Judgment is a form of thinking in which something is affirmed or denied about objects, their properties or relations between them.

Judgment is characterized by content and form. Content of judgment- this is what it is about, its meaning.

Logical form of judgment- its structure, the method of connection of its constituent parts.

A judgment is always a declarative sentence. According to the structure,

can be simple or complex.

In the judgment, the subject S is singled out ( logical subject) is the concept referred to in the judgment; the predicate P ( logical predicate)

- this is a concept with the help of which something is affirmed or denied about the subject and a connection - the word is, is, is called (often absent).

A proposition is called simple if it contains only one subject and one predicate.

A judgment is called complex if it is formed from simple ones with the help of logical operations (connections).

By quality, simple judgments are divided into affirmative ones (a bunch

is ) and negative (the link is not ).

Example 1. Given the judgment "Earth is a planet".

In it, the subject S is "Earth", the predicate P is "planet", the connective is the word "is". Therefore, the judgment is simple, affirmative.

Example 2. Judgment "The lecture on logic will not take place today."

The subject S is "a lecture on logic", the predicate P is "it will take place today", the link in the judgment is omitted, there is a particle not . Therefore, this judgment is simple, negative.

By the number of judgments are divided into general, private. The quantity is determined by the volume of the subject of judgment. The volume of the subject can be half-

nym (all, none) or partial (some).

Example 3. All students are students (general). Some animals are predators (private). The sun is a celestial body (general, since we are talking about the entire scope of the concept of "sun", the specific Sun). A simple judgment can be written as a formula. The quantitative characteristic of judgments is conveyed with the help of quantifiers. Singular judgments are general.

is the general quantifierreplaces the words all”, “any”, “every”, etc.

S P(S) means that "for every S, P(S) is true", "All S are P".

is the existential quantifier replaces the words some" , " exists", "part" etc.

S P(S) means that "there is an S for which P(S) is true", "Some S are P" .

Example 4. Given the proposition "Some students take exams before-

urgent ". This is a simple judgment, we single out the logical subject and the logical predicate in it. S - "student", P - "passing exams ahead of schedule." The quality judgment is affirmative, since the nature of the relationship between the subject and the predicate is expressed by the verb without the particle "not By quantity, the judgment is particular, since the word " some" is used. Therefore, the judgment with the help of logical symbols will be written in the form of a formula S P (S).

Table 2. Classification of simple judgments

Type of judgment, designation, formula and structure

general affirmative(A):S

All S's are P's

general negative(E):S

No S is P

private affirmative(J):S

Some S's are P's

private negative(O):S

Some S's are not P's

Relationships of the volumes of concepts

S and P

All violets (S) are flowers (P) Rainy days (S) are boring (P)

No person (S)

dislikes moralizing (P) Musketeers (S)

do not evade duels (P)

Some people (S)

playing chess (P)

Among people (S)

there are phlegmatic people (P)

Some people (S)

do not know the taste of trout (P) Many musketeers (S) did not like

cardinal (P)

Rejection of simple judgments. To construct the negation of a judgment with a quantifier, it is enough to replace the quantifier with its opposite, and transfer the negation to the predicate.

Example 6. Initial judgment " All books are donated to the library". Required

dimo construct its negation. We define the type of judgment and write down its formula. S - "books", P - "delivered to the library." There is a word “everything”, there is no “not”. We get that the judgment is general in quantity and affirmative in quality: general affirmative(view A).

We take data from table 2 and write down its formula:

We build negation first in a symbolic form, and then we write it down in words. We work according to the above rule.

We change the quantifier to the opposite: was, became. The negation goes to the predicate.

Chain of transformations:

Let us write the judgment with the words: “Some books are not donated to the library».

Example 7. The judgment "Some students do not attend lectures" is given.

Build its negation.

S - "students", P - "those who attend lectures". Judgment by quantity is private (“some”), by quality it is negative (particle “not”). We get private negative(view O).

Let's write the formula

We build negation according to the rule. quantifier me-

we take from to. A double negation appeared above the predicate: one was according to the formula, the second appeared as a result of the transformation. The double negative is simply removed.

S P(S) SP(S) SP(S)

Now with the words: All students attend lectures.

As can be seen from the examples, judgments (A) and (O) are in relation to contradiction. That is, negating a judgment of one kind, we always get a narrowing of another kind. The picture is similar for judgments (E) and (J).

According to the logical value, any judgment can be true, or it can be false. If the original judgment is true, then the judgment resulting from the negation of the original will be false and vice versa. This is clearly seen from the examples above.

If we consider all four types of judgments (A, E, J, O) formed on one pair of concepts "subject-predicate", then knowing the logical value of one of them, one can often indicate the values ​​of the other three judgments. This relationship between values ​​in logic is called the "logical square". It is a system of pairwise relations between logical values:

Pairs A-O and J-E are in relation to contradiction, as already noted above, their logical values ​​are always opposite, i.e. if one is "true", then the other is "false" and vice versa.

A pair of general propositions A-E - in relation to the opposite, which means the impossibility of simultaneously taking on the value "true", but does not exclude the simultaneous "falsehood".

A couple of private judgments J-O - regarding subopposites (subopposites), which, opposite to the previous relation, means the impossibility of simultaneous "falsehood", but allows simultaneous "truth". Pairs of affirmative propositions A-J and negative propositions E-O are in a subordination relationship: if the first is "true", then the second is also "true" and vice versa, if the second is "false", then the first is also "false".

These six pairs of relationships can be depicted on the diagram as a 4-vertex complete graph.

Task 2. Determine the logical subject, the logical predicate and the type of this judgment. Write down the formula of judgment. Construct a formula for negating a given judgment, write down the resulting judgment in words, determine the type of judgment received. Determine the logical meaning of two other types of judgments formed with the same subject and predicate on the basis of a logical square.

2.1. No egoist can be generous.

2.2. Every surgeon is a doctor by training.

2.3. Among the students there are initiative people.

2.4. Some of the posts are not true.

2.5. All people have to take risks.

2.6. Some students do not play sports.

2.7. Not a single word should be left without attention.

2.8. Some people speak several foreign languages.

2.9. Some patients do not have a temperature.

2.10. Not all entrepreneurs have higher education.

2.11. Some oceans have fresh water.

2.12. Some students are not excellent students.

2.13. Not a single student of our group lives in a hostel.

2.14. Every soldier dreams of becoming a general.

2.15. All electrons are elementary particles.

2.16. No person is immune from failure.

2.17. Every student of KuzGTU studies mathematics.

2.18. Some of the military personnel are officers.

2.19. No prosecutor is a lawyer.

2.20. All students are happy about the end of the session.

2.21. Some plants do not tolerate dry soil.

2.22. All athletes need training.

2.23. There are singers with great voices.

2.24. Every mathematician should understand logic.

2.25. Some politicians are writers.

2.26. Some residents of our country have dual citizenship.

2.27. Some animals are insects.

2.28. No fan will refuse to meet an idol.

2.29. Some plants do not bloom in Siberia.

2.30. No parent wants harm for their children.

The purpose of studying the topic: the formation of basic ideas about judgment as a form of thinking, understanding the basics of their classification, the establishment of those operations that are carried out on judgments. Translation of a complex judgment into the language of propositional logic, their verification for truth using a truth table.

The importance of studying the topic for the practical activities of police officers: knowledge of logical unions, the ability to identify the logical form of one or another legal law can greatly help in interpreting legislation. It can even be argued that without knowledge of the logical form of the law, it is impossible to find out its meaning at all. If a lawyer wants not only to read the law, to learn it, but also to understand what it is about, then attention should be paid not only to the analysis of the content of this document. No less important is the role of logical examination of any legal documents, since only a logical law included in a legal law will enable the latter to be fully implemented.

Basic terms and concepts: attributive proposition, disjunction, implication, true proposition, quantifier, general quantifier, existential quantifier, conjunction, false proposition, modal proposition, incompatible propositions, incomparable propositions, general negative proposition, generally affirmative proposition, relation of opposite (contrary), relation of sub-opposite (subcontrary) ), relation of contradiction (contradiction), predicate judgment, simple judgment, link, compound judgment, compatible judgments, comparable judgments, subject of knowledge, judgment, judgment with relation, judgment of existence, private negative judgment, private affirmative judgment, equivalence.

Main content: when starting to study the topic “Judgments, types, composition, logical relations”, first of all, one should define the concept of “judgment” as a form of thinking.

