or not.

For example, the statements are considered true: “If there is life on the Sun, then twice two equals four”, “If the Volga is a lake, then Tokyo is a big village”, etc. The conditional statement is also true when A false, and yet again indifferent, true V or not, and it is related in content to A or not. The following statements are true: “If the Sun is a cube, then the Earth is a triangle”, “If twice two equals five, then Tokyo is a small city”, etc.

In ordinary reasoning, all of these statements are unlikely to be regarded as meaningful, and even less so as true.

While implication is useful for many purposes, it is not entirely consistent with conventional understanding of conditional communication. The implication covers many important features of the logical behavior of a conditional statement, but at the same time it is not a sufficiently adequate description of it.

In the last half century, there have been vigorous attempts to reform the theory of implication. In this case, it was not about rejecting the described concept of implication, but about introducing along with it another concept that takes into account not only the truth values of statements, but also their connection in content.

Closely related to implication equivalence, sometimes called "double implication".

Equivalence is a complex statement "A if and only if B", formed from the statements of Lie B and decomposed into two implications: "if A, then B ", and" if B, then A". For example: "A triangle is equilateral if and only if it is conformal." The term "equivalence" also denotes the link "... if and only if ...", with the help of which a given complex statement is formed from two statements. Instead of “if and only if” for this purpose can be used “if and only if”, “if and only if”, etc.

If logical connectives are defined in terms of truth and falsehood, equivalence is true if and only if both statements of it have the same truth value, i.e. when they are both true or both are false. Accordingly, an equivalence is false when one of the statements included in it is true and the other is false.

Expressing denial

Among the statements of negation, statements with external and internal negation are distinguished. Depending on the objectives of the study, the statement of negation can be considered either as a simple or as a complex statement.

When considering the statement of negation as a simple statement, an important task is to determine the correct logical form of the statement:

A simple statement containing internal negation is usually referred to as negative statements (see "Types of attributive statements by quality"). For example: " Some residents of the Republic of Belarus do not use bank loans "," Not a single hare is a predator ";

The correct logical form of a simple statement with external negation is a statement that contradicts the given statement (see "Logical relations between statements. Logical square"). For example: a statement "Not all people are greedy" matches the saying “Some people are not greedy».

Considering the statement of negation as a complex statement, it is necessary to determine its logical meaning.

Original saying: The sun is shining(R).

Expressing denial: The sun is not shining(┐p).

Saying double denial: It's not true that the sun doesn't shine(┐┐p).

R	┐p	┐┐p
AND	L	AND
L	AND	L
Rice. 16

A negative statement is true only when the original statement is false, and vice versa. The law of double negation is associated with the statement of negation: the double negation of an arbitrary statement is equivalent to the statement itself. The conditions for the truth of the statement of negation are shown in Fig. 16.

Complicated a statement is considered, consisting of several simple statements, connected using logical conjunctions "and", "or", "if ..., then ...", etc. Complex statements include connecting, separating, conditional, equivalent statements, as well as statements denial.

Connecting utterance (conjunction)- This is a complex statement, consisting of simple, connected using the logical connective "and". The logical union "and" (conjunction) can be expressed in natural language by the grammatical unions "and", "but", "however", "as well", etc. For example: "The clouds came, and it began to rain", "Both big and small rejoice in a good day"... In the symbolic language of logic, these statements are written as follows: p∧q... A conjunction is true only when all its constituent simple statements are true (Fig. 17).

Separating statement (disjunction). Distinguish between weak and strong disjunction. Weak disjunction corresponds to the use of the conjunction "or" in the connecting-separating sense (either one or the other, or both together). For instance: "This student is an athlete or an excellent student" (p⋁q), "Hereditary factors, poor ecology and bad habits are the causes of most diseases."(p⋁q⋁r). A weak disjunction is true when at least one of its simple statements is true (see Fig. 17).

Strong disjunction corresponds to the use of the conjunction "either" in the exclusive-separative sense (either one or the other, but not both together). For instance: "In the evening I will be in class or go to a disco", "A man is either alive or dead"... Symbolic notation p⊻q... A strong disjunction is true when only one of its simple statements is true (see Fig. 17).

Conditional statement (implication)- This is a complex statement, consisting of two parts, connected by a logical union "if ... then ...". The statement after the "if" particle is called the base, and the statement after the "then" is called the effect. In the logical analysis of conditional statements, the basis of the implication is always put at the beginning. In natural language, this rule is often not followed. An example of a conditional statement: "If the swallows fly low, it will rain" (p → q). The implication is false only in one case when its basis is true and the effect is false (see Fig. 17).

Equivalent statement- This is a statement consisting of simple ones connected by a logical union “if and only if” (“if and only if…, then…). An equivalent statement implies the simultaneous presence or absence of two situations. In natural language, the equivalent can be expressed by grammatical unions "if ... then ...", "only in the case when ...", etc. For example: “Our team will only win if they prepare well» ( p↔q). An equivalent statement will be true when its constituent statements are either simultaneously true or simultaneously false (see Fig. 17).

