We love the wise sayings of great people. Those whose names are inscribed in golden letters in the history of the world. But even ordinary people, our friends, buddies, classmates, sometimes they will “soak” this - even stand, even fall. On this page we have collected for you a kind of mix of the most, in our opinion, interesting statements 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 is not an indicator of the mind. Beautiful words are not a sign of love. Beautiful appearance is not an indicator of a beautiful person. 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…
In life, you need to have one important principle - always pick up the phone if a loved one calls you. 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.
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Those who stubbornly test their life for strength, sooner or later achieve their goal and end it spectacularly.
I realized that in order to understand the meaning of life, it is necessary first of all that life be not meaningless and evil, and then the mind in order to understand it. Tolstoy L.N.
The stronger the love, the more defenseless it is. Duchess Diana (Marie de Bosack)
Once in a lifetime, fortune knocks on the door of every person, but at this time a person often sits in the nearest pub and does not hear any knock. Mark Twain
I am not afraid of someone who learns 10,000 different strokes. I fear the one who learns one punch 10,000 times.
I dream about you every day, I think about you at night!
He who cannot have 2/3 of the day for himself should be called a slave. Friedrich Nietzsche
I was one of those who agree to talk about the meaning of life in order to be ready to edit the layout on this topic. Eco W.
Desinit in piscem mulier formosa superne - a beautiful woman from above ends in a fish tail.
We are slaves of our habits. Change your habits, your life will change. Robert Kiyosaki
You could reach out and grab happiness. It's right next to it! But you always look back
You can always forgive yourself for mistakes, if only you have the courage to admit them. Bruce Lee
The first breath of love is the last breath of wisdom. Anthony Bret.
Friendship is love without wings. Byron
If a person can say what love is, it means that he did not love anyone.
What you fell in love with, then kiss.
because of a few people I can overcome my pride and my fear ...
Our love began at first sight.
Jealousy is treason by suspicion of treason. V. Krotov
With a unique man - I want to repeat!
A romantic woman is disgusted by sex without love. Therefore, she hurries to fall in love at first sight. Lydia Yasinskaya
Love is inside everyone, but it is worth showing it only to those who are open to you.
The secret of love for a person begins at the moment when we look at him without the desire to possess him, without the desire to dominate him, without the desire in any way to use his gifts or his personality - we just look and are amazed at the beauty that has been revealed to us . Anthony, Metropolitan of Sourozh
I would like to be in a primitive society. No need to think about money, about the army, about some titles and scientific degrees. Only females, cattle and slaves are important.
When it is uncomfortable for a person to lie on one side, he rolls over to the other, and when it is uncomfortable for him to live, he only complains. And you make an effort to roll over. Maksim Gorky
The slow hand of time smoothes the mountains. Voltaire
Women have the whole heart, even the head. Jean Paul
Your kiss was so sweet that I just winged with happiness!
A person stretches, like a sprout, to the Luminary and becomes taller. Dreaming of unrealizable dreams, reaches sky-high heights.
Real friendship is better than fake love!
We cannot be deprived of self-respect unless we ourselves give it to Gandhi
Love is selfishness together.
Knowledge makes a person more significant, and actions give him brilliance. But many people tend to look but not weigh. T. Carlyle
Only in Russia they call their loved ones ... Woe is mine!
Unrequited love is not love, but torture!
Adequacy is the ability to do two things: keep silent at the right time and speak at the right time.
Happiness comes with right judgments, right judgments come with experience, and experience comes with wrong judgments.
Don't expect it to get easier, easier, better. It won't. There will always be difficulties. Learn to be happy right now. Otherwise, you won't be able to.
Life, happy or unhappy, successful or unsuccessful, is still extremely interesting. B. Show
Do not consider yourself wise, otherwise your soul will be lifted up with pride, and you will fall into the hands of your enemies. Anthony the Great
Courting his wife seemed to him as absurd as hunting for roast game. Emil Krotky
Letters and gifts and glossy pictures expressing tenderness are important. But it is even more important to listen to each other face to face, this is a great and rare art. T. Jansson.
