Formal logic in solving problems of diagnosis, treatment and prevention of diseases. Fundamentals 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 with a great mood.

Life gives us a sea of ​​​​chances, but, most often, we are just too lazy to swim.

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

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

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

What began with a lie must have ended with a lie; it is the 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 - your choice. Being respectful is your upbringing.

rule hot air balloon: Throw away everything superfluous 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 can't lose what isn't there. You can't destroy what hasn't been built. You can only dispel the illusion of what seems real.

To achieve complete harmony in life, you need to swap only two things: at 7 in the morning you should want to eat, and at one in the morning you should sleep.

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

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

If you don't make mistakes, then you decide 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 from me, I will fall into their slavery.

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

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

Not better way revenge than to erase from memory. Janusz Leon Wisniewski

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)

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

There will always be people who will hurt you. You have to keep trusting people, just be a little more careful.

Love dies from fatigue, and oblivion buries it.

Words placed differently take on a different meaning; otherwise thoughts placed produce a different impression.

He who wages 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 a person than he allows, and do not let a person get closer than he deserves.

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

Don't expect miracles, miracle yourself. And run, run from the pessimists, skeptics, whiners, move them away. They destroy expectation and faith in the miracles of life.

In life, one should strive to overtake not others, but oneself.

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

Learn to tell no one anything. That's when everything will be fine.

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

Keep your spiritual light. Against all odds, no matter what. This is the light by which the same bright souls will find you.

People don't always need advice. Sometimes they need a hand to support them. An ear that listens and a heart that understands.

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

If there are rumors about you, you are a person. Remember: never discuss and do not envy the bad. Envy the best, discuss the best.

Give a man a purpose worth living for, and he 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, but you are already misunderstood.

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

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

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

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

Being offended and indignant is like drinking poison in the hope that it will kill your enemies.

The worst thing is not “fail again”. The worst thing is “I don’t want to try anymore”.

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

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

You don't sing a song to get to the last note. Singing brings joy. The same goes for life. 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 the work of his life every evening does not need time.

Who never swore allegiance, he will never break it. August von Platen

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

Make your work a full life, not life a full work.

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

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

When you see an eagle, you see an example of Perfection - so often look at the sky.

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

The only difference between a good meal and life is that sweets are served at the end.

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

The past can't be fixed, but you can push yourself and change the future.

A strong person is not the one who is doing well, but the one who is doing well, DESPITE 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.

It's not about predicting the future, but about creating it.

Be with those who make you happy.

You have to do the impossible. 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 another as a thing is absurd.

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

The only way to survive is to constantly challenge yourself with new challenges.

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

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

Every person has a sun. Just let it shine. Socrates

I still don't regret anything, if only because it's pointless.

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

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

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 goal or find a new dream.

It is not always the one who is guilty who asks for forgiveness. Most often, this is done by someone who values ​​​​relationships ...

Appreciate those who can see in you three things: 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 talk about themselves.

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

It doesn't matter if you're walking 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 a melancholic when you think about life, and a 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. This is a boring and useless activity. I am much more interested in choosing vegetables in the market. Friends are gifts of fate.

The basic (undefined) concept of mathematical logic is the concept of "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 given conditions of place and time. The logical values ​​of statements are "true" and "false".

Here are some examples of statements:

1) Novgorod stands on the Volkhov.

2) Paris is the capital of England.

3) Carp is not a fish.

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

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

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, which is a single statement, is usually called simple or elementary. Examples of elementary propositions are propositions 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” with the help of the negation of “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 matriculation certificate" using the grammatical connective "if ...,
then …". Similarly, complex statements can be obtained from simple statements using the grammatical connectives "or", "if and only then".

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

In what follows, elementary statements will be denoted 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 true, we will write a=1, if false, then a=0.

Logical statements are usually divided into two types: elementary logical statements and compound logical statements.

Compound logical statement is a statement formed from other statements with the help of logical connectives.

logical link 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 links.

Elementary logical statements These are non-compound statements.

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 to each other by means of a bunch of "and".

46. ​​Elements of the algebra of logic

The 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. The algebra of logic operates with logical statements.

