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 the relationships between objects.

Examples of propositions: “Astronauts exist,” “Paris is larger than Marseille,” “Some numbers appear even.” If what is said in the judgment corresponds to the actual state of affairs, 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 – types of multivalued logics – a proposition can be either true, false, or indeterminate. For example, the proposition “There is life on Mars” is currently neither true nor false, but indeterminate. Many judgments about future single events are uncertain. Aristotle wrote about this, giving an example of such a vague judgment: “Tomorrow it will be necessary naval battle».

The linguistic form of expressing a judgment is a sentence. A judgment is expressed by a declarative sentence, always containing either an affirmation or a negation. Judgment and proposal differ in their composition. Every simple judgment consists of three elements:

1)subject of judgment – this is the concept of the subject of judgment. The subject of the judgment is designated by the letter S (from the Latin word subjectum);

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

3)ligaments, expressed in Russian with the words “is”, “is”, “essence”.

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

TYPES OF SIMPLE JUDGMENTS

1. Property judgments (attributive):

they affirm or deny that the object belongs to known properties, states, and types of activity.

Scheme this type of judgment: « S There is R" or « S do not eat R".

Examples : “Sweet honey,” “Chopin is not a playwright.”

2. Judgments with relationships:

judgments reflecting relationships 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 a proposition with a relation, something can be affirmed or denied not only about two, but also about three, four or more objects, for example: “Moscow is located between St. Petersburg and Kiev.” Such judgments are expressed by the formula R (a,a,a ,…,a).

Examples: “Every proton is heavier than an electron,” “The 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 subject of judgment.

Scheme this type 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 connective (“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 second-hand books" And " No rabbit is a carnivore" – negative. The connective “is” in an affirmative judgment reflects the inherent nature of the object (objects) of certain properties. The connective “is not” reflects the fact that an object (objects) does not have a certain property.

Some logicians believed that negative judgments do not reflect reality. In fact, the absence of certain characteristics also constitutes a valid characteristic that has objective significance. In a negative true judgment, our thought separates (separates) what is separated in the objective world.

In cognition, an affirmative judgment generally has greater significance than a negative one, because it is more important to reveal what attribute 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.).

Depending on whether the subject is talking about the entire class of objects, a part of this class, or one object, judgments are divided into general, private And single.

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

Structure general judgments: “All S are (are not) R". Single judgments will be treated as general, since their subject is a single-element class.

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

Among the general judgments there are exclusive judgments, for example: “All metals at a temperature of 20°C, with the exception of mercury, are solid.” Exclusive judgments also include those that express exceptions to certain rules of Russian or other languages, rules of logic, mathematics, and other sciences.

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

IN single In judgment, the subject is a single concept. Single judgments have a structure: “This S is (is not) P.” Examples of single propositions: “Lake Victoria is not located in the USA”; "Aristotle – teacher of Alexander the Great"; "Hermitage – one of the world's largest art, cultural and historical museums."

Thus, a special place in the classification of judgments is occupied by singling out, excluding and definitely-particular judgments, built on the basis of attributive judgments and representing some complicated versions of the latter:

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

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

2. Place the quantifier words “all” (“none”) or “some” at the beginning of the statement.

3. Place the subject of the statement after the quantifier word.

4.Place the logical connective “is” (“essence”) or “is not” (“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 one 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 quantitative and qualitative characteristics. Therefore, logic uses a combined classification of judgments by quantity and quality, on the basis of which the following are distinguished: four types of judgments :

1. A – general assertion.

Structure: "All S essence R".

Example: "All people want happiness."

2. I– private affirmative judgment.

Structure: "Some S's are R".

Example: “Some lessons stimulate students' creativity.”

ü Conventions for affirmative judgments are taken from the word affirmo, or I approve; in this case, the first two vowels are taken: A – to denote a generally affirmative and I – to denote a private affirmative proposition.

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's aren't 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 be either distributed or undistributed. The term is considered distributed, if its scope is completely included in the scope of another term or is completely excluded from it. The term will be unallocated, 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 A – universal . Its structure: " All S is P ».

Let's consider two cases:

Example 1 . In the judgment “All crucian carp – fish" the subject is the concept "crucian carp", and the predicate – the concept of "fish". General quantifier – "All". The subject is distributed because we're talking about about all crucian carp, i.e. its scope is completely included in the scope of the predicate. The predicate is not distributed, since only part of the fish that coincide with crucian carp is thought of in it; we are talking only about that part of the volume of the predicate that coincides with the volume of the subject.

Example 2 . In the proposition “All squares are equilateral rectangles” the terms are: S- "square", R– “equilateral rectangle” and the general quantifier – “all”. In this judgment S distributed and P distributed, because their volumes completely coincide. If S equal in volume R, That R distributed This happens in definitions and in distinguishing general judgments.

2. Judgment I – private affirmative . Its structure: " 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”, existence quantifier – “some”. The subject is not distributed, since only a part of teenagers is thought of 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 proposition “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 the writers is thought of 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. Thus, R distributed if the volume R less than volume S , what happens in particular distinguishing judgments.

3. Judgment E – general negative . Its structure: " None S is not P » . For example : “No lion is a herbivore.” The terms in it 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 ABOUT –partial negative . Its structure: " Some S is not P ». For example : “Some students are not athletes.” It contains the following terms: S – “student”, R– “athlete” and the existential quantifier – “some”. The subject is not distributed, since only a part of the students is thought of, but the predicate is distributed, because all the athletes are thought of in it, none of whom is included in that part of the students that is thought of in the subject

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

Let's imagine this 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 judgments. In distinguishing judgments the predicate is distributed.

Legend: +– term distribution;

– – non-distribution of the term

· JUDGMENTS WITH RELATIONS 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 relationship in which something is affirmed or denied in relation to two (several) terms. In this type of judgment, the predicate is a relation, and the subject is two (or several) concepts. The location of the relationship is determined by the number of concepts included in the subject.

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

• The relationship is called symmetrical(from Latin “proportionality”), if it occurs between objects x And y , and between objects y and x (If X equal to (similar to, at the same time) y , then y equal to (similar to, at the same time) X .
• The relationship is called reflective(from Latin “reflection”), if each member of the relation is in the same relation to itself (if X =at , That X =X And at =at ).
• The relationship is called transitive(from Latin “transition”), if it takes place between X And z , then when it occurs between X And at and between at And z (If X equals at And at equals z , That 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 narrative sentences, as the grammatical equivalent of a judgment, are a completely complete thought in which the connection between an object and its attribute, the relationship between objects, the fact of the existence of an object is affirmed or denied, and which can be either true or false.

Ø Interrogative sentences do not contain judgments, 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 expresses a rhetorical question, – for example: “Who doesn’t want happiness?”, “Which of you hasn’t loved?” or “Is there anything more monstrous than an ungrateful person?” (W. Shakespeare), or “Is there a person who looks at a river in a moment of reflection and does not remember the constant movement of all things?” (R. Emerson) – then it contains a judgment, since there is a statement, a certainty that “Everyone wants happiness” or “All people love,” etc.

Ø Interrogative rhetorical sentences contain judgments in their composition, since they affirm or deny something. They can be either true or false.

Incentive offers do not contain judgments: (“Take care of your health”; “Don’t light fires in the forest”; “Go to school, not to the skating rink!”). But the sentences in which military commands and orders, appeals or slogans are formulated express judgments, but not assertoric, but modal (modal judgments include modal operators expressed in the words: possible, necessary, prohibited, proven, etc.). For example: “Take care of the world!”, “Get ready to start!”, “My friend! Let us dedicate our souls to our homeland with wonderful impulses” (A.S. Pushkin). These sentences express judgments, but modal judgments that include modal words. As noted by A.I. Uemov, express judgments and such incentive sentences: “Take care of the world!”, “Don’t smoke!”, “Fulfill your obligations!” “Before any meal, eat a salad of raw vegetables or raw fruits” and “Don’t harm yourself by overeating” – These tips (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 join forces 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 a sentence is the presence of a moment of affirmation or negation, leading to the assessment of the judgment for truth or falsity.

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

II. Types and logical probability of complex judgments

Complex judgments are formed from simple ones, as well as from other complex judgments with the help of conjunctions “if..., then...”, “or”, “and”, etc., with the help of negation “it is not true that”, modal terms “it is possible that”, “it is necessary that”, “it is accidental that”, etc. These conjunctions, negation "it is not true that", modal terms in everyday language are used in different senses. IN scientific languages they are given a precise meaning, as a result of which they stand out different kinds judgments formed from other judgments through, for example, the same grammatical conjunction.

I.Connecting are judgments that assert the existence of two or more situations. Most often, these judgments are expressed in language by sentences containing the conjunction “and”.

The conjunction "and" is used in different meanings. For example, the sentences “Petrov studied English language, 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 graduate school and graduated from the university" express different judgments.

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

1. (Vague) conjunctive judgments are formed from two judgments through a conjunction, denoted by the symbol & (read “and”) and called the sign (indefinite) conjunctions. The definition of a 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 propositions assert the sequential occurrence or existence of two or more situations. They are formed from two or more propositions using conjunctions, denoted by the symbols & ® 2, & ® 3, etc., depending on the number of propositions from which they are formed. These symbols are called sequential conjunction signs and accordingly read “..., and then..”, “..., then..., and then...”, etc. Indexes 2,3, etc. indicate the location of the union. Form of a judgment with the sign of a two-place sequential conjunction: & ® 2 (A,B) or (A&® 2 IN). Example judgments of this form: “The buyer paid the cost of the goods, and then the seller released the goods.” Instead of the expression “and then,” the conjunction “and” is most often used: “The buyer paid the cost of the goods, and the seller delivered the goods.” Form of judgment with a three-place conjunction. Example: “Petrov mortgaged the apartment, then contributed money to the pyramid, and then became a person without a fixed place of residence.”
3. Simultaneously, conjunctive judgments. These judgments are formed from two judgments by means of the conjunction “and”, called the sign simultaneous conjunction. Notation - & = . These judgments assert the simultaneous existence of two situations. Example: "It rains and the sun shines."

1. Disjunctive, or non-strictly dividing, or connecting-dividing, judgments. These judgments assert the existence of at least one of two situations. They are formed from two judgments by means of the conjunction “or”, denoted by the sign v (read “or”), called the sign of weak disjunction (or simply the sign of disjunction).
2. Strictly disjunctive, or strictly dividing judgments. These judgments assert the existence of exactly one of two, three, or more situations. They are formed from two, three, etc. judgments through conjunctions "or..., or..." ("either..., or..."), "or..., or..., or...", etc. Sometimes the conjunction "or..., or..." is replaced by the conjunction "or", and its dividing meaning is determined by the context. Conjunctions through which strictly disjunctive judgments are formed are indicated by the sign v.

III. Conditional propositions are usually expressed in sentences with the conjunction “if..., then...”. They assert that the presence of one situation determines the presence of another. Example: “If the sun is at its zenith, then its shadows are the shortest.” In a conditional proposition, there is a basis and a consequence. The basis is that part of a conditional proposition that is located between the word “if” and the word “then”. The part of the conditional proposition that is located after the word “that” is called consequence. In the judgment “If it rains, then the roofs of the houses are wet,” the basis is the simple judgment “it is raining,” and the consequence is “the roofs of the houses are wet.”

A more strictly conditional proposition is defined through the concept of a sufficient condition. Condition is sufficient for any event, any situation, if, and only if, whenever this condition exists, 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 a judgment in which the situation described by the reason is a sufficient condition for the situation described by the consequence. The conditional conjunction “if..., then...” is indicated by an arrow (®).

IV. Counterfactual propositions. Example: “If Petrov were president, he would not travel around the city by bus.” As in conditional propositions, in these judgments there is a basis and a consequence. The conjunction "if..., then..." is denoted by the sign É, which is called the sign counterfactual implications. The proposition has this meaning: the situation described by the reason does not take place, but if it existed, then the consequence would exist

V. Equivalent judgments. Equivalence judgments assert the mutual conditionality of two situations. These judgments are expressed, as a rule, through sentences with the conjunction “if, and only if, ..., then...” (“then, and only then, ..., when...”). They can also highlight reasons and consequences. The basis 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 conjunction “if, and only if, ..., then,” used in the described sense, is denoted by the symbol º

In an equivalence judgment, 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 states 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” in front of 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 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 component judgments and on the type of their connection. Identically true formula is a formula that, for any combination of values ​​for the variables included in it, takes the value “true”. Identity-false formula– one that (accordingly) takes only the value “false”. The formula being executed can be either true or false.

