A statement is a more complex formation than a name. When decomposing statements into simpler parts, we always get one or another name. Let's say the statement "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 exact definition, equally applicable in its various sections.
A statement is considered true if the description given by it corresponds to the real situation, and false if it does not correspond to it. "True" and "false" are called "truth-values of propositions".
From individual statements different ways you can create new sentences. For example, from the statements “The wind is blowing” and “It is raining”, more complex statements can be formed “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 statement is called simple, if it does not include other statements as its parts.
The statement is called complicated if it is received with the help of logical connectives from other more simple sayings.
Consider the most important ways to build complex statements.
negative statement consists of the original statement and negation, usually expressed by the words "not", "it is not true that". A negative proposition is thus a compound proposition: it includes as its part a proposition distinct 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's denote the statements by letters A, B, C,... The full meaning of the concept of negation of a statement is given by the condition: if the statement BUT is true, its negation is false, and if BUT false, its negation 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.
Combining two statements with the word "and" gives a compound statement called conjunction. Statements connected in this way are called "terms of conjunction".
For example, if the statements “Today it is hot” and “Yesterday it was cold” are combined in this way, the conjunction “Today is hot and yesterday it was cold” is obtained.
A conjunction is true only if both statements in it are true; if at least one of its terms is false, then the whole conjunction is false.
In ordinary language, two statements are connected by the union "and" when they are related 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 went to the coat, and I went to the university" as an expression that makes sense 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 components of these statements are not related in meaning. Simplifying the meaning of the conjunction and other logical connectives and for this, abandoning the vague concept of "connection of statements by meaning", logic makes the meaning of these connectives both broader and more specific.
Connecting two sentences with 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 together." For example, the statement “This season I want to go to the Queen of Spades or to Aida” allows for the possibility of visiting the honorary twice. In the statement “He studies at Moscow or Yaroslavl University”, it is implied that the mentioned person studies only at one of these universities.
The first sense of "or" is called non-exclusive. Taken in this sense, the disjunction of two statements means that at least one of these statements is true, whether they are both true or not. Taken in the second exclusive or in a strict sense, the disjunction of two propositions states that one of the propositions is true and the other is false.
A non-exclusive disjunction is true when at least one of its statements 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 using the link "if ..., then ..." and establishing that one event, state, etc. is in one sense or another the basis or condition for the other.
For example: “If there is fire, then there is smoke”, “If a number is divisible by 9, it is divisible by 3”, etc.
A conditional statement is made up of two simpler statements. The one to which the word "if" is prefixed is called foundation, or antecedent(previous), the statement that comes after the word "that" is called consequence, or consequential(subsequent).
By asserting a conditional statement, we first of all mean that it cannot be that what is said in its foundation takes place, but what is said in the consequence is absent. In other words, it cannot happen that the antecedent is true and the consequent false.
In terms of a conditional statement, the concepts of sufficient and necessary conditions are usually defined: the antecedent (base) 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 option, and that choosing such an option is a necessary condition for its rationality.
A typical function of a conditional statement is to substantiate one statement by referring 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 the justified (grounds and consequences) expressed by the conditional statement is difficult to characterize in general view, and only sometimes its 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 conclusion (“If all living multicellular creatures are mortal, and the jellyfish is such a creature, then it is mortal”); secondly, by the law of nature (“If the body is subjected to friction, it will begin to heat up”); thirdly, by causality (“If the Moon at the new moon is at the node of its orbit, solar eclipse»); fourthly, social regularity, rule, tradition, etc. (“If society changes, the person also changes”, “If the advice is reasonable, it must be carried out”).
The connection expressed by the conditional statement is usually connected with the conviction that the consequence necessarily "follows" from the reason and that there is some general law, having been able to formulate which, we could logically deduce the consequence from the reason.
For example, the conditional statement “If bismuth is a metal is plastic”, as it were, implies the general law “Here 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 related to any implied general law or rule (“If I want, I will cut my cloak”); fix any sequence (“If last summer was dry, then this year it is rainy”); to express disbelief in a peculiar form (“If you solve this problem, I will prove Fermat’s last theorem”); opposition (“If elderberry grows in the garden, then an uncle lives in Kyiv”), etc. The multiplicity and heterogeneity of the functions of a 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 a metal, it is an electrical conductor").