To formalize the reasoning, it is necessary:

1) find and mark with small consonant letters of the Latin alphabet simple sentences that are part of a complex one. Variables are assigned arbitrarily, but if the same simple statement occurs several times, then the corresponding variable is used the same number of times;

2) find and designate logical unions (∧, ⋁, ⊻, →. ↔, ┐) with logical constants;

3) if necessary, place technical signs [...], (...).

In fig. 18 shows an example of the formalization of a complex statement .

I have already freed myself (p) and (∧), if me not delay (┐q) or (⋁) not car breaks down (┐r), then (→) I will come soon (s) .

p ∧ ((┐q ⋁ ┐r) → s

Rice. eighteen

After the statement is written in symbolic form, you can determine the type of formula. In logic, there are identically true, identically false and neutral formulas. Identically true formulas, regardless of the values of the variables included in their composition, always take the value "true", and identically false ones - the value "false". Neutral formulas accept both true and false.

To determine the type of a formula, a tabular method is used, an abbreviated way of checking a formula for truth by the method of "reducing to absurdity" and reducing the formula to normal form. The normal form of a certain formula is its expression that meets the following conditions:

Does not contain signs of implication, equivalence, strict disjunction and double negation;

Negative signs are found only for variables.

Tabular way of defining the type of a formula:

1. Build columns of input values for each of the available variables. These columns are called free (independent), they take into account all possible combinations of variable values. If there are two variables in the formula, then two free columns are built, if there are three variables, then three columns, etc.

2. For each subformula, that is, a part of a formula containing at least one union, a column of its values is built. This takes into account the values of free columns and the features of the logical union (see Fig. 17).

3. Build a column of output values for the entire formula as a whole. The values obtained in the output column determine the type of the formula. So, if the output column contains only the value "true", then the formula will refer to identically true, and so on.

Truth table for a formula(p ^ q) → r
p	q	r	p ^ q	(p ^ q) → r
AND	AND	AND	AND	AND
L	AND	L	L	AND
L	L	AND	L	AND
AND	L	L	L	AND
AND	AND	L	AND	L
AND	L	AND	L	AND
L	AND	AND	L	AND
L	L	L	L	AND
Rice. 19

The number of columns in the table is equal to the sum of the variables included in the formula and the unions present in it. (For example: in the formula in Fig. 18 there are four variables and five unions, therefore, the table will have nine columns).

The number of rows in the table is calculated by the formula C = 2 n, where n- the number of variables. (The formula table in Figure 18 should have sixteen rows.)

In fig. 19 shows an example of a truth table.

An abbreviated way to test a formula for truth by the method of reduction to absurdity:

((p⋁q) ⋁r) → (p⋁ (q⋁r))

1. Suppose that the given formula is not identically true. Therefore, for a certain set of values, it takes on the value "false".

2. This formula can take on the value “false” only if the base of the implication (p⋁q) ⋁r is “true”, and the consequence p⋁ (q⋁r) is “false”.

3. The consequence of the implication p⋁ (q⋁r) will be false if p is “false” and q⋁r is “false” (see the meaning of weak disjunction in Fig. 17).

4. If q⋁r is “false”, then both q and r are “false”.

5. We have established that p is "false", q is "false" and r is "false". The base of the implication (p⋁q) ⋁r is a weak disjunction of these variables. Since a weak disjunction takes on the value “false” when all of its components are false, then the implication base (p⋁q) ⋁r will also be “false”.

6. In item 2 it was established that the base of the implication (p⋁q) ⋁r is “true”, and in item 5 that it is “false”. The contradiction that has arisen indicates that the assumption we made in Section 1 is erroneous.

7. Since this formula does not assume the value "false" for any set of values of its variables, it is identically true.

3.8. Logical relationships between statements
(logical square)

Connections are established between statements that have a similar meaning. Consider the relationship between simple and complex statements.

In logic, the whole set of statements is divided into comparable and incomparable. Incomparable among simple statements are statements that have different subjects or predicates. For instance: "All students are students" and "Some students are excellent students".

Comparable are statements with the same subjects and predicates and differing in connectiveness and quantifier. For instance: "All citizens of the Republic of Belarus have the right to rest" and "Not a single citizen of the Republic of Belarus has the right to rest."

Rice. twenty

The relationship between comparable statements is expressed using a model called logical square (fig. 20).

Among comparable statements, a distinction is made between compatible and incompatible.

Compatibility relation

1.Equivalence (full compatibility)- statements that have the same logical characteristics: the same subjects and predicates, the same type of affirmative or negative link, the same logical characteristic. Equivalent statements differ in the verbal expression of the same thought. The logical square is not used to illustrate the relationship between these statements.

2... Partial compatibility (subcontrast, subcontracted). In this respect, there are partly affirmative and partly negative statements (I and O). This means that two such statements can be true at the same time, but cannot be false at the same time. If one of them is false, then the other is necessarily true. If one of them is true, then the other is indefinite.