Judgment - is a thought that affirms the presence or absence of properties of objects, relationships between objects, connections between situations, or judgment - it is a form of thinking that reveals the connection between an object and its attribute.

Like a concept, a judgment has its own specific structure, which can be represented using the following formula:

For example: All cadets are people who know logic.

Thus: the logical structure of the judgment consists of the subject "S", the predicate "P" and the logical connective "is / is not" or "are / are not".

Schematically, this is written as the following formula:

"S is P" or "S is not P" - Where: "S" and "P" are called propositional terms.

Drawing an analogy with the concept, it can be argued that the judgment has several types. When we speak about the outside world or talk about the inner world, we, so to speak, "judge" about it - hence the name of this logical form. A descriptive proposition can be true or false. The true judgment corresponds to reality: "The Russian Federation is a federation." The false judgment is not true: "The Russian Federation is a monarchy." Logic does not determine the truth or falsity of judgments - this is a matter of specific sciences or practice. The task of logic is to provide formal conditions and methods for maintaining truth throughout the entire process of reasoning.

The first of these conditions is the differentiation between simple and complex propositions.

Consider the whole variety of simple judgments that can be classified according to the following grounds:

1. According to the volume of the subject: single, general and private.

3. By the number of bundles: negative, affirmative and negative.

4. By modality: objective (judgments of reality, judgments of possibility, judgments of necessity) and logical (judgments are reliable and judgments are problematic).

Let us consider in more detail the judgments by the nature of the features that are represented by the judgment predicate.

attributive is a simple proposition whose predicate represents a property. You can also define an attributive judgment in this way: “Attributive judgment is a type of simple judgments in which we are talking about the presence of some properties in an object, or their absence in an object” ( For example: The crime must be solved.

Judgment with attitude This type of simple proposition is called, in which the predicate is the relation ( For example: My friend doesn't know my brother. In the judgment there is a negation of the relationship of knowledge between my friend and my brother).

judgment of existence a type of simple proposition is called, in which the predicate expresses the presence (being) of an object ( For example: There are people who can predict the future. There is no life on the moon).

Let us dwell on the analysis of attributive judgments. Interest in attributive judgments in traditional logic was caused by the fact that they were the source material in Aristotle's construction of the first theory of logical inference - syllogistics. To a large extent, this predetermined the fact that simple judgments (judgments with relations and judgments of existence) after appropriate syntactic reconstructions were interpreted as attributive.

Attributive judgments are divided into types according to quantity and quality.

By quality allocate: affirmative and negative attribute judgments ( For example: A crime is a socially dangerous act - an affirmative judgment).

In count distinguish: single, general and private attribute judgments.

single such an attributive judgment is called, in which the subject is a single concept. ( For example: Investigator Petrov is a good person).

General such an attributive judgment is called, in which the subject is a general concept ( For example: Crime is a socially dangerous act).

Private called an attributive judgment, in which the subject represents part of the class of objects under study ( For example: Some sentences are unfair.)

These two typologies of attributive judgments are singled out for methodological purposes. In the practice of reasoning, they exist in interaction, therefore, a typology of attributive judgments is specially distinguished according to the combined feature of a qualitative-quantitative characteristic:

· general affirmative,

· private affirmative,

· general negative,

· private negative attributive judgments.

	+	-
All	Asp	Esp
Some	isp	osp

generally affirmative is called a judgment which is general in quantity and affirmative in quality. For example: All students are doing well. The logical structure of a general affirmative judgment is as follows: "All S are P." This type of judgment is denoted by the letter "A").

private affirmative a judgment is such an attributive judgment, which is partial in quantity, and affirmative in quality ( For example: Some crimes are official. The logical structure of a particular affirmative judgment is as follows: "Some S are P." This judgment is denoted by the letter "I"). All-negative called an attributive judgment, which is general in quantity, and negative in quality ( For example: None of my friends were among the participants in the crime. Logical structure about A general negative proposition has the following form:"No S is P." This judgment is denoted by the letter "E").

Judgment - this is a form of thinking in which something is affirmed or denied about the connection between an object and its attribute or about the relationship between objects. Main logical characteristic proposition is its truth value - every proposition is either true or false. A proposition is true if and only if the situation described in it actually takes place, otherwise it is false.

By a simple judgment called proposition expressing the relationship of two terms. The terms in a simple judgment are called subject and predicate judgments. The subject of judgment (S ) is what is said in the judgment, i.e. subject of thought. Judgment predicate ( R) what is said about the subject, what signs are attributed to him or not are called. In addition to the subject and the predicate, the structure of the judgment includes the quantifier and the connective. The judgment quantifier indicates the amount of judgment, i.e. indicates the total, partial or singular quantity of the subject of judgment (expressed by the words "all", "none", "some", "this"). A copula denotes a relationship between a subject ( S ) and predicate ( R ) judgments, due to which thought takes the form of a judgment. The link indicates the quality of the judgment. (Expressed by the words "is", "is not", "is", "is not").

Unified classification of simple categorical propositions. Depending on the quantity and quality, there are generally affirmative, general negative, particular affirmative and particular negative judgments.

Affirmative ( BUT) called a judgment that is general in quantity and affirmative in quality. Canonical form "All S's are P's" .

General negative ( E) called a judgment that is general in quantity and negative in quality. Canonical form "No S is a P" .

private affirmative (I ) called a judgment that is partial in quantity and affirmative in quality. Canonical form "Some S's are P's" .

private negative ( O) called a judgment that is partial in quantity and negative in quality. Canonical form "Some S's are not P's» .

Distribution of terms in simple categorical judgments. In simple judgments, terms can be distributed ( S+ , R + ), or not distributed ( S- , R - ). A term is called distributed if it is taken in full in the judgment. A term is called undistributed if in the judgment it is taken in terms of volume. The distribution of terms in a judgment is derived from the definition of relations between concepts that express the terms of a judgment. When determining the distribution of terms in simple categorical judgments, one should be guided by the following rules:

a) B general affirmative judgments ( BUT) : subject ( S R ) is always undistributed in the case of a subordination relation between the subject and the predicate of the judgment; subject ( S ) is always distributed and the predicate ( R ) is always distributed in the case of an equivalence relation between the subject and the predicate of the judgment;

b) B general negative judgments ( E): subject ( S ) and predicate ( R ) judgments are always distributed;

c) B private affirmative judgments (I ) : subject ( S ) and predicate ( R ) are undistributed in the case of an intersection relation between the subject and the predicate of the judgment; and subject ( S ) is undistributed, and the predicate ( R) is distributed in the case of a subordination relationship between the predicate and the subject of the judgment;

d) B private negative judgments ( O) : subject of judgment ( S ) is always undistributed, and the judgment predicate ( R ) is always distributed.

Complicated judgment is called a proposition, consisting of several simple, connected by logical connectives. Writing a complex proposition in the symbolic language of logic, in which simple propositions are replaced by symbols p, q, r, s, t ..., and logical unions to the symbols that replace them Ù, v, → , ↔ is called the logical form of a compound proposition. There are five main types of logical connection:

Asserting the presence of several situations at the same time - conjunction (Ù );

Statement of the presence of at least one of several situations - weak disjunction(v);

Asserting the existence of only one of several situations - strong disjunction ();

One situation is a sufficient condition for the occurrence of another situation - implication (→);

One situation is a sufficient and necessary condition for the occurrence of another situation - equivalent (↔).

Depending on the type of logical connection, the following complex judgments are distinguished:

- connecting judgments- judgments in which simple judgments are interconnected by a logical connective conjunction ( Ù ). Boolean form: ( R Ù q );

- disjunctive judgments- judgments in which simple judgments are interconnected by a logical connective weak disjunction ( v) or strong disjunction (). Boolean form: ( R v q ); (pq );

- conditional propositions- judgments in which simple judgments are interconnected by a logical link implication ( → ) or the equivalent ( ↔ ). Boolean form: ( R → q ), (R ↔ q ), where R - basis of judgment q - a consequence of judgment. In conditional propositions in the correct logical form, the base always comes first, and the conclusion at the end of the formula.