The conditional statement finds a very wide application in all areas of reasoning. In logic, it is usually represented by 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, depending on the context, the connection between the ground and the consequence, which is characteristic of a conditional statement, can be expressed with the help of not only “if ..., then ...”, but also other language tools. For example, “Since water is a liquid, it transfers pressure evenly in all directions”, “Although plasticine is not a metal, it is plastic”, “If a tree were a 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.
In asserting the implication, we assert that it cannot happen that its foundation takes place and its consequence does not exist. In other words, an implication is false only if the reason is true and the consequence is false.
This definition assumes, like the previous definitions of connectives, that every proposition is either true or false, and that the truth value of a compound proposition depends only on the truth values of its component propositions and on the way they are connected.
An implication is true when both its reason and its consequence are true or false; it is true if its reason is false and its consequence is true. Only in the fourth case, when the reason is true and the consequence false, is the implication false.
The implication does not imply that statements BUT and AT somehow related to each other in terms of content. In case of truth AT saying "if BUT, then AT" true regardless of whether BUT true or false, and it is connected in meaning with AT or not.
For example, the following 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 is also true when BUT false, and yet again indifferent, true AT or not, and it is related in content to BUT 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 these statements are unlikely to be considered as meaningful, and even less so as true.
Although implication is useful for many purposes, it does not quite fit in with the usual understanding of conditional association. The implication covers many important features of the logical behavior of the conditional statement, but at the same time it is not a sufficiently adequate description of it.
In the last half century, vigorous attempts have been made to reform the theory of implication. At the same time, it was not a question of abandoning the described concept of implication, but of 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 "L if and only if B", formed from the statements of Lee V and decomposed into two implications: "if BUT, then B", and "if B, then BUT". For example: "A triangle is equilateral if and only if it is equiangular." The term "equivalence" also denotes the link "..., if and only if ...", with the help of which this complex statement is formed from two statements. Instead of “if and only if”, “if and only if”, “if and only if”, etc. can be used for this purpose.
If logical connectives are defined in terms of true and false, an equivalence is true if and only if both of its constituent statements have the same truth value, i.e. when both are true or both are false. Accordingly, an equivalence is false when one of its statements is true and the other is false.
Simple and complex sentences. Denial of a statement
Mathematical logic, the foundations of which were laid by G. Leibniz back in the 17th century, was formed as a scientific discipline only in the middle of the 19th century thanks to the work of mathematicians J. Boole and O. Morgan, who created the algebra of logic.
1. A statement is any declarative sentence, which is known to be either true or false. Statements 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 number 8 (false statement);
d) II + VI > VII (true statement);
e) within our Galaxy there are extraterrestrial civilizations (this statement is undoubtedly either true or false, but it is not yet known which of these possibilities is true).
It is clear that statements b) and c) mean the same thing, but they are expressed differently. In general, we will write statements like this: a: (The moon is a satellite of the Earth); b:(there is a real number x such that 2x+5=15); c: (all triangles are isosceles).
Not every sentence is a statement. For example, exclamatory 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 vertex of the triangle with the midpoint of the opposite side." Here, only the name of some object is established. Thus, definitions, but can be true or false, they only fix the accepted use of terms.The sentences "He is gray-eyed" or "x 2 - 4x + 3 \u003d 0" are not statements - they do not indicate which person in question or for what x equality is considered. Such sentences with an unknown member (variable) are called indefinite 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 a statement that cannot be decomposed further - simple. For example, the statement "Today at 4 pm I was at school, and by 6 pm I went to the skating rink" consists of two parts "Today at 4 pm I was at school" and "Today by 6 pm I went to the skating rink ". Or this statement: "the function y \u003d 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 given numbers using the operations of addition, subtraction, multiplication and division, so new statements are obtained from given statements using operations that have special names: conjunction, disjunction, implication, equivalence, negation. Although these names sound unusual, they only mean the well-known connection of individual sentences with connectives "and", "or", "if ... then ...", "if and only if ...", as well as the addition of the particles "not" to the statement,
3. The negation of a proposition a is such a proposition a that a is false if a is true and a is true if a is false. The designation a is read 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 skating rink);
b: (Today I was at the skating rink not at 12 noon);
c: (I was at the skating rink at 12 noon, not today);
d:(Today at 12 noon I was at school);
e: (Today I was at the skating rink at 3 pm);
f:(Today at 12 noon I was not at the skating rink);
At first glance, all propositions b - f negate proposition a. But actually it is not. If you carefully read 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 was not at the skating rink at all. The same applies to the statements a and c, a and a. And statements a and e can turn out to be both true (if, for example, I was skating from 11 to 4 in the afternoon), and at the same time false (if today I was not at the rink at all). And only the proposition f has the following property: it is true if the proposition a is false, and false if the proposition 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 "i" and "l" are abbreviations for the words "true" and "false", respectively. These words in logic are called truth values. The table is called a truth table.
