Formal logic in solving problems of diagnostics, treatment and prevention of diseases. Basics of Propositional Logic

The source of fatigue is not in the body, but in the mind. You can do much more than you think.

Relying only on yourself is a great way to stop being disappointed in people and live in a great mood.

Life gives us a lot of chances, but, more often than not, we are just too lazy to swim.

All people are two-faced. The first person is kind, sincere, responsive. The second appears when the first is misused.

Hell? Unfortunately it doesn't exist. There is only Paradise and ... Earth.

The most important thing in communication is to hear what has not been said. Peter Drucker

What began with a lie, it must have ended with a lie; it is a law of nature. Fedor Dostoevsky

Words are like keys. With the right choice, you can open any soul and close any mouth.

Respect or not respect a person is your choice. Being respectful is your upbringing.

The rule hot air balloon: throw away all unnecessary things to gain height.

We are not mistaken in people, we are just in a hurry to see them the way we want them to be.

You cannot lose what is not. You cannot destroy what is not built. You can only dispel the illusion of what seems real.

To achieve complete harmony in life, only two things need to be reversed: at 7 in the morning you should be hungry, and at 1 in the morning - to sleep.

Music is life. As long as it sounds, nothing dies forever. A musician, playing music, lives with memories as if they were real events.

I will never deceive someone who sincerely trusts me. But I will not prove the truth to someone who does not believe me.

If you don’t make mistakes, then you’re making a decision too. simple tasks... And this is a big mistake.

There is nothing accidental in life, and everything that happens to us happens at the right time and in the right place.

If I do exactly the things that people expect of me, I will fall into their slavery.

Time is an amazing phenomenon. There is so little of it when you are late and so much when you wait.

What you think is what you feel. What you feel is what you radiate. What you radiate is what you receive.

No better way revenge than erase from memory. Janusz Leon Vishnevsky

Treat everyone with kindness and respect, even those who are rude to you. Not because they are worthy people, but because you are a worthy person. (Confucius)

Family is what it is worth waking up for every day, breathing every second, and praying to God every moment to protect and protect them.

There will always be people who will hurt you. You need to continue to trust people, just be a little more careful.

Love dies of fatigue, but oblivion buries it.

Otherwise, the spaced words take on a different meaning, otherwise the spaced thoughts make a different impression.

Waging war with others has not made peace with himself.

Trusting a person unconditionally, you end up with one of two things: either a person for life, or a lesson for life.

Do not come closer to the person than he will allow, and do not let the person closer than he deserves.

To discover new parts of the world, one must have the courage to lose sight of the old shores.

Do not expect a miracle, do your own freak. And run, run from the pessimists, skeptics, whiners, push them away. They destroy expectation and faith in the miracles of life.

In life, you need to strive to overtake not others, but yourself.

There are three golden qualities in a person's character: patience, a sense of proportion and the ability to remain silent. Sometimes they help in life more than intelligence, talent and beauty.

Learn not to tell anyone. Then everything will be fine.

A diamond that has fallen into the mud is still a diamond, and dust that has risen to the skies remains dust.

Maintain soul light. Despite everything, no matter what. This is the light through which the same bright souls will find you.

People don't always need advice. Sometimes they need a hand to support. An ear that will listen and a heart that will understand.

It is not the strongest or the smartest that survives, but the one who adapts best to change.

If there are rumors about you, You are a person. Remember: they never discuss or envy bad things. They envy the best, discuss the best.

Give the person a purpose worth living for and they can survive in any situation.

Never criticize the actions of another person if you do not know why he did them. Perhaps, under the same circumstances, you would have done the same.

Conscience usually torments those who are not guilty. Erich Maria Remarque

Know how to say "thank you" to what is left behind. It certainly taught us something important.

You are silent, and you have not been understood correctly.

Society often forgives the offender. But not a dreamer. Oscar Wilde

In our precarious world, nothing is more difficult to achieve and fragile than trust.

The past is always with us, it is waiting to turn the present.

The end of lies does not mean the beginning of truth. Frederic Beigbeder

Taking offense and resentment is like drinking poison in the hope that it will kill your enemies.

The worst thing is not "failing again." The worst thing is "I don't want to try anymore."

You need to arrange your life until life begins to suit you.

No one in the world will meet you halfway. If you need something - take it yourself, always do only what you decide.