3... Subordination (subordination). In this regard, there are generally affirmative and partially affirmative statements (A and I), as well as general negative and partially negative statements (E and O).

The truth of the particular always follows from the truth of a general statement. While the truth of a particular statement indicates the uncertainty of a general statement.

From the falsity of a particular statement always follows the falsity of a general statement, but not vice versa.

The relationship of incompatibility. Incompatible statements are statements that cannot be true at the same time:

1. Opposition (opposite, contra)- in this respect there are generally affirmative and generally negative statements (A and E). This relationship means that two such statements cannot be true at the same time, but they can be false at the same time. If one of them is true, then the second is necessarily false. If one of them is false, then the other is indefinite.

2.Controversy (contradictory)- it contains general affirmative and partial negative statements (A and O), as well as general negative and partially affirmative statements (E and I). Two contradictory statements can be neither simultaneously false, nor simultaneously true. One is necessarily true and the other is false.

Comparable among complex statements are statements that have at least one identical component. Otherwise, complex statements are incomparable.

Comparable complex statements may be compatible or incompatible.

Compatibility relation means that statements can be true at the same time:

2.Partial compatibility means that statements can be true at the same time, but cannot be false at the same time (Fig. 22).

p	q	p → q	q → p
AND	AND	AND	AND
AND	L	L	AND
L	AND	AND	L
L	L	AND	AND
Rice. 22

3.The attitude of following (submission) means that the truth of one statement implies the truth of another, but not vice versa (Fig. 23).

p	q	r	(p → q) ∧ (q → r)	p↔r
AND	AND	AND	AND	AND
AND	AND	L	L	L
AND	L	AND	L	AND
L	AND	AND	AND	AND
AND	L	L	L	L
L	AND	L	L	AND
L	L	AND	AND	AND
L	L	L	AND	AND
Rice. 23

4... Grip ratio means that the truth (falsity) of one statement does not exclude the falsity (truth) of another (Fig. 24).

p	q	p → q	┐p → q
AND	AND	AND	AND
AND	L	L	AND
L	AND	AND	AND
L	L	AND	L
Rice. 24

Incompatibility relation means that statements cannot be true at the same time:

2.Contradiction- the relationship between statements that can be neither true at the same time, nor at the same time false (Fig. 26).

p	q	p → q	p∧┐q
AND	AND	AND	L
AND	L	L	AND
L	AND	AND	L
L	L	AND	L
Rice. 26

Utterance- a grammatically correct sentence, taken together with the meaning (content) expressed by it and which is true or false.

The concept of an utterance is one of the initial, key concepts of logic. As such, it does not admit of a precise definition that is equally applicable in its various sections.

From individual statements, you can build new statements in different ways.

For example, from the statements “The wind is blowing” and “It is raining”, you can form more complex statements “The wind is blowing and it is raining”, “Either the wind is blowing or it is raining”, “If it is raining, then the wind is blowing”, etc. ...

The saying is called simple, unless it includes other utterances as its parts.

The statement is called I'm challenging if it is obtained using logical connectives from other simpler statements.

Let's consider the most important ways to build complex statements.

Let us denote statements by the letters A, B, C, ... The full meaning of the concept of denial of a statement is given by the condition: if statement A is true, its negation is false, and if A is false, its negation is true. For example, since “1 is a positive integer” is true, its negation “1 is not a positive integer” is false, and since “1 is a prime” is false, its negation “1 is not a prime” is true.

The combination of two statements using the word "and" gives a complex statement called conjunction... Statements put together in this way are called "conjunction terms."

For example, if the statements “Today is hot” and “Yesterday was cold” are combined in this way, the conjunction “Today is hot and yesterday it was cold”.

A conjunction is true only if both statements included in it are true; if at least one of its members is false, then the whole conjunction is false.

In ordinary language, two statements are connected by the conjunction "and" when they are related to each other in content, or meaning. The nature of this connection is not entirely clear, but it is clear that we would not consider the conjunction “He wore a coat and I went to university” as an expression that has meaning and can be true or false. Although the statements "2 is a prime number" and "Moscow is a big city" are true, we are not inclined to consider their conjunction "2 is a prime number and Moscow is a big city" as true either, since the statements that make it up are not related in meaning. Simplifying the meaning of conjunction and other logical connectives and rejecting the vague concept of “connection of statements by meaning,” logic makes the meaning of these connectives both broader and clearer.

The combination of two statements using the word "or" gives disjunction these statements. Statements that form a disjunction are called "members of the disjunction" .

The word "or" in everyday language has two different meanings. Sometimes it means "one or the other, or both," and sometimes "one or the other, but not both." For example, the statement “This season I want to go to The Queen of Spades or Aida” allows for the possibility of two visits to the opera. The statement "He studies at Moscow or Yaroslavl University" implies that the person mentioned is studying at only one of these universities.

The first meaning of "or" is called non-exclusive. Taken in this sense, a disjunction of two statements means that at least one of these statements is true, regardless of whether they are both true or not. Taken in the second, excluding, or in the strict sense, a disjunction of two statements asserts that one of the statements is true and the other is false.