We love very much wise sayings great people. Those whose names are inscribed in golden letters in the history of the world. But also ordinary people, our friends, buddies, classmates, sometimes they will “soak” such a thing - even stand, even fall. On this page we have collected for you a kind of mix of the most, in our opinion, interesting sayings about life, fate, love. Creative, humorous, wise, impressive, touching, soul-catching, positive... for every color and taste)
1. About work and salary
2. About lies and truth
Lies... have a wide road... Truth has a narrow path... Lies... has many languages... But the truth... is stingy with words... Lies... these are slippery words... but they will creep into any ears... And the truth... a thin string... but it breaks through souls!!!
3. Inscrutable are the ways of the Lord...
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 to make you who you are meant to be.
4. Cool!!!
So cool! Back to work in 20 years!
5. Calculation system…
It just seems like 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 threw 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. Who is ready to let go with a smile - they try to keep him.
8. The difference between a dog and a human…
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 human.
9. Only SO!
10. Road of fate
Every person has to go through this in their life. Break someone else's heart. Break yours. And then learn to take care of both your own and someone else's heart.
11. What is the strength of character?
The 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 puff of a cigarette and a sip of beer, happiness is when you come to the doctor and they tell you: “Your baby is developing well, there are no deviations!”
13. From Mother Teresa, a vital thought...
To create a family, it is enough to fall in love. And to save - you need to learn to endure and forgive.
14. It seemed)
As a child, it seemed that after thirty it was old age ... Thank God it seemed!
15. Separate the wheat from the chaff...
Learn to distinguish between the important and the unimportant. Higher education- not an indicator of the mind. Beautiful words- not a sign 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 starts smiling and being indifferent - you have lost her.
17. About children ...
Deciding to have a baby is a big deal. It means deciding that from now on and forever your heart will roam outside your body.
18. Very wise Portuguese proverb
A tent where they laugh is more precious than a palace where they cry.
19. Listen…
One must have in life important principle- always pick up the phone if someone calls you close person. Even if you are offended by him, even if you don’t want to talk, and even more so if you just want to teach him a lesson. You should definitely pick up the phone and listen to what he wants to tell you. Maybe it will be something really important. And life is too unpredictable, and who knows if you will ever hear this person again.
20. Everything can be experienced
Everything can be experienced in this life, as long as there is something to live for, someone to love, someone to take care of and someone to believe.
21. Mistakes... 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 hug me with your wing... Hold me tight so that I don't fall... And if I stumble, You pick me up...
23. From the gorgeous Marilyn Monroe)
Of course, my character is not angelic, not everyone can stand it. Well, I'm sorry ... and I'm not for everyone!
24. Communicate…
It is foolish not to communicate with a person who is dear to you. And it doesn't matter what happened. He may be gone at any moment. Can you imagine? Forever and ever. And you won't get anything back.
25. Life dimension
You can't do anything about the length of your life, but you can do a lot about its breadth and depth.
Negative statements
Among 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 the statement that contradicts the given statement (see "Logical relations between statements. Logical square"). For example: statement "Not all people are greedy" corresponds to the statement "Some people are not greedy».
Considering the negative statement as a complex statement, it is necessary to determine its logical meaning.
Original statement: The sun is shining(R).
Negative statement: The sun is not shining(┐p).
