Simple quotes. Compound statements

Negation, conjunction, disjunction.

Our reasoning is made up of statements. For example, in the conclusion “Some birds fly; therefore, some flying - birds " includes two different statements.

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.

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

The concept of a statement is one of the initial, key concepts of logic. As such, it does not allow for a precise definition that is equally applicable in its various sections. It is clear that any statement describes a certain situation, asserting or denying something about it, and is true or false.

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 the truth-values ​​of the proposition.

From individual statements in different ways, you can build new statements. So, 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, the wind is blowing”, etc. Expressions "and", "either, or", "if, then", etc., which serve to form complex statements, are called logical connectives.

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

A statement is complex if it is obtained with the help of logical connectives from other, simpler statements.

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

Negation - a logical connective, with the help of which a new statement is obtained from a given statement, such that if the original statement is true, its negation is false, and vice versa. A negative statement consists of 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").

As a result of connecting two statements with the help of the word "and", we get a complex statement called a conjunction. Statements connected in this way are called members of the 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 entire conjunction is false.

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

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

the truth value of a compound statement depends only on the truth values ​​of the statements included in it and the way they are logically connected with each other.

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

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 wore a 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 its constituent statements are not related in meaning.

By simplifying the meaning of the conjunction and other logical connectives and for this, abandoning the obscure concept of "connection of statements by meaning", logic makes the meaning of these connectives both broader and clearer at the same time.

By connecting two statements with the word "or", we get a disjunction of 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." The statement “This season I want to go to the Queen of Spades or to Aida” allows for the possibility of visiting the opera twice. In the statement “He studies at Moscow or Leningrad University”, it is understood that the person in question studies at only one of these universities.

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

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

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

The central task of logic is the separation of correct reasoning schemes from incorrect ones and the systematization of the former. Logical correctness is determined by the logical form. To reveal it, one must abstract from the meaningful parts of the reasoning (proper symbols) and focus on improper symbols that represent this form in its purest form. Hence the interest of formal logic in words that usually do not attract attention, such as "and", "or", "if, then", etc.

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...

Love is enough to create a family. 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 hut 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.

Clever thoughts come only when stupid things have already been done.

Only those who make absurd attempts can achieve the impossible. Albert Einstein

Good friends, good books, and a sleeping conscience are the ideal life. Mark Twain

You can't go back in time and change your start, but you can start now and change your finish.

Upon closer examination, it generally becomes clear to me that those changes that seem to come with the passage of time are, in fact, no changes at all: only my view of things changes. (Franz Kafka)

And although the temptation is great to go along two roads at once, you can’t play with the same deck of cards with both the devil and God ...

Appreciate those with whom you can be yourself.
Without masks, omissions and ambitions.
And take care of them, they are sent to you by fate.
After all, in your life there are only a few of them

For an affirmative answer, only one word is enough - “yes”. All other words are invented to say no. Don Aminado

Ask a person: "What is happiness?" and you will find out what he misses the most.

If you want to understand life, then stop believing what they say and write, but observe and feel. Anton Chekhov

There is nothing in the world more destructive, more unbearable than inaction and waiting.

Turn your dreams into reality, work on ideas. Those who laughed at you before will begin to envy.

Records are there to be broken.

Don't waste time, invest in it.

The history of mankind is the history of a fairly small number of people who believed in themselves.

Did you push yourself to the limit? Don't you see the point in living anymore? So, you are already close ... Close to the decision to reach the bottom in order to push off from it and decide to be happy forever .. So do not be afraid of the bottom - use it ....

If you are honest and frank, then people will deceive you; still be honest and frank.

A person rarely succeeds in anything if his occupation does not give him joy. Dale Carnegie

If at least one flowering branch remains in your soul, a singing bird will always sit on it. (Eastern wisdom)

One of the laws of life says that as soon as one door closes, another one opens. But the whole trouble is that we look at the locked door and do not pay attention to the opened one. André Gide

Don't judge a person until you talk to them in person, because everything you hear is hearsay. Michael Jackson.

First they ignore you, then they laugh at you, then they fight you, then you win. Mahatma Gandhi

Human life is divided into two halves: during the first half they strive forward to the second, and during the second back to the first.

If you don't do anything yourself, how can you be helped? You can only drive a moving car

All will be. Only when you decide to do it.

In this world, you can look for everything except love and death... They will find you when the time comes.

Inner satisfaction in spite of the surrounding world of suffering is a very valuable asset. Sridhar Maharaj

Start now to live the life that you would like to see it in the end. Marcus Aurelius

We must live every day as if it were the last moment. We do not have a rehearsal - we have a life. We don't start it from Monday - we live today.