The truth values ​​of complex judgments depend on the truth values ​​of the constituent judgments and on the type of their connection, which is determined by compiling truth tables:

- conjunction (Ù ) takes the value " True» only in the case of simultaneous truth of all variables; in other cases, the conjunction takes the value " Lie» (See: Fig. 18);

- weak (nonstrict) disjunction(v) takes on the value " Lie» only in the case of simultaneous falsity of all variables; in other cases, the weak disjunction takes the value " True» (See: Fig. 19);

- strong (strict) disjunction() takes the value " Lie» in case of simultaneous truth or falsity of all variables; in other cases, strong disjunction takes the value " True» (See: Fig. 20);

- implication (→ ) takes the value " Lie"only in the case of the truth of the basis of the judgment and the falsity of the consequence of the judgment; in other cases, the implication takes the value " True» (See: Fig. 21);

- equivalent (↔ ) takes the value " Lie"in the case of the truth of the foundation and the falsity of the consequence of the judgment, or vice versa, the falsity of the foundation and the truth of the consequence of the judgment; in other cases, the equivalent takes the value " True» (See: Fig. 22).

negation of judgment- this is an operation consisting in the transformation of the logical content of the negated judgment, the end result of which is the formulation of a new judgment, which is in relation to the contradiction to the original judgment. The negation of a simple attributive judgment is made according to the following equivalences: A = O; E = I; I = E; O = A - where A, E, I, O - types of simple categorical judgments, - a sign of external negation.

The negation of a complex judgment is made according to the following equivalences:

(p Ù q) ↔ (p v q)– 1st De Morgan's law

(p v q) ↔ (p Ù q)– 2nd De Morgan's law

(p q) ↔ (p ↔ q)

(p → q) ↔ (p Ù q)

(p ↔ q) ↔ (p Ù q) v (p Ù q)

We express the above in the form of complex schemes:

Rice. 23-24

Rice. 27.

Typical examples on the topic "Judgment"

Task 6. Bring the statement to the correct logical form, give a unified classification of judgments, give their schemes and the designations A, E, I, O accepted in logic.

To solve the problem, we use an algorithm for reducing natural language sentences to the canonical form of categorical judgments and analysis of simple judgments.

1. Determine subject and predicate statements, naming them accordingly S and R (composite S and R emphasize with one solid line).

2. When defining a predicate, keep in mind the following:

If the predicate is expressed noun or phrase with noun, then in this case predicate remains unchanged.

sample 1:

« Some lawyers (S) - lawyers (R) ».

If the predicate is expressed adjective or communion, which can be represented , then in this case .

Sample 2:

« Some roses (S) beautiful (R) ». → « Some roses (S) - beautiful flowers (R) ».

If the predicate is expressed verb, which can be represented one word or phrase, then in this case a generic concept for the subject of the statement should be added to the predicate, a turn the verb into its corresponding participle.

Sample 3:

« Some students of our group (S) handed over today logically (R) ». → "Some students of our group (S) there is students who passed the logic test today (R) ».

3. Determine quantifier word ("all", "some", "none", "this").

4. Determine logical link("is", "is not")

5. Record judgment in canonical form: quantifier - subject ( S) - connective - predicate ( R) .

6. Record judgment formula, to determine the quantitative and qualitative characteristics of the judgment.

7. Graphically portray relations between terms of judgment.

8. Determine distribution terms.

Example 1:

"The ancient Greeks made a great contribution to the development of philosophy."

Solution:

1. In this sentence, only the subject is logically defined - "ancient Greeks" ( S ). The predicate is expressed by the phrase "made a great contribution to the development of philosophy" ( R ).

2. Bring the predicate to canonical form. To do this, we select to the subject of the judgment ( "Ancient Greeks") generic concept ( "People"). AT canonical predicate form will be expressed as a phrase "People who have made a great contribution to the development of philosophy".

3. quantifier word in a sentence missing, but from the analysis of the meaning of the sentence it is clear that we are talking only about some of the ancient Greeks. Judgment quantifier - " Some».

4. The proposal states that the subject « Ancient Greeks» ( S Made a great contribution to the development of philosophy» ( R ). Means logical connective affirmative (« there is»).

5. Canonical judgment form: Some ancient Greeks (S) there is people. who made a great contribution to the development of philosophy (R) ».

6. Formula judgments - Some S's are P's . Quantitative-qualitative characteristic of the judgment - private affirmative

7. We graphically depict the relationship between the terms of the judgment. We define the relationship between the concept " Ancient Greeks» ( S ) and the concept " People who made a great contribution to the development of philosophy» ( R ) as a ratio crossing .

8. Define distribution terms: both terms are taken in terms of volume, which means they are undistributed ( S - , R - ) (Fig. 28).

Example 2:

"No one can be held criminally responsible twice for the same crime."

Solution:

1. In this offer subject is not explicitly defined. From an analysis of the meaning of the statement, it is clear that It's about the concept of Human» (S ) . Predicate expressed by the phrase "" ( R ).

2. Bring the predicate to canonical Human"") generic concept (" Creature"). In canonical form predicate will be expressed by the phrase "" ( R ).

3. quantifier word in a sentence missing, but from the analysis of the meaning of the sentence it is clear that it is about the whole volume the concept of "person" S ). quantifier judgments - None».

4. The sentence denies that the subject has “ Human» ( S ) property expressed in the predicate " Can be held criminally responsible twice for the same crime» ( R). (« do not eat»).

5. Write down the judgment in canonical form: " No one human (S) do not eat a living being that can be criminally responsible twice for the same crime (R) ».

6. Recording formula judgments - No S is P general negative (E ).

7. Graphically depict the relationship between the terms of the judgment. We define the relationship between the concept " Human» ( S ) and the concept " A living being that can be criminally responsible twice for the same crime» ( R ) as a ratio incomparability .

8. Define distribution terms: both terms are taken in full, which means they are distributed (S+ , R + ) (Fig. 29).

Example 3:

"Some mushrooms are not edible."

Solution:

1. In this sentence, logically only the subject is defined - " Mushrooms" ( S ) . Predicate expressed by the word edible» ( R ).

2. Bring the predicate to canonical form. To do this, we select the subject of the judgment (“ Mushrooms"") generic concept (" Living organisms"). In canonical form, the predicate will be expressed by the phrase " edible living organisms» ( R ).

3. quantifier the word is present in the sentence we are talking about part of the scope of the concept " Mushrooms» (S ). quantifier judgment word - " Some».

4. Offer denied availability subject « Mushrooms» ( S ) property expressed in predicate « Edible» ( R ). Logical connective is negative (« do not eat»).

5. Write down the judgment in canonical form: " Some mushrooms (S) do not eat edible living organisms (R) ».

6. Recording formula judgments - Some S's are not P's . We determine the quantitative and qualitative characteristics of the judgment - private negative (O ).

7. Graphically depict the relationship between the terms of the judgment. We define the relationship of the relationship between the concept " Mushrooms» ( S ) and the concept " edible living organism» ( R ) as a ratio crossing .

8. Define distribution terms: S taken in terms of volume, a R taken in full, means, distribution theirs is: S - , R + (Fig. 30).

Task 7. Consider complex judgments, express them in symbolic notation. Indicate the antecedent and consequent in implicative judgments.

Example 1:

Their labor rights, freedoms and legitimate interests in all not prohibited ways.

Solution:

a) " The worker has the right to protection their labor rights R);

b) "The worker has the right to protection their freedoms by all means not prohibited" - ( q);

in) "The worker has the right to protection their legitimate interests by all means not prohibited" - ( r).

conjunction (Ù );

r u qÙ r

4. p, q, r are conjuncts.

Example 2:

"Humanity can die either from the depletion of earth's resources, or from an environmental catastrophe, or as a result of the third world war."

Solution:

1. We divide this complex judgment into simple ones and express them in the correct notation adopted in Russian, i.e. in the relation of the subject and the predicate and denote these simple judgments in the form adopted in formal logic:

a) "Humanity can die from the depletion of earth's resources" - ( R);

b) "Humanity can die from an ecological catastrophe" - ( q);

in) "Humanity may perish as a result of the third world war" - ( r).

weak disjunction(v);

3. The formula for this complex judgment looks like this:

R v q v r

4. p, q, r are clauses.

Example 3:

“A citizen, due to a physical disability, illness or illiteracy, cannot sign with his own hand, then at his request another citizen can sign the transaction.”

Solution:

1. We divide this complex judgment into simple ones and express them in the correct notation adopted in Russian, i.e. in the relation of the subject and the predicate and denote these simple judgments in the form adopted in formal logic:

a) “A citizen, due to a physical handicap, cannot sign with his own hand” - ( R);

b) “A citizen, due to illness, cannot sign with his own hand” - ( q);

in) “A citizen, due to illiteracy, cannot sign with his own hand” - ( r);

G) “At the request of this citizen, another citizen can sign the transaction” - ( s).