So, conjunction(a b ) true when both simple propositions are true. Strict disjunction ( a b ) true when only one simple proposition is true. Loose disjunction ( a b ) true when at least one simple proposition is true. Implication ( a É b ) true in all cases except one - when A - true, b- false. Equivalence ( a º b ) true when both propositions are true or both are false. Negation (ù a) lies give truth, and vice versa.

Ø Any linguistic construction consisting of a certain set of judgments can be translated into 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 conjunction 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 evaluate to “true” in all rows of the resulting column of truth tables are logically necessary (logically true) propositions. The logical form of a logically necessary judgment is expressed by an identically true formula, which, for any truth value of the variables, takes on the value “true”, that is, its resulting column consists only of “AND”. Identically 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 propositions that evaluate to “false” on all sides of the resulting truth table column are logically impossible (logically false) propositions. The logical form of a logically impossible proposition 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”. Identically 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 ​​of both “true” and “false”, is called logically random. The logical form of a logically random proposition is expressed by a neutral (actually satisfiable) 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 this depends on which simple propositions are true and which are false.

III. Denial of judgments

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

When negating 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.

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

ù A equivalent ABOUT ù ABOUT equivalent A

ù E equivalent I ù I equivalent E

Negation of complex judgments is made according to the following equivalences:

ù (A& IN) equivalent ù Avù B; according to de Morgan's law

ù (AvB) equivalent ù A& ù B;

ù (AÉ B) equivalent A& ù B;

ù (Aº B) equivalent (ù A& IN)v(A& ù B);

ù (Av IN) equivalent Aº IN

IV. Relationship between judgments

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

LOGICAL SQUARE

RELATIONS 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. Dependents – 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 simultaneously be true) and incompatible (judgments that cannot be true at the same time).

Relationship

V. Modality of judgments

MODALITY – is expressed in a judgment Additional Information about the logical or factual status of a 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 connections between an object and its characteristics. An object S can simply be said to have property P. Such an attributive judgment is simply an assertion. Along with simple affirmation (negation), there are so-called strong and weak statements and negations, which are modal judgments.

MAIN TYPES OF MODALITIES:

Ø ALETHIC MODALITY– expressed in a judgment through the modal concepts “necessary”, “mandatory”, “certainly”, “accidentally”, “possibly”, “maybe”, “not excluded”, “allowed” and other information about the logical or factual determinacy of the judgment . In the alethic group there are ontological (actual ) modality, which associated with the objective determinism of judgments, when their truth or falsity is determined by the situation taking place in reality, And logical modality , which associated with the logical determinism of a judgment, when truth or falsity is determined by the form or structure of the judgment.

Ø EPISTEMIC MODALITY– this is expressed in a judgment through the modal operators “known”, “unknown”, “provable”, “refutable”, “assumed”, etc. information about 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 behavior or order, prompting a person to take specific actions. Legal norms are also considered deontic (here the following operators can be distinguished: “obligated”, “must”, “must”, “recognized”, “prohibited”, “cannot”, “not allowed”, “has the right”, “may” have”, “can accept”, etc.).

Modality of judgment ( R) is represented using the operator M, according to the scheme Mr(for example, “possibly P”). The truth of a modal proposition depends on the truth of the proposition under the modal operator and on the type of modal operator.

Modal simple propositions

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,” one can obtain the modal statement “It is physically necessary that if the temperature is above 100 degrees, then water turns into steam.”

VI. Concept of logical law

Correct thinking must meet the following requirements: to be specific, consistent, consistent and justified. Certain thinking is precise and strict, free from any confusion. Consistent thinking is free from internal contradictions that destroy the necessary connections between thoughts. Consistency is associated with the avoidance of mutually exclusive thoughts as equally acceptable in one respect or another. Well-founded 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 of connecting its component parts. Thus, there is a connection between thoughts, the logical forms of which are represented by the expressions “All S are P” and “all P are S”: if one of these thoughts is true, then the second is true, regardless of the specific content of these thoughts. The connections between thoughts, in which the truth of some necessarily determine the truth of others, are determined by formal logical laws, or laws of logic.

§ LAWS OF LOGIC- these are 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, a logical law is the logical form itself, which guarantees the truth of an 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 (thus, any law of propositional logic or predicate logic is an example of a logical law). These are the so-called laws of connection between thoughts. Logical laws are also called tautologies.

§ LOGICAL TAUTOLOGY- this is an “always true expression”, that is, it remains true regardless of what area of ​​​​objects we are talking about. Any law of logic is a logical tautology.

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

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

аº а (in propositional logic) and Аº А (in class logic, in which classes are identified with the volumes 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.). The thing yesterday and today are both identical and different. For example, a person's appearance changes over time, but we recognize him and consider him to be the same person. Abstract, absolute identity does not really exist, but within certain limits we can abstract from 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, and different ones as identical.

For example, three such concepts will be identical in scope: “a scientist on whose initiative Moscow University was founded”; “a scientist who formulated the principle of conservation of matter and motion”; “a scientist who became the first Russian academician of the St. Petersburg Academy in 1745” - 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 present was complete without history” (N.V. Gogol). “Strive to pay your debt, and you will achieve a double goal, for by doing so you will fulfill it” (Kozma Prutkov). The play on words in these examples is based on the use of homonyms.

In thinking, a 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, in identifying objects, people, identifying handwriting, documents, signatures on a document, identifying fingerprints.

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

A thought is contradictory if we affirm something about the same object at the same time and in the same relation and deny the same thing. For example: “The Kama is a tributary of the Volga” and “The 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: “In autumn, rain is good for mushrooms” and “in autumn, rain is not good for harvesting.” 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 relations 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 (contrary) judgments A And E, which can both be false, and therefore are not mutually negating, and cannot be designated as a and ā; 2) contradictory (contradictory) judgments A And ABOUT, E And I, as well as the singular propositions “This S is P” and “This S is not P”, which are negating, since if one of them is true, then the other is necessarily false, so 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 substantive Aristotelian law of non-contradiction, since it applies only to contradictory judgments (a and not-a) and does not apply to contrary (contrary judgments). Therefore, the formula a∧ ā inadequately and does not fully represent the meaningful law of non-contradiction. Following tradition, we retain the name “law of non-contradiction” for the formula a∧ ā, although it is much broader than this formula.

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

3. Law of the excluded middle: Of 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 asserted about an object, and in the other the same thing about the same object is denied, therefore they cannot both be true and both false at the same time; one of them is true and the other is necessarily false. Such judgments are called mutually negating. If one of the contradictory judgments is designated by a variable A, then something else should be designated ā . Thus, of the two propositions: “James Fenimore Cooper is the author of the Leatherstocking series of novels, which were created over almost 20 years” and “James Fenimore Cooper is not the author of the Leatherstocking series of novels, which were created over almost 20 years,” the first is true, the second false, and there cannot be a 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 A And ABOUT).

3) “No S is P” and “Some S are P” (judgments E And I).

In relation to contradictory (contradictory) judgments ( A And ABOUT, E And I) both the law of the excluded middle and the law of non-contradiction operate – this is one of the similarities between these laws.

The difference in the areas of definition (i.e., application) of these laws is that in relation to opposing (counter) judgments A And E(for example: “All mushrooms are edible” and “No mushroom is edible”), which both cannot be true, but both can be false, are subject only to the law of non-contradiction and not to the law of excluded middle. So, the scope of action of the substantive law of non-contradiction is wider (these are contradictory and contradictory judgments) than the scope of action of the substantive law of excluded middle (only contradictory, i.e. judgments like A And nope). Indeed, one of two propositions is true: “All houses in this village are electrified” or “Some houses in this village are not electrified” and there is no third option.

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

In thinking, the law of excluded middle presupposes a clear choice of one of two mutually exclusive alternatives. To conduct a discussion correctly, fulfilling this requirement is mandatory.

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

        The main concept of mathematical logic is the concept of a “simple statement”. A statement is usually understood as any declarative sentence that states 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 meanings of statements are “true” and “false”.

        Examples of statements.
        1) Moscow stands on the Neva.
        2) London is the capital of England.
        3) A falcon is not a fish.
        4) The number 6 is divisible by 2 and 3.

        Statements 2), 3), 4) are true, and statement 1) is false.
        Obviously, the sentence “Long live Russia!” is not a statement.
        There are two types of statements.
        A statement that is one statement is usually called simple or elementary. Examples of elementary statements are statements 1) and 2).
        Statements that are obtained from elementary ones using grammatical connectives “not”, “and”, “or”, “if.... then...”, “then and only then” are usually called complex or compound .
        Thus, statement 3) is obtained from the simple statement “Falcon is a fish” using the negation “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".
        Similarly, complex statements can be obtained from simple statements using the grammatical connectives “or”, “then and only then”.
        In the algebra of logic, all statements are considered only from the point of view of their logical meaning, and their everyday content is abstracted. It is believed that every statement is either true or false and no statement can be both true and false.
        Elementary statements are indicated by small letters of the Latin alphabet: x, y, z, ..., a, b, c, ...; the true meaning of a statement is indicated by the number 1, and the false meaning is indicated by the letter number 0.
        If the statement A true, then we will write a = 1, and if A false, then a = 0.

Logical operations on statements

Negation.

        Negation of statement x called a new statement x, which is true if the statement X false, and false if the statement X true.
        Negation of the statement X denoted by x read "not X" or “it’s not true that x”.
        Logical meanings of the statement x can be described using a table.

        Tables of this type are usually called truth tables.
        Let X statement. Because x is also a statement, then we can form the negation of the statement x, that is, a statement that is called the double negation of a statement X. It is clear that the logical meanings of statements X and match.
        For example, for the statement “Putin is the President of Russia,” the negation will be the statement “Putin is not the President of Russia,” and the double negation will be the statement “It is not true that Putin is not the President of Russia.”

Conjunction.

        The conjunction (logical multiplication) of two statements x and y a new statement is called, which is considered true if both statements x and y true, and false if at least one of them is false.
        Conjunction of statements x and y indicated by the symbol x&y (x∧y, xy), read "x and y". Statements x and y are called members of the conjunction.
        The logical values ​​of the conjunction are described by the following truth table:

        For example, for the statements “6 is divided by 2”, “6 is divided by 3”, their conjunction will be the statement “6 is divided by 2 and 6 is divided by 3”, which is obviously true.
        From the definition of the conjunction operation it is clear that the conjunction “and” in the algebra of logic is used in the same sense as in everyday speech. But in ordinary speech it is not customary to connect two statements that are far from each other in content with the conjunction “and,” but in the algebra of logic the conjunction of any two statements is considered.

Disjunction

        Disjunction (logical addition) of two statements x and y a new statement is called, which is considered true if at least one of the statements x, y true, and false if they are both false. Disjunction of propositions x, y indicated by the symbol "x V y", read "x or y". Statements x, y are called terms of the disjunction.
        The logical values ​​of the disjunction are described by the following truth table:

        In everyday speech, the conjunction “or” is used in different senses: exclusive and non-exclusive. In the algebra of logic, the conjunction “or” is always used in a non-exclusive sense.

Implication.

        By implication of two statements x and y is a new statement that is considered false if x is true and y is false, and true in all other cases.
        Statement implication x, y indicated by the symbol x→y, read “if x then y” or “from x follows y.” Statement X called a condition or premise, statement at- consequence or conclusion, statement x→y by implication or implication.
        The logical values ​​of the implication operation are described by the following truth table:

        The use of the words “if.... then...” in the algebra of logic differs from their use in everyday speech, where we, as a rule, believe that if the statement X is false, then the statement "If x then y" doesn't make sense at all. In addition, constructing a sentence of the form "if x then y" in everyday speech, we always mean that a sentence at follows from the sentence X. The use of the words “if..., then...” in mathematical logic does not require this, since it does not consider the meaning of statements.
        Implication plays important role in mathematical proofs, since many theorems are formulated in conditional form “If x, then y.” If it is known that X true and the implication has been proven true x→y, then we have the right to draw a conclusion about the truth of the conclusion at.

Equivalence.