Life is arranged so devilishly skillfully that, without knowing how to hate, it is impossible to sincerely love. M. Gorky
It's nice when a loved one gives you a huge bouquet just like that, because it's nice, damn it!
Without fear, people turn into reckless fools who often lose their lives. Isaac Asimov Fantastic Journey II
A friend is one soul living in two bodies. Aristotle
Being a person who thinks only of himself does not mean doing whatever you want. It means wanting the whole world to live the way you want. — O. Wilde
Every mother should carve out for herself a few minutes of free time to wash the dishes.
Under saying a linguistic expression is understood, about which only one of two things can be said: it is true or false. The statement, unlike judgments, does not have a personal character.
Questions, requests, orders, exclamations, individual words (except when they act as representatives of statements such as "it's getting evening", "it's getting colder", etc.) are not statements. The truth and falsity of propositions are their boolean values.
Statements are divided into attributive, existential and relational.
attributive are called statements in which a property or state of an object is affirmed or denied.
existential are called statements that affirm or deny the fact of existence.
relational are called statements expressing relations between objects.
Statements, like their logical forms, are simple and complex. complex statements can be broken down into simple ones. Simple statements are not divided into simpler ones.
A simple attributive statement has a structure that includes a subject, a predicate, and a connective.
Subject statements (S) - this is the part of the statement that expresses the subject of thought.
Predicate statements (P) - this is a part of the statement, which displays the sign of the subject of thought, its property, state, attitude.
Subject (S) and predicate (P) are called terms. Bundle indicates the relationship between the terms (S and P).
Attributive statements often use existential and general quantifiers.
Attributive statements are divided according to quality and quantity.
By quality, they are divided into affirmative and negative. AT affirmative indicates the belonging (presence) of the sign, conceivable in the predicate, to the subject of the statement: "S is P". For example: "Plato is an idealist philosopher." AT negative indicates that the predicate does not belong to its subject: "S is not P".
According to the number of statements are divided into single, private and general. This refers to the totality (number, quantity) of individual items that make up the name of the subject class.
AT single In utterances, the subject consists of one object.
Private statements are of the form: "Some S are (are not) P".
AT general In utterances, the subject embraces all objects. Such statements have the form: "All S is (is not) P".
Statements are classified according to quality and quantity. There are 4 classes of statements:
1) general affirmative (BUT) - general in quantity and affirmative in quality (“All S is P”);
2) private affirmative (J)- private in quantity and affirmative in quality ("Some S are R");
3) common negative (E) - general in quantity and negative in quality (“Not a single S is P”);
4) private negative (O)- private in quantity and negative in quality ("Some S are not P").
In each class of statements, the ratio of the volumes of S and P (terms) is different. In logic, the problem of the ratio of volumes S and P is called term distribution problem. A term is distributed if it is completely included in the scope of another term or completely excluded from it.
In class A | All S is P | the subject is fully distributed in the predicate, and the predicate is not distributed.
Simple and complex statements, logical variables and logical constants, logical negation, logical multiplication, logical addition, truth tables for logical operations
To automate information processes, it is necessary to be able not only to uniformly present information of various types (numerical, textual, graphic, sound) in the form of sequences of zeros and ones, but also to determine the actions that can be performed on information. The performance of such actions is carried out in accordance with the rules that govern the process of thinking. In other words, in accordance with the laws of logic. The term "logic" is derived from the ancient Greek word1 about§08 , meaning "thought, reasoning, law." The sciencelogicsstudies the laws and forms of thinking, methods of proof.
To describe the reasoning and the rules for performing actions with information, a special language is used, adopted in mathematical logic. Reasoning is based on special sentences called propositions. In statements, something is always affirmed or denied about objects, their properties and relations between objects. A proposition is any proposition that can be said to be true or false. Statements can only be declarative sentences. Interrogative or imperative sentences are not statements.
statement - a proposition formulated as a declarative sentence, about which it can be said whether it is true or false.