You're not singing a song to get to the last note. Singing itself brings joy. The same goes for life. The joy is to live.

The greatest flaw in life is its eternal incompleteness due to our habit of procrastinating from day to day. Whoever finishes his life's work every evening does not need time.

He who has never sworn an oath of fidelity will never break it. August von Platen

He who is able to tame his heart, the whole world will submit. Paulo Coelho

Make your work life-filled, not work-filled life.

Our actions can lift us to heaven and throw us into the deepest abyss. We are the children of our deeds. Victor Hugo.

If you want to achieve a goal, you need to move at least a little towards it every day.

When you see an eagle, you see a model of Perfection - so often look into the sky.

Wanting to change is the first step. But the second is to get them!

The only difference between a good dinner and life is that the sweet is served at the end.

Sometimes the person you don't notice becomes the one you need the most.

The past cannot be fixed, but you can strain and change the future.

A strong person is not the one who is doing well, but the one who is doing well, IN SPITE OF ANYTHING!

Remember the rule. Treat a woman like a person. Then like a princess. Then as a Greek goddess, and then again as a person.

When you have nothing to lose, you can risk everything ...

I hear and forget. I see and remember. I do and understand. Confucius

You can wait, if there is something to wait.

No need to think about what life was or will be. There is no past and there will be no future. Everything happens here and now.

This is not about anticipating the future, but about creating it.

Be with those who make you happy.

The impossible has to be done. You have to work hard. And if you succeed, you are positive, you have a ray of hope.

Love lives only when there is respect for each other and freedom. The desire to possess others as a thing is absurd.

Look at what you are thinking now, this will become your future. Think good things, love, success, luck, abundance and joy. And enjoy it in the future.

The only way to survive is to constantly challenge yourself.

The less expectations there are in the head, the more surprises there are in life.

Those who do not understand their past are forced to relive it again.

There is a sun in every person. Just let it shine. Socrates

I do not regret anything anyway - if only because it is pointless.

I don't like being alone. I just don’t make extra acquaintances, so as not to be disappointed in people once again.

Do not promise if you are not sure that you will fulfill what you promised, because the pain that you cause to another will sooner or later return to you.

Those who have succeeded in this world come and find the circumstances they need. If they cannot find them, then they create them themselves.

It's never too late to deliver new target or find a new dream.

The one who is to blame does not always ask for forgiveness. Most often, this is done by someone who values ​​the relationship ...

Appreciate those who know how to see three things in you, the sadness behind the smile, the love behind the anger, and the reason for your silence.

Any problem ceases to be a problem with the right attitude.

Nobody can say anything about you. Whatever people say, they are talking about themselves.

Always choose the most difficult path - you will not meet any competitors on it.

It doesn't matter that you walk slowly ... the main thing is don't stop.

If you decide to act, close the doors to doubt. Friedrich Nietzsche

People most often use the word "nothing" in order to hide a very important "something" behind it.

You become melancholic when you think about life, and cynic when you see what most people make of it. Remarque

Do not forget that the position - from the word "duty", work - from the word "slave", and dismissal - from the word "will".

I don’t choose friends. It’s boring and useless. It is much more interesting for me to choose vegetables on the market. Friends are gifts of fate.

The main (undefined) concept of mathematical logic is the concept of a "simple statement".

A statement is usually understood as any declarative sentence that asserts something about something, and at the same time we can say whether it is true or false in the given conditions of place and time. The logical meanings of statements are "true" and "false".

Here are some examples of statements:

1) Novgorod is on the Volkhov.

2) Paris is the capital of England.

3) Crucian carp is not a fish.

4) The number 6 is divisible by 2 and 3.

5) If a young man graduated from high school, then he receives a certificate of maturity.

Statements 1), 4), 5) are true, and 2) and 3) are false.

Obviously, the sentence "Long live our athletes!" is not a statement.

A statement that is one statement is usually called simple or elementary. Examples of elementary statements are statements 1) and 2).

Statements that are obtained from elementary ones with the help of grammatical connectives "not", "and", "or", "if ... then ...", "then and only then" are usually called complex or compound. So, statement 3) is obtained from a simple statement “Crucian is a fish” by negating “not”, statement 4) is formed from elementary statements “The number 6 is divided by 2”, “The number 6 is divided by 3”, connected by the union “and”. Statement 5) is obtained from simple statements "The young man graduated from high school", "The young man receives a certificate of maturity" using the grammatical link "if ...,
then …". Similarly, complex utterances can be obtained from simple utterances using the grammatical links "or", "then and only then."