2. In this case, there is a statement of the presence of at least one of several situations, but other situations can also be present at the same time - weak disjunction(v); one of these situations or all of them at the same time is a sufficient condition for the occurrence of another situation - implication(→); thus, we have jointly weak disjunction and implication;

3. The formula for this complex judgment looks like this:

(R v q v r) → s

4. p, q, r are disjuncts; (R v q v r) – antecedent; s is the consequent.

Example 4:

“The marriage is terminated if the court finds that the further life of the spouses and the preservation of the family have become impossible.”

Solution:

1. We divide this complex judgment into simple ones and express them in the correct notation adopted in Russian, i.e. in the relation of the subject and the predicate and denote these simple judgments in the form adopted in formal logic:

a) “The court found that the further joint life of the spouses became impossible” - ( R);

b) “The court found that the preservation of the family became impossible” - ( q);

in) "Marriage is dissolved" - ( r).

2. In this case, there is a statement of the simultaneous presence of several situations - conjunction (Ù ); both of these situations are a sufficient condition for the occurrence of another situation - implication(→); thus takes place jointly conjunction and implication;

3. The formula for this complex judgment looks like this:

(r u q) → r

4. р, q – conjuncts; (R v q) – antecedent; r is the consequent.

Task 8. Write down the logical formulas of complex judgments in the language of propositional logic and construct truth tables for them.

To solve the problem, we use the algorithm for analyzing complex statements:

1. Identify and write down all the simple propositions that make up the sentence. Label them with symbols.

2. Determine the logical connection between simple judgments.

3. Write down the formula for a complex judgment. If the judgment is conditional, then it is necessary to determine the reason and the consequence.

4. Compile and fill in the truth table of a complex judgment.

Example 1

"Insult can be inflicted accidentally or intentionally"

Solution:

a) "Insult can be inflicted by accident" - (R)

b) "Insult may be intentional" – (q)

2. Union " or» in the statement asserts the presence of only one of the two situations. The logical connection in this judgment is strong disjunction ().

3. The formula of a complex judgment: p q.

4. We build a truth table for the judgment of this form.

To build a truth table, you need to know the number of columns when entering the table (the number of variables) and the number of rows in the table ( x = 2n , where X - the number of rows in the table, n - the number of variables in the formula). This table has three columns ( R , q, p q) and four lines (2 2 = 4). In the first column we write down all the truth options for R (I and L). In the second column, against each of the values ​​​​of the first column, it fixes the values ​​\u200b\u200bfirst both times as AND, and then both times as L. Under the logical union sign, a strong disjunction () writes the final result, focusing on the truth table placed on page 3, fig. 20. The formula of this judgment is feasible, since it takes both the value of I and the value of L.

R	q	p q
And	And	L
L	And	And
And	L	And
L	L	L

The system for constructing truth tables for any number of propositional ones can be understood from the following considerations:

AT general case number of all possible sets of values n variables is 2n. For example, the number of valid interpretations for a single variable is 2 1 = 1 ; for two variables - 2 2 = 4 ; for three variables - 2 3 = 8; for four variables is 16 , for five - 32 etc.

For example, let the sequence of propositional variables р 1 , р 2 , …p n consists only of one variable ( n= 1). Then there is only two value set:<and > and<l >:

Let the sequence of propositional variables р 1 , р 2 , …p n comprises two variables ( n= 2). In this case, the sets of specified values ​​will be such pairs (there are four):

<and , and >, <l , and >, <and , l >, <l , l >.

If this sequence contains three variables, then the sets of such values ​​will be such combinations ( eight triplets):

<и, и, и>, <л, и, и>, <и, л, и>, <л, л, и>,

<и, и, л>, <л, и, л>, <и, л, л>, <л, л, л>

Formal logic uses the following propositional connectives: , ^, v, →, ↔, where

Symbol denial(additions);

^ - symbol conjunctions(associations);

v - character non-strict disjunction(separation-unification);

- symbol strict disjunction(separation-exclusions);

→ - character implications(logical consequence).

↔ - symbol equivalences(logical identity).

When denial(additions) statement ( BUT) takes the value "true" only if BUT false. And vice versa, if BUT true, then ( BUT)- false.

Example 2

"Turning your back on the most intriguing events of history, it is impossible to understand the logic of this story."

Solution:

1. Define and write down simple judgments:

a) "Man has turned his back on the most intriguing events in history" - R (base)

b) “A person cannot understand the logic of this story” - q (consequence)

2. Union " if, ... then ..." means that the situation expressed by the base ( "man has turned his back on the most intriguing events in history") is sufficient condition for the occurrence of the situation expressed by the consequence ( “a person cannot understand the logic of this story”). The logical connection in this judgment is implication (→ )

3. Judgment formula: p → q

4. We build a truth table for a judgment of this form (see p. 4, Fig. 21).

Under the sign of the logical union, the implication ( → ) we write down its truth values. The formula of this judgment is feasible, since it takes both the value of I and the value of L.

R	q	p → q
And	And	And
L	And	And
And	L	And
L	L	And

Example 3

“If a student is in this faculty, then he is capable or very diligent.”

Solution:

1. Define and write down simple judgments:

a) "The student is studying at this faculty" - R(base)

b) "This student is capable" - q(consequence)

in) "This student is diligent" - r(consequence)

2. Union " if..then.." means that the situation expressed by the reason ("the person studies at this faculty") is a sufficient condition for the occurrence of the situation expressed by the consequence ("he is capable or very diligent"). The logical connection in the judgment is the implication ( → ). As a result, there is a union “or” between the judgments, which means the statement of the presence of at least one of the two situations. Logical connection - weak disjunction (v).

3. Judgment formula: p → (q v r)

4. We build a truth table for the judgment of this form. The number of columns in the input to the table is three (the variables in the formula are 3), and the number of rows in the table is 8. In order to determine the truth values ​​of this formula, it is necessary to determine the procedure. The first step is to find the truth value of the weak disjunction (v), and then the truth value of the implication ( → ).

The truth values ​​of the implication ( → ) are the truth values ​​of the given formula. The formula of this judgment is feasible, since it takes both the value of I and the value of L.

Task 9. Determine the modality of the judgment, write judgments using modal operators:

Modality(from lat. modus - measure, method) is explicitly or implicitly expressed in the judgment judgment characteristic, additional information about the logical and actual status of the judgment, about its regulatory, evaluative, temporal and other characteristics, about the degree of its validity.

initial information in judgment express, as we already know, subject, predicate, quantifier word and mode of expression this information is the formula (S-P) .

Concerning additional information, it can be very different. So, for example, the logician of the middle of the XIII century. William Sherwood counted six types of modal forms: true, false, Maybe, impossible, by chance and necessary. AT contemporary In logical thinking, modalities that appear under the names are used more often than others. alethic, deontic and epistemic.

The concept of "alethic"(from Greek aletheia - truth) means "true". Alethic modality in this sense is a relation to basic requirement of logic- to express criteria true and false statements.

Alethic modality is expressed in judgments and terms necessity-accident or possibilities-impossibility information about the features of the logical or actual determinism of judgments.

Asserting the existence of something, as true to reality , denoted symbolically as p.

Likewise, affirmation of the non-existence of something, as a negative reality , denoted by -ÿ ù p.

Example:

“The presence of a causal relationship between the actions committed by this person and the socially dangerous consequences that have occurred ( p) is an indispensable condition for bringing him to criminal responsibility ( q)».

ÿ (p ® q).

As opposed to "necessity", "chance" is not associated with inevitability, but fixes only private events in their arbitrary occurrence and existence.

Example:

p) sometimes contributes to the occurrence of cardiovascular diseases ( q)».

In terms of alethic modality, this statement looks like this:

ù ÿ (p ® q).

As for the "possibility" of something, then she is always bound with the compatibility of the phenomenon under consideration with other phenomena, components for this phenomenon environment of his existence.

Example:

"Environmental pollution ( p) may contribute to the occurrence of cardiovascular and pulmonary diseases ( q)».

In terms of alethic modality, this statement looks like this:

à (p ® q).

In turn, the "impossibility" of something always tied With the incompatibility of a given phenomenon with others that are its environment for it.

Along with the concept, judgment is one of the main forms of thinking. Judgment - a form of thinking in which something is affirmed or denied about the existence of objects, the connections between an object and its properties, or about the relationship between objects.