        The equivalence of two statements x and y is a new statement that is considered true when both statements x, y either simultaneously true or simultaneously false, and false in all other cases.
        Equivalence of statements x, y indicated by the symbol x↔y, read “in order for x, it is necessary and sufficient that y” or “x if and only if y.” Statements x, y are called equivalence terms.
        The logical values ​​of the equivalence operation are described by the following truth table:

        Equivalence plays an important role in mathematical proofs. It is known that a significant number of theorems are formulated in the form of necessary and sufficient conditions, that is, in the form of equivalence. In this case, knowing the truth or falsity of one of the two terms of equivalence and proving the truth of the equivalence itself, we conclude the truth or falsity of the second term of equivalence.

Affirming or denying something about the existence of objects, about the connections between them and their properties, as well as about the relationships between objects.

Examples of judgments: “The Volga flows into the Caspian Sea”, “A.S. Pushkin wrote the poem “The Bronze Horseman”, “The Ussuri Tiger is Listed in the Red Book”, etc.

Structure of judgment

A proposition includes the following elements: subject, predicate, connective and quantifier.

1. The subject (lat. subjektum - “underlying”) is what is said in this judgment, its subject (“S”).
2. Predicate (Latin praedicatum - “said”) is a reflection of the attribute of an object, what is said about the subject of the judgment (“P”).
3. A connective is a relationship between a subject (“S”) and a predicate (“P”). Determines the presence/absence of the subject of any property expressed in the predicate. It can be either implied or indicated by the “dash” sign or the words “is” (“is not”), “is”, “is”, “essence”, etc.
4. A quantifier (quantifier word) determines the scope of the concept to which the subject of the judgment belongs. Stands before the subject, but may also be absent from the judgment. Denoted by words such as “all”, “many”, “some”, “none”, “no one”, etc.

True and false propositions

A judgment is true in the case when the presence of signs, properties and relationships of objects affirmed/denied in the judgment corresponds to reality. For example: “All swallows are birds”, “9 is more than 2”, etc.

If the statement contained in the judgment is not true, we are dealing with a false proposition: “The sun revolves around the Earth,” “A kilogram of iron is heavier than a kilogram of cotton wool,” etc. Correct judgments form the basis of correct conclusions.

However, in addition to two-valued logic, in which a proposition can be either true or false, there is also multidimensional logic. According to its terms, the judgment may also be indefinite. This is especially true for future individual judgments: “Tomorrow there will be/will not be a sea battle” (Aristotle, “On Interpretation”). If we assume that this is a true proposition, then a naval battle cannot but happen tomorrow. Therefore, it is necessary for it to happen. Or vice versa: by asserting that a given judgment is false at the present moment, we thereby make necessary the impossibility of tomorrow

Judgments by type of statement

As you know, according to the type of statement, three types are distinguished: incentive and interrogative. For example, the sentence “I remember a wonderful moment” belongs to the narrative type. It is useful to propose that such a judgment will also be narrative. It contains certain information and reports a certain event.

In its turn, interrogative sentence contains a question that implies an answer: “What does the coming day have in store for me?” At the same time, it neither states nor denies anything. Accordingly, the assertion that such a judgment is interrogative is erroneous. An interrogative sentence, in principle, does not contain a judgment, since the question cannot be differentiated according to the principle of truth/falsity.

The incentive type of sentences is formed in the case when there is a certain incentive to action, a request or a prohibition: “Arise, prophet, and see and hear.” As for judgments, according to some researchers, they are not contained in sentences of this type. Others believe that we are talking about a type of modal judgment.

Quality of judgment

From the point of view of quality, judgments can be either affirmative (S is P) or negative (S is not P). In the case of an affirmative proposition, with the help of a predicate the subject is given a certain property(s). For example: “Leonardo da Vinci is an Italian painter, architect, sculptor, scientist, naturalist, as well as inventor and writer, the largest representative of Renaissance art.”

In a negative judgment, on the contrary, the property is taken away from the subject: “James Vickery's theory of the 25th frame has no experimental confirmation.”

Quantitative characteristics

Judgments in logic can be of a general nature (applying to all objects of a given class), particular (to some of them) and individual (when we are talking about an object that exists in a single copy). For example, one could argue that a proposition such as “All cats are gray at night” would refer to general appearance, since it affects all felines (subject of judgment). The statement “Some snakes are not poisonous” is an example of a private judgment. In turn, the judgment “Wonderful is the Dnieper in calm weather” is isolated, since we are talking about one specific river that exists in a single form.

Simple and complex judgments

Depending on the structure, the judgment can be of the simple or complex type. The structure of a simple judgment includes two related concepts (S-P): “A book is a source of knowledge.” There are also judgments with one concept - when the second is only implied: “It was getting dark” (P).

A complex form is formed by combining several simple propositions.

Classification of simple judgments

Simple judgments in logic can be of the following types: attributive, judgments with relations, existential, modal.

Attributive (judgment-properties) are aimed at affirming/denying the presence of certain properties (attributes) in an object. These judgments have a categorical form and are not questioned: “ Nervous system mammals consists of the brain and outgoing nerve tracts.”

In relational judgments, certain relationships between objects are considered. They can have a spatio-temporal context, cause-and-effect, etc. For example: “An old friend is better than two new ones,” “Hydrogen is 22 times lighter than carbon dioxide.”

An existential judgment is a statement of the existence/non-existence of an object (both material and ideal): “There is no prophet in his own country,” “The moon is a satellite of the Earth.”

A modal proposition is a form of statement that contains a certain modal operator (necessary, good/bad; proven, known/unknown, prohibited, believe, etc.). For example:

• “In Russia it is necessary to carry out educational reform” (alethic modality - possibility, necessity of something).
• “Everyone has the right to personal integrity” (deontic modality - moral norms of public behavior).
• “A careless attitude towards state property leads to its loss” (axiological modality - attitude towards material and spiritual values).
• “We believe in your innocence” (epistemic modality - the degree of reliability of knowledge).

Complex judgments and types of logical connectives

As already noted, complex judgments consist of several simple ones. The following techniques serve as logical connections between them:

Judgment (statement) is a form of thinking in which something is affirmed or denied. For example: “All pines are trees”, “Some people are athletes”, “No whale is a fish”, “Some animals are not predators”.

Let us consider several important properties of a judgment, which at the same time distinguish it from a concept:

1. Any judgment consists of concepts interconnected.

For example, if we connect the concepts “ crucian carp" And " fish", then the following judgments may result: " All crucian carp are fish”, “Some fish are crucian carp”.

2. Any judgment is expressed in the form of a sentence (remember, a concept is expressed in a word or phrase). However, not every sentence can express a judgment. As you know, sentences can be declarative, interrogative and exclamatory. In interrogative and exclamatory sentences, nothing is affirmed or denied, so they cannot express a judgment. A declarative sentence, on the contrary, always affirms or denies something, due to which the judgment is expressed in the form of a declarative sentence. Nevertheless, there are interrogative and exclamatory sentences that are questions and exclamations only in form, but in meaning they affirm or deny something. They're called rhetorical. For example, the famous saying: “ And what Russian doesn’t like driving fast?“- is a rhetorical interrogative sentence (rhetorical question), because it states in the form of a question that every Russian loves driving fast.

There is a judgment in a question like this. The same can be said about rhetorical exclamations. For example, in the statement: “ Try to find a black cat in a dark room if it is not there!“- in the form of an exclamatory sentence, the idea of ​​​​the impossibility of the proposed action is stated, due to which this exclamation expresses a judgment. It is clear that this is not a rhetorical, but a real question, for example: “ What is your name?" - does not express a judgment, just as a real and not rhetorical exclamation does not express it, for example: " Farewell, free elements!

3. Any judgment is true or false. If a judgment corresponds to reality, it is true, and if it does not correspond, it is false. For example, the judgment: “ All roses are flowers", is true, and the proposition: " All flies are birds" - false. It should be noted that concepts, unlike judgments, cannot be true or false. It is impossible, for example, to assert that the concept “ school" is true, and the concept " institute" - false, concept " star" is true, and the concept " planet" - false, etc. But is the concept " Dragon», « Koschei the Deathless», « perpetual motion machine"Aren't they false? No, these concepts are null (empty), but not true or false. Let us remember that a concept is a form of thinking that designates an object, and that is why it cannot be true or false. Truth or falsity is always a characteristic of some statement, affirmation or negation, therefore it applies only to judgments, but not to concepts. Since any judgment takes one of two meanings - truth or falsehood - Aristotelian logic is also often called two-valued logic.

4. Judgments can be simple or complex. Complex propositions consist of simple ones connected by some kind of conjunction.

As we see, a judgment is a more complex form of thinking compared to a concept. It is not surprising, therefore, that the judgment has a certain structure, in which four parts can be distinguished:

1. Subject S) is what the judgment is about. For example, in the judgment: “ ", - we are talking about textbooks, so the subject of this judgment is the concept " textbooks».

2. Predicate(denoted by the Latin letter R) is what is said about the subject. For example, in the same judgment: “ All textbooks are books", - it is said about the subject (about textbooks) that they are books, therefore the predicate of this judgment is the concept " books».

3. Bunch- This is what connects the subject and the predicate. The connectives can be the words “is”, “is”, “this”, etc.

4. Quantifier– this is a pointer to the volume of the subject. The quantifier can be the words “all”, “some”, “none”, etc.

Consider the proposition: “ Some people are athletes" In it the subject is the concept “ People", the predicate is the concept " athletes", the role of the connective is played by the word " are", and the word " some" represents a quantifier. If some judgment lacks a copula or quantifier, then they are still implied. For example, in the judgment: “ Tigers are predators“, - the quantifier is missing, but it is implied - this is the word “all”. By using symbols subject and predicate, one can discard the content of the judgment and leave only its logical form.

For example, if the judgment: “ All rectangles are geometric shapes", - discard the content and leave the form, then it turns out: "Everything S There is R" Logical form of judgment: “ Some animals are not mammals", - "Some S do not eat R».

The subject and predicate of any judgment always represent some concepts that, as we already know, can be in various relationships between themselves. The following relations can exist between the subject and the predicate of a judgment.

1. Equivalence. In the judgment: " All squares are equilateral rectangles", - subject " squares" and the predicate " equilateral rectangles"are in a relationship of equivalence because they represent equivalent concepts (a square is necessarily an equilateral rectangle, S = P and an equilateral rectangle is necessarily a square) (Fig. 18).

2. Intersection. In judgment:

« Some writers are American", - subject " writers" and the predicate " Americans“are in a relation of intersection, because they are intersecting concepts (a writer may be an American and may not be, and an American may be a writer, but may also not be one) (Fig. 19).

3. Subordination. In judgment:

« All tigers are predators", - subject " tigers" and the predicate " predators"are in a relationship of subordination because they represent species and generic concepts (a tiger is necessarily a predator, but a predator is not necessarily a tiger). Also in the judgment: “ Some predators are tigers", - subject " predators" and the predicate " tigers"are in a relationship of subordination, being generic and specific concepts. So, in the case of subordination between the subject and the predicate of a judgment, two types of relationships are possible: the scope of the subject is completely included in the scope of the predicate (Fig. 20, a), or vice versa (Fig. 20, b).

4. Incompatibility. In the judgment: " ", - subject " planets" and the predicate " stars"are in a relation of incompatibility, because they are incompatible (subordinate) concepts (no planet can be a star, and no star can be a planet) (Fig. 21).

To establish the relationship between the subject and the predicate of a given judgment, we must first establish which concept of a given judgment is the subject and which is the predicate. For example, it is necessary to determine the relationship between the subject and the predicate in a judgment: “ Some military personnel are Russians" First we find the subject of judgment - this is the concept “ military personnel"; then we establish its predicate - this concept “ Russians" Concepts " military personnel" And " Russians» are in relation to intersection (a serviceman may or may not be a Russian, and a Russian may or may not be a serviceman). Consequently, in the indicated judgment the subject and the predicate intersect. Likewise in the judgment: “ All planets are celestial bodies", - the subject and predicate are in a relationship of subordination, and in the judgment: " No whale is a fish

As a rule, all judgments are divided into three types:

1. Attributive judgments(from lat. attributum– attribute) are judgments in which the predicate represents any essential, integral feature of the subject. For example, the judgment: “ All sparrows are birds”, - attributive, because its predicate is an integral feature of the subject: being a bird is the main feature of a sparrow, its attribute, without which it would not be itself (if a certain object is not a bird, then it is certainly not a sparrow). It should be noted that in an attributive judgment the predicate is not necessarily an attribute of the subject; it may be the other way around - the subject is an attribute of the predicate. For example, in the judgment: “ Some birds are sparrows"(as we see, in comparison with the above example, the subject and predicate have swapped places), the subject is an integral feature (attribute) of the predicate. However, these judgments can always be formally modified in such a way that the predicate becomes an attribute of the subject. Therefore, those judgments in which the predicate is an attribute of the subject are usually called attributive.