In the algebra of logic, all statements are considered only from the point of view of their logical meaning, and they are abstracted from their everyday content. It is believed that every statement is either true or false, and no statement can be both true and false at the same time.

In what follows, we will denote elementary statements by letters of the Latin alphabet: a, b, c,…, x, y, z,…; the true value is the letter I or the number 1, and the false value is the letter L or the number 0.

If the statement a is true, then we will write a = 1 if false, then a = 0.

It is customary to subdivide logical statements into two types: elementary logical statements and compound logical statements.

Compound logical statement is a statement formed from other statements using logical connectives.

Logical bundle is any logical operation on a statement. For example, words and phrases used in ordinary speech "Not", "and", "or", "if ... then", "then and only then" are logical connectives.

Elementary logical statements- these are statements that are not compound.

Examples: "Ivanov is a football player" - elementary logical statements. "Ivanov is a football player and a chess player" is a compound logical statement, consisting of two elementary statements connected with each other by means of the "and" link.

46. ​​Elements of the algebra of logic

Algebra of logic is a section of mathematical logic, the values ​​of all elements (functions and arguments) of which are defined in a two-element set: 0 and 1. Algebra of logic operates with logical statements.

Saying - it is any sentence in respect of which a statement about its truth or falsity makes sense. In this case, it is considered that the statement satisfies the law of the excluded middle, that is, each statement is either true or false and cannot be both true and false at the same time.

Statements:

- “It is snowing now” - this statement can be true or false;

- “Washington is the capital of the USA” is a true statement;

- "The quotient of dividing 10 by 2 is 3" – false statement.

In algebra of logic, all statements are denoted by letters a, b, c it. e. The content of statements is taken into account only when they are introduced. letter designations, and in the future, any actions provided by this algebra can be performed on them. Moreover, if some operations allowed in the algebra of logic are performed on the initial elements of the algebra, then the results of the operations will also be elements of this algebra.

The simplest operations in the algebra of logic are the operations logical addition(otherwise: operation OR(OR), disjunction operation) and logical multiplication(otherwise: operation AND (AND),conjunction operation). To denote the operation of logical addition, the symbols + or V are used, and logical multiplication - symbols or The rules for performing operations in the algebra of logic are determined by a number of axioms, theorems and consequences. In particular, the following laws apply to the algebra of logic:

1. Combined:

47. (a + b) + c = a +(b + c),

48. (a b) with= a(b with).

2. Traveling:

49. (a + b) = (b + a),

50. (a b)= (b a).

3. Distribution:

51. a (b + c) = a b + (a with),

52. (a + b) c = a c + b c.

The relations are valid, in particular:

53. a + a = aa + b = b, if a ≤ b,

54. a a = aa b= a, if a ≤ b,

a + a b = aa b = b, if a≥ b,

a + b = a, if a≥ b.

The smallest element of the algebra of logic is 0, the largest element is 1. In the algebra of logic, one more operation is also introduced - denials(operation NOT, inversion), denoted by a bar above the element.

A-priory

Function in Boolean Algebra - an expression containing elements of Boolean Algebra a, b, c and others related to the operations defined in this algebra. Examples of logical functions:

etc. These relationships are used to synthesize logic functions and computational circuits.

Explanatory dictionary of the Russian language. D.N. Ushakov

utterance

statements, cf. (book).

only units Action by verb. express. Expressing your opinion.

The expressed judgment, remark, opinion. Collect the statements of the classics of Marxism about the language.

Explanatory dictionary of the Russian language. S.I.Ozhegov, N.Yu.Shvedova.

New explanatory and derivational dictionary of the Russian language, T. F. Efremova.

Encyclopedic Dictionary, 1998

utterance

a thought expressed in a declarative sentence and can be true or false; in linguistics - a unit of verbal communication, formalized according to the laws of a given language.