A non-exclusive disjunction is true when at least one of the statements included in it is true, and false only when both of its terms are false.

An exclusive disjunction is true when only one of its terms is true, and it is false when both of its terms are true or both are false.

In logic and mathematics, the word "or" is almost always used in a non-exclusive sense.

Conditional statement - a complex statement, usually formulated with the help of the connective "if ..., then ..." and establishing that one event, state, etc. is in one sense or another a basis or condition for another.

For example: "If there is fire, then there is smoke", "If the number is divisible by 9, it is divisible by 3", etc.

The connection between the substantiating and the substantiated (grounds and consequences) expressed by a conditional statement is difficult to characterize in general terms, and only sometimes its nature is relatively clear. This connection can be, firstly, a connection of logical consequence that takes place between the premises and the conclusion of the correct inference ("If all living multicellular creatures are mortal, and the medusa is such a creature, then it is mortal"); secondly, by the law of nature ("If a body is subjected to friction, it will begin to heat up"); third, by a causal connection (“If the Moon is in the node of its orbit on a new moon, a solar eclipse occurs”); fourth, a social pattern, rule, tradition (“If society changes, the person also changes”, “If advice is reasonable, it must be followed”), etc.

For example, the conditional statement “If bismuth is a metal, it is plastic”, as it were, presupposes the general law “All metals are plastic”, which makes the consequent of a given statement a logical consequence of its antecedent.

Both in ordinary language and in the language of science, a conditional statement, in addition to the function of justification, can also perform a number of other tasks: to formulate a condition that is not associated with any implied general law or rule (“If I want, I will cut my cloak”); to fix some sequence (“If last summer was dry, then this year it was rainy”); express disbelief in a peculiar form ("If you solve this problem, I will prove the great Fermat's theorem"); opposition ("If an elderberry grows in the garden, then an uncle lives in Kiev"), etc. The multiplicity and heterogeneity of the functions of the conditional statement significantly complicates its analysis.

The use of a conditional statement is associated with certain psychological factors. We usually formulate such a statement only if we do not know with certainty whether its antecedent and consequent are true or not. Otherwise, its use seems unnatural ("If cotton wool is metal, it is electrically conductive").

The conditional statement finds very wide application in all areas of reasoning. In logic, it is represented, as a rule, by means of implicative statement, or implications... At the same time, logic clarifies, systematizes and simplifies the use of "if ... then ...", frees it from the influence of psychological factors.

Logic is distracted, in particular, from the fact that the connection of reason and effect, characteristic of a conditional statement, depending on the context, can be expressed using not only "if ... then ...", but also other linguistic means.

For example, "Since water is liquid, it transfers pressure in all directions evenly", "Although plasticine is not a metal, it is plastic", "If wood were metal, it would be electrically conductive", etc. These and similar statements are represented in the language of logic by means of implication, although the use of "if ... then ..." in them would not be entirely natural.

By asserting an implication, we are asserting that it cannot happen that its foundation takes place, and the effect is absent. In other words, the implication is false only if its basis is true and the effect is false.

The implication does not imply that statements A and B are somehow related to each other in content. If B is true, the statement “if A, then B” is true regardless of whether A is true or false and it is connected in meaning with B or not.

For example, the statements are considered true: “If there is life on the Sun, then twice two equals four”, “If the Volga is a lake, then Tokyo is a big village”, etc. The conditional statement is also true when A is false, and at the same time again, it makes no difference whether B is true or not, and whether it is related in content to A or not. The statements are true: “If the Sun is a cube, then the Earth is a triangle”, “If two times two equals five, then Tokyo is a small city”, etc.

In ordinary reasoning, all of these statements are unlikely to be regarded as meaningful, and even less so as true.

Closely related to implication equivalence sometimes called "double implication".

Equivalence- a complex statement "A if and only if B", formed from statements A and B and decomposed into two implications: "if A, then B", and "if B, then A". For example: "A triangle is equilateral if and only if it is conformal." The term "equivalence" also denotes the link "... if and only if ...", with the help of which a given complex statement is formed from two statements. Instead of “if and only if” for this purpose can be used “if and only if”, “if and only if”, etc.

If logical connectives are defined in terms of truth and falsehood, equivalence is true if and only if both statements of it have the same truth value, that is, when they are both true and both are false. Accordingly, the equivalence is false when one of the statements included in it is true and the other is false.

When considering the methods of forming complex statements from simple ones, the internal structure of simple statements was not taken into account. They were taken as indecomposable particles with only one property: to be true or false. Simple sayings

it is not by chance that they are sometimes called atomic: from them, as from elementary bricks, with the help of logical connectives "and", "or", etc., various complex ("molecular") statements are constructed.

Now we should dwell on the question of the internal structure, or internal structure, of the simple statements themselves: from what concrete parts they are composed and how these parts are related to each other.

It should be emphasized right away that simple statements can be decomposed into their component parts in different ways. The result of the decomposition depends on the purpose for which it is carried out, that is, on the concept of logical inference (logical consequence), within the framework of which such statements are analyzed.