2. Existential judgments(from lat. existentia– existence) are judgments in which the predicate indicates the existence or non-existence of the subject. For example, the judgment: “ There are no perpetual motion machines", - is existential, because its predicate " can not be“testifies to the non-existence of the subject (or rather, the object that is designated by the subject).

3. Relative judgments(from lat. relativus– relative) are judgments in which the predicate expresses some kind of relationship to the subject. For example, the judgment: “ Moscow was founded before St. Petersburg" - is relative because its predicate " founded before St. Petersburg" indicates the temporary (age) relationship of one city and the corresponding concept to another city and the corresponding concept, which is the subject of judgment.

Test yourself:

1. What is a judgment? What are its main properties and differences from the concept?

2. In what linguistic forms is the judgment expressed? Why can't interrogative and exclamatory sentences express judgments? What are rhetorical questions and rhetorical exclamations? Can they be a form of expressing judgments?

3. Find the linguistic forms of judgments in the expressions below:

1) Didn't you know that the Earth revolves around the Sun?

2) Farewell, unwashed Russia!

3) Who wrote the philosophical treatise "Critique of Pure Reason"?

4) Logic appeared around the 5th century. BC e. in Ancient Greece.

5) America's first president.

6) Turn around and march!

7) We all learned a little...

8) Try moving at the speed of light!

4. Why concepts, unlike judgments, cannot be true or false? What is two-valued logic?

5. What is the structure of the judgment? Come up with five propositions and indicate in each of them the subject, predicate, connective and quantifier.

6. In what relationships can the subject and predicate of a judgment exist? Give three examples for each case of relations between a subject and a predicate: equivalence, intersection, subordination, incompatibility.

7. Define the relationship between the subject and the predicate and depict it using Euler's circle diagrams for the following propositions:

1) All bacteria are living organisms.

2) Some Russian writers are world famous people.

3) Textbooks cannot be entertaining books.

4) Antarctica is an ice continent.

5) Some mushrooms are inedible.

8. What are attributive, existential and relative judgments? Give, independently selecting, five examples each for attributive, existential and relative judgments.

2.2. Simple judgments

If a judgment contains one subject and one predicate, then it is simple. All simple judgments based on the volume of the subject and the quality of the connective are divided into four types. The scope of the subject can be general (“all”) and particular (“some”), and the connective can be affirmative (“is”) and negative (“is not”):

Volume of the subject……………… “all” “some”

The quality of the ligament……………… “is” “is not”

As we see, based on the volume of the subject and the quality of the connective, only four combinations can be distinguished, which exhaust all types of simple judgments: “all are”, “some are”, “all are not”, “some are not”. Each of these types has its own name and symbol:

1. General affirmative propositions A) are judgments with the general volume of the subject and the affirmative connective: “Everything S There is R" For example: " All schoolchildren are students».

2. Particularly affirmative judgments(denoted by a Latin letter I) are judgments with a particular subject and an affirmative connective: “Some S There is R" For example: " Some animals are predators».

3. General negative judgments(denoted by a Latin letter E) are judgments with the total volume of the subject and a negative connective: “All S do not eat R(or "None S do not eat R"). For example: " All planets are not stars», « No planet is a star».

4. Partial negative judgments(denoted by a Latin letter O) are judgments with a partial volume of the subject and a negative connective: “Some S do not eat R" For example: " ».

Next, you should answer the question of which judgments - general or particular - should be classified as judgments with a single volume of the subject (i.e. those judgments in which the subject is a single concept), for example: “ The sun is heavenly body", "Moscow was founded in 1147", "Antarctica is one of the continents of the Earth." A judgment is general if it concerns the entire volume of the subject, and particular if we are talking about part of the volume of the subject. In judgments with a single volume of the subject, we are talking about the entire volume of the subject (in the above examples - about the entire Sun, about all of Moscow, about all of Antarctica). Thus, judgments in which the subject is a single concept are considered general (generally affirmative or generally negative). Thus, the three propositions given above are generally affirmative, and the proposition: “ The famous Italian Renaissance scientist Galileo Galilei is not the author of the theory electromagnetic field " - generally negative.

In the future we will talk about the types of simple judgments, without using their long names, using symbols - Latin letters A, I, E, O. These letters are taken from two Latin words: a ff i rmo– assert and n e g o - to deny, were proposed as a designation for types of simple judgments back in the Middle Ages.

It is important to note that in each type of simple judgment the subject and the predicate are in certain relationships. Thus, the total volume of the subject and the affirmative copula of judgments of the form A lead to the fact that in them the subject and the predicate can be in relations of equivalence or subordination (other relations between the subject and the predicate in judgments of the form A it can not be). For example, in the judgment: “ All squares (S) are equilateral rectangles (P)", - the subject and the predicate are in a relationship of equivalence, and in a judgment: " All whales (S) are mammals (P)" - in relation to submission.

Partial volume of the subject and the affirmative copula of judgments of the form I determine that in them the subject and the predicate can be in relations of intersection or subordination (but not in others). For example, in the judgment: “ Some athletes (S) are blacks (P)", - the subject and the predicate are in a relation of intersection, and in a judgment: " Some trees (S) are pine trees (P)" - in relation to submission.

The total volume of the subject and the negative connective of judgments of the form E lead to the fact that in them the subject and predicate are only in a relation of incompatibility. For example, in judgments: “ All whales (S) are not fish (P)”, “All planets (S) are not stars (P)”, “All triangles (S) are not squares (P)", - subject and predicate are incompatible.

Partial volume of the subject and the negative connective of judgments of the form O determine that there is a subject and a predicate in them, as well as in judgments of the form I, can only exist in relationships of intersection and subordination. The reader can easily find examples of judgments of the form O, in which the subject and predicate are in these relations.

Test yourself:

1. What is a simple proposition?

2. On what basis are simple judgments divided into types? Why are they divided into four types?

3. Describe all types of simple propositions: name, structure, symbol. Come up with an example for each of them. Which judgments – general or particular – are judgments with a unit volume of the subject?

4. Where did the letters come from to designate types of simple judgments?

5. In what relationships can there be a subject and a predicate in each type of simple judgment? Think about why in judgments like A subject and predicate cannot intersect or be incompatible? Why in judgments of the form I subject and predicate cannot be in a relationship of equivalence or incompatibility? Why in judgments of the form E subject and predicate cannot be equivalent, intersecting or subordinate? Why in judgments of the form O subject and predicate cannot be in a relationship of equivalence or incompatibility? Draw Euler circles for possible relationships between subject and predicate in all types of simple propositions.

2.3. Allocated and unallocated terms

In terms of judgment its subject and predicate are called.

The term is considered distributed(expanded, exhausted, taken in full), if the judgment deals with all objects included in the scope of this term. A distributed term is denoted by a “+” sign, and in Euler’s diagrams it is depicted as a complete circle (a circle that does not contain another circle and does not intersect with another circle) (Fig. 22).

The term is considered unallocated(unexpanded, unexhausted, not taken in full), if the judgment does not deal with all objects included in the scope of this term. An undistributed term is indicated by a “–” sign, and in Euler’s diagrams it is depicted as an incomplete circle (a circle that contains another circle (Fig. 23, a) or intersects with another circle (Fig. 23, b).

For example, in the judgment: “ All sharks (S) are predators (P)“, - we are talking about all sharks, which means that the subject of this judgment is distributed.

However, in this judgment we are not talking about all predators, but only about some of the predators (namely those that are sharks), therefore, the predicate of this judgment is undistributed. Having depicted the relationship between the subject and the predicate (which are in the relation of subordination) of the considered judgment with Euler’s schemes, we see that the distributed term (subject “ sharks") corresponds to a full circle, and undistributed (the predicate " predators") - incomplete (the circle of the subject falling into it seems to cut out some part from it):

The distribution of terms in simple judgments can be different depending on the type of judgment and the nature of the relationship between its subject and predicate. In table 4 presents all cases of distribution of terms in simple judgments:

All four types of simple judgments and all possible cases of relations between the subject and the predicate in them are considered here (see section 2.2). Pay attention to judgments like O, in which the subject and predicate are in an intersection relationship. Despite the intersecting circles in Euler's diagram, the subject of this judgment is undistributed, but the predicate is distributed. Why does this happen? We said above that the Euler circles intersecting in the diagram indicate undistributed terms. The shading shows that part of the subject that is being discussed in the judgment (in this case, about schoolchildren who are not athletes), due to which the circle denoting the predicate in Euler’s diagram remained complete (the circle denoting the subject does not cut off any part from it -part, as it happens in a judgment of the form I, where the subject and predicate are in an intersection relationship).

So, we see that the subject is always distributed in judgments of the form A And E and is always not distributed in judgments of the form I And O, and the predicate is always distributed in judgments of the form E And O, but in judgments of the form A And I it can be either distributed or undistributed, depending on the nature of the relationship between it and the subject in these judgments.

The easiest way to establish the distribution of terms in simple propositions is with the help of Euler schemes (it is not at all necessary to remember all cases of distribution from the table). It is enough to be able to determine the type of relationship between the subject and the predicate in the proposed judgment and depict them with circular diagrams. Further, it is even simpler - a complete circle, as already mentioned, corresponds to a distributed term, and an incomplete circle corresponds to an undistributed term. For example, it is required to establish the distribution of terms in a judgment: “ Some Russian writers are world famous people" First, let’s find the subject and predicate in this judgment: “ Russian writers" – subject, " world famous people" is a predicate. Now let's establish in what relation they are. A Russian writer may or may not be a world famous person, and a famous person may or may not be a Russian writer, therefore, the subject and predicate of the said judgment are in a relation of intersection. Let's depict this relationship on Euler's diagram, shading the part that is discussed in the judgment (Fig. 25):

Both the subject and the predicate are depicted as incomplete circles (each of them seems to have some part cut off), therefore, both terms of the proposed judgment are undistributed ( S –, P –).

Let's look at another example. It is necessary to establish the distribution of terms in the judgment: “ " Having found a subject and a predicate in this judgment: “ People" – subject, " athletes" is a predicate, and having established the relationship between them - subordination, we depict it on Euler’s diagram, shading the part that is discussed in the judgment (Fig. 26):

The circle denoting the predicate is complete, and the circle corresponding to the subject is incomplete (the circle of the predicate seems to cut out some part from it). Thus, in this judgment the subject is undistributed, and the predicate is distributed ( S –, P –).

Test yourself:

1. In what case is the term of judgment considered distributed, and in what case is it considered undistributed? How can we use Euler's circular diagrams to establish the distribution of terms in a simple proposition?

2. What is the distribution of terms in all types of simple judgments and in all cases of relations between their subject and predicate?

3. Using Euler schemes, establish the distribution of terms in the following judgments:

1) All insects are living organisms.

2) Some books are textbooks.

3) Some students are not achieving.

4) All cities are populated areas.

5) No fish is a mammal.

6) Some ancient Greeks are famous scientists.

7) Some celestial bodies are stars.

8) All rhombuses with right angles are squares.

2.4. Transformation of a simple proposition

There are three ways of transformation, i.e. changing the form, of simple judgments: conversion, transformation and opposition to a predicate.