Utterance

a declarative sentence considered, together with its content (meaning), as true or false. V. understood in this way are usually opposed to imperative, interrogative, and in general any sentences, the assessment of the truth or falsity of which is impossible. Examples Q: "Moscow is the capital", "5 is less than 3 and more than 2", "All engineers studied the strength of materials." Of these V., the first and third are true, and the second is false. "Truth" and "false" are called the truth values ​​of V. (or the values ​​of its truth). By definition, any V. has grammatical and logical aspects. The grammatical aspect of V. is expressed by a declarative sentence (simple or complex), and the logical one is expressed by its meaning and truth value. V., differing as grammatical sentences(for example, belonging to different languages) can express the same idea. This idea, common to grammatically different V., is called the content, or meaning, V.; it is also often called judgment. However, the terminology related to V. has not been established, and the terms "V.", "sentence", "judgment" are sometimes used as synonyms or they are assigned meanings that differ from those described above.

In connection with language practice, there are different ways usage of V. It is said that V. is used in the affirmative if it is used for the purpose of affirming the truth of the thought expressed in it. The affirmative use of V. is their most frequent use: when expressing their thoughts, people usually claim their truth. (In logic, in order to distinguish B. as a sentence, which can be either true or false, from the truth statement B., in some cases, a special sign is used; A means the statement of A.) In the case when the truth of the content of B . is not approved, they speak of the non-affirmative use of V. (for example, in the class dictation V. is used non-affirmatively). One of the ways of non-affirmative use of V. is their indirect use. It aims not to assert the truth of thought, but only to convey the content of V. That is how, for example, V. is used "the orbits of the planets have the shape of a circle" in the composition of V. "Kepler believed that the orbits of the planets have the shape of a circle." In asserting the latter, we do not at all want to say that it is true that the orbits of the planets have the shape of a circle, but only inform what V. Kepler asserted; this V. itself can be both true and false (the latter actually takes place). From different types V.'s use should be distinguished by their mention (citation).

In logic, we are dealing mainly with the use of logical calculus in a particular area of ​​objects. In the formulas of the so-called "pure" logical calculi themselves, the variables V. and V. forms (utterance forms) appear mainly. Variable V. is not V. in the true sense, but a variable for V., that is, a variable, in place of which concrete ("constant") V. (of a given type) or their names can be substituted. Form V. is an expression that contains variables (in particular, perhaps, variables for V.) and turns into V. after substitution of any values ​​≈ from the corresponding permissible ranges of values ​​≈ instead of all variables included in it. For example, the form B. is the formula x + y> 2 (x, y ≈ variables that take values, for example, from the range of real numbers; for x = 1, y = 2 this formula turns into true B. 1 + 2> 2) ...

Lit .: A. Tarskiy, Introduction to the logic and methodology of deductive sciences, trans. from English, M., 1948; Church A., Introduction to Mathematical Logic, trans. from English, vol. 1, M., 1960.

B.V.Biryukov.

In linguistics, linguistics is a unit of linguistic communication. Segmentation of linguistic material according to intonation and semantic features allows you to highlight the communicative units of speech, sometimes called phrases. Segmentation of linguistic material according to formal characteristics allows you to distinguish syntactic units language, often called sentences (there are other correlative pairs of terms). A sentence and a phrase are units of the same (commutative) level, but belong to different aspects of the language material. V. as a real unit of communication is a synthesis of correlative units of language and speech - sentences and phrases. In modern linguistics, there are other interpretations of the concept of "V."

Lit .: Vannikov Yu. V., Utterance as a synthetic unit, in the book: Questions of grammar and word formation, M., 1968; Hausenblas K., On the characterization and classification of discourses, Travaux linguistiques de Prague, 1966, ╧ 1.

Yu.V. Vannikov.

Wikipedia

Statement (logic)

Utterance- a sentence expressing a judgment. If the judgment that makes up the content of a certain statement is true, then this statement is also said to be true. Similarly, a statement that is an expression of a false judgment is called false. Truth and falsity are called logical or truth values ​​of statements.

The utterance should be a declarative sentence. Statements are usually opposed to imperative, interrogative and any other sentences, the assessment of the truth or falsity of which is impossible.

Utterance

Utterance:

• A statement - in logic, a sentence that can be true or false.
• Utterance - in linguistics, a sentence in a specific speech situation.

Examples of the use of the word utterance in literature.

Graves was silent for so long that Eisenberg felt embarrassed by the excessive pathos of his utterances.