The special interest in categorical statements is primarily due to the fact that the development of logic as a science began with the study of their logical connections. In addition, statements of this type are widely used in our reasoning. The theory of logical connections of categorical statements is usually called syllogistics.

For example, in the saying "All dinosaurs are extinct" dinosaurs are attributed the attribute "to be extinct." In the statement "Some dinosaurs flew", the ability to fly is attributed to certain types of dinosaurs. The judgment “All comets are not asteroids” denies the presence of the sign “to be an asteroid” in each of the comets. The statement "Some animals are not herbivores" denies that some animals are herbivores.

If we ignore the quantitative characteristics contained in a categorical statement and expressed by the words “all” and “some”, then we get two versions of such statements: positive and negative. Their structure:

"S is P" and "S is not P",

where the letter S represents the name of the subject referred to in the statement, and the letter P is the name of a feature inherent or not inherent in this subject.

The name of the subject referred to in a categorical statement is called subject, and the name of its feature is predicate... The subject and predicate are named terms categorical statements and are connected with each other by the bundles "is" or "is not" ("is" or "is not", etc.). For example, in the statement "The sun is a star" the terms are the names "Sun" and "star" (the first of them is the subject of the statement, the second is its predicate), and the word "is" is a bundle.

Simple statements of the type “S is (is not) P” are called attributive: in them, the attribution (assignment) of some property to an object is carried out.

Attributive statements are opposed by statements about relationships in which relationships are established between two or more objects: "Three less than five", "Kiev is more than Odessa", "Spring is better than autumn", "Paris is between Moscow and New York", etc. Statements about relationships play an essential role in science, especially in mathematics. They are not reduced to categorical statements, since the relationship between several objects (such as "equal", "loves", "warmer", "is between", etc.) are not reduced to the properties of individual objects. One of the significant shortcomings of traditional logic was that it considered judgments about relations to be reducible to judgments about properties.

A categorical statement not only establishes a connection between an object and a feature, but also gives a certain quantitative characteristic of the subject of the statement. In statements like “All S is (is not) P” the word “all” means “each of the objects of the corresponding class”. In statements like “Some S are (are not) P” the word “some” is used in a non-exclusive sense and means “some, and maybe all”. In an exclusive sense, the word "some" means "only some" or "some, but not all." The difference between the two meanings of this word can be demonstrated by the example of the saying "Some stars are stars." In a non-exclusive sense, it means "Some, and possibly all, stars are stars," and is obviously true. In an exclusive sense, this statement means "Only a few stars are stars" and is clearly false.

In categorical statements, the belonging of some signs to the objects under consideration is affirmed or denied and it is indicated whether we are talking about all these objects or some of them.

Thus, four types of categorical statements are possible:

All S is P - a generally affirmative statement,

Some S are P - a particular affirmative statement,

All S is not P - a generally negative statement,

Some S are not P - a partial negative statement.

Categorical statements can be viewed as the results of substitution of some names in the following expressions with spaces (ellipses): “Everything… is…”, “Some… is…”, “All… is not…” and “Some… is not…”. Each of these expressions is a logical constant (logical operation) that allows you to get a statement from two names. For example, substituting the names "flying" and "birds" instead of dots, we obtain, respectively, the following statements: "All flying are birds", "Some flying birds are",

Inferences

"All that fly are not birds" and "Some that fly are not birds." The first and third statements are false, and the second and fourth are true.

Inferences

“A person who can think logically can draw a conclusion about the existence of the Atlantic Ocean or Niagara Falls by one drop of water, even if he has never seen either one or the other and never heard of them ... By the nails of a person, by his hands, shoes, the fold of his trousers on the knees, by the thickening of the skin on the thumb and forefinger, by the expression on his face and the cuffs of his shirt - from such trifles it is easy to guess his profession. And there is no doubt that all this, taken together, will prompt a competent observer to the correct conclusions. "

This is a quote from a keynote article by the world's most famous detective consultant, Sherlock Holmes. Based on the smallest details, he built logically flawless chains of reasoning and solved intricate crimes, often from the comfort of his apartment on Baker Street. Holmes used a deductive method that he himself created, which, as his friend Dr. Watson believed, put crime solving on the brink of an exact science.

Of course, Holmes somewhat exaggerated the importance of deduction in forensic science, but his reasoning about the deductive method did the trick. "Deduction" from a special and known only to a few term has become a commonly used and even fashionable concept. The popularization of the art of correct reasoning, and above all deductive reasoning, is no less a merit of Holmes than all the crimes he disclosed. He managed to "give logic the charm of a dream, making its way through the crystal labyrinth of possible deductions to a single shining conclusion" (V. Nabokov).

Deduction is a special case of inference.

V broad senseinference - a logical operation, as a result of which a new statement is obtained from one or several accepted statements (premises) - a conclusion (conclusion, consequence).

Depending on whether there is a connection between the premises and the conclusion logical consequence, there are two types of inferences.

At the heart of deductive inference there is a logical law, by virtue of which the conclusion with logical necessity follows from the accepted premises.

