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.
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 allow precise definition, equally applicable in its different 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 different ways you can build new statements. 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 more simple statements.
Consider the most important ways to build difficult 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 an integer positive number"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 a necessary condition are usually defined: the antecedent (reason) is a sufficient condition for the consequent (consequence), and the consequent is necessary condition for the 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 a causal connection (“If the moon on a new moon is in the node of its orbit, solar eclipse"); 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 distracted, 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.
Simple and complex statements. Denial of a statement
Mathematical logic, the foundations of which were laid by G. Leibniz in the 17th century, was formed as a scientific discipline only in the middle of the 19th century thanks to the works of mathematicians J. Boole and O. Morgan, who created the algebra of logic.
1. Any statement is called declarative sentence that is known to be either true or false. Expressions can be expressed using words, as well as mathematical, chemical and other signs. Here are some examples:
b) 2 + 6> 8 (false statement),
c) the sum of the numbers 2 and 6 more numbers 8 (false statement);
d) II + VI> VII (true statement);
e) extraterrestrial civilizations exist within our Galaxy (this statement is undoubtedly either true or false, but it is not yet known which of these possibilities is fulfilled).
It is clear that statements b) and c) mean the same thing, but they are expressed in different ways. In general, we will write the statements as follows: a: (The moon is a satellite of the Earth); b: (there is such a real number x that 2x + 5 = 15); c: (all triangles are isosceles).
Not every sentence is a statement. For example, exclamation and interrogative sentences are not statements ("What color is this house?", "Drink tomato juice! "," Stop! ", Etc.). Definitions are not statements, for example," Let's call the median the segment connecting the apex of the triangle with the middle of the opposite side. " be true or false, they only record the accepted use of terms. They are not statements and sentences "He has a gray-eyed" or "x 2 - 4x + 3 = 0" - they do not indicate which person in question or for what x is equality considered. Such sentences with an unknown member (variable) are called vague statements. Note that the sentence "Some people are gray-eyed" or "" For all x, the equality x 2 - 4x + 3 = 0 "is already a statement (the first of them is true, and the second is false).
2. A statement that can be decomposed into parts will be called complex, and an indecomposable statement will be called simple. For example, the statement "Today at 4 pm I was at school, and by 6 pm I went to the rink" consists of two parts "Today at 4 pm I was at school" and "Today at 6 pm I went to the ice rink ". Or such a statement:" the function y = ax 2 + bx + c is continuous and differentiable for all values X" consists of two simple statements: "The function y = ax 2 + bx + c is continuous for all values of x" and "the function y = ax 2 + bx + c is differentiable for all values of x".
Just as other numbers can be obtained from the given numbers using the operations of addition, subtraction, multiplication and division, so new statements are obtained from the given statements using operations that have special names: conjunction, disjunction, implication, equivalence, negation. Although these names sound unusual, they only mean the well-known connections of individual sentences with the ligaments "and", "or", "if ... then ...", "if and only if ...", as well as joining the particle "not" to the statement,
3. The negation of a statement a is a statement a such that a is false if a is true, and a is true if a is false. The designation a reads like this: "Not a", or "It is not true that a". Let's try to understand this definition with examples. Consider the following statements:
a: (Today at 12 noon I was at the rink);
b: (Today I was at the rink not at 12 noon);
c: (I was at the rink at 12 noon not today);
d: (I was at school today at 12 noon);
e: (Today I was at the rink at 3 o'clock in the afternoon);
f: (I was not at the rink today at 12 noon);
At first glance, all statements b - f negate statement a. But actually it is not. If you carefully read into the meaning of statement b, you will notice that both statements a and b can turn out to be false at the same time - this will be so if today I have not been at the rink at all. The same applies to statements a and c, a and a. And statements a and e can be both true (if, for example, I skated from 11 to 4 pm), and at the same time false (if today I was not at the rink at all). And only the statement f has the following property: it is true in the case when the statement a is false, and false in the case when the statement a is true. Hence, the statement f is the negation of the statement a, that is, f = a. The following table shows the relationship between the statements a and;
The letters "and" and "l" are abbreviations for the words "true" and "false", respectively. These words in logic are called truth values. The table is called the truth table.