For example, interrogative sentences “In what year was the first chronicle mention of Moscow?” and "What is a computer's external memory?" or the incentive sentence "Observe safety regulations in the computer lab" are not statements. The declarative sentences “The first annalistic mention of Moscow was in 1812”, “Random Access Memory is an external memory of a computer” and “In a computer class you do not need to follow safety rules” are statements, since these are judgments, each of which can be said, that it is false. True statements will be the judgments "The first annalistic mention of Moscow was in 1147", "A hard magnetic disk is an external memory of a computer."
Each statement corresponds to only one of two values: either "true" or "false", which areboolean constants.The true value is usually denoted by the number 1, and the false value by the number 0. Statements can be denoted usingboolean variables,which are used as capital Latin letters. Boolean variables can only take one of two possible values: "true" or "false". For example, the statement "Information in a computer is encoded using two characters" can be denoted by a logical variableBUT,and the statement "The printer is an information storage device" can be denoted by a logical variableAT.Since the first statement is true, thenBUT= 1. This notation means that the statementBUTtrue. Since the second statement is not true, thenB =0. Such a notation means that the statement in is false.
Statements can be simple or complex. The statement is calledsimple,if no part of it is a statement. So far, examples of simple statements have been given, which are denoted by logical changes. Building a chain of reasoning, a person using logical operations combines simple statements intoharder" statements.To find out the meaning of a complex statement, there is no need to think about its content. It is enough to know the meaning of simple statements that make up a complex statement, and the rules for performing logical operations.
Boolean operation - an action that allows you to make a complex statement from simple statements.
All human reasoning, as well as the operation of modern technical devices, are based on typical actions with information - three logical operations: logical negation (inversion), logical multiplication (conjunction) and logical addition (disjunction).
Logical negation a simple statement is obtained by adding words"It's not true" at the beginning of a simple sentence.
■ EXAMPLE 1.There is a simple saying "Crocodiles can fly". The result of logical negation is the statement"It is not true that crocodiles can fly. The value of the original statement is "false" and the value of the new one is "true".
■ EXAMPLE 2.There is a simple saying "The file must have a name". The result of logical negation is the statement"It is not true that the file must have a name. The value of the original statement is "true" and the value of the new statement is "false".
It can be seen that the logical negation of the statement is true when the original statement is false, and vice versa, the logical negation of the statement is false when the original statement is true.
Logical negation (inversion) - a logical operation that associates a simple statement with a new statement, the meaning of which is opposite to the value of the original statement.
Denote a simple statement by a boolean variableBUT.Then the logical negation of this statement will be denoted NOTBUT. Let's write down all possible values of the boolean variableBUTand the corresponding results of the logical negation NOTBUT in the form of a table calledtruth table for logical negation (Table 40).
TRUTH TABLE FOR LOGICAL NEGATIVE
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If /1 = 0, thenNOT A= 1 (see Example 1). If aBUT= 1, thenNOT A= 0 (see example 2) |
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You can see that in the truth table for logical negation, zero changes to one, and one changes to zero.
Boolean multiplicationtwo simple propositions are obtained by combining these propositions using the unionand.Let's look at examples 3-6, which will be the result of logical multiplication.
■ EXAMPLE3. There are two simple statements. One statement - "Carlson lives in the basement." Another statement is "Carlson is treated with ice cream."
The result of the logical multiplication of these simple statements will be the complex statement “Carlson lives in the basement,andCarlson is treated with ice cream. You can formulate a new statement more briefly: “Carlson lives in the basementandtreated with ice cream. Both original statements are false. The meaning of the new compound statement is also "false".
■ EXAMPLE 4.There are two simple statements. The first statement is "Carlson lives in the basement." The second statement is "Carlson is treated with jam."
The result of the logical multiplication of these simple statements will be the complex statement “Carlson lives in the basementandcured with jam. The first original statement is false, and the second is true. The meaning of the new compound statement is "false".
■ EXAMPLE 5.There are two simple statements. The first statement is "Carlson lives on the roof." The second statement is "Carlson is treated with ice cream."
The result of the logical multiplication of these simple statements will be the complex statement "Carlson lives on the roofandtreated with ice cream. The first original statement is true, and the second is false. The meaning of the new compound statement is "false".