A distinctive feature of such a conclusion is that it always leads from true premises to a true conclusion.

V inductive inference the connection between premises and conclusions is based not on the law of logic, but on some factual or psychological foundations that do not have a purely formal character.

In such a conclusion, the conclusion does not follow logically from the premises and may contain information that is absent in them. The reliability of the premises does not mean, therefore, the reliability of the statement derived from them inductively. Induction gives only probable, or plausible, conclusions requiring further verification.

For example, deductive conclusions include:

If it rains, the ground is wet. It's raining.

The ground is wet.

If helium is a metal, it is electrically conductive. Helium is not electrically conductive.

Helium is not a metal.

The line separating premises from conclusion replaces, as usual, the word "therefore."

Examples of induction are the following reasoning:

Argentina is a republic; Brazil is a republic; Venezuela is a republic; Ecuador is a republic.

Argentina, Brazil, Venezuela, Ecuador are Latin American states.

All Latin American states are republics .

Italy is a republic, Portugal is a republic, Finland is a republic, France is a republic.

Italy, Portugal, Finland, France - Western European countries.

All Western European countries are republics.

Induction does not give a complete guarantee of obtaining a new truth from the existing ones. The maximum that can be talked about is a certain degree of probability of the statement being inferred. So the premises of both the first and second inductive inference are true, but the conclusion of the first of them is true, and the second is false. Indeed, all Latin American states are republics; but among the Western European countries there are not only republics, but also monarchies, for example England, Belgium and Spain.

Inferences

Especially characteristic deductions are logical transitions from general knowledge to particular knowledge, such as:

All metals are ductile. Copper is a metal.

Copper is ductile.

In all cases when it is required to consider a certain phenomenon on the basis of an already known general rule and to draw the necessary conclusion in relation to these phenomena, we reason in the form of deduction. Reasoning leading from knowledge about a part of objects (private knowledge) to knowledge about all objects of a certain class (general knowledge) are typical inductions. There is always the possibility that the generalization will be hasty and unfounded ("Napoleon is a commander; Suvorov is a commander; hence, every person is a commander").

At the same time, one cannot identify deduction with the transition from the general to the particular, and induction with the transition from the particular to the general.

In the discourse “Shakespeare wrote sonnets; therefore, it is not true that Shakespeare did not write sonnets "there is deduction, but there is no transition from the general to the particular. The reasoning "If aluminum is plastic or clay is plastic, then aluminum is plastic" is, as it is commonly thought, inductive, but there is no transition from the particular to the general.

Deduction is the derivation of conclusions that are as reliable as the accepted premises, induction is the derivation of probable (plausible) conclusions. Inductive inferences include both transitions from the particular to the general and analogy, methods for establishing causal relationships, confirmation of consequences, purposeful justification, etc.

The particular interest in deductive reasoning is understandable. They allow one to obtain new truths from existing knowledge, and moreover, with the help of pure reasoning, without resorting to experience, intuition, common sense, etc. - the probability of a true conclusion. Starting from true premises and reasoning deductively, we will definitely get reliable knowledge in all cases.

While emphasizing the importance of deduction in the process of developing and substantiating knowledge, one should not, however, separate it from induction and underestimate the latter. Almost all general provisions, including scientific laws, are the results of inductive generalization. In this sense, induction is the basis of our knowledge. By itself, it does not guarantee its truth and validity, but it generates assumptions, connects them with experience and thereby gives them a certain likelihood, a more or less high degree of probability. Experience is the source and foundation of human knowledge. Induction, starting from what is comprehended in experience, is a necessary means of its generalization and systematization.

LOGICAL LAWS

Chapter

The concept of a logical law

Logical laws form the basis of human thinking. They determine when other statements logically follow from some statements, and represent that invisible iron frame on which consistent reasoning is held and without which it turns into chaotic, incoherent speech. Without a logical law, it is impossible to understand what a logical consequence is, and thus what a proof is.

Correct, or, as they usually say, logical, thinking is thinking according to the laws of logic, according to those abstract schemes that are fixed by them. Hence, the importance of these laws is clear.

Homogeneous logical laws are combined into logical systems, which are also usually called "logics". Each of them gives a description of the logical structure of a certain fragment, or type, of our reasoning.

For example, the laws describing the logical connections of statements that do not depend on the internal structure of the latter are combined into a system called the "logic of statements." The logical laws that determine the connections of categorical statements form a logical system called the "logic of categorical statements", or "syllogistics", etc.

Logical laws are objective and do not depend on the will and consciousness of a person. They are not the result of an agreement between people, some specially developed or spontaneous convention. They are not a product of some kind of "world spirit", as Plato once believed. The power of the laws of logic over a person, their force obligatory for correct thinking is due to the fact that they represent a reflection in human thinking of the real world and the centuries-old experience of its cognition and transformation by a person.

Like all other scientific laws, logical laws are universal and necessary. They operate always and everywhere, extending equally to all people and to any era. Representatives

The concept of a logical law

different nations and different cultures, men and women, ancient Egyptians and modern Polynesians from the point of view of the logic of their reasoning do not differ from each other.