Double negative statement: It's not true that the sun doesn't shine(┐┐p).
| R | ┐p | ┐┐r |
| And | L | And |
| L | And | L |
| Rice. sixteen |
A negative statement is true only if the original statement is false, and vice versa. The law of double negation is connected with the statement of negation: the double negation of an arbitrary statement is equivalent to this statement itself. The conditions for the truth of the negative statement are shown in Fig. sixteen.
complex a statement is considered to consist of several simple statements connected using the logical unions “and”, “or”, “if ..., then ...”, etc. Compound statements include connecting, separating, conditional, equivalent statements, as well as statements denial.
Connective statement (conjunction)- this is a complex statement, consisting of simple ones, connected using the logical connective "and". The logical union "and" (conjunction) can be expressed in natural language by the grammatical unions "and", "but", "however", "and also", etc. For example: “Clouds came running, and it started to rain”, “Both big and small rejoice have a nice 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 union "or" in the connective-dividing sense (either one or the other, or both together). For example: "This student is an athlete or an excellent student" (p⋁q), "Hereditary factors, poor ecology and bad habits are the cause of most diseases(p⋁q⋁r). A weak disjunction is true when at least one of the simple statements included in it is true (see Fig. 17).
Strong disjunction corresponds to the use of the union "either" in the exclusive dividing sense (either one or the other, but not both together). For example: “In the evening I will be in class or go to a disco”, “A person is either alive or dead”. Symbolic notation p⊻q. A strong disjunction is true when only one of its constituent simple statements is true (see Fig. 17).
Conditional statement (implication)- this is a complex statement consisting of two parts connected using the logical union "if ..., then ...". The statement after the “if” particle is called the basis, and the statement after the “then” is called the consequence. In the logical analysis of conditional statements, the basis of the implication is always placed first. In natural language, this rule is often not respected. An example of a conditional statement: "If the swallows fly low, it will rain" (p→q). An implication is false only in one case, when its basis is true, and the consequence is false (see Fig. 17).
equivalent statement- this is a statement consisting of simple ones, connected using the logical union "if and only if, when" ("if and only if ..., then ..."). An equivalent statement implies the simultaneous presence or absence of two situations. In a natural language, equivalence can be expressed by grammatical unions “if ..., then ...”, “only if ...”, etc. For example: “Our team will win only if it prepares well» ( p↔q). An equivalent statement will be true when the statements that make it up are either both true or both false (see Fig. 17).
To formalize the reasoning, it is necessary:
1) find and designate in small consonants of the Latin alphabet simple statements 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 (∧, ⋁, ⊻, →. ↔, ┐) as logical constants;
3) if necessary, place technical signs [...], (...).
On fig. 18 shows an example of the formalization of a complex statement .
I'm already free (p) and (∧), if me not will detain (┐q) or (⋁)not the car breaks down (┐r), then(→) I will come soon (s) .
p ∧ ((┐q ⋁ ┐r) → s
Rice. eighteen
After the statement is written in symbolic form, it is possible to determine the type of the formula. In logic, there are identically true, identically false and neutral formulas. Identical-true formulas, regardless of the values of the variables included in their composition, always take the value "true", and identically-false formulas always take the value "false". Neutral formulas accept both true and false values.
To determine the type of formula, a tabular method is used, an abbreviated method for checking the formula for truth by the “reduction to absurdity” method and reducing the formula to normal form. The normal form of a formula is its expression, which corresponds to the following conditions:
Does not contain signs of implication, equivalence, strict disjunction and double negation;
Negative signs are found only with variables.
Tabular way to determine the type of 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 the formula containing at least one union, a column of its values is built. This takes into account the values of free columns and 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 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 in it. (For example: in the formula in Fig. 18 there are four variables and five unions, therefore, there will be nine columns in the table).
The number of rows in the table is calculated by the formula С = 2n, where n is the number of variables. (There should be sixteen rows in the table according to the formula in Figure 18.)
On fig. 19 shows an example of a truth table.
An abbreviated way to test a formula for truth by reducing to absurdity:
((p⋁q)⋁r)→(p⋁(q⋁r))
1. Suppose that given formula is not identically true. Therefore, for a certain set of values, it takes the value "false".