We love very much wise sayings great people. Those whose names are written in golden letters in the history of the world. But also ordinary people, our friends, acquaintances, classmates, sometimes they will “soak off” such a thing - even if you stand, even if you fall. On this page, we have collected for you a kind of mix of the most, in our opinion, interesting statements about life, destiny, love. Creative, humorous, wise, impressive, touching, grabbing for the soul, positive ... for every color and taste)
1. About work and salary
2. About lies and truth
Lies ... have a wide road ... Truth ... a narrow path ... Lies ... have many languages ... But the truth ... stingy words ... Lies ... these are slippery words ... but crawl into any ears ... But the truth ... a thin string ... but breaks through souls !!!
3. The Lord's Ways are inscrutable ...
God doesn't give you the people you want. He gives you the people you need. They hurt you, they love you, they teach you, they break you down to make you who you should be.
4. Cool !!!
So cool! To work only after 20 years!)
5. Payment system ...
It only seems that everything is paid for with money. For everything really important, they pay with pieces of the soul ...
6. You need to see the positive in everything)
If fate has thrown you a sour lemon - think about where to get tequila and have great fun.
7. From Erich Maria Remarque
Who wants to keep - he loses. Those who are ready to let go with a smile - they try to keep them.
8. The difference between a dog and a man ...
If you pick up a hungry dog and make his life full, he will never bite you. This is the fundamental difference between a dog and a person.
9. Only THIS!
10. The road of fate
Every person in his life must go through this. Break someone else's heart. Break your own. And then learn to take care of your own and someone else's heart.
11. What is the strength of character?
Strength of character is not in the ability to break through walls, but in the ability to find doors.
12. Your baby is developing well)
Girls, happiness is not a drag on a cigarette and a sip of beer, happiness is when you come to the doctor and they say to you: “Your baby is developing well, there are no deviations!”
13. From Mother Teresa, vital thought ...
To create a family, it is enough to love. And to preserve, you need to learn to endure and forgive.
14. It seemed)
In childhood it seemed that after thirty - this is old age ... Thank God it seemed!
15. Separate the wheat from the chaff ...
Learn to distinguish between the important and the secondary. Higher education- not an indicator of intelligence. Beautiful words- not an indicator of love. Beautiful appearance is not an indicator handsome man... Learn to appreciate the soul, believe in actions, look at things.
16. From the great Faina Ranevskaya
Take care of your beloved women. After all, while she scolds, worries and freaks out - she loves, but as soon as she begins to smile and be indifferent, you have lost her.
17. About children ...
Deciding to have a child is not a joke. It means deciding that from now on and forever your heart is out of your body.
18. A very wise Portuguese proverb
The hut where they laugh is dearer than the palace where they cry.
19. Listen ...
In life you need to have one important principle- always pick up the phone if it calls you close person... Even if you are offended at him, even if you don’t want to talk, and even more so if you just want to teach a lesson. You must definitely pick up the phone and listen to what he wants to tell you. Perhaps it will be something really important. And life is too unpredictable, and who knows if you will ever hear this person again.
20. Anything can be survived
Everything can be experienced in this life, as long as there is something to live for, someone to love, someone to care for and someone to believe.
21. Errors ... who doesn't have them?
Your mistakes, your strength. On crooked roots, trees stand stronger.
22. Simple prayer
My Guardian Angel ... I'm tired again ... Give me your hand, please, and embrace your wing ... Hold me tight so that I don't fall ... And if I stumble, You lift me up ...
23. From the gorgeous Merlin Monroe)
Of course, my character is not angelic, not everyone can stand it. Well, excuse me ... and I'm not for everyone!
24. Communicate ...
It is foolish not to communicate with a person who is dear to you. It doesn't matter what happened. It may not be at any moment. Can you imagine? Forever. And you won't get anything back.
25. Life dimension
You cannot do anything with the length of your life, but you can do a lot with its width and depth.
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 instance: " 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. sixteen |
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. sixteen.