* EXAMPLEb. There are two simple statements. One statement - "Carlson lives on the roof." Another statement is "Carlson is treated with jam."
The result of the logical multiplication of these simple statements will be the complex statement "Carlson lives on the roof and is treated with jam." Both original statements are true. The meaning of the new compound statement is also "true".
It can be seen that the logical multiplication of two statements is true only in one case - when both original statements are true.s.
Boolean multiplication (conjunction) - a logical operation that associates two simple propositions with a new proposition whose value is true if and only if both original propositions are true.
TRUTH TABLE FOR LOGICAL MULTIPLICATION
Table 41
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AandB |
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If aBUT = 0, AT =0, then A and B0 (see example 3). If aA = 07? = 1, thenBUTAndAT -0 (see example 4). If /1 = 1,B =0, thenBUTAnd d=0 (see example 5). If L= \, B = \, then A\\ B = \(see example 6).
You can see that the results of logical multiplication are the same as the results of the usual multiplication of zeros and ones.
Boolean additiontwo simple propositions are obtained by combining these propositions using the unionor.Let's look at examples 7-10, which will be the result of logical addition.
EXAMPLE 7 . There are two simple statements. One statement - "The comedy" The Inspector General "was written by M. Yu. Lermontov." Another statement - "The comedy" Inspector General "was written by I. A. Krylov."
The result of the logical addition of these simple statements will be a complex statement “The comedy“ Inspector General ”was written by M. Yu. LermontovorI. A. Krylov. Both original statements are false. The meaning of the new compound statement is also "false".
EXAMPLE 8. There are two simple statements. The first statement - "The comedy" Inspector General "was written by M. Yu. Lermontov." The second statement - "The comedy" The Inspector General "was written by N. V. Gogol."
The result of the logical addition of these simple propositionsnythere will be a complex statement “The comedy“ Inspector General ”was written by M, K). LermontovorN. V. Gogol. The first initialthe statement is false and the second is true. The meaning of the new compound statement is "truth".
EXAMPLE 9 . There are two simple statements. The first statement - "The poem" Mtsyri "was written by M. Yu. Lermontov." The second statement - "The poem" Mtsyri "was written by N. V. Gogol". The result of the logical addition of these simple statements will be a complex statement "The poem" Mtsyri "was written by M. Yu. Lermontov or N. V. Gogol." The first statement is true and the second is false. The meaning of the new compound statement is "truth".
EXAMPLE 10 . There are two simple statements. One sentence - "A. S. Pushkin wrote poetry” Another statement is “A. S. Pushkin wrote prose.” The result of the logical addition of these simple statements will be the complex statement “A. S. Pushkin wrote poetry or prose.” Both original statements are true. The meaning of the new compound statement is also "true".
It can be seen that the logical addition of two statements is false only in one case - when both original statements are false.
Logical addition (disjunction)- a logical operation that associates two simple statements with a new statement, the value of which is false if and only if both original statements are false.
Denote one simple proposition by the boolean variable A, and the other simple proposition by the boolean variable B.
Then the logical addition of these statements will be denoted BUT OR AT
Let's write down all possible values of logical variables A , B , as well as the corresponding result of logical addition A OR B in the form of a table called the truth table.
Operations with binary signs are performed according to the truth tables for logical addition
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If A=0, B=0, then A OR B=0 (see example 7) If A \u003d 0, B \u003d 1, then A OR B \u003d 1 (see example 8) If A=1, B=0, then A OR B=1 (see example 9) If A=1, B=1, then A OR B=1 (see example 10) |
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You can see that the results of the logical addition, except for the last line, are the same as the results of the usual addition of zeros and ones.