Appeal (conversion) is a transformation of a simple proposition in which the subject and predicate change places. For example, the judgment: “ All sharks are fish", - is transformed by turning into a judgment: " " Here the question may arise as to why the original proposition begins with the quantifier " All", and new - with the quantifier " some"? This question, at first glance, seems strange, because one cannot say: “ All fish are sharks", - therefore, the only thing that remains is: " Some fish are sharks" However, in this case, we turned to the content of the judgment and changed the quantifier “ All"to the quantifier" some"; and logic, as already mentioned, is abstracted from the content of thinking and deals only with its form. Therefore, the reversal of the judgment: “ All sharks are fish”, - can be performed formally, without referring to its content (meaning). To do this, let us establish the distribution of terms in this judgment using a circular diagram. Terms of judgment, i.e. subject " sharks" and the predicate " fish", are in this case in relation to subordination (Fig. 27):

The circular diagram shows that the subject is distributed (full circle), and the predicate is undistributed (incomplete circle). Remembering that the term is distributed when we are talking about all the objects included in it, and undistributed when we are not talking about all of them, we automatically mentally put before the term “ sharks"quantifier" All", and before the term " fish"quantifier" some" By reversing the indicated judgment, i.e., swapping its subject and predicate and starting a new judgment with the term “ fish", we again automatically supply it with the quantifier " some", without thinking about the content of the original and new judgments, and we get the error-free version: " Some fish are sharks" Perhaps all this may seem like an excessive complication of an elementary operation, however, as we will see later, in other cases the transformation of judgments is not easy to do without using the distribution of terms and circular schemes.

Let us pay attention to the fact that in the example considered above, the initial judgment was of the form A, and the new one is of the form I, i.e., the operation of reversal led to a change in the type of simple judgment. At the same time, of course, its form changed, but the content did not change, because in the judgments: “ All sharks are fish" And " Some fish are sharks“, - we are talking about the same thing. In table 5 presents all cases of address depending on the type of simple judgment and the nature of the relationship between its subject and predicate:

Judgment of the form A I. Judgment of the form I turns either into itself or into a judgment of the form A. Judgment of the form E always turns into itself, and a judgment of the form O cannot be handled.

The second method of transforming simple judgments, called transformation (obversion), lies in the fact that the judgment changes the copula: positive to negative, or vice versa. In this case, the predicate of the judgment is replaced by a contradictory concept (i.e., the particle “not” is placed before the predicate). For example, the same judgment that we considered as an example for appeal: “ All sharks are fish", - is transformed by turning into a judgment: " " This judgment may seem strange, because this is not usually said, although in fact we have a shorter formulation of the idea that no shark can be a creature that is not a fish, or that the set of all sharks is excluded from the set of all creatures, which are not fish. Subject " sharks" and the predicate " not fish“The judgments resulting from the transformation are in a relation of incompatibility.

The given example of transformation demonstrates an important logical pattern: any statement is equal to a double negative, and vice versa. As we see, the initial judgment of the form A as a result of the transformation it became a judgment of the form E. Unlike conversion, transformation does not depend on the nature of the relationship between the subject and the predicate of a simple judgment. Therefore, a judgment of the form A E, and a judgment of the form E- into a judgment of the form A. Judgment of the form I always turns into a judgment of the form O, and a judgment of the form O- into a judgment of the form I(Fig. 28).

The third way to transform simple judgments is opposition to predicate- consists in the fact that first the judgment undergoes transformation, and then conversion. For example, in order to transform a judgment by contrasting a predicate: “ All sharks are fish“, - you must first subject it to transformation. It will turn out: “ All sharks are not fish" Now we need to reverse the resulting judgment, i.e., swap its subject “ sharks" and the predicate " not fish" In order not to be mistaken, we will again resort to establishing the distribution of terms using a circular diagram (the subject and predicate in this judgment are in a relation of incompatibility) (Fig. 29):

The circular diagram shows that both the subject and the predicate are distributed (both terms correspond to a full circle), therefore, we must accompany both the subject and the predicate with a quantifier " All" After this, we will make an appeal with a judgment: “ All sharks are not fish" It will turn out: “ All non-fish are not sharks" The proposition sounds unusual, but it is a shorter formulation of the idea that if some creature is not a fish, then it cannot possibly be a shark, or that all creatures that are not fish automatically cannot be sharks as well . The appeal could have been made simpler by looking at the table. 5 for treatment, which is given above. Seeing that a judgment of the form E always turns into itself, we could, without using a circular scheme and without establishing the distribution of terms, immediately put “ not fish"quantifier" All" In this case, another method was proposed to show that it is quite possible to do without the table. for circulation, and memorizing it is not at all necessary. Here, roughly the same thing happens as in mathematics: you can memorize various formulas, but you can do without memorizing, since any formula is not difficult to derive on your own.

All three operations of transforming simple judgments are most easily performed using circular diagrams. To do this, you need to depict three terms: subject, predicate and concept that contradicts the predicate (non-predicate). Then their distribution should be established, and four judgments will follow from the resulting Euler scheme - one initial and three results of transformations. The main thing to remember is that a distributed term corresponds to the quantifier “ All", and unallocated - to the quantifier " some"; that the circles touching in Euler’s diagram correspond to the connective “ is", and non-contacting ones - to the ligament " is not" For example, it is required to perform three transformation operations with a judgment: “ All textbooks are books" Let's depict the subject " textbooks", predicate " books" and the nonpredicate " not books» circular diagram and establish the distribution of these terms (Fig. 30):

1. All textbooks are books(initial judgment).

2. Some books are textbooks(appeal).

3. All textbooks are not books(transformation).

4. All non books are not textbooks

Let's look at another example. It is necessary to transform the judgment in three ways: “ All planets are not stars" Let us depict the subject " planets", predicate " stars" and the nonpredicate " not stars" Please note that the concepts " planets" And " not stars"are in a relationship of subordination: a planet is not necessarily a star, but a celestial body that is not a star is not necessarily a planet. Let us establish the distribution of these terms (Fig. 31):

1. All planets are not stars(initial judgment).

2. All stars are not planets(appeal).

3. All planets are not stars(transformation).

4. Some non-stars are planets(opposite to predicate).

Test yourself:

1. How is the circulation operation carried out? Take three of any judgments and make an appeal to each of them. How does conversion occur in all types of simple propositions and in all cases of relations between their subject and predicate? What judgments cannot be reversed?

2. What is transformation? Take any three judgments and perform a transformation operation with each of them.

3. What is the operation of contrasting a predicate? Take three propositions and transform each of them by contrasting them with a predicate.

4. How can knowledge about the distribution of terms in simple judgments and the ability to establish it using circular diagrams help in carrying out operations of transforming judgments?

5. Take some judgment of the form A and perform all transformation operations with it using circular schemes and establishing the distribution of terms. Do the same with some proposition like E.

2.5. Logical square

Simple judgments are divided into comparable and incomparable.

Comparable (identical in material) judgments have the same subjects and predicates, but may differ in quantifiers and connectives. For example, judgments: “ », « Some students don't study math”, - are comparable: their subjects and predicates are the same, but their quantifiers and connectives are different. Incomparable judgments have different subjects and predicates. For example, judgments: “ All schoolchildren study mathematics», « Some athletes are Olympic champions”, – are incomparable: their subjects and predicates do not coincide.

Comparable judgments, like concepts, can be compatible or incompatible and can be in different relationships with each other.

Compatible propositions that can be true at the same time are called. For example, judgments: “ Some people are athletes», « Some people are not athletes”, are both true and compatible propositions.

Incompatible are judgments that cannot be simultaneously true: the truth of one of them necessarily means the falsity of the other. For example, judgments: “ All schoolchildren study mathematics", "Some schoolchildren do not study mathematics”, – cannot be both true and are incompatible (the truth of the first judgment inevitably leads to the falsity of the second).

Compatible judgments can be in the following relations:

1. Equivalence is a relationship between two judgments in which the subjects, predicates, connectives, and quantifiers coincide. For example, judgments: “ Moscow is an ancient city»,

« The capital of Russia is an ancient city,” are in a relationship of equivalence.

2. Subordination- this is a relationship between two judgments in which the predicates and connectives coincide, and the subjects are in the relation of aspect and gender. For example, judgments: “ All plants are living organisms», « All flowers (some plants) are living organisms" - are in a relationship of subordination.

3. Partial match (subcontrary) Some mushrooms are edible», « Some mushrooms are not edible,” are in a partial match relationship. It should be noted that in this respect there are only private judgments - private affirmative ( I) and partial negatives ( O).

Incompatible judgments can be in the following relations.

1. Opposite (contrary) is a relationship between two propositions in which the subjects and predicates coincide, but the connectives differ. For example, judgments: “ All people are truthful», « ”, – are in a relationship of opposites. In this regard, there can only be general judgments - generally affirmative ( A) and general negative ( E). An important feature of opposing propositions is that they cannot be simultaneously true, but can be false at the same time. Thus, the two opposing propositions given cannot be simultaneously true, but can be false at the same time: it is not true that all people are truthful, but it is also not true that all people are not truthful.

Opposite judgments can be false at the same time, because between them, indicating some extreme options, there is always a third, middle, intermediate option. If this middle option is true, then the two extreme ones will be false. Between opposite (extreme) judgments: “ All people are truthful», « All people are not truthful", - there is a third, middle option: " Some people are truthful and some are not”, - which, being a true judgment, determines the simultaneous falsity of two extreme, opposing judgments.

2. Contradiction (contradictory)- this is the relationship between two judgments in which the predicates coincide, the connectives are different, and the subjects differ in their volumes, that is, they are in a relationship of subordination (type and gender). For example, judgments: “ All people are truthful", "Some people are not truthful", – are in a relation of contradiction. An important feature of contradictory judgments, in contrast to opposite ones, is that between them there cannot be a third, middle, intermediate option. Because of this, two contradictory propositions cannot be simultaneously true and cannot be false at the same time: the truth of one of them necessarily means the falsity of the other, and vice versa - the falsity of one determines the truth of the other. We will return to opposite and contradictory judgments when we talk about the logical laws of contradiction and the excluded middle.

The considered relationships between simple comparable judgments are depicted schematically using a logical square (Fig. 32), which was developed by medieval logicians:

The vertices of the square represent four types of simple propositions, and its sides and diagonals represent the relationships between them. Thus, judgments of the form A and type I, as well as judgments of the form E and type O are in a relationship of subordination. Judgments of the form A and type E are in a relation of opposition, and judgments of the form I and type O– partial coincidence. Judgments of the form A and type O, as well as judgments of the form E and type I are in a relationship of contradiction. It is not surprising that the logical square does not depict a relation of equivalence, because in this relation there are judgments of the same type, i.e. equivalence is a relation between judgments A And A, I And I, E And E, O And O. To establish the relationship between two judgments, it is enough to determine what type each of them belongs to. For example, it is necessary to find out in what relation the judgments are: “ All people studied logic», « Some people haven't studied logic" Seeing that the first judgment is generally affirmative ( A), and the second is a partial negative ( O), we can easily establish the relationship between them using a logical square - a contradiction. Judgments: " All people studied logic (A)», « Some people studied logic (I)", are in a relationship of subordination, and the judgments: " All people studied logic (A)», « All people have not studied logic (E)”, – are in a relationship of opposites.

As already mentioned, an important property of judgments, in contrast to concepts, is that they can be true or false.

As for comparable judgments, the truth values ​​of each of them are connected in a certain way with the truth values ​​of the others. So, if a judgment of the form A is true or false, then the other three ( I, E, O), judgments comparable to it (having subjects and predicates similar to it), depending on this (on the truth or falsity of a judgment of the form A) are also true or false. For example, if a judgment is of the form A: « All tigers are predators", is true, then a judgment of the form I: « Some tigers are predators”, – is also true (if all tigers are predators, then some of them, i.e., some tigers are also predators), a judgment of the form E: « All tigers are not predators" – is false, and a judgment of the form O: « Some tigers are not predators,” is also false. Thus, in this case, from the truth of a proposition of the form A the truth of a proposition of the form follows I and the falsity of judgments of the form E and type O(of course, we are talking about comparable judgments, that is, having the same subjects and predicates).

Test yourself:

1. Which judgments are called comparable and which are called incomparable?

2. What are compatible and incompatible judgments? Give three examples of compatible and incompatible judgments.

3. In what relationships can there be compatible judgments? Give two examples each for the relationships of equivalence, subordination, and partial coincidence.

4. In what respects can there be incompatible judgments?

Give three examples each of opposite and contradictory relationships. Why can opposing propositions be simultaneously false, but contradictory ones cannot?

5. What is a logical square? How does he depict the relations between judgments? Why doesn't a logical square represent an equivalence relationship? How to use a logical square to determine the relationship between two simple comparable propositions?