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", "And great and small rejoice have a good day» ... On the symbolic language 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'll be 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 corresponds to 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 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. nineteen |
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 this 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, the base of implication (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."
|
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:
| 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). |
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| 4... Grip ratio means that the truth (falsity) of one statement does not exclude the falsity (truth) of another (Fig. 24). |
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Incompatibility relation means that statements cannot be true at the same time:
| p | q | p → q | p∧┐q |
| AND | AND | AND | L |
| AND | L | L | AND |
| L | AND | AND | L |
| L | L | AND | L |
| Rice. 26 |
Propositional logic , also called propositional logic, is a branch of mathematics and logic that studies the logical forms of complex statements built from simple or elementary statements using logical operations.
The logic of statements is distracted from the content of statements and studies their truth value, that is, whether the statement is true or false.
The picture above is an illustration of a phenomenon known as the Liar's Paradox. At the same time, in the opinion of the author of the project, such paradoxes are possible only in environments that are not free from political problems, where someone can a priori be branded as a liar. In a natural layered world on the subject of "truth" or "falsehood" is evaluated only for individual statements ... And further in this lesson you will be presented with the opportunity to evaluate on this subject a lot of statements (and then see the correct answers). Including complex statements, in which simpler ones are connected by signs of logical operations. But first let us consider these operations on statements themselves.
Propositional logic is used in computer science and programming in the form of declaring logical variables and assigning them logical values "false" or "true", on which the course of further program execution depends. In small programs where only one boolean variable is involved, this boolean variable is often given a name such as "flag" and is assumed to be "flag raised" when the value of this variable is "true" and "flag is off" when the value of this variable is false. In programs of a large volume, in which there are several or even a lot of logical variables, professionals are required to come up with names of logical variables that have the form of statements and a semantic load that distinguishes them from other logical variables and understandable to other professionals who will read the text of this program.
Thus, a boolean variable with the name "UserRegistered" (or its English-language analogue) can be declared, which has the form of a statement, which can be assigned a boolean value "true" if the conditions are met that the data for registration was sent by the user and this data is recognized as suitable by the program. In further calculations, the values of the variables may change depending on which boolean value ("true" or "false") the variable "UserRegistered" has. In other cases, a variable, for example, with the name "Up to a DayXThere are more than three days left", can be assigned the value "True" up to a certain block of calculations, and in the course of further execution of the program this value can be saved or changed to "false" and the course of further execution depends on the value of this variable. programs.
If a program uses several logical variables, whose names are in the form of statements, and more complex statements are constructed from them, then it is much easier to develop a program if, before developing it, we write down all operations from statements in the form of formulas used in statement logic than we do in the course of this lesson and let's do it.
Logical operations on statements
For mathematical statements, you can always make a choice between two different alternatives "true" and "false", and for statements made in "verbal" language, the concepts of "truth" and "falsity" are somewhat more vague. However, for example, verbal forms such as "Go home" and "Is it raining?" Are not utterances. Therefore, it is clear that statements are such verbal forms in which something is stated ... Interrogative or exclamatory sentences, appeals, as well as wishes or demands are not statements. They cannot be evaluated with the meanings "true" and "false".
Statements, on the other hand, can be viewed as a quantity that can take on two meanings: "true" and "false".
For example, the following judgments are given: "a dog is an animal", "Paris is the capital of Italy", "3
The first of these statements can be evaluated with the symbol "true", the second - "false", the third - "true" and the fourth - "false". This interpretation of propositions is the subject of propositional algebra. We will denote statements with capital Latin letters A, B, ..., and their values, that is, true and false, respectively AND and L... In ordinary speech, connections are used between the statements "and", "or" and others.
These connections allow, connecting various statements with each other, to form new statements - difficult statements ... For example, a bunch of "and". Let the statements be given: " π more than 3 "and saying" π less than 4 ". You can organize a new - a complex statement" π more than 3 and π less than 4 ". Saying" if π irrational, then π ² is also irrational “is obtained by linking two statements with the link“ if - then. ”Finally, we can get from any statement a new - a complex statement - by denying the original statement.
Considering statements as quantities taking values AND and L, we will define further logical operations on statements that allow you to get new ones from these statements - complex statements.