Thus, using the language of logic, reasoning can be replaced by actions with statements. Statements, in turn, can be assigned a binary sign - 0 or 1. Actions with binary signs are performed in accordance with the truth tables for the basic logical operations of logical negation, logical multiplication and logical addition (see Tables 40-42)
23. Statements. Boolean operations
Logical addition (disjunction) of two statements is false
1) if and only if both statements are true
2) if and only if both statements are false
3) when at least one statement is true
4) when at least one statement is false
Boolean expressions. Performing Boolean Operations
Recording logical expressions, priority of executing logical operations, finding the value of a logical expression, performing logical operations with various types of information Logical negation, logical multiplication and logical addition form a complete system of logical operations with which you can compose any complex statement and determine its truth. When describing reasoning using the language of mathematical logic, simple statements are denoted by logical variables (Latin letters), the values of statements are denoted by logical constants (zeros or ones), and logical operations are denoted by special connectives (NOT, AND, OR). The record, compiled with the help of such variables, constants and connectives, is called a logical expression.
Logical expression - a symbolic notation in the language of mathematical logic, composed of logical variables or logical constants, united by logical operations (connections).
When finding the value of a logical expression, logical operations are performed in a certain order, according to their priority - first, logical negation, then logical multiplication, and only then logical addition. Logical operations that have the same priority are executed from left to right. Parentheses are used to change the order in which logical operations are performed.
■ EXAMPLE 1. A simple true statement A = "Aristotle is an ancient Greek philosopher" and a simple false statement B = "Aristotle is an ancient Russian philosopher" are given.
Actions on information. Basic operations
meanings of compound statements that correspond to the following logical expressions:
1) NOT A;
2) A OR B;
3) A AND (NEV).
Decision. 1) The result of the logical negation of statement A will be the statement "It is not true that Aristotle is an ancient Greek philosopher." Since the value of the original statement "true" A = 1, then the value of the logical negation of this statement "false" is NOT A = 0 (see Table 40). 2) The result of the logical addition of two statements will be the statement "Aristotle is an ancient Greek or Aristotle is an ancient Russian philosopher." Since the value of the first initial statement "true" A = 1, and the value of the second initial statement "false" B = 0, then the value of the logical addition of these statements "true" A OR B = 1 (see Table 42). 3) The result of the logical multiplication of statement A and the logical negation of statement B will be the statement "Aristotle is an ancient Greek philosopher, and it is not true that Aristotle is an ancient Russian philosopher." First, we perform the logical negation of statement B. Since the value of the original statement “false” B = 0, then the value of the logical negation of this statement “true” is NOT B = 1 (see Table 40). Since the value of the first original statement "true" A = 1 and the value of the logical negation of the second original statement "true" NOT B =1, then the value of the logical multiplication of these statements "true" A AND (NOT B) =1
(see tab. 41)
Answer. 1) "Lie"; 2) "truth"; 3) "truth". To find the meaning of a complex statement, it is sufficient to know the meanings of the simple statements included in the complex statement and the rules for performing logical operations that combine these simple statements.
■ EXAMPLE 2. Find the value of the logical expression NOT A OR (0 OR 1) AND (NOT B AND 1), if the values of the logical variables A =1, B =0.
Decision. 1) Let's replace the logical variables in the logical expression with logical constants. NAIOR(0OR 1) AND(NEVI 1)==NOT1OR(0OR1) AND(NOT0AND1).
2) Let's determine the sequence of execution of logical operations in accordance with their priority. NOT4 1 OR6 (0 OR1 1) AND5 (HOT 0 AND3 1).
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 for a precise definition that is 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 in different ways, you can build new statements. 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 obtained with the help of logical connectives from other simpler statements.
Let's consider the most important ways of constructing 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 a positive integer" 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 at Yaroslavl University" it is understood 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, according to 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 a sufficient and necessary condition are usually defined: an antecedent (base) is a sufficient condition for a consequent (consequence), and a consequent is a necessary condition for an antecedent. For example, the truth of the conditional statement "If the choice is rational, then the best available alternative is chosen" means that rationality is a sufficient reason for choosing the best available 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 justifier and the justified (grounds and consequences) expressed by the conditional statement is difficult to characterize in a general way, and only sometimes the nature of it 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 is at the node of its orbit at the new moon, a solar eclipse occurs”); 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 reason and the consequence, which is characteristic of a conditional statement, can be expressed using not only “if ..., then ...”, but also other linguistic means. 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.