And this is his utterance clearly shows that by the names of diseases that the allopaths operate on, they mean only the gross outward manifestations of the disorder of the vital force.

Not prone to software statements, Annensky in his social aspirations is extremely close to the position expressed by P.

This action was carried out despite the fact that Igor Dobrovolsky was well acquainted with all statements Sergei Prokofiev and Christian Lazarides about the many screaming contradictions, both in the worldview of Tomberg himself and in the worldview of this Dutch anthroposophist - Harry Zalman.

Fantasia, Trio, and many other instrumental and vocal pieces by Arensky, not being very deep in terms of the emotional and intellectual content inherent in them, not differing in innovation, at the same time attract with the sincerity of the lyrical - often elegiac - utterances, generous melody.

Why the disengagement is made is also clear: this is done in order to deprive the philosophical discourse of its initially inherent atonality, polemical acuteness of some statements against others.

After all these years of careful censorship of their own statements Bergen felt satisfaction as he spoke these words, speaking truthfully and without diplomatic embellishment.

These utterances Charlotte Brontë, as well as the satirical images of English priests she created, show how false the statements of some bourgeois literary scholars who claim that the main source of her work is.

Booby's impressive advice to Joseph and Fanny's encounter with the seducer Habit, my good reader, has such power over the human mind that no utterances about her should not seem too strange or too strong.

What is in simple peeping the severity utterances may be absent, does not give the right to deny this simple vision in any articulating interpretation and thus in a as-structure.

Wittgenstein gave the first formulation of the requirement of verification as a criterion for the meaningfulness of scientific statements.

The texts of telegrams, notes and statements Rasputin are partly taken from documents discovered after February 1917 in the affairs of the close associates, including Goremykin, Sturmer and Voeikov, partly from the correspondence of the Romanovs, memoirs and notes of contemporaries.

Only in this way is this being in itself capable of binding every possible utterance, T.

Any initially gleaned phenomenological concept and position as communicated utterances subject to the possibility of degeneration.

However, the memories of Alexander Pavlovich coincide with statements Chekhov himself both in letters and in his stories to contemporaries.

Types of statements

It is customary to subdivide logical statements into two types: elementary logical statements and compound logical statements.

Compound logical statement is a statement formed from other statements using logical connectives.

Logical bundle is any logical operation on a statement. For example, words and phrases used in ordinary speech "Not", "and", "or", "if ... then", "then and only then" are logical connectives.

Elementary logical statements- these are statements that are not compound.

Examples: "Petrov is a doctor", "Petrov is a chess player" - elementary logical statements. "Petrov is a doctor and a chess player" is a compound logical statement consisting of two elementary statements connected with each other by means of the "and" link.

Relationship with mathematical logic

Conventional logic is two-valued, that is, it ascribes only two possible meanings to statements: true or false.

Let it be a statement. If it is true, then write, if false, then.

Basic operations on logical statements

Negation logical statement - a logical statement that takes on the meaning "true" if the original statement is false, and vice versa.

Conjunction two logical statements - a logical statement, true only when they are simultaneously true.

Disjunction two logical statements - a logical statement, true only if at least one of them is true.

Implication two logical statements A and B is a logical statement that is false only if B is false and A is true.

Equivalence(equivalence) of two logical statements - a logical statement, true only when they are simultaneously true or false.

Quantifier universality() is a logical statement that is true only if for each object x from a given set the statement A (x) is true.

Quantifier logical statement with a quantifier existence() is a logical proposition that is true only if there is an object x in a given set such that the proposition A (x) is true.

see also

• Statement

Notes (edit)

Literature

• Karpenko, A.S. Contemporary research in philosophical logic // Logical studies. Issue 10. - M .: Nauka, 2003. ISBN 5-02-006257-X - S. 61-93.
• Kripke, S. A. Wittgenstein on the rules and individual language / Per. V. A. Ladova, V. A. Surovtseva. Under total. ed. V. A. Surovtseva. - Tomsk: Publishing house of Vol. University, 2005 .-- 152 p. - (Library of Analytical Philosophy). ISBN 5-7511-1906-1
• Kurbatov, V.I. Logics. Systematic course. - Rostov n / a: Phoenix, 2001 .-- 512 p. ISBN 5-222-01850-4
• Schumann, A. N. Modern logic: theory and practice. - Minsk: Econompress, 2004 .-- 416 p. ISBN 985-6479-35-5
• Makarova, N.V. Informatics and ICT. - St. Petersburg: Peter Press, 2007 ISBN 978-5-91180-198-4 - S. 343-345.
• N. I. Kondakov Logical dictionary / Gorskiy D.P .. - M .: Nauka, 1971. - 656 p.