Let two arbitrary statements be given A and B.
1 ... The first logical operation on these statements - conjunction - is the formation of a new statement, which we will denote A ∧ B and which is true if and only if A and B are true. In ordinary speech, this operation corresponds to the connection of utterances by the link "and".
Truth table for conjunction:
| A | B | A ∧ B |
| AND | AND | AND |
| AND | L | L |
| L | AND | L |
| L | L | L |
2 ... The second logical operation on statements A and B- disjunction, expressed as A ∨ B, is defined as follows: it is true if and only if at least one of the original statements is true. In ordinary speech, this operation corresponds to the combination of utterances with the link "or". However, here we have no separating "or", which is understood in the sense of "either-or" when A and B both cannot be true. In the definition of the logic of statements A ∨ B true if only one of the statements is true, and if both statements are true A and B.
Truth table for disjunction:
| A | B | A ∨ B |
| AND | AND | AND |
| AND | L | AND |
| L | AND | AND |
| L | L | L |
3 ... The third logical operation on statements A and B expressed as A → B; the statement thus obtained is false if and only if A true, and B false. A called parcel , B - consequence and the statement A → B - following , also called implication. In ordinary speech, this operation corresponds to the conjunction "if - then": "if A, then B". But in the definition of the logic of statements, this statement is always true, regardless of whether the statement is true or false. B... This circumstance can be briefly formulated as follows: "anything follows from the false." In turn, if A true, and B false, then the whole statement A → B false. It will be true if and only if and A, and B are true. Briefly, it can be formulated as follows: "false cannot follow from the true."
Truth table for following (implication):
| A | B | A → B |
| AND | AND | AND |
| AND | L | L |
| L | AND | AND |
| L | L | AND |
4 ... The fourth logical operation on statements, more precisely on one statement, is called the negation of the statement A and denoted by ~ A(you can also find the use of not the ~ symbol, but the ¬ symbol, as well as the upper overscore above A). ~ A there is a saying that is false when A true and true when A false.
Truth table for negation:
| A | ~ A |
| L | AND |
| AND | L |
5 ... And, finally, the fifth logical operation on statements is called equivalence and is denoted A ↔ B... The resulting statement A ↔ B is a true statement if and only if A and B both are true or both are false.
Truth table for equivalence:
| A | B | A → B | B → A | A ↔ B |
| AND | AND | AND | AND | AND |
| AND | L | L | AND | L |
| L | AND | AND | L | L |
| L | L | AND | AND | AND |
Most programming languages have special characters to denote logical values of statements, they are written in almost all languages as true and false.
Let's summarize the above. Propositional logic studies connections, which are completely determined by the way in which some statements are constructed from others, called elementary. In this case, elementary statements are considered as whole, not decomposable into parts.
Let us systematize in the table below the names, designations and meaning of logical operations on statements (we will soon need them again to solve examples).
| Bunch | Designation | Operation name |
| not | negation | |
| and | conjunction | |
| or | disjunction | |
| if ... then ... | implication | |
| then and only then | equivalence |
For logical operations, the correct ones are laws of logic algebra that can be used to simplify Boolean expressions. It should be noted that in the logic of statements, they are distracted from the semantic content of the statement and are limited to considering it from the position that it is either true or false.
Example 1.
1) (2 = 2) AND (7 = 7);
2) Not (15;
3) ("Pine" = "Oak") OR ("Cherry" = "Maple");
4) Not ("Pine" = "Oak");
5) (Not (15 20);
6) ("Eyes are given to see") AND ("Under the third floor is the second floor");
7) (6/2 = 3) OR (7 * 5 = 20).
1) The value of the statement in the first brackets is "true", the value of the expression in the second brackets is also true. Both statements are connected by the logical operation "AND" (see the rules for this operation above), therefore the logical meaning of this entire statement is "true".
2) The meaning of the statement in brackets is "false". This statement is preceded by a logical operation of negation, therefore the logical meaning of the entire given statement is "truth".