The necessity inherent in logical laws is in some sense even more urgent and immutable than natural, or physical, necessity. It is impossible even to imagine that the logically necessary was different. If something contradicts the laws of nature and is physically impossible, then no engineer, for all his giftedness, will be able to realize it. But if something contradicts the laws of logic and is logically impossible, then not only an engineer - even an omnipotent being, if it suddenly appeared, would not be able to bring it to life.

As mentioned earlier, in correct reasoning, the conclusion follows from premises with logical necessity, and the general scheme of such reasoning is a logical law.

The number of schemes of correct reasoning (logical laws) is infinite. Many of these schemes are known to us from the practice of reasoning. We apply them intuitively, without realizing that in every inference we correctly draw, one or another logical law is used.

Before introducing the general concept of a logical law, we will give several examples of reasoning schemes that are logical laws. Instead of variables A, B, C, ..., usually used to designate statements, we will use, as was done in antiquity, the words "first" and "second", replacing variables.

“If there is the first, then there is the second; there is the first; therefore, there is a second. " This scheme of reasoning allows from the statement of the conditional statement ("If there is the first, then there is the second") and the statement of its basis ("There is the first") to the statement of the consequence ("There is the second"). According to this scheme, in particular, the reasoning proceeds: “If ice is heated, it melts; the ice is heated; therefore it melts. "

Another scheme of correct reasoning: “Either the first takes place, or the second; there is the first; so there is no second. " Through this scheme, from two mutually exclusive alternatives and establishing which of them takes place, a transition is made to the negation of the second alternative. For example: “Either Dostoevsky was born in Moscow, or he was born in St. Petersburg. Dostoevsky was born in Moscow. It means that it is not true that he was born in St. Petersburg. " In the American western The Good, the Bad and the Ugly, one bad guy says to another: “Remember, the world is divided into two parts: those who hold the revolver, and those who dig. I have the revolver now, so take the shovel. " This reasoning is also based on the indicated scheme.

And a final preliminary example of a logical law, or general scheme of correct reasoning: “The first or the second takes place. But the first is not there. This means that the second takes place. " Let's substitute for the expression "the first" the statement "It is day", and instead of the "second" - the statement "Now is the night". From the abstract scheme we get the reasoning: “It is day or now night. But it is not true that it is day.

So it’s night now. ”

These are some simple schemes of correct reasoning to illustrate the concept of a logical law. Hundreds and hundreds of such schemes are sitting in our heads, although we do not realize it. Based on them, we reason logically, or correctly.

Law of logic (logical law)- an expression that includes only logical constants and variables instead of substantial parts and is true in any area of reasoning.

Let us take as an example an expression consisting only of variables and logical constants, the expression: “If A, then B; then, if notA, then notB. " The logical constants here are the propositional connectives "if, then" and "not." Variables A and B represent statements of some kind. Let's say A is the statement “There is a reason”, and B is the statement “There is a consequence”. With this specific content, we get the reasoning: “If there is a cause, then there is a consequence; it means that if there is no effect, then there is no reason either. " Suppose, further, that instead of A, the statement “The number is divisible by six” is substituted, and instead of B, the statement “The number is divisible by three”. With this specific content, on the basis of the considered scheme, we get the reasoning: “If a number is divisible by six, it is divisible by three. Therefore, if a number is not divisible by three, it is not divisible by six. " Whatever other statements are substituted for variables A and B, if these statements are true, then the conclusion drawn from them will be true.

In logic, a reservation is usually made that the area of objects about which the reasoning is conducted and about which the statements substituted into the logical law speak cannot be empty: it must contain at least one object. Otherwise, reasoning according to a scheme that is a law of logic can lead from true premises to a false conclusion.

For example, from the true premises “All elephants are animals” and “All elephants have a trunk”, according to the law of logic, the true conclusion “Some animals have a trunk” follows. But if the area of objects in question is empty, following the law of logic does not guarantee a true conclusion with true premises. We will argue according to the same scheme, but about mountains of gold. Let's build a conclusion: “All golden mountains are mountains; all mountains of gold are golden; therefore, some mountains are golden. " Both premises of this inference are true. But his conclusion "Some mountains are golden" is clearly false: not a single golden mountain exists.

The concept of a logical law

Thus, for reasoning based on the law of logic, two features are characteristic:

Such reasoning always leads from true premises to true conclusions;

The corollary follows from premises with logical necessity.

The logical law is also called logical tautology.

Logical tautology- an expression that remains true, regardless of what objects are in question, or an "always true" expression.

For example, all the results of substitutions into the logical law of double negation "If A, then it is not true that it is not A" are true statements: "If the soot is black, then it is not true that it is not black." he does not tremble with fear, "and so on.