6. Take some true or false proposition of the form A and draw conclusions from it about the truth of comparable types of judgments E, I, O. Take some true or false proposition of the form E and draw conclusions from it about the truth of judgments comparable to it A, I, O.

2.6. Complex judgment

Depending on the conjunction with which simple judgments are combined into complex ones, five types of complex judgments are distinguished:

1. Conjunctive proposition (conjunction) is a complex proposition with the connecting conjunction “and”, which is denoted in logic by the conventional sign “?”. Using this sign, a conjunctive judgment consisting of two simple judgments can be represented as a formula: a ? b(reads " a And b"), Where a And b– these are two simple judgments. For example, a complex judgment: “ Lightning flashed and thunder roared", is a conjunction (combination) of two simple propositions: “Lightning flashed”, “Thunder rumbled”. A conjunction can consist not only of two, but also of a larger number of simple propositions. For example: " Lightning flashed and thunder rumbled and rain began to fall (a ? b ? c)».

2. Disjunctive (disjunction) is a complex judgment with the disjunctive conjunction “or”. Let us remember that, speaking about the logical operations of addition and multiplication of concepts, we noted the ambiguity of this union - it can be used both in a non-strict (non-exclusive) meaning and in a strict (exclusive) meaning. It is not surprising, therefore, that disjunctive judgments are divided into two types:

1. Loose disjunction is a complex judgment with the disjunctive conjunction “or” in its non-strict (non-exclusive) meaning, which is indicated by the conventional sign “?”. Using this sign, a non-strict disjunctive judgment, consisting of two simple judgments, can be represented as a formula: a ? b(reads " a or b"), Where a And b Is he studying English, or is he studying German", is a non-strict disjunction (separation) of two simple propositions: “He is studying English”, “He is studying German”. These judgments do not exclude each other, because it is possible to study both English and German at the same time, so this disjunction is not strict.

2. Strict disjunction is a complex judgment with the dividing conjunction “or” in its strict (exclusive) meaning, which is indicated by the conventional sign “”. Using this sign, a strict disjunctive judgment, consisting of two simple judgments, can be represented as a formula: a b(reads "or a, or b"), Where a And b– these are two simple judgments. For example, a complex judgment: “ He is in 9th grade, or he is in 11th grade", is a strict disjunction (separation) of two simple propositions: “He is in 9th grade”, “He is in 11th grade”. Let us pay attention to the fact that these judgments exclude each other, because it is impossible to simultaneously study in both the 9th and 11th grades (if he studies in the 9th grade, then he certainly does not study in the 11th grade, and vice versa), due to which this disjunction is strict.

Both non-strict and strict disjunctions can consist not only of two, but also of a larger number of simple propositions. For example: " He is studying English, or he is studying German, or he is studying French (a ? b ? c)», « He is in 9th grade, or he is in 10th grade, or he is in 11th grade (a b c)».

3. Implicative proposition (implication) is a complex judgment with a conditional conjunction “if ... then”, which is indicated by the symbol “>”. Using this sign, an implicative proposition, consisting of two simple propositions, can be represented as a formula: a > b(reads “if a, That b"), Where a And b– these are two simple judgments. For example, a complex judgment: “ If a substance is a metal, then it is electrically conductive“, – represents an implicative proposition (cause-and-effect relationship) of two simple propositions: “The substance is a metal”, “The substance is electrically conductive”. In this case, these two judgments are connected in such a way that the second follows from the first (if a substance is a metal, then it is necessarily electrically conductive), but the first does not follow from the second (if a substance is electrically conductive, this does not mean at all that it is a metal). The first part of the implication is called basis, and the second – consequence; a consequence follows from a foundation, but a foundation does not follow from a consequence. Implication formula: a > b, can be read as follows: “if a, then definitely b, but if b, then not necessarily a».

4. Equivalent judgment (equivalence)- this is a complex judgment with the conjunction “if ... then” not in its conditional meaning (as in the case of implication), but in its identical (equivalent) meaning. In this case, this union is denoted by the symbol “”, with the help of which an equivalent judgment consisting of two simple judgments can be represented as a formula: a b(reads “if a, That b, and if b, That a"), Where a And b– these are two simple judgments. For example, a complex judgment: “ If the number is even, then it is divisible by 2 without a remainder.“, – represents an equivalent judgment (equality, identity) of two simple propositions: “The number is even”, “The number is divisible by 2 without a remainder”. It is easy to notice that in this case the two propositions are connected in such a way that the second follows from the first, and the first follows from the second: if a number is even, then it is necessarily divisible by 2 without a remainder, and if a number is divisible by 2 without a remainder, then it is necessarily even . It is clear that in equivalence, unlike implication, there can be neither a reason nor a consequence, since its two parts are equivalent judgments.

5. Negative judgment (negation) is a complex judgment with the conjunction “it is not true that...”, which is denoted by the symbol “¬”. Using this sign, a negative judgment can be represented as a formula: ¬ a(reads “it is not true that a"), Where a- this is a simple judgment. Here the question may arise: where is the second part of a complex proposition, which we usually denoted by the symbol b? In the entry: ¬ a, two simple propositions are already present: a- this is some kind of statement, and the sign “¬” is its negation. Before us are, as it were, two simple judgments - one affirmative, the other negative. An example of a negative judgment: “ It is not true that all flies are birds».

So, we examined five types of complex judgments: conjunction, disjunction (non-strict and strict), implication, equivalence and negation.

There are many conjunctions in natural language, but in meaning they all boil down to the five types considered, and any complex judgment belongs to one of them. For example, a complex judgment: “ Midnight is approaching, but Herman is still not there", is a conjunction because it contains the conjunction " A" is used as a connecting conjunction "and". A complex proposition in which there is no conjunction at all: “ Sow the wind, reap the storm”, is an implication, because two simple propositions in it are connected in meaning by the conditional conjunction “if... then”.

Any complex proposition is true or false depending on the truth or falsity of the simple propositions included in it. The table is given. 6 the truth of all types of complex judgments depending on all possible sets of truth values ​​of the two simple judgments included in them (there are only four such sets): both simple judgments are true; the first proposition is true and the second is false; the first proposition is false, and the second is true; both statements are false).

As we see, a conjunction is true only if both simple propositions included in it are true. It should be noted that a conjunction, consisting not of two, but of a larger number of simple judgments, is also true only if all the judgments included in it are true. In all other cases it is false. A weak disjunction, on the contrary, is true in all cases except when both simple propositions included in it are false. A loose disjunction, consisting not of two, but of a larger number of simple propositions, is also false only if all simple propositions included in it are false. A strict disjunction is true only if one simple proposition included in it is true and the other is false. A strict disjunction, consisting not of two, but of a larger number of simple propositions, is true only if only one of the simple propositions included in it is true, and all the others are false. An implication is false only in one case - when its basis is true and its consequence is false. In all other cases it is true. An equivalence is true when two of its constituent simple propositions are true or when both are false. If one part of the equivalence is true and the other is false, then the equivalence is false. The simplest way to determine the truth of a negation is: when a statement is true, its negation is false; when a statement is false, its negation is true.

Test yourself:

1. On what basis are the types of complex judgments distinguished?

2. Describe all types of complex propositions: name, conjunction, symbol, formula, example. What is the difference between a non-strict disjunction and a strict one? How to distinguish implication from equivalence?

3. How can one determine the type of complex judgment if instead of the conjunctions “and”, “or”, “if... then” some other conjunctions are used?

4. Give three examples for each type of complex judgment, without using the conjunctions “and”, “or”, “if...then”.

5. Determine what type the following complex judgments belong to:

1. Creature is a person only when it has thinking.

2. Humanity may die either from the depletion of the earth's resources, or from an environmental disaster, or as a result of the third world war.

3. Yesterday he received a D not only in mathematics, but also in Russian.

4. A conductor heats up when electric current passes through it.

5. The world around us is either knowable or not.

6. Either he is completely untalented, or he is a complete lazy person.

7. When a person flatters, he lies.

8. Water turns into ice only at temperatures of 0 °C and below.

6. What determines the truth of complex judgments? What truth values ​​do conjunction, loose and strict disjunction, implication, equivalence and negation take, depending on all sets of truth values ​​of the simple judgments included in them?

2.7. Logical formulas

Any statement or whole argument can be formalized. This means discarding its content and leaving only its logical form, expressing it using the already familiar symbols of conjunction, non-strict and strict disjunction, implication, equivalence and negation.

For example, to formalize the following statement: “ He is engaged in painting, or music, or literature“, - you must first highlight the simple judgments included in it and establish the type of logical connection between them. The above statement includes three simple propositions: “He is engaged in painting”, “He is engaged in music”, “He is engaged in literature”.

These judgments are united by a dividing connection, but they do not exclude each other (you can engage in painting, music, and literature), therefore, we have before us a loose disjunction, the form of which can be represented by the following conditional notation: a ? b ? c, Where a, b, c– the above simple judgments. Shape: a ? b ? c, can be filled with any content, for example: “ Cicero was a politician, or an orator, or a writer", "He studies English, or German, or French", "People travel by land, or air, or water transport».

Let us formalize the reasoning: “ He is in 9th grade, or 10th grade, or 11th grade. However, it is known that he is not studying in either the 10th or 11th grade. Therefore, he is in 9th grade" Let us highlight the simple statements included in this reasoning and denote them in small letters of the Latin alphabet: “He studies in the 9th grade (a)”, “He studies in the 10th grade (b)”, “He studies in the 11th grade (c)”. The first part of the argument is a strict disjunction of these three statements: a ? b ? c. The second part of the argument is a negation of the second: ¬ b, and third: ¬ c, statements, and these two negations are connected, that is, they are connected conjunctively: ¬ b ? ¬ c. The conjunction of negations is added to the above-mentioned strict disjunction of three simple propositions: ( a ? b ? c) ? (¬ b ? ¬ c), and from this new conjunction, as a consequence, the statement of the first simple proposition follows: “ He is in 9th grade" Logical consequence, as we already know, is an implication. Thus, the result of formalizing our reasoning is expressed by the formula: (( a ? b ? c) ? (¬ b ?¬ c)) > a. This logical form can be filled with any content. For example: " The first man flew into space was in 1957, or 1959, or 1961. However, it is known that the first man flew into space was not in 1957 or 1959. Therefore, the first man flew into space in 1961"Another option: " The philosophical treatise “Critique of Pure Reason” was written either by Immanuel Kant, or Georg Hegel, or Karl Marx. However, neither Hegel nor Marx are the authors of this treatise. Therefore, it was written by Kant».

The result of the formalization of any reasoning, as we have seen, is some kind of formula consisting of small letters of the Latin alphabet, expressing the simple statements included in the reasoning, and symbols of the logical connections between them (conjunction, disjunction, etc.). All formulas are divided into three types in logic:

1. Identically true formulas are true for all sets of truth values ​​of the variables (simple judgments) included in them. Any identically true formula is a logical law.

2. Identity-false formulas are false for all sets of truth values ​​of the variables included in them.

Identically false formulas are the negation of identically true formulas and are a violation of logical laws.

3. Doable (neutral) formulas for different sets of truth values, the variables included in them are either true or false.

If, as a result of the formalization of any reasoning, an identically true formula is obtained, then such reasoning is logically flawless. If the result of formalization is an identically false formula, then the reasoning should be recognized as logically incorrect (erroneous). A feasible (neutral) formula indicates the logical correctness of the reasoning of which it is a formalization.