3) The meaning of the statement in the first brackets is "false", the meaning of the statement in the second brackets is also "false". Statements are connected by logical operation "OR" and none of the statements has the value "true". Therefore, the logical meaning of this entire statement is "false".
4) The meaning of the statement in brackets is "false". This statement is preceded by the logical operation of negation. Therefore, the logical meaning of this whole statement is "truth".
5) In the first brackets, the statement in the inner brackets is negated. This statement in the inner brackets has the meaning of "false", therefore, its negation will have the logical meaning of "true". The statement in the second parentheses has the meaning "false". These two statements are connected by the logical operation "AND", that is, "true AND false" is obtained. Therefore, the logical meaning of the entire given statement is "false".
6) The meaning of the statement in the first brackets is "true", the meaning of the statement in the second brackets is also "true". These two statements are connected by the logical operation "AND", that is, "truth AND truth" is obtained. Consequently, the logical meaning of the entire given statement is "truth."
7) The meaning of the statement in the first brackets is "true". The meaning of the statement in the second parentheses is "false". These two statements are connected by the logical operation "OR", that is, "true OR false" is obtained. Consequently, the logical meaning of the entire given statement is "truth."
Example 2. Write down the following complex statements using logical operations:
1) "User is not registered";
2) "Today is Sunday and some employees are at work";
3) "The user is registered if and only if the data sent by the user is found to be valid."
1) p- single statement "User is registered", logical operation:;
2) p- a single statement "Today is Sunday", q- "Some employees are at work", logical operation:;
3) p- a single statement "The user is registered", q- "The data sent by the user is validated", logical operation:.
Solve the examples on the logic of statements yourself, and then see the solutions
Example 3. Calculate the logical values of the following statements:
1) ("There are 70 seconds in a minute") OR ("The running clock shows the time");
2) (28> 7) AND (300/5 = 60);
3) ("TV set - electrical appliance") And (" Glass - wood ");
4) Not ((300> 100) OR ("Thirst can be quenched with water"));
5) (75 < 81) → (88 = 88) .
Example 4. Using logical operations, write down the following complex statements and calculate their logical values:
1) "If the clock does not show the time correctly, then you may not come to class at the wrong time";
2) "In the mirror you can see your reflection and Paris is the capital of the United States";
Example 5. Determine Boolean Expression
(p → q) ↔ (r → s) ,
p = "278 > 5" ,
q= "Apple = Orange",
p = "0 = 9" ,
s= "A hat covers the head".
Propositional logic formulas
The concept of the logical form of a complex statement is clarified using the concept propositional logic formulas .
In examples 1 and 2, we learned to write complex statements using logical operations. In fact, they are called formulas of propositional logic.
To denote statements, as in the above example, we will continue to use the letters
p, q, r, ..., p 1 , q 1 , r 1 , ...
These letters will play the role of variables that take the truth values "true" and "false" as values. These variables are also called propositional variables. We will further call them elementary formulas or atoms .
To construct formulas for the logic of statements, in addition to the above letters, signs of logical operations are used
~, ∧, ∨, →, ↔,
as well as symbols that provide the ability to unambiguously read formulas - left and right brackets.
Concept propositional logic formulas we define as follows:
1) elementary formulas (atoms) are formulas of propositional logic;
2) if A and B- formulas of the logic of statements, then ~ A , (A ∧ B) , (A ∨ B) , (A → B) , (A ↔ B) are also formulas of the logic of statements;
3) only those expressions are formulas of the logic of propositions for which it follows from 1) and 2).
The definition of a propositional logic formula contains an enumeration of the rules for the formation of these formulas. According to the definition, any formula of the logic of statements is either an atom, or is formed from atoms as a result of the consistent application of rule 2).
Example 6. Let p- a single statement (atom) "All rational numbers are real", q- "Some real numbers are rational numbers", r- "some rational numbers are real". Convert the following formulas of the logic of statements into the form of verbal statements:
6) 
.