Examples of judgments: "Astronauts exist", "Paris is bigger than Marseille", "Some numbers appear even". If what is said in the judgment corresponds to the actual state of things, then the judgment is true. The above judgments are true, since they adequately (correctly) reflect what takes place in reality. Otherwise, the proposition is false ("All plants are edible").

Traditional logic is two-valued because in it a proposition has one of two truth values: it is either true or false. In three-valued logics – varieties of multivalued logics – a proposition can be either true or false or indeterminate. For example, the proposition "There is life on Mars" is currently neither true nor false, but uncertain. Many judgments about future single events are uncertain. Aristotle wrote about this, giving an example of such an indefinite judgment: "Tomorrow a sea battle will be necessary."

The language form of expression of a judgment is a sentence. A judgment is expressed by a declarative sentence, which always contains either an affirmation or a negation. Judgment and proposition differ in their composition. Every simple proposition consists of three elements:

1)the subject of judgment - This is the concept of the subject matter. The subject of judgment is denoted by the letter S (from the Latin word subjectum);

2)judgment predicate – concept of the attribute of the object referred to in the judgment. The predicate is denoted by the letter R (from lat. praedicatum);

3)bundles, expressed in Russian by the words "is", "is", "essence".

The subject and the predicate are called terms of judgment. The structure of some judgments also includes the so-called quantifier words (“some”, “all”, “none”, “sometimes”, etc.). The quantified word indicates whether the judgment refers to the entire scope of the concept expressing the subject, or to a part of it.

TYPES OF SIMPLE JUDGMENTS

1. Property judgments (attributive):

they affirm or deny belonging to the subject of known properties, states, activities.

Scheme this kind of judgment: « S there is R" or « S do not eat R".

Examples : "Honey is sweet", "Chopin is not a playwright."

2. Relationship Judgments:

judgments reflecting the relationship between objects.

Formula , expressing a judgment with a two-place relation, is written as aRb or R(a,b ), where a and b- names of objects (members of the relation), and R – relation name. In an attitude judgment, something can be affirmed or denied not only about two, but also about three, four or more objects, for example: "Moscow is between St. Petersburg and Kyiv." Such judgments are expressed by the formula R(a ,a ,a ,…,a).

Examples: “Every proton is heavier than an electron”, “French writer Victor Hugo was born later than the French writer Stendhal”, “Fathers are older than their children”.

3. Judgments of existence (existential):

they express the very fact of the existence or non-existence of the object of judgment.

Scheme this kind of judgment: « S there is R" or « S do not eat R".

Examples of these judgments: "There are nuclear power plants", "There are no causeless phenomena."

In traditional logic, all three of these types of judgments are simple categorical judgments. According to the quality of the link (“is” or “is not”), categorical judgments are divided into affirmative and negative . Judgments: " Some teachers are talented educators" and " All hedgehogs are prickly"- affirmative. Judgments: " Some books are not secondhand" and " No rabbit is a carnivore' are negative. The link "is" in an affirmative judgment reflects the inherent nature of the object (objects) of certain properties. The link “is not” reflects the fact that a certain property is not inherent in the object (objects).

Some logicians believed that there is no reflection of reality in negative judgments. In fact, the absence of certain features is also a real feature that has objective significance. In a negative true judgment, our thought disunites (separates) that which is divided in the objective world.

In cognition, an affirmative judgment is generally more important than a negative one, because it is more important to reveal what feature an object has than what it does not have, since any object does not have very many properties (for example, a dolphin is not a fish, not an insect, not a plant, not a reptile, etc.).

Judgments are divided into general, private and single.

For example: "All sable – valuable fur animals "and" All sane people want a long, happy and useful life "(P. Bragg) – general judgments ; "Some animals – waterfowl" – private ; Vesuvius – active volcano" – singular .

Structure general judgments: "All S are (not the essence) R". Singular judgments will be treated as general ones, since their subject is a one-element class.

Among the general statements there are highlighting judgments, which include the quantified word "only". Examples of highlighting judgments: "Bragg only drank distilled water"; “A brave man is not afraid of the truth. Only a coward is afraid of her ”(A.K. Doyle).

Among the general statements are exclusive judgments, for example: "All metals at a temperature of 20 ° C, with the exception of mercury, are solid." Exceptional judgments also include those in which exceptions are expressed from certain rules of Russian or other languages, rules of logic, mathematics, and other sciences.

Private judgments have structure: "Some S essence (not essence) R". They are divided into indefinite and definite. For example, "Some berries are poisonous" – indefinite private judgment. We have not established whether all berries have a sign of toxicity, but we have not established that some berries do not have a sign of toxicity. If we have established that "only some S have the attribute R", then it will be a certain private judgment, the structure of which is: “Only some S essence (not essence) R". Examples: "Only some berries are poisonous"; "Only some figures are spherical"; "Only some bodies are lighter than water." Quantifier words are often used in certain private judgments: most, minority, many, not all, many, almost all, a few, etc.

AT single in judgment, the subject is a single concept. Singular judgments have a structure: "This S is (is not) P." Examples of singular judgments: "Lake Victoria is not in the USA"; "Aristotle – educator of Alexander the Great"; "Hermitage – one of the world's largest art and cultural-historical museums.

Thus, a special place in the classification of judgments is occupied by distinguishing, excluding and definitely particular judgments, which are built on the basis of attributive judgments and represent some complicated variants of the latter:

The procedure for reducing natural language sentences to the canonical form of categorical propositions

1. Determine the quantifier, subject and predicate of the statement.

2. Put the quantifier words "all" ("none") or "some" at the beginning of the statement.

3. Put the subject of the statement after the quantified word.

4. Put the logical connective "is" ("essence") or "is not" ("is not the essence") after the subject of the statement.

5. Put the predicate of the statement after the logical connective.

When performing the last operation, keep the following in mind:

Firstly, if the predicate is expressed by a noun that can be represented by a single word or phrase, then in this case the predicate remains unchanged;

Secondly, if the predicate is expressed by an adjective (participle), which can be represented by one word or phrase, then in this case a generic concept for the subject of the statement should be added to the predicate;

Thirdly, if the predicate is expressed by a verb that can be represented by one word or phrase, then in this case a generic concept for the subject of the statement should be added to the predicate, and the verb should be turned into the corresponding participle.

Each judgment has both quantitative and qualitative characteristics. Therefore, in logic, a combined classification of judgments by quantity and quality is used, on the basis of which the following are distinguished four types of judgments :

1. BUT – general assertion.

Structure: "All S essence R".

Example: "All people want happiness."

2. I– private statement.

Structure: "Some S's are R".

Example: "Some lessons stimulate the creative activity of students."

ü Conventions for affirmative judgments are taken from the word affirmo, or affirm; in this case, the first two vowels are taken: BUT – to denote a general affirmative and I – to denote a particular affirmative judgment.

3. E – general negative judgment.

Structure: "None S do not eat R".

Example: "No ocean is freshwater."

4. O– private negative judgment.

Structure: "Some S don't eat R".

Example: "Some athletes are not Olympic champions."

ü The symbol for negative judgments is taken from the word nego , or I deny.

In judgments, the terms S and R may or may not be allocated. The term is considered distributed, if its scope is fully included in the scope of another term or completely excluded from it. The term will undistributed, if its scope is partially included in the scope of another term or partially excluded from it. Let's analyze four types of judgments: A, I, E, O(we consider typical cases).

1. Judgment BUT – general affirmative . Its structure is: All S is P ».

Consider two cases:

Example 1 . In the judgment "All carp – fish" the subject is the concept of "crucian", and the predicate – the concept of fish. General quantifier – "all". The subject is distributed, since we are talking about all crucian carp, i.e. its scope is fully included in the scope of the predicate. The predicate is not distributed, since only a part of the fishes that coincide with crucian carp are conceived in it; we are talking only about that part of the scope of the predicate, which coincides with the scope of the subject.

Example 2 . In the proposition "All squares are equilateral rectangles" the terms are: S- "square", R- "equilateral rectangle" and the quantifier of generality - "all". In this judgment S is distributed and P is distributed, because their volumes are exactly the same. If a S equal in volume R, then R distributed. This happens in definitions and in singling out general judgments.

2. Judgment I – private affirmative . Its structure is: Some S is P ». Let's consider two cases.

Example 1 . In the judgment “Some teenagers are philatelists”, the terms are: S - "teenager", R– “philatelist”, existential quantifier – “some”. The subject is not distributed, since only a part of adolescents is conceived in it, i.e. the scope of the subject is only partially included in the scope of the predicate. The predicate is also not distributed, since it is also only partially included in the scope of the subject (only some philatelists are teenagers). If concepts S and R cross, then R not distributed.