Statement - it is any sentence in relation to which the statement about its truth or falsity makes sense. At the same time, 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.

Sayings:

- "It's 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 the algebra of logic, all statements are denoted by letters a, b, c it. e. The content of statements is taken into account only when introducing them. letters, and in the future, any actions provided by this algebra can be performed on them. Moreover, if certain operations allowed in the logic algebra 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 the symbols or symbols are used for logical multiplication. The rules for performing operations in the algebra of logic are determined by a number of axioms, theorems, and corollaries. In particular, for the algebra of logic, the following laws apply:

1. Combined:

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

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

2. Relocatable:

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 ratios 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 - denial(operation NOT (NOT), inversion), denoted by a bar above the element.

By definition

A function in the algebra of logic is an expression containing elements of the algebra of logic a, b, c and others connected by 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

statement

sayings, cf. (book).

only ed. Action on verb. express. Expressing your opinion.

Expressed judgment, remark, opinion. Collect the statements of the classics of Marxism about 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

statement

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

statement

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 of V.: “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. "True" 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), while the logical aspect is expressed by its meaning and truth value. V., differing as grammar 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.; often it is also called judgment. However, the terminology related to V. has not been established, and the terms "V.", "sentence", "judgment" are sometimes used as synonyms or assigned meanings that differ from those described above.

In connection with language practice, there are various ways use of V. It is said that V. is used in the affirmative if it is used for the purpose of asserting 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 of B., in some cases a special sign is used; ═A means the statement of A's statement.) In the case when the truth of the content of B. . is not affirmed, they speak of the non-affirmative use of V. (for example, in 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 affirm the truth of thought, but only to convey the content of V. This is exactly how, for example, V. used “the orbits of the planets are in the shape of a circle” in the composition of V. “Kepler believed that the orbits of the planets are in 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 are in the form of a circle, but only to report what V. Kepler claimed; this V. itself can be both true and false (the latter actually takes place). From various kinds the use of V. should be distinguished by their mention (citation).

In logic, one deals with V. mainly in the application of logical calculus in any particular area of ​​objects. In the formulas of the so-called "pure" logical calculi themselves, V. variables and V. forms (propositional forms) mainly appear. Variable V. is not V. in the true sense, but a variable for V., i.e., a variable, in place of which specific (“constant”) V. (of a given type) or their names can be substituted. The form V. ≈ is an expression containing variables (in particular, perhaps, variables for V.) and turning into V. after substituting any values ​​≈ from the corresponding admissible ranges of values ​​≈ instead of all the variables included in it. For example, the form of V. is the formula x + y > 2 (x, y ≈ variables that take values, for example, from the region of real numbers; for x = 1, y = 2, this formula turns into true V. 1 + 2 > 2) .

Lit .: Tarsky A., 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, language is a unit of linguistic communication. Segmentation of linguistic material according to intonational and semantic features allows us to distinguish communicative units of speech, sometimes called phrases. Segmentation of linguistic material according to formal features makes it possible to single out 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 linguistic 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 "B.".

Lit .: Vannikov Yu. V., Statement 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

Proposition (logic)

statement- a sentence expressing a judgment. If the proposition that constitutes the content of some statement is true, then this statement is also said to be true. Similarly, a statement is said to be false if it is an expression of a false proposition. Truth and falsity are called logical or truth-values ​​of propositions.

The statement must 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.

statement

statement:

• A proposition is, in logic, a sentence that can be true or false.
• Statement - in linguistics, a sentence in a specific speech situation.

Examples of the use of the word statement in the literature.

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

And this is his statement clearly shows that by the names of the diseases that allopaths operate with, they mean only the gross outward manifestations of a 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 glaring contradictions, both in the worldview of Tomberg himself and in the worldview of this Dutch anthroposophist - Harry Zalman.

And Fantasia, Trio, and many other instrumental and vocal pieces by Arensky, not being very deep in their emotional and intellectual content, not distinguished by innovation, at the same time attract with the sincerity of the lyrical - often elegiac - statements, generous melody.