Every moment of life is another opportunity.

A year later, you will look at the world with different eyes, and even this tree that grows near your house will seem different to you.

Happiness does not need to be sought - it must be. Osho

Almost every success story that I know of started with a man lying on his back, defeated by failure. Jim Rohn

Every long journey begins with one, the first step.

Nobody is better than you. Nobody is smarter than you. They just started early. Brian Tracy

The one who runs falls. He who crawls does not fall. Pliny the Elder

It is enough just to understand that you live in the future, as soon as you find yourself there.

I choose to live rather than exist. James Alan Hetfield

When you appreciate what you have, and not live in search of ideals, then you will truly become happy ..

Only those who are worse than us think badly about us, and those who are better than us are simply not up to us. Omar Khayyam

Sometimes one call separates us from happiness… One conversation… One confession…

By admitting one's weakness, one becomes strong. Honre Balzac

He who humbles his spirit is stronger than he who conquers cities.

When an opportunity comes up, you have to grab it. And when you grabbed it, achieved success - enjoy it. Feel the joy. And let everyone around suck your hose for being goats, when they didn’t even give you a penny. And then - leave. Handsomely. And leave everyone in shock.

Never despair. And if you have already fallen into despair, then continue to work in despair.

A decisive step forward is the result of a good kick from behind!

In Russia, you have to be either famous or rich to be treated the way anyone in Europe is treated. Konstantin Raikin

It all depends on your attitude. (Chuck Norris)

No reasoning can show a man the path he does not want to see Romain Rolland

What you believe in becomes your world. Richard Matheson

It's good where we're not. We are no longer in the past, and therefore it seems beautiful. Anton Chekhov

The rich get richer because they learn to overcome financial hardship. They see them as an opportunity to learn, grow, develop, and prosper.

Everyone has their own hell - it's not necessarily fire and tar! Our hell is a wasted life! Where Dreams Lead

It doesn't matter how hard you work, the main thing is the result.

Only mother has the most affectionate hands, the most tender smile and the most loving heart ...

Winners in life always think in the spirit: I can, I want, I. Losers, on the other hand, focus their scattered thoughts on what they could have, could do, or what they could not do. In other words, the winners always take responsibility for themselves, and the losers blame their failures on circumstances or other people. Denis Waitley.

Life is a mountain climbing slowly, descending quickly. Guy de Maupassant

People are so afraid to take a step towards a new life that they are ready to close their eyes to everything that does not suit them. But it’s even scarier: to wake up one day and realize that everything is not right, wrong, wrong ... Bernard Shaw

Friendship and trust cannot be bought or sold.

Always, in every minute of your life, even when you are absolutely happy, have one attitude towards the people around you: - In any case, I will do what I want, with or without you.

In the world, only one can choose between loneliness and vulgarity. Arthur Schopenhauer

One has only to look at things differently, and life will flow in a different direction.

The iron said to the magnet: Most of all I hate you because you attract, not having enough strength to drag you along! Friedrich Nietzsche

Know how to live even when life becomes unbearable. N. Ostrovsky

The picture you see in your mind will eventually become your life.

“The first half of your life you ask yourself what you are capable of, but the second - and who needs it?”

It's never too late to set a new goal or find a new dream.

Control your destiny or someone else will.

see beauty in the ugly
to see rivers in the streams…
who knows how to be happy on weekdays,
he really is a lucky man! E. Asadov

The sage was asked:

How many types of friendship are there?

Four, he replied.
There are friends, like food - every day you need them.
There are friends, like medicine, you look for them when you feel bad.
There are friends, like a disease, they themselves are looking for you.
But there are such friends as air - they are not visible, but they are always with you.

I will become the person I want to be - if I believe that I will become one. Gandhi

Open your heart and listen to what it dreams about. Follow your dreams, because only through the one who is not ashamed of himself will the glory of the Lord be manifested. Paulo Coelho

To be refuted is nothing to fear; one should be afraid of another - to be misunderstood. Immanuel Kant

Be realistic - demand the impossible! Che Guevara

Don't put off your plans if it's raining outside.
Don't give up on your dreams if people don't believe in you.
Go against nature, people. You are a person. You are strong.
And remember - there are no unattainable goals - there is a high coefficient of laziness, a lack of ingenuity and a stock of excuses.

Either you create the world, or the world creates you. Jack Nicholson

I love it when people just smile. You go, for example, on a bus and you see a person looking out the window or texting and smiling. It feels so good to the soul. And I want to smile too.

Even under the most severe and rough shell, a tender soul and a sensitive heart are sometimes hidden. Stephen Covey

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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 knocking. 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: to be silent at the right time and to 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.