1) "there are no real numbers that are rational";
2) "if not all rational numbers are real, then no rational numbers being valid ";
3) "if all rational numbers are real, then some real numbers are rational numbers and some rational numbers are real";
4) "all real numbers are rational numbers and some real numbers are rational numbers and some rational numbers are real numbers";
5) "all rational numbers are real if and only if it is not the case that not all rational numbers are real";
6) "there is no place to be, that there is no place to be, that not all rational numbers are real and there are no real numbers that are rational or there are no rational numbers that are real."
Example 7. Make a truth table for a propositional logic formula 
, which in the table can be denoted f
.
Solution. We start compiling a truth table by recording the values ("true" or "false") for single statements (atoms) p , q and r... All possible values are recorded in eight rows of the table. Further, determining the values of the implication operation, and moving to the right in the table, remember that the value is equal to "false" when "false" follows from "truth".
| p | q | r | f | ||||
| AND | AND | AND | AND | AND | AND | AND | AND |
| AND | AND | L | AND | AND | AND | L | AND |
| AND | L | AND | AND | L | L | L | L |
| AND | L | L | AND | L | L | AND | AND |
| L | AND | AND | L | AND | L | AND | AND |
| L | AND | L | L | AND | L | AND | L |
| L | L | AND | AND | AND | AND | AND | AND |
| L | L | L | AND | AND | AND | L | AND |
Note that no atom has the form ~ A , (A ∧ B) , (A ∨ B) , (A → B) , (A ↔ B). Complex formulas have this form.
The number of parentheses in propositional logic formulas can be reduced by assuming that
1) in a complex formula, we will omit the outer pair of parentheses;
2) let's order the signs of logical operations "by seniority":
↔, →, ∨, ∧, ~ .
In this list, the ↔ sign has the most large area action, and ~ is the smallest. The scope of the operation sign is understood as those parts of the propositional logic formula to which the considered occurrence of this sign is applied (to which acts). Thus, it is possible to omit in any formula those pairs of parentheses that can be restored, taking into account the "order of precedence". And when restoring parentheses, all parentheses related to all occurrences of the ~ sign are first placed (in this case, we move from left to right), then to all occurrences of the ∧, and so on.
Example 8. Repair the parentheses in the propositional logic formula B ↔ ~ C ∨ D ∧ A .
Solution. The brackets are restored step by step as follows:
B ↔ (~ C) ∨ D ∧ A
B ↔ (~ C) ∨ (D ∧ A)
B ↔ ((~ C) ∨ (D ∧ A))
(B ↔ ((~ C) ∨ (D ∧ A)))
Not every propositional logic formula can be written without parentheses. For example, in the formulas A → (B → C) and ~ ( A → B) further elimination of parentheses is not possible.
Tautologies and contradictions
Logical tautologies (or simply tautologies) are such formulas of the logic of propositions that if letters are arbitrarily replaced with propositions (true or false), then the result will always be a true proposition.
Since the truth or falsity of complex statements depends only on the meanings, and not on the content of statements, each of which corresponds to a certain letter, then the check of whether a given statement is a tautology can be substituted in the following way... In the expression under study, the values 1 and 0 (respectively "true" and "false") are substituted in place of the letters in all possible ways, and the logical values of the expressions are calculated using logical operations. If all these values are equal to 1, then the expression under study is a tautology, and if at least one substitution gives 0, then this is not a tautology.
Thus, the formula of propositional logic, which takes on the value "true" for any distribution of the values of the atoms included in this formula, is called identically to the true formula or tautology .
The opposite meaning has a logical contradiction. If all the values of the statements are equal to 0, then the expression is a logical contradiction.
Thus, the formula of propositional logic, which takes on the value "false" for any distribution of the values of the atoms included in this formula, is called identically false formula or contradiction .
In addition to tautologies and logical contradictions, there are formulas of the logic of statements that are neither tautologies nor contradictions.
Example 9. Make a truth table for the propositional logic formula and determine if it is a tautology, a contradiction, or neither.
Solution. We compose a truth table:
| AND | AND | AND | AND | AND |
| AND | L | L | L | AND |
| L | AND | L | AND | AND |
| L | L | L | L | AND |
In the values of the implication, we do not find a line in which from "truth" follows "false". All meanings of the original statement are equal to "truth". Consequently, this formula of propositional logic is a tautology.