Why the disengagement is being carried out is also clear: this is done in order to deprive the philosophical discourse of its inherent atonality, the polemical sharpness of some statements against others.

After all these years of carefully censoring my own statements, Bergen felt satisfied when he said these words, speaking truthfully and without diplomatic embellishments.

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

Mr. Booby's imposing advice to Joseph, and Fanny's meeting with the seducer Habit, my good reader, has such power over the human mind that no statements about it should not seem too strange or too strong.

What is in a simple peering expression statements may be absent, does not give the right to deny this simple vision any articulate interpretation and thus as-structure.

Wittgenstein gave the first formulation of the verification requirement 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 files of close associates of the guy, including Goremykin, Stürmer and Voeikov, partly from the correspondence of the Romanovs, memoirs and records of contemporaries.

Only in this way can this being in itself be capable of binding every possible statement, T.

Every originally learned phenomenological concept and position as communicated statements subject to the possibility of degeneration.

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

Types of statements

Logical statements are usually divided into two types: elementary logical statements and compound logical statements.

Compound logical statement is a statement formed from other statements with the help of logical connectives.

logical link 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 links.

Elementary logical statements These are non-compound statements.

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 a bunch of “and”.

Relationship with mathematical logic

Ordinary logic is two-valued, that is, it assigns only two possible values ​​to propositions: it is true or false.

Let 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 the value "true" if the original statement is false, and vice versa.

Conjunction two logical propositions - a logical proposition that is true only if they are both true at the same time.

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

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

equivalence(equivalence) of two logical statements - a logical statement that is true only if they are both true or false.

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

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

see also

• Statement

Notes

Literature

• Karpenko, A. S. Modern research in philosophical logic // Logical research. Issue. 10. - M.: Nauka, 2003. ISBN 5-02-006257-X - S. 61-93.
• Kripke, S. A. Wittgenstein on rules and individual language / Per. V. A. Ladova, V. A. Surovtseva. Under total ed. V. A. Surovtseva. - Tomsk: Publishing House Vol. un-ta, 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
• Shuman, A. N. Modern logic: theory and practice. - Minsk: Ekonompress, 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.
• Kondakov N.I. Logical dictionary / Gorsky D.P.. - M .: Nauka, 1971. - 656 p.

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

Proposition: A proposition (logic) is a sentence that can be true or false. An utterance (linguistics) is a sentence in a specific speech situation. See also Judgment ... Wikipedia

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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 connected with the practice of theoretical thinking and, above all, with the development of science. Concrete reasoning gives material to logic... Philosophical Encyclopedia

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.

Statement and judgment

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

Types of statements

Logical statements are usually divided into compound (or complex) and elementary. 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 by those logical constants with which it is built.

Elementary logical propositions are propositions that are not related to compound ones. 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.

Boolean constants

Logical constant (logical constant, logical operation) - the name of a term that retains the same value in all statements and does not depend on the specific content of the statement. Logical constants are used to connect simple statements into complex ones. Logical constants are divided into quantifiers and logical unions (bundles). The words: not; it is not true that; and; or; if…then; if and only if; or either; incompatible; 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 happens that(general quantifier);
• ∃ (\displaystyle \exists )- logical constants there is one that..., for some... it is the case that(existence quantifier);
• ∧ (\displaystyle \land ), & (\displaystyle \And )- union and(conjunction);
• ∨ (\displaystyle\vee )- union or when it appears in a connecting-separating meaning (disjunction);
• ∨ ˙ (\displaystyle (\dot (\vee ))), ∨ ∨ (\displaystyle \vee \vee )- union or when it appears in a strictly divisive 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 further introduced into the language of predicate logic, which is an extension of the language of propositional logic.

Logical subject and logical predicate

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

The role of logical subjects is played by simple and complex names, the role of logical predicates is played by predicators (or predicates). The latter include properties and relationships. Predicators act as a subject-truth mapping, giving objects of a certain class an assessment of "true" or "false". At the same time, properties are single-place predictors, characterizing one separate object, and relations are multi-place, characterizing a pair, triple, etc. of objects. The statement itself in the case of a multiplace predicator contains several logical subjects.

Statement forms

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