Example 2 . In the judgment "Some writers are playwrights" the terms are: S - "writer", P - "playwright" and the existential quantifier - "some". The subject is not distributed, since only a part of writers is conceived in it, i.e. the scope of the subject is only partially included in the scope of the predicate. The predicate is distributed, because the scope of the predicate is completely included in the scope of the subject. In this way, R distributed if the volume R less than the volume S , what happens in particular highlighting judgments.

3. Judgment E – general negative . Its structure is: None S is not P » . For example : "No lion is a herbivore." In it, the terms are: S - "lion", R- "herbivore" and the quantifier word - "none". Here the scope of the subject is completely excluded from the scope of the predicate, and vice versa. Therefore, S , and R distributed.

4. Judgment O –private negative . Its structure is: Some S is not P ». For example : "Some students are not athletes." It contains the following terms: S - "student", R– "sportsman" and the existential quantifier are "some". The subject is not distributed, since only a part of the students is conceived, and the predicate is distributed, because all athletes are conceived in it, none of which is included in that part of the students that is conceived in the subject

So, S is distributed in general judgments and not distributed in particular; P is always distributed in negative judgments, while in affirmative ones it is distributed when, in terms of volume, P ≤S.

Imagine it in the term distribution table:

Terms / Type of judgment	A	E	I	O
S
P
P highlighting judgments

The subject is distributed in general and not distributed in particular judgments. The predicate is distributed in negative and not distributed in affirmative propositions. In distinguishing propositions, the predicate is distributed.

Designations: +– distribution of the term;

– – undistributed term

· JUDGMENTS WITH RELATIONSHIPS are such judgments in which the relationship between two terms - the subject and the predicate is expressed not with the help of a connective (“is”, “is”, etc.), but with the help of a relation in which something is affirmed or denied in relation to two (multiple) terms. In this type of judgment, the predicate is a relation, and the subject is two (or more) concepts. The locality of the relationship is determined by the number of concepts included in the subject.

· Judgments with relations are divided by quality into affirmative and negative. Judgments with relations are divided by number. The most common are judgments with two-place relations. Two-place relations have a number of properties on the basis of which one can draw conclusions from judgments about relations. These are the properties of symmetry, reflexivity and transitivity.

• The relation is called symmetrical(from Latin “proportionality”), if it takes place both between objects x and y , and between objects y and x (if X equal to (similar to, at the same time) y , then and y equal to (similar to, at the same time) X .
• The relation is called reflective(from Latin “reflection”), if each member of the relation is in the same relation to itself (if X =at , then X =X and at =at ).
• The relation is called transitive(from Latin "transition"), if it takes place between X and z , when it occurs between X and at and between at and z (if X equals at and at equals z , then X equals z ).

Every judgment is expressed in a sentence, but not every sentence expresses a judgment.

Ø Judgments are expressed through declarative sentences, which always contain either an affirmation or a negation. That is why declarative sentences, as the grammatical equivalent of a judgment, are a completely complete thought, which affirms or denies the connection between an object and its attribute, the relationship between objects, the fact of the existence of an object, and which can be either true or false.

Ø Interrogative sentences do not contain judgments in their composition, since nothing is affirmed or denied in them. They are neither true nor false. For example: “When will you start gardening?” or “Is this method of learning a foreign language effective?”. If the sentence is a rhetorical question, – for example: “Who does not want happiness?”, “Which of you did not love?” or “Is there anything more monstrous than an ungrateful person?” (W. Shakespeare), or “Is there a person who looks at the river in a moment of thought and does not remember the constant movement of all things?” (R. Emerson), – then it contains a judgment, since there is an assertion, a certainty that "Everyone wants happiness" or "All people love", etc.

Ø Interrogative-rhetorical sentences contain judgments in their composition, since something is affirmed or denied in them. They can be either true or false.

Incentive Offers do not contain judgments in their composition: (“Take care of your health”; “Do not make fires in the forest”, “Go not to the skating rink, but to school!”). But sentences in which military commands and orders, calls or slogans are formulated express judgments, however, not assertoric, but modal (modal judgments include modal operators expressed in the words: perhaps, necessary, forbidden, proved, etc.). For example: “Take care of the world!”, “Get ready to start!”, “My friend! Let us dedicate our souls to the Fatherland with wonderful impulses ”(A.S. Pushkin). These sentences express judgments, but the judgments are modal, including modal words. As A.I. Uyomov, express judgments and such incentive sentences: “Protect the world!”, “Do not smoke!”, “Fulfill your obligations!”. "Before any meal, eat raw vegetable salad or raw fruit" and "Do not harm yourself by overeating" – these advices (calls) of the famous American scientist Paul Bragg, taken from his book "The Miracle of Fasting", are judgments. It is a judgment and a call: “People of the world! Let's unite our efforts in solving universal, global problems!

Ø One-part impersonal sentences and nominal are judgments only when considered in context and with appropriate clarification.

The criterion for the presence of a judgment in the composition of a sentence is the presence of a moment of affirmation or negation, leading to an assessment of the judgment for truth or falsity.

In natural language, the same proposition can be expressed in different sentences. Therefore, in logic, in order to avoid ambiguity and the multiplicity of different meaningful interpretations of the sentence, the term "statement" is used, meaning by it some formalized expression of thought, which can have only one logical meaning. A judgment considered together with the sentence expressing it is a proposition. The latter is a grammatically correct declarative sentence, taken together with the meaning unambiguously expressed by it; it can be either true or false.

II. Types and logical probability of complex judgments

Compound judgments are formed from simple ones, as well as from other complex judgments with the help of the unions "if ..., then ...", "or", "and", etc., with the help of the negation of "it is not true that", modal the terms "it is possible that", "it is necessary that", "accidentally that", etc. These conjunctions, the negation of "it is not true that", modal terms in everyday language are used in various senses. In scientific languages, they are given a precise meaning, as a result of which different types of judgments are distinguished, formed from other judgments by means of, for example, the same grammatical union.

I.connecting are called judgments in which the existence of two or more situations is affirmed. Most often, these judgments are expressed in the language by sentences containing the union "and".

The union "and" is used in different meanings. For example, the sentences "Petrov studied English and he studied French" and "Petrov studied French and he studied English" express the same proposition, while the sentences "Petrov graduated from the university and entered graduate school" and "Petrov entered to graduate school and graduated from the university" express different opinions.

Thus, there are different types of statements about the presence of two or more situations, i.e. different types of connecting propositions: (indefinitely) conjunctive, sequentially conjunctive, simultaneously conjunctive.

1. (Indefinitely) conjunctive propositions are formed from two judgments by means of a union, denoted by the symbol & (read "and") and called the sign (indefinite) conjunctions. The definition of the conjunction sign is a table showing the dependence of the truth of a conjunctive judgment on the truth of its constituent judgments.
2. Consistently conjunctive judgments. These judgments assert the successive occurrence or existence of two or more situations. They are formed from two or more propositions with the help of unions, denoted by the symbols & ® 2 , & ® 3, etc., depending on the number of propositions from which they are formed. These characters are called signs of sequential conjunction and are respectively read "..., and then ..", "..., then..., and then ...", etc. Indices 2,3 etc. indicate the area of ​​the union. The form of the judgment with the sign of the double consecutive conjunction: & ® 2 (A, B) or (BUT&® 2 AT). Example judgments of this form: "The buyer paid the cost of the goods, and then the seller issued the goods." Instead of the expression "and then" the union "and" is most often used: "The buyer paid the cost of the goods, and the seller issued the goods." A form of judgment with a tripartite conjunction. Example: "Petrov mortgaged the apartment, then contributed money to the pyramid, and then became a man of no fixed abode."
3. Simultaneously conjunctive judgments. These judgments are formed from two judgments by means of the union "and", called the sign simultaneous conjunction. Notation - & = . These judgments assert the simultaneous existence of two situations. Example: "It's raining and the sun is shining."