In order to determine what type a particular formula belongs to, and, accordingly, evaluate the logical correctness of some reasoning, a special truth table is usually compiled for this formula. Consider the following reasoning: “ Vladimir Vladimirovich Mayakovsky was born in 1891 or 1893. However, it is known that he was not born in 1891. Therefore, he was born in 1893.”. Formalizing this reasoning, let us highlight the simple statements included in it: “Vladimir Vladimirovich Mayakovsky was born in 1891.” “Vladimir Vladimirovich Mayakovsky was born in 1893.”. The first part of our argument is undoubtedly a strict disjunction of these two simple statements: a ? b. Next, the negation of the first simple statement is added to the disjunction, and a conjunction is obtained: ( a ? b) ? ¬ a. And finally, the statement of the second simple proposition follows from this conjunction, and the implication is obtained: (( a ? b) ? ¬ a) > b, which is the result of the formalization of this reasoning. Now we need to create a table. 7 truths for the resulting formula:

The number of rows in the table is determined by the rule: 2 n, where n is the number of variables (simple statements) in the formula. Since there are only two variables in our formula, the table should have four rows. The number of columns in the table is equal to the sum of the number of variables and the number of logical conjunctions included in the formula. The formula in question contains two variables and four logical union(?, ?, ¬, >), which means the table should have six columns. The first two columns represent all possible sets of truth values ​​of the variables (there are only four such sets: both variables are true; the first variable is true and the second is false; the first variable is false and the second is true; both variables are false). The third column is the truth values ​​of the strict disjunction, which it takes depending on all (four) sets of truth values ​​of the variables. The fourth column is the truth values ​​of the negation of the first simple statement: ¬ a. The fifth column is the truth values ​​of the conjunction consisting of the above strict disjunction and negation, and finally the sixth column is the truth values ​​of the entire formula, or implication. We have divided the entire formula into its component parts, each of which is a binomial complex proposition, i.e., consisting of two elements (in the previous paragraph it was said that negation is also a binomial complex proposition):

The last four columns of the table present the truth values ​​of each of these binomial complex propositions that form the formula. First, fill in the third column of the table. To do this, we need to return to the previous paragraph, where the truth table of complex judgments was presented ( see table 6), which in this case will be basic for us (like the multiplication table in mathematics). In this table we see that a strict disjunction is false when both parts are true or both parts are false; when one part of it is true and the other is false, then the strict disjunction is true. Therefore, the values ​​of the strict disjunction in the table to be filled in (from top to bottom) are: “false”, “true”, “true”, “false”. Next, fill in the fourth column of the table: ¬a: when a statement is twice true and twice false, then a negation ¬a, on the contrary, is twice false and twice true. The fifth column is a conjunction. Knowing the truth values ​​of strict disjunction and negation, we can establish the truth values ​​of a conjunction, which is true only if all its elements are true. The strict disjunction and negation that form this conjunction are simultaneously true only in one case, therefore the conjunction takes on the value “true” once, and “false” in other cases. Finally, you need to fill in the last column: for the implication, which will represent the truth values ​​of the entire formula. Returning to the basic table of the truth of complex propositions, let us remember that an implication is false only in one case: when its basis is true and its consequence is false. The basis of our implication is the conjunction presented in the fifth column of the table, and the consequence is a simple proposition ( b), presented in the second column. Some inconvenience in this case is that from left to right the consequence comes before the base, but we can always mentally swap them. In the first case (the first line of the table, not counting the “header”), the basis of the implication is false, but the consequence is true, which means the implication is true. In the second case, both the reason and the consequence are false, which means the implication is true. In the third case, both the reason and the consequence are true, which means the implication is true. In the fourth case, as in the second, both the reason and the consequence are false, which means the implication is true.

The formula in question takes the value “true” for all sets of truth values ​​of the variables included in it, therefore, it is identically true, and the reasoning, the formalization of which it serves, is logically flawless.

Let's look at another example. It is required to formalize the following reasoning and establish what type the formula expressing it belongs to: “ If any building is old, then it needs major renovation. This building is in need of major renovation. Therefore this building is old" Let us highlight simple statements included in this reasoning: “Some building is old”, “Some building needs major repairs”. The first part of the argument is an implication: a > b, these simple statements (the first is its basis, and the second is its consequence). Next, the statement of the second simple statement is added to the implication, and the conjunction is obtained: ( a > b) ? b. And finally, the statement of the first simple statement follows from this conjunction, and a new implication is obtained: (( a > b) ? b) > a, which is the result of the formalization of the reasoning under consideration. To determine the type of the resulting formula, let's make a table. 8 its truth.

There are two variables in the formula, which means there will be four lines in the table; There are also three conjunctions in the formula (>, ?, >), which means the table will have five columns. The first two columns are the truth values ​​of the variables. The third column is the truth values ​​of the implication.

The fourth column is the truth values ​​of the conjunction. The fifth and last column is the truth values ​​of the entire formula - the final implication. Thus, we have divided the formula into three components, which are two-term complex propositions:

Let us fill in the last three columns of the table sequentially according to the same principle as in the previous example, i.e., based on the basic truth table of complex judgments (see Table 6).

The formula in question takes both the value “true” and the value “false” for different sets of truth values ​​of the variables included in it, therefore, it is feasible (neutral), and the reasoning, the formalization of which it serves, is logically correct, but not flawless: otherwise the content of the argument, such a form of its construction could lead to an error, for example: “ If a word comes at the beginning of a sentence, it is written with capital letters. The word "Moscow" is always written with a capital letter. Therefore, the word “Moscow” always appears at the beginning of the sentence».

Test yourself:

1. What is the formalization of a statement or reasoning? Come up with some reasoning and formalize it.

2. Formalize the following reasoning:

1) If a substance is a metal, then it is electrically conductive. Copper is a metal. Therefore, copper is electrically conductive.

2) The famous English philosopher Francis Bacon lived in the 17th century, or in the 15th century, or in the 13th century. Francis Bacon lived in the 17th century. Consequently, he did not live either in the 15th century or in the 13th century.

3) If you are not stubborn, then you can change your mind. If you can change your mind, then you are able to recognize this judgment as false. Therefore, if you are not stubborn, then you are able to recognize this judgment as false.

4) If the sum of the interior angles of a geometric figure is 180°, then such a figure is a triangle. The sum of the internal angles of a given geometric figure is not equal to 180°. Therefore, this geometric figure is not a triangle.

5) Forests can be coniferous, or deciduous, or mixed. This forest is neither deciduous nor coniferous. Therefore, this forest is mixed.

3. What are identically true, identically false and satisfiable formulas? What can be said about reasoning if the result of its formalization is an identically true formula? What will the reasoning be like if its formalization is expressed by an identically false formula? From the point of view of logical correctness, what are the reasonings that, when formalized, lead to feasible formulas?

4. How can one determine the type of a particular formula that expresses the result of the formalization of a certain reasoning?

What algorithm is used to construct and fill truth tables for logical formulas? Come up with some reasoning, formalize it and, using a truth table, determine the type of the resulting formula.

2.8. Types and rules of question

The question is very close to a judgment. This is manifested in the fact that any judgment can be considered as an answer to a certain question.

Therefore, the question can be characterized as a logical form, as if preceding the judgment, representing a kind of “prejudice”. Thus, a question is a logical form (construction) that is aimed at obtaining an answer in the form of some judgment.

Questions are divided into research and informational.

Research questions are aimed at obtaining new knowledge. These are questions that have no answers yet. For example, the question: “ How was the Universe born?” – is research.

Information questions are aimed at acquiring (transferring from one person to another) existing knowledge (information). For example, the question: “ What is the melting point of lead?” – is informational.

Questions are also divided into categorical and propositional.

Categorical (replenishing, special) questions include interrogative words “who”, “what”, “where”, “when”, “why”, “how”, etc., indicating the direction of searching for answers and, accordingly, the category of objects, properties or phenomena , where you should look for the answers you need.

Propositional(from lat. propositio– judgment, proposal) ( clarifying, are common) questions, also often called, are aimed at confirming or denying some already existing information. In these questions, the answer seems to be already laid down in the form of a ready-made judgment, which only needs to be confirmed or rejected. For example, the question: “ Who created periodic table chemical elements? " is categorical, and the question: " Is studying mathematics useful?» – propositional.

It is clear that both research and information questions can be either categorical or propositional. One could put it the other way around: both categorical and propositional questions can be both exploratory and informational. For example: " How to create a universal proof of Fermat's theorem?» – research categorical question:

« Are there planets in the Universe that, like the Earth, are inhabited by intelligent beings?” – research propositional question:

« When did logic appear?" – informational categorical question: " Is it true that the number ? Is it the ratio of the circumference of a circle to its diameter?” is an informational propositional question.

Any question has a certain structure, which consists of two parts. The first part represents some information (expressed, as a rule, by some kind of judgment), and the second part indicates its insufficiency and the need to supplement it with some kind of answer. The first part is called basic (basic)(it is also sometimes called premise of the question), and the second part is the one you are looking for. For example, in an informational categorical question: “ When was the theory of the electromagnetic field created?" - the main (basic) part is an affirmative proposition: " The theory of the electromagnetic field was created", - and the desired part, represented by the question word " When", indicates the insufficiency of information contained in the basic part of the question, and requires its addition, which should be sought in the area (category) of temporary phenomena. In a propositional research question: “ Is it possible for earthlings to fly to other galaxies?", - the main (basic) part is represented by the judgment: " Flights of earthlings to other galaxies are possible", - and the desired part, expressed by the particle " whether", indicates the need to confirm or deny this judgment. In this case, the sought part of the question does not indicate the absence of some information contained in its basic part, but the absence of knowledge about its truth or falsity and requires obtaining this knowledge.

The most important logical requirement for posing a question is that its main (basic) part be a true proposition. In this case, the question is considered logically correct. If the main part of the question is a false proposition, then the question should be considered logically incorrect. Such questions do not require an answer and must be rejected.

For example, the question: “ When was the first attempt made? trip around the world? " - is logically correct, since its main part is expressed by a true proposition: " The first trip around the world took place in human history" Question: " In what year did the famous English scientist Isaac Newton complete his work on the general theory of relativity?" – is logically incorrect, because its main part is represented by a false proposition: " By general theory relativity is the famous English scientist Isaac Newton».

So, the main (basic part) of the question must be true and must not be false. However, there are logically correct questions, the main parts of which are false propositions. For example, questions: “Is it possible to create a perpetual motion machine?”, “Is there intelligent life on Mars?”, “Will a time machine be invented?”– undoubtedly should be recognized as logically correct, despite the fact that their basic parts are false propositions: “ . The fact is that the required parts of these questions are aimed at clarifying the truth values ​​of their main, basic parts, that is, it is required to find out whether the judgments are true or false: “ It is possible to create a perpetual motion machine”, “There is intelligent life on Mars”, “They will invent a time machine”. In this case, the questions are logically correct. If the sought parts of the questions under consideration were not aimed at clarifying the truth of their main parts, but had something else as their goal, these questions would be logically incorrect, for example: “ Where was the first perpetual motion machine created?”, “When did intelligent life appear on Mars?”, “How much will it cost to travel in a time machine?”. Thus, the main rule for posing a question should be expanded and clarified: the main (basic) part of a correct question must be a true judgment; if it is a false proposition, then its sought-after part should be aimed at clarifying the truth value of the main part; otherwise the question will be logically incorrect. It is not difficult to guess that the requirement for the main part to be true is primarily a matter of categorical questions, while the requirement for the main part to be true is primarily a matter of propositional questions.

It should be noted that correct categorical and propositional questions are similar to each other in that they can always be given a true answer (as well as a false one). For example, to a categorical question: “ When did the first one end? World War? " - can be given as a true answer: " In 1918", - and false: " In 1916" To a propositional question: “ Does the Earth revolve around the Sun?" - can also be given as true: " Yes, it rotates", - and false: " No, it doesn't rotate", - answer. Both of the above questions are logically correct. So, the fundamental possibility of obtaining true answers is the main feature of correct questions. If it is fundamentally impossible to obtain true answers to certain questions, then they are incorrect. For example, one cannot obtain a true answer to a propositional question: “ Will World War I ever end?" - just as it is impossible to get it in response to a categorical question: " At what speed does the Sun rotate around a stationary Earth?».

Any answers to these questions will need to be considered unsatisfactory, and the questions themselves - logically incorrect and subject to rejection.

Test yourself:

1. What is a question? What is the similarity between question and judgment?

2. How do research questions differ from information questions? Give five examples each of research and information questions.

3. What are categorical and propositional questions? Give five examples each of categorical and propositional questions.

4. Characterize the questions below in terms of their belonging to research or informational, as well as categorical or propositional:

1) When was the law of universal gravitation discovered?

2) Will the inhabitants of the Earth be able to settle on other planets of the solar system?

3) In what year was Bonaparte Napoleon born?

4) What is the future of humanity?

5) Is it possible to prevent World War III?

5. What is the logical structure of the question? Give an example of a categorical research question and highlight the main (basic) and sought parts in it. Do the same with the categorical information question, the propositional inquiry question, and the propositional information question.