2. This formula can take 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 implication p⋁(q⋁r) will be false if p is “false” and q⋁r is “false” (see the meaning of the 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 implication base (p⋁q)⋁r is a weak disjunction of these variables. Since a weak disjunction takes the value "false" when all its components are false, the base of the implication (p⋁q)⋁r will also be "false".
6. In paragraph 2, it was established that the base of the implication (p⋁q)⋁r is “true”, and in paragraph 5 that it is “false”. The resulting contradiction indicates that the assumption made by us in Section 1 is erroneous.
7. Since this formula does not take 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 entire set of statements is divided into comparable and incomparable. Incomparable among simple propositions are propositions having different subjects or predicates. For example: “All students are students” and “Some students are excellent students”.
Comparable are statements with the same subjects and predicates and differing in connective and quantifier. For example: “All citizens of the Republic of Belarus have the right to rest” and “No citizen of the Republic of Belarus has the right to rest.”
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Among comparable statements, compatible and incompatible are distinguished.
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 connective, the same logical characteristic. Equivalent statements differ in the verbal expression of the same thought. With the help of the logical square, the relations between these statements are not illustrated.
2. Partial compatibility (oppositeness, subcontrarality). In this relation there are particular affirmative and particular negative statements (I and O). This means that two such statements can be both true, but cannot be both false. If one of them is false, then the other must be true. If one of them is true, then the other is indefinite.
3. subordination (subordination). In this respect, there are general affirmative and particular affirmative statements (A and I), as well as general negative and particular negative statements (E and O).
The truth of a particular statement always follows from the truth of a general statement. While the truth of a particular statement indicates the uncertainty of the general statement.
The falsity of a particular statement always implies the falsity of a general statement, but not vice versa.
Incompatibility relation. Incompatible statements are statements that cannot be both true:
1. Opposite (opposition, contrariety)- in this respect there are generally affirmative and generally negative statements (A and E). This relation means that two such statements cannot be both true, but they can be both false. If one of them is true, then the other must be false. If one of them is false, then the other is indefinite.
2.Contradiction (contradiction)- it contains general affirmative and particular negative statements (A and O), as well as general negative and particular affirmative statements (E and I). Two contradictory statements can neither be both 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, compound statements are incomparable.
Comparable compound statements can be compatible or incompatible.
Compatibility relation means that statements can be both true:
| 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 relation of following (subordination) means that the truth of one statement implies the truth of another, but not vice versa (Fig. 23). |
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| 4. Clutch ratio means that the truth (falsity) of one statement does not exclude the falsity (truth) of another (Fig. 24). |
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Relationship of incompatibility means that statements cannot be both true:
| 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 - 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 propositions is abstracted from the meaningful load of propositions and studies their truth value, that is, whether the proposition is true or false.
The figure above is an illustration of a phenomenon known as the Liar 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 be branded a liar a priori. In the natural layered world on the subject of "truth" or "falsehood" is evaluated only separately taken statements . And later in this lesson, you will be introduced to the opportunity to evaluate many statements on this subject (and then look at the correct answers). Including complex statements in which simpler ones are interconnected by signs of logical operations. But first let us consider these operations on propositions themselves.
Propositional logic is used in computer science and programming in the form of declaring logical variables and assigning them the logical values "false" or "true", on which the course of further execution of the program depends. In small programs where only one boolean variable is involved, that boolean variable is often given a name, such as "flag" and the "flag" is implied when that variable's value is "true" and "flag is down" when the value of this variable is "false". In large programs, 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 is understandable to other professionals who will read the text of this program.
So, a logical variable with the name "UserRegistered" (or its English equivalent) can be declared, having the form of a statement, which can be assigned the logical value "true" if the conditions are met that the data for registration is sent by the user and this data is recognized by the program as valid. In further calculations, the values of the variables may change depending on what logical value ("true" or "false") the "UserLogged in" variable has. In other cases, a variable, for example, with the name "More than Three Days Until Day", can be assigned the value "True" up to a certain block of calculations, and during the 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 have the form of propositions, and more complex propositions are built from them, then it is much easier to develop a program if, before developing it, all operations from propositions are written in the form of formulas used in propositional logic than we do in the course of this lesson and let's do it.