1. disjunctive, or not strictly separating, or connecting-separating, judgments. These judgments assert the existence of at least one of two situations. They are formed from two propositions by means of the union "or", denoted by the sign v (read "or"), called the non-strict disjunction sign (or simply the disjunction sign).
2. Strictly disjunctive, or strictly dividing, judgments. These judgments assert the presence of exactly one of two, three or more situations. They are formed from two, three, etc. judgments through the unions "or ..., or ..." ("either ..., or ..."), "or ..., or ..., or ...", etc. Sometimes the union "or ..., or ..." is replaced by the union "or", and its divisive meaning is determined by the context. The conjunctions by means of which strict disjunctive judgments are formed are denoted by the sign v.

III. Conditional propositions are expressed, as a rule, by sentences with the union "if ..., then ...". They argue that the presence of one situation determines the presence of another. Example: "If the sun is at its zenith, then the shadows from it are the shortest." In a conditional proposition, a reason and a consequence are distinguished. foundation the part of the conditional proposition that is between the word "if" and the word "then" is called. The part of the conditional proposition that comes after the word "that" is called consequence. In the proposition "If it rains, then the roofs of the houses are wet," the basis is the simple proposition "it is raining," and the consequence is "the roofs of the houses are wet."

A more strictly conditional proposition is defined by means of the notion of a sufficient condition. Condition is sufficient for any event, any situation, if, and only if, always, when there is this condition, there is also an event (situation). Thus, the presence of free electrons in a substance is a sufficient condition for the substance to be electrically conductive. conditional is called a judgment in which the situation described by the reason is a sufficient condition for the situation described by the consequence. The conditional union "if ... then ..." is indicated by an arrow (®).

IV. Counterfactual statements. Example: "If Petrov were president, he would not travel around the city by bus." As in conditional propositions, in these propositions a reason and a consequence are distinguished. The union "if ..., then ..." is indicated by the sign É, which is called the sign counterfactual implications. The judgment has such a meaning, the situation described by the reason does not take place, but if it existed, then the consequence would exist.

V. equivalent judgments. Judgments of equivalence assert the mutual conditionality of two situations. These judgments are expressed, as a rule, by means of sentences with the union "if, and only if, ..., then ..." ("then, and only then, ..., when ..."). They also have reasons and consequences. The reason in them expresses a sufficient and necessary condition for the situation described by the consequence ( The condition is called necessary for a given event (situation, action, etc.), if, and only if, in its absence, this event does not occur.) The union "if, and only if, ... then", used in the sense described, is denoted by the symbol º

In the judgment of equivalence, the event described by the consequence is also a sufficient and necessary condition for the event described by the reason.

VI. Judgment with external negation. This is a statement that asserts the absence of a certain situation.

External negation is indicated by the symbol "l" (negation sign). This sign in natural language corresponds to the negation “not” or the expression “it is not true that”, which usually appear at the beginning of a sentence. By placing the expression “it is not true that” before an arbitrary false statement, we obtain a true statement, and from a true statement by substituting the expression “it is not true that” to it, we form a false statement. A judgment with an external negation refers to complex judgments and is formed from a simple one through negation.

The truth values ​​of complex judgments depend on the truth values ​​of the constituent judgments and on the type of their connection. The identically true formula A formula is called which, for any combination of values ​​for the variables included in it, takes the value "true". Identical-false formula- one that (respectively) takes only the value "false". The formula to be executed can be either true or false.

So, conjunction(and b ) is true when both propositions are true. Strict disjunction ( a b ) is true when only one simple proposition is true. Nonstrict disjunction ( a b ) is true when at least one simple proposition is true. implication ( a e b ) true in all cases except one - when a - true, b- false. Equivalence ( a º b ) true when both statements are true or both are false. Negation (ù a) false gives truth, and vice versa.

Ø Any language construction consisting of a certain set of judgments can be translated into a symbolic language. To do this, you need to replace judgments with logical variables, and the connection between them with logical unions. The logical feature of a complex judgment, its form, depends on the union with which the variables are connected.

Ø A complex proposition, the logical form of which takes the value "true" for all sets of values ​​of its constituent variables, is called logically necessary. In other words, complex propositions that take the value "true" in all rows of the resulting column of truth tables are logically necessary (logically true) propositions. The logical form of a logically necessary proposition is expressed by an identically true formula, which, for any truth value of the variables, takes the value "true", that is, its resulting column consists only of "AND". Identical-true formulas are the basis of logically correct statements. Each such formula is considered as a law of logic (logical tautology).

Ø A complex proposition, the logical form of which takes the value "false" for all sets of values ​​of its constituent variables, is called logically impossible. In other words, complex judgments that take the value “false” from all sides of the resulting column of the truth table are logically impossible (logically false) judgments. The logical form of a logically impossible judgment is expressed by an identically false formula, which takes the value "false" for any truth value of the variables, that is, its resulting column consists only of "L". Identical false formulas are called contradictions.

Ø A complex proposition, the logical form of which in the resulting column of the truth table takes on the values ​​both "true" and "false", is called logically random. The logical form of a logically random proposition is expressed by a neutral (actually feasible) formula, the resulting column of which consists of both "I" and "L".

Ø The peculiarity of the first two types of complex judgments is that their truth and falsity do not depend on the truth and falsity of the simple judgments that make them up. Logically random propositions are sometimes true, sometimes false. And it depends on which simple propositions are true and which are false.

III. Negation of judgments

NEGATIVE JUDGMENT - this is an operation consisting in transforming the logical content of the negated judgment, the end result of which is the formulation of a new judgment, which is in relation to the contradiction to the original judgment.

When denying simple attributive judgments:

1) a general judgment changes to a particular one, and vice versa;

2) an affirmative judgment changes to a negative one, and vice versa.

Attributive judgments are negated according to the following equivalences:

ù BUT is tantamount to O ù O is tantamount to BUT

ù E is tantamount to I ù I is tantamount to E

The negation of complex judgments is made according to the following equivalences:

u (A& AT) is tantamount to ù Avù B; according to de Morgan's law

u (AvB) is tantamount to ù A& ù B;

u (AÉ B) is tantamount to BUT& ù B;

u (Aº B) is tantamount to (ù A& AT)v(A& ù B);

u (Av AT) is tantamount to BUTº AT

IV. Relationship between judgments

The relationship between judgments of truth is usually depicted schematically in the form of a "logical square":

LOGICAL SQUARE

RELATIONSHIPS BETWEEN COMPLEX JUDGMENTS

Relations between complex judgments are divided into dependent (comparable) and independent (incomparable). Independent - judgments that do not have common components; they are characterized by all combinations of true values. Dependent - these are judgments that have the same components and can differ in logical connectives, including negation. Dependents, in turn, are divided into compatible (judgments that can be true at the same time) and incompatible (statements that cannot be true at the same time).

Relations

V. Modality of judgments

MODALITY - this is additional information expressed in the judgment about the logical or actual status of the judgment, about its regulatory, evaluative, temporal and other characteristics.

Assertoric judgments, that is, attributive and relational judgments, as well as complex statements formed from them, can be considered as judgments with incomplete information. The main function of an attributive judgment is to reflect the links between an object and its features. An object S can simply be said to have property P. Such an attributive judgment is simply a statement. Along with a simple statement (negation), the so-called strong and weak statements and negations, which are modal judgments, are distinguished.

MAIN TYPES OF MODALITIES:

Ø ALETIC MODALITY- expressed in the judgment through the modal concepts "necessary", "mandatory", "certainly", "accidentally", "possibly", "maybe", "not excluded", "allowed" and other information about the logical or factual determinism of the judgment . In the aletic group, there are ontological (actual ) modality, which associated with the objective determinism of judgments, when their truth or falsity is determined by the situation that takes place in reality, and logical modality , which associated with the logical determinism of the judgment, when the truth or falsity is determined by the form or structure of the judgment.

Ø EPISTEMIC MODALITY- it is expressed in a judgment by means of modal operators “known”, “unknown”, “provable”, “refutable”, “assumed”, etc. information on the grounds for acceptance and the degree of its validity.

Ø DEONTIC MODALITY- an instruction expressed in a judgment in the form of advice, wishes, rules of conduct or an order that encourages a person to take specific actions. The norms of law also belong to the deontic ones (the following operators can be distinguished here: “obliged”, “must”, “should”, “recognized”, “forbidden”, “cannot”, “not allowed”, “has the right”, “may have", "can accept", etc.).

Judgment modality ( R) is represented using the operator M, according to the scheme Mr(e.g. "possibly R"). The truth of a modal judgment depends on the truth of the judgment under the modal operator and on the type of the modal operator.

Modal simple judgments

Simple judgments expressing the nature of the connection between the subject and the predicate using modal operators (modal concepts)

pÉ q);M(pº q ).