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See what "Statement (logic)" is in other dictionaries:

Utterance: Utterance (logic) a sentence that can be true or false. Utterance (linguistics) a sentence in a specific speech situation. See also Judgment ... Wikipedia

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A section of logic that studies the truth relationships between statements. Within the framework of this section, statements (propositions, sentences) are considered only from the point of view. their truth or falsity, regardless of their internal subjective ... Philosophical Encyclopedia

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Or the Logic of Science, the application of ideas, methods and apparatus of logic in the analysis of scientific knowledge. The development of logic has always been closely associated with the practice of theoretical thinking and, above all, with the development of science. Concrete reasoning gives logic material ... Philosophical Encyclopedia

It should be a declarative sentence, and are opposed to imperative, interrogative and any other sentences, the assessment of the truth or falsity of which is impossible.

Utterance and judgment

One and the same judgment can be expressed in different languages ​​and in different sign forms within one language. When a judgment is considered in connection with some particular form of its linguistic expression, it is called a statement. The term "judgment" is used when they are distracted from what exactly its sign form is.

Types of statements

It is customary to subdivide logical statements into compound (or complex) and elementary ones. Compound logical statements are statements containing logical constants. Compound statements are built on the basis of other statements. The logical meaning of a complex statement is determined by the logical meaning of the statements included in it and those logical constants with the help of which it is constructed.

Elementary logical statements are statements that are not compound statements. An example of an elementary statement is 5 < 7 {\displaystyle 5<7} ... An example of a compound logical statement is if 5 < 7 {\displaystyle 5<7} , then 5 (\ displaystyle 5)- even number.

Logical constants

Logical constant (logical constant, logical operation) is the name of a term that retains the same meaning in all statements and does not depend on the specific content of the statement. Boolean constants are used to combine simple statements into complex ones. Logical constants are divided into quantifiers and logical unions (connectives). The words: not; it is not true that; and; or; if ... then; if and only if; or either; inconsistent; no no; not ... but; but and their closest synonyms are logical connectives, words for all ... it is the case that; for some ... it is the case that and their closest synonyms are quantifiers. Logical constants serve both to express thoughts in everyday reasoning and in scientific evidence.

• ∀ (\ displaystyle \ forall)- logical constants all, for everyone ... it is the case that(quantifier of generality);
• ∃ (\ displaystyle \ exists)- logical constants there is such that ..., for some ... it is the case that(existential quantifier);
• ∧ (\ displaystyle \ land), & (\ displaystyle \ And)- union and(conjunction);
• ∨ (\ displaystyle \ vee)- union or when it acts in a connecting-separating meaning (disjunction);
• ∨ ˙ (\ displaystyle (\ dot (\ vee))), ∨ ∨ (\ displaystyle \ vee \ vee)- union or when it appears in a strictly separating exclusive meaning (disjunction);
• → (\ displaystyle \ rightarrow), ⊃ (\ displaystyle \ supset)- union if ... then(implication);
• ¬ (\ displaystyle \ neg)- the words not, wrong(negation).

Logical conjunctions are part of the language of propositional logic; quantifiers have been additionally introduced into the language of predicate logic, which is an extension of the language of propositional logic.

Boolean subject and boolean predicate

A logical subject is what is said in a sentence (statement), what the statements or negations contained in the sentences refer to. Logical predicate - information contained in a sentence (statement) about a logical subject.

Simple and complex names play the role of logical subjects, predicators (or predicates) play the role of logical predicates. The latter include properties and relationships. Predicators perform the role of subject-truth display, giving the objects of a certain class the assessment "true" or "false". At the same time, properties are one-place predictors, characterizing one separate object, and relations are multi-place, characterizing a pair, three, etc. of objects. The statement itself in the case of a multiplace predictor contains several logical subjects.

Forms of statements

The form of the statement requires an addition, whether the affirmation or negation in the judgment applies to all or not all objects of the class that the given common name represents. The function of such pointers is performed by express or implied