Logical operations on statements
For mathematical statements, one can always choose between two different alternatives "true" and "false", but for statements made in "verbal" language, the concepts of "true" and "false" are somewhat more vague. However, for example, such verbal forms as "Go home" and "Is it raining?" are not utterances. Therefore, it is clear that utterances are 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 by the values "true" and "false".
Propositions, on the other hand, can be viewed as a quantity that can take on two values: "true" and "false".
For example, 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". Such an interpretation of propositions is the subject of propositional algebra. We will denote statements in 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 make it possible, by combining various statements, to form new statements - complex statements . For example, a bunch of "and". Let the statements be given: π greater than 3" and the statement " π less than 4. You can organize a new - complex statement " π more than 3 and π less than 4". The statement "if π irrational, then π ² is also irrational" is obtained by linking two statements with the link "if - then". Finally, we can get a new - complex statement - from any statement - negating the original statement.
Considering propositions as quantities taking on the values And and L, we define further logical operations on statements , which allow us to obtain new - complex statements from these 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 true. In ordinary speech, this operation corresponds to the connection of statements with a bunch of "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 connection of statements with a bunch of "or". However, here we have a non-separative "or", which is understood in the sense of "either-or" when A and B both cannot be true. In the definition of propositional logic 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 resulting statement is false if and only if A true, and B false. A called parcel , B - consequence , and the statement A → B - following , also called an implication. In ordinary speech, this operation corresponds to the link "if - then": "if A, then B". But in the definition of propositional logic, this proposition is always true, regardless of whether the proposition is true or false B. This circumstance can be briefly formulated as follows: "anything you like 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 A, and B true. Briefly, this can be formulated as follows: "false cannot follow from the true."
Truth table to follow (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 a statement. A and denoted by ~ A(you can also find the use of not the symbol ~, but the symbol ¬, as well as the overline over A). ~ A there is a statement 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 propositions 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 true or both 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 symbols for logical values of statements, they are written in almost all languages as true (true) and false (false).
Let's summarize the above. propositional logic studies connections that are completely determined by the way in which some statements are built from others, called elementary ones. Elementary statements are considered as whole, not decomposable into parts.
We systematize in the table below the names, designations and meaning of logical operations on statements (we will need them again soon to solve examples).
| Bundle | Designation | Operation name |
| not | negation | |
| and | conjunction | |
| or | disjunction | |
| if...then... | implication | |
| then and only then | equivalence |
For logical operations are true laws of the algebra of logic, which can be used to simplify boolean expressions. At the same time, it should be noted that in the logic of propositions they are abstracted from the semantic content of the proposition 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), so the logical value of this entire statement is "true".
2) The meaning of the statement in brackets is "false". This proposition is preceded by a logical negation operation, so the logical value of this entire proposition is "true".
3) The meaning of the statement in the first brackets is "false", the meaning of the statement in the second brackets is also "false". The statements are connected by the logical operation "OR" and none of the statements has the value "true". Therefore, the logical meaning of this whole statement is "false".
4) The meaning of the statement in brackets is "false". This statement is preceded by a logical negation operation. Therefore, the logical meaning of the whole given statement is "true".
5) In the first brackets, the statement in the inner brackets is negated. This statement in parentheses evaluates to "false", so its negation will evaluate to the logical value "true". The statement in the second brackets has the value "false". These two statements are connected by the logical operation "AND", that is, "true AND false" is obtained. Therefore, the logical meaning of the whole 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, "true AND truth" is obtained. Therefore, the logical meaning of the whole given statement is "true".
7) The meaning of the statement in the first brackets is "true". The meaning of the statement in the second brackets is "false". These two statements are connected by the logical operation "OR", that is, "true OR false" is obtained. Therefore, the logical meaning of the whole given statement is "true".
Example 2 Write down the following complex statements using logical operations:
1) "User not registered";
2) "Today is Sunday and some employees are at work";
3) "The user is registered when and only when the data sent by the user is found to be valid."
1) p- single statement "User is registered", logical operation: ;
2) p- single statement "Today is Sunday", q- "Some employees are at work", logical operation: ;
3) p- single statement "User is registered", q- "Data sent by the user is valid", logical operation: .