Example: From the complex statement "If the temperature is above 100 degrees, then water turns into steam" you can get the modal statement "It is physically necessary that if the temperature is above 100 degrees, then water turns into steam."

VI. The concept of a logical law

Correct thinking must meet the following requirements: to be definite, consistent, consistent and justified. Certain thinking is precise and strict, free from any inconsistency. Consistent thinking is free from internal contradictions that destroy the necessary connections between thoughts. Consistency is associated with the non-admission of mutually exclusive, as equally acceptable, in one way or another, thoughts. Reasonable thinking is not just formulating the truth, but at the same time indicating the grounds on which it should be recognized as truth.

Since the features of certainty, consistency, consistency and validity are necessary properties of any thinking, they have the force of laws over thinking. Where thinking turns out to be correct, it obeys certain logical laws in all its actions and operations.

As already noted, the logical form of thought is the structure of thought, that is, the way its components are connected. So, between the thoughts, the logical forms of which are represented by the expressions “All S are P” and “All P are S” there is a connection: if one of these thoughts is true, then the second one is true, regardless of the specific content of these thoughts. Connections between thoughts, in which the truth of some necessarily determine the truth of others, determine the formal logical laws, or the laws of logic.

§ LAWS OF LOGIC- these are such expressions that are true only by virtue of their logical form, that is, only on the basis of the connection of their components. In other words, the logical law is the logical form itself, which guarantees the truth of the expression for any content.

§ LAW OF LOGIC is an expression that contains only constants and variables and is true in any (non-empty) subject area (for example, any law of propositional logic or predicate logic is an example of a logical law). These are the so-called laws of communication between thoughts. The laws of logic are also called tautologies.

§ LOGICAL TAUTOLOGY is an "always true expression", that is, it remains true no matter what domain of objects it is. Any law of logic is a logical tautology.

§ A special role is played by the so-called laws (principles) defining the necessary general conditions, which our thoughts and logical operations with thoughts must satisfy. In traditional logic, these are considered:

In mathematical logic, the law of identity is expressed by the following formulas:

aº a (in propositional logic) and Aº A (in class logic, in which classes are identified with scopes of concepts).

Identity is equality, the similarity of objects in some respect. For example, all liquids are identical in that they are thermally conductive and elastic. Each object is identical to itself. But in reality identity exists in connection with difference. There are not and cannot be two absolutely identical things (for example, two leaves of a tree, twins, etc.). A thing yesterday and today is both identical and different. For example, a person's appearance changes over time, but we recognize him and consider him the same person. Abstract, absolute identity does not really exist, but within certain limits we can abstract from the existing differences and fix our attention on the identity of objects or their properties alone.

In thinking, the law of identity acts as a normative rule (principle). It means that in the process of reasoning it is impossible to replace one thought with another, one concept with another. It is impossible to pass off identical thoughts as different ones, and different ones as identical ones.

For example, three such concepts will be identical in scope: “scientist, on whose initiative Moscow University was founded”; "a scientist who formulated the principle of conservation of matter and motion"; “a scientist who, since 1745, became the first Russian academician of the St. Petersburg Academy” - they all refer to the same person (M.V. Lomonosov), but give different information about him.

Violation of the law of identity leads to ambiguities, which can be seen, for example, in the following reasoning: “Nozdryov was in some respects a historical person. Not a single meeting where he was could do without history ”(N.V. Gogol). “Strive to pay your debt, and you will achieve a double goal, for in doing so you will fulfill it” (Kozma Prutkov). The play on words in these examples is based on the use of homonyms.

In thinking, the violation of the law of identity manifests itself when a person speaks not on the topic under discussion, arbitrarily replaces one subject of discussion with another, uses terms and concepts in a different sense than is customary, without warning about it.

Identification (or identification) is widely used in investigative practice, for example, when identifying objects, people, identifying handwriting, documents, signatures on a document, identifying fingerprints.

2. Law of non-contradiction: If the subject BUT has a certain property, then in judgments about BUT people should affirm this property, not deny it. If a person, stating something, denies the same thing or asserts something incompatible with the first, there is a logical contradiction. Formal-logical contradictions are the contradictions of confused, incorrect reasoning. Such contradictions make it difficult to understand the world.

Thought is contradictory if we affirm and deny something about the same object at the same time and in the same respect. For example: “Kama is a tributary of the Volga” and “Kama is not a tributary of the Volga”. Or: “Leo Tolstoy is the author of the novel “Resurrection” and “Leo Tolstoy is not the author of the novel “Resurrection”.

There will be no contradiction if we are talking about different subjects or about the same subject, taken at different times or in different respects. There will be no contradiction if we say: “Rain is good for mushrooms in autumn” and “Rain is not good for harvesting in autumn”. The judgments "This bouquet of roses is fresh" and "This bouquet of roses is not fresh" also do not contradict each other, because the objects of thought in these judgments are taken in different relationships or at different times.

The following four types of simple propositions cannot be true at the same time:

∧ā. The law of non-contradiction reads as follows: "Two opposing propositions cannot be true at the same time and in the same respect." Opposite judgments include: 1) opposite (contrarian) judgments BUT and E, which can both be false, so they are not negating each other and cannot be denoted as a and ā; 2) contradictory (contradictor) judgments BUT and O, E and I, as well as singular judgments "This S is P" and "This S is not P", which are negative, since if one of them is true, then the other is necessarily false, therefore they are denoted by a and ā.

The formula of the law of non-contradiction in two-valued classical logic a ∧ ā reflects only part of the meaningful Aristotelian law of non-contradiction, since it applies only to contradictory judgments (a and not-a) and does not apply to the opposite (contrarian judgments). Therefore, the formula a∧ ā is inadequate, does not fully represent the substantive law of non-contradiction. Following tradition, we retain the name “law of non-contradiction” behind the formula a∧ ā, although it is much broader than this formula.

If a formal-logical contradiction is found in the thinking (and speech) of a person, then such thinking is considered incorrect, and the judgment from which the contradiction follows is denied and considered false. Therefore, in the controversy, when refuting the opponent's opinion, the method of "reduction to absurdity" is widely used.

3. Law of the excluded middle: Of the two contradictory propositions, one is true, the other is false, and the third is not given.. Contradictory (contradictory) are such two judgments, in one of which something is affirmed about the subject, and in the other the same is denied about the same subject, therefore they cannot be both true and both false at the same time; one of them is true and the other is necessarily false. Such judgments are called negating each other. If one of the contradictory judgments is denoted by the variable a, then the other should be denoted ā . Thus, of the two statements: "James Fenimore Cooper is the author of a series of novels about Leather Stocking, created over a period of almost 20 years" and "James Fenimore Cooper is not the author of a series of novels about Leather Stocking, created over a period of almost 20 years," the first is true, the second is false, and there can be no third - intermediate - judgment.

The following pairs of propositions are negative:

1) "This S is P" and "This S is not P" (single judgments).

2) "All S are P" and "Some S are not P" (judgments BUT and O).

3) "No S is P" and "Some S are P" (judgments E and I).

With regard to contradictory (contradictor) judgments ( BUT and O, E and I) operates both the law of the excluded middle and the law of non-contradiction - this is one of the similarities of these laws.

The difference in the areas of definition (i.e., application) of these laws is that in relation to contrary (contrarian) judgments BUT and E(for example: "All mushrooms are edible" and "No mushroom is edible"), which cannot both be true, but both can be false, only the law of non-contradiction applies and the law of the excluded middle does not apply. So, the scope of the substantive law of non-contradiction is wider (these are contradictory and contradictory judgments) than the scope of the substantive law of the excluded middle (only contradictory, i.e., judgments of the type a and nope). Indeed, one of the two propositions is true: "All the houses in this village are electrified" or "Some houses in this village are not electrified" and there is no third.

The law of the excluded middle, both in a meaningful and formalized form, covers the same circle of judgments - contradictory, i.e. denying each other. Formula of the law of the excluded middle: BUT v ù A

In thinking, the law of the excluded middle implies a clear choice of one of two mutually exclusive alternatives. For the correct conduct of the discussion, the fulfillment of this requirement is mandatory.

4. Law of sufficient reason:Every true thought must be sufficiently substantiated. We are talking about justifying only true thoughts: false thoughts cannot be justified, and there is no point in trying to “justify” a lie, although often individuals try to do so. There is a good Latin proverb: “To err is common to all people, but only fools tend to insist on their mistakes.”