As already mentioned, the concept of a logical law is directly related to the concept of logical consequence: the conclusion logically follows from the accepted premises, if it is connected with them by a logical law. For example, from the premises “If A, then B” and “If B, then C” the conclusion “If A, then C” logically follows, since the expression “If A, then B, and if B, then C, then if A , then C "is a logical law, namely transitivity law(transitivity). For example, from the premises "If a person is a father, then he is a parent" and "If a person is a parent, then he is a father or mother", according to this law, follows the consequence "If a person is a father, then he is a father or mother."

Logical follow-up- the relationship between the premises and the conclusion of inference, the general scheme of which is a logical law.

Since the connection of logical consequence is based on a logical law, it is characterized by two features:

Logical following leads from true premises only to a true conclusion;

The conclusion following from the premises follows from them with logical necessity.

Not all logical laws directly determine the concept of logical consequence. There are laws that describe other logical connections: "and", "or", "it is not true that", etc. and are only indirectly related to the relationship of logical consequence. This is, in particular, the law of contradiction considered below: “It is not true that an arbitrarily taken statement and

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Those who stubbornly test their life for strength, sooner or later achieve their goal effectively end it.

I realized that in order to understand the meaning of life, it is necessary, first of all, that life is not meaningless and evil, and then only the mind in order to understand it. Tolstoy L. N.

The stronger the love, the more defenseless it is. Duchess Diana (Marie de Bosac)

Once in a lifetime, fortune knocks on the door of every person, but at this time a person often sits in the nearest pub and does not hear any knocking. Mark Twain

I'm not afraid of someone who studies 10,000 different strokes. I am afraid of someone who studies one hit 10,000 times.

I dream about you every day, I think about you at night!

Anyone who cannot dispose of 2/3 of the day for himself should be called a slave. Friedrich Nietzsche

I was one of those who agree to talk about the meaning of life in order to be ready to edit the layout on this topic. Eco W.

Desinit in piscem mulier formosa superne - a beautiful woman on top ends in a fish tail.

We are slaves to our habits. Change your habits, your life will change. Robert Kiyosaki

You could reach forward and grab happiness. It's quite close! But you always only look back

You can always forgive yourself for mistakes, if you only have the courage to admit them. Bruce Lee

The first breath of love is the last breath of wisdom. Anthony Bret.

Friendship is love without wings. Byron

If a person can say what love is, then he did not love anyone.

What you fell in love with, then kiss.

because of a few people I can step over my pride and my fear ...

Our love began at first sight.

Jealousy is treason by suspicion of treason. V. Krotov

With a unique man - I want to repeat!

A romantic woman disgusts sex without love. Therefore, she is in a hurry to fall in love at first sight. Lydia Yasinskaya

Love is inside everyone, but it is worth showing it only to those who are open to you.

The mystery of love for a person begins at the moment when we look at him without a desire to possess him, without a desire to dominate him, without a desire in any way to use his gifts or his personality - we just look and are amazed at the beauty that has opened up to us ... Anthony, Metropolitan of Surozh

I would like to be in a primitive society. You don't need to think about money, about the army, about some titles and scientific degrees. Only females, livestock and slaves are important.

When a person is uncomfortable lying on one side, he rolls over onto the other, and when he is uncomfortable to live, he only complains. And you make an effort to roll over. Maksim Gorky

The slow hand of time smooths the mountains. Voltaire

Women have their whole heart, even their head. Jean Paul

Your kiss was so sweet that I was simply inspired by happiness!

A person stretches, like a sprout, to the Luminary and becomes higher. Dreaming of unrealizable dreams, reaches transcendental heights.

Better real friendship than fake love!

We cannot be deprived of our self-esteem, unless we ourselves give it to Gandhi.

Love is selfishness together.

Knowledge makes a person more important, and actions make him shine. But many people tend to take a look, but not weigh. T. Carlyle

Only in Russia are loved ones called ... Woe, you are mine!

Unrequited love is not love, but torture!

Adequacy is the ability to do two things: to be silent on time and to speak on time.

Happiness comes with right judgments, right judgments come with experience, and experience comes with wrong judgments.

Don't expect it to get easier, easier, better. It won't. There will always be difficulties. Learn to be happy right now. Otherwise, you will not be in time.

Life, happy or unhappy, good or bad, is still extremely interesting. B. Shaw

Do not consider yourself wise; otherwise your soul will be lifted up by pride, and you will fall into the hands of your enemies. Anthony the Great

Caring for his wife seemed to him as ridiculous as hunting roast game. Emil the Meek

Letters and gifts and glossy pictures expressing affection are important. But it is even more important to listen to each other face to face, this is a great and rare art. T. Jansson.

Life is arranged so devilishly skillfully that, not knowing how to hate, it is impossible to sincerely love. M. Gorky

It's nice when your beloved just gives you a huge bouquet, it's nice, damn it!

Without fear, people turn into reckless fools who often give up their lives. Isaac Asimov Fantastic Journey II

A friend is one soul living in two bodies. Aristotrel

Being a person who thinks only of yourself does not mean doing whatever you want. It means wanting the whole world to live the way you want. - O. Wilde

Every mother should carve out a few minutes of free time for herself to wash the dishes.