Solve propositional logic examples on your own and then look at the solutions
Example 3 Calculate the boolean values of the following statements:
1) ("There are 70 seconds in a minute") OR ("A 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 Write down the following complex statements using logical operations and calculate their logical values:
1) "If the clock does not show the time correctly, then you can come to class at the wrong time";
2) "In the mirror you can see your reflection and Paris - the capital of the USA";
Example 5 Determine Boolean Expression
(p → q) ↔ (r → s) ,
p = "278 > 5" ,
q= "Apple = Orange",
p = "0 = 9" ,
s= "The hat covers the head".
Propositional logic formulas
The concept of the logical form of a complex statement is specified with the help of the concept propositional logic formulas .
In examples 1 and 2, we learned how to write complex statements using logical operations. In fact, they are called propositional logic formulas.
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 henceforth call them elementary formulas or atoms .
To construct propositional logic formulas, in addition to the above letters, the signs of logical operations are used
~, ∧, ∨, →, ↔,
as well as symbols that provide the possibility of unambiguous reading of formulas - left and right brackets.
concept propositional logic formulas define as follows:
1) elementary formulas (atoms) are formulas of propositional logic;
2) if A and B- propositional logic formulas, then ~ A , (A ∧ B) , (A ∨ B) , (A → B) , (A ↔ B) are also formulas of propositional logic;
3) only those expressions are propositional logic formulas for which this 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, every formula of propositional logic is either an atom or is formed from atoms as a result of the successive application of rule 2).
Example 6 Let be p- single statement (atom) "All rational numbers are real", q- "Some real numbers are rational numbers", r- "some rational numbers are real". Translate into the form of verbal propositions the following formulas of propositional logic:
6) 
.
1) "there are no real numbers that are rational";
2) "if not all rational numbers are real, then no rational numbers, which are 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) "it is not the case that it is not the case that not all rational numbers are real and there are no real numbers that are rational or no rational numbers that are real."
Example 7 Make a truth table for the propositional logic formula 
, which in the table can be denoted f
.
Decision. We start compiling the truth table by recording the values ("true" or "false") for single statements (atoms) p , q and r. All possible values are written in eight rows of the table. Further, when defining the values of the implication operation, and moving to the right in the table, remember that the value is equal to "false" when "true" implies "false".
| 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) . These are complex formulas.
The number of brackets in propositional logic formulas can be reduced if we assume that
1) in a complex formula, we will omit the outer pair of brackets;
2) order the signs of logical operations "by seniority":
↔, →, ∨, ∧, ~ .
In this list, the ↔ sign has the most large area actions, and the ~ sign is the smallest. The scope of an operation sign is understood as those parts of the propositional logic formula to which the considered occurrence of this sign is applied (acted). Thus, it is possible to omit in any formula those pairs of brackets that can be restored, taking into account the "order of precedence". And when restoring brackets, first all brackets are placed that refer to all occurrences of the ~ sign (in this case, we move from left to right), then to all occurrences of the ∧ sign, and so on.
Example 8 Restore parentheses in propositional logic formula B ↔ ~ C ∨ D ∧ A .
Decision. 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 brackets. For example, in formulas BUT → (B → C) and ~( A → B) no further deletion of brackets is possible.
Tautologies and contradictions
Logical tautologies (or simply tautologies) are such formulas of propositional logic that if letters are arbitrarily replaced by 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 test 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 for the letters in all possible ways, and using logical operations, the logical values of the expressions are calculated. 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, a propositional logic formula that takes the value "true" for any distribution of the values of the atoms included in this formula is called identically true formula or tautology .
The opposite meaning is a logical contradiction. If all proposition values are 0, then the expression is a logical contradiction.
Thus, a propositional logic formula that takes 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 propositional logic that are neither tautologies nor contradictions.
Example 9 Make a truth table for the propositional logic formula and determine whether it is a tautology, a contradiction, or neither.
Decision. We make 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 meanings of the implication, we do not find a line in which "true" implies "false". All values of the original statement are equal to "true". Therefore, this propositional logic formula is a tautology.