6. Which questions are logically correct and which are incorrect? Give five examples of logically correct and incorrect questions. Can a logically correct question have a false main part? Is the requirement of the truth of its main part sufficient to determine a correct question?

What do logically correct categorical and propositional questions have in common?

7. Answer which of the following questions are logically correct and which are incorrect:

1) How many times larger is the planet Jupiter than the Sun?

2) What is the area of ​​the Pacific Ocean?

3) In what year did Vladimir Vladimirovich Mayakovsky write the poem “A Cloud in Pants”?

4) How long did the fruitful collaboration last? scientific work Isaac Newton and Albert Einstein?

5) What is the length of the earth's equator?

Propositional logic , also called propositional logic - a branch of mathematics and logic that studies the logical forms of complex statements constructed from simple or elementary statements using logical operations.

Propositional logic abstracts from the content of statements and studies their truth value, that is, whether the statement is true or false.

The picture above is an illustration of a phenomenon known as the Liar Paradox. At the same time, in the opinion of the author of the project, such paradoxes are possible only in environments that are not free from political problems, where someone can a priori be labeled a liar. In the natural multi-layered world the subject of “truth” or “false” only individual statements are evaluated . And later in this lesson you will be introduced to the opportunity to evaluate many statements on this subject for yourself (and then look at the correct answers). Including complex statements in which simpler ones are interconnected by signs of logical operations. But first, let’s consider these operations on statements themselves.

Propositional logic is used in computer science and programming in the form of declaring logical variables and assigning them logical values ​​“false” or “true”, on which the course of further execution of the program depends. In small programs where only one boolean variable is involved, the boolean variable is often given a name such as "flag" and the meaning is "flag is up" when the variable's value is "true" and "flag is down." , when the value of this variable is "false". In large programs, in which there are several or even many logical variables, professionals are required to come up with names for logical variables that have a form of statements and a semantic meaning that distinguishes them from other logical variables and is understandable to other professionals who will read the text of this program.

Thus, a logical variable with the name “UserRegistered” (or its English-language analogue) can be declared in the form of a statement, which can be assigned the logical value “true” if the conditions are met that the registration data was sent by the user and this data is recognized as valid by the program. In further calculations, the values ​​of the variables may change depending on the logical value (true or false) of the UserRegistered variable. In other cases, a variable, for example, with the name “More than Three Days Left Before the Day”, can be assigned the value “True” before a certain block of calculations, and during further execution of the program this value can be saved or changed to “false” and the progress of further execution depends on the value of this variable programs.

If a program uses several logical variables, the names of which have the form of statements, and more complex statements are built from them, then it is much easier to develop the program if, before developing it, we write down all the operations from statements in the form of formulas used in statement logic than we do during This lesson is what we will do.

Logical operations on statements

For mathematical statements one can always make a choice between two different alternatives, “true” and “false,” but for statements made in “verbal” language, the concepts of “truth” and “false” are somewhat more vague. However, for example, verbal forms such as “Go home” and “Is it raining?” are not statements. Therefore it is clear that statements are verbal forms in which something is stated . Interrogative or exclamatory sentences, appeals, as well as wishes or demands are not statements. They cannot be evaluated with the values ​​"true" and "false".

Statements, on the contrary, can be considered as quantities that can take on two meanings: “true” and “false”.

For example, the following judgments are given: “a dog is an animal”, “Paris is the capital of Italy”, “3

The first of these statements can be evaluated with the symbol “true”, the second with “false”, the third with “true” and the fourth with “false”. This interpretation of statements is the subject of propositional algebra. We will denote statements in capital letters A, B, ..., and their meanings, that is, true and false, respectively AND And L. In ordinary speech, connections between statements “and”, “or” and others are used.

These connections allow, by connecting different statements with each other, to form new statements - complex statements . For example, the connective "and". Let the statements be given: " π more than 3" and the statement " π less than 4". You can organize a new - complex statement " π more than 3 and π less than 4". Statement "if π irrational then π ² is also irrational" is obtained by connecting two statements with the connective "if - then". Finally, we can obtain from any statement a new one - a complex statement - by denying the original statement.

Considering statements as quantities that take on meanings AND And L, we will define further logical operations on statements , which allow us to obtain new complex statements from these statements.

Let two arbitrary statements be given A And B.

1 . The first logical operation on these statements - conjunction - represents the formation of a new statement, which we will denote A ∧ B and which is true if and only if A And B are true. In ordinary speech, this operation corresponds to the connection of statements with the connective “and”.

Truth table for conjunction:

A	B	A ∧ B
AND	AND	AND
AND	L	L
L	AND	L
L	L	L

2 . Second logical operation on statements A And B- disjunction expressed as A ∨ B, is defined as follows: it is true if and only if at least one of the original statements is true. In ordinary speech, this operation corresponds to connecting statements with the connective “or”. However, here we have a non-dividing “or”, which is understood in the sense of “either or” when A And B both cannot be true. In defining propositional logic A ∨ B true both if only one of the statements is true, and if both statements are true A And B.

Truth table for disjunction:

A	B	A ∨ B
AND	AND	AND
AND	L	AND
L	AND	AND
L	L	L

3 . The third logical operation on statements A And B, expressed as A → B; the statement thus obtained is false if and only if A true, but B false. A called by parcel , B - consequence , and the statement A → B - following , also called implication. In ordinary speech, this operation corresponds to the “if-then” connective: “if A, That B". But in the definition of propositional logic, this statement is always true regardless of whether the statement is true or false B. This circumstance can be briefly formulated as follows: “from the false everything follows.” In turn, if A true, but B is false, then the entire statement A → B false. It will be true if and only if A, And B are true. Briefly, this can be formulated as follows: “false cannot follow from the true.”

Truth table to follow (implication):

A	B	A → B
AND	AND	AND
AND	L	L
L	AND	AND
L	L	AND

4 . The fourth logical operation on statements, more precisely on one statement, is called the negation of a statement A and is denoted by ~ A(you can also find the use of not the symbol ~, but the symbol ¬, as well as an overscore above A). ~ A there is a statement that is false when A true, and true when A false.

Truth table for negation:

A	~ A
L	AND
AND	L

5 . And finally, the fifth logical operation on statements is called equivalence and is denoted A ↔ B. The resulting statement A ↔ B a statement is true if and only if A And B both are true or both are false.

Truth table for equivalence:

A	B	A → B	B → A	A ↔ B
AND	AND	AND	AND	AND
AND	L	L	AND	L
L	AND	AND	L	L
L	L	AND	AND	AND

Most programming languages ​​have special symbols to denote the logical meanings of statements; they are written in almost all languages ​​as true and false.

Let's summarize the above. Propositional logic studies connections that are completely determined by the way in which some statements are built from others, called elementary. In this case, elementary statements are considered as wholes and cannot be decomposed into parts.

Let us systematize in the table below the names, notations and meaning of logical operations on statements (we will soon need them again to solve examples).

Bunch	Designation	Operation name
Not		negation
And		conjunction
or		disjunction
if... then...		implication
then and only then		equivalence

True for logical operations laws of algebra logic, which can be used to simplify Boolean expressions. It should be noted that in propositional logic one abstracts from the semantic content of a statement and limits itself to considering it from the position that it is either true or false.

Example 1.

1) (2 = 2) AND (7 = 7) ;

2) Not(15;

3) ("Pine" = "Oak") OR ("Cherry" = "Maple");

4) Not("Pine" = "Oak") ;

5) (Not(15 20) ;

6) (“Eyes are given to see”) And (“Under the third floor is the second floor”);

7) (6/2 = 3) OR (7*5 = 20) .

1) The meaning of the statement in the first brackets is “true”, the meaning of the expression in the second brackets is also true. Both statements are connected by the logical operation “AND” (see the rules for this operation above), therefore the logical value of this entire statement is “true”.

2) The meaning of the statement in brackets is “false”. Before this statement there is a logical operation of negation, therefore the logical meaning of this entire statement is “true”.

3) The meaning of the statement in the first brackets is “false”, the meaning of the statement in the second brackets is also “false”. Statements are connected by the logical operation "OR" and none of the statements has the value "true". Therefore, the logical meaning of this entire statement is “false.”

4) The meaning of the statement in brackets is “false”. This statement is preceded by the logical operation of negation. Therefore, the logical meaning of this entire statement is “true”.

5) The statement in the inner brackets is negated in the first brackets. This statement in inner brackets has the meaning "false", therefore its negation will have the logical meaning "true". The statement in the second brackets means "false". These two statements are connected by the logical operation “AND”, that is, “true AND false” is obtained. Therefore, the logical meaning of this entire statement is “false.”

6) The meaning of the statement in the first brackets is “true”, the meaning of the statement in the second brackets is also “true”. These two statements are connected by the logical operation “AND”, that is, “true AND truth” is obtained. Therefore, the logical meaning of the entire given statement is “true.”

7) The meaning of the statement in the first brackets is “true”. The meaning of the statement in the second brackets is "false". These two statements are connected by the logical operation “OR”, that is, “true OR false”. Therefore, the logical meaning of the entire given statement is “true.”

Example 2. Write the following complex statements using logical operations:

1) "User is not registered";

2) “Today is Sunday and some employees are at work”;

3) “The user is registered if and only if the data submitted by the user is considered valid.”

1) p- single statement “User is registered”, logical operation: ;

2) p- single statement “Today is Sunday”, q- "Some employees are at work", logical operation: ;

3) p- single statement “User is registered”, q- “The data sent by the user was found valid”, logical operation: .

Solve examples of propositional logic yourself, and then look at the solutions

Example 3. Compute the logical values ​​of the following statements:

1) (“There are 70 seconds in a minute”) OR (“A running clock tells the time”);

2) (28 > 7) AND (300/5 = 60) ;

3) (“TV is an electrical appliance”) AND (“Glass is wood”);

4) Not((300 > 100) OR ("You can quench your thirst with water"));

5) (75 < 81) → (88 = 88) .

Example 4. Write down the following complex statements using logical operations and calculate their logical values:

1) “If the clock shows the time incorrectly, then you may arrive at class at the wrong time”;

2) “In the mirror you can see your reflection and Paris, the capital of the USA”;

Example 5. Determine the Boolean Value of an Expression

(p → q) ↔ (r → s) ,

p = "278 > 5" ,

q= "Apple = Orange",

p = "0 = 9" ,

s= "The hat covers the head".

Propositional logic formulas

The concept of the logical form of a complex statement is clarified using the concept propositional logic formulas .

In examples 1 and 2 we learned to write complex statements using logical operations. Actually, they are called propositional logic formulas.

To denote statements, as in the mentioned example, we will continue to use the letters

p, q, r, ..., p 1 , q 1 , r 1 , ...

These letters will play the role of variables that take the truth values ​​“true” and “false” as values. These variables are also called propositional variables. We will further call them elementary formulas or atoms .

To construct propositional logic formulas, in addition to the letters indicated above, signs of logical operations are used

~, ∧, ∨, →, ↔,

as well as symbols that provide the possibility of unambiguous reading of formulas - left and right brackets.

Concept propositional logic formulas let's define it as follows:

1) elementary formulas (atoms) are formulas of propositional logic;

2) if A And B- propositional logic formulas, then ~ A , (A ∧ B) , (A ∨ B) , (A → B) , (A ↔ B) are also formulas of propositional logic;

3) only those expressions are propositional logic formulas for which this follows from 1) and 2).

The definition of a propositional logic formula contains a listing of the rules for the formation of these formulas. According to the definition, every propositional logic formula is either an atom or is formed from atoms as a result of the consistent application of rule 2).

Example 6. Let p- single statement (atom) “All rational numbers are real”, q- "Some real numbers are rational numbers" r- "some rational numbers are real." Translate the following formulas of propositional logic into the form of verbal statements:

6) .

1) “there are no real numbers that are rational”;

2) "if not all rational numbers are real, then no rational numbers, which are valid";

3) “if all rational numbers are real, then some real numbers are rational numbers and some rational numbers are real”;

4) “all real numbers are rational numbers and some real numbers are rational numbers and some rational numbers are real numbers”;

5) “all rational numbers are real if and only if it is not the case that not all rational numbers are real”;

6) “it is not the case that it is not the case that not all rational numbers are real and there are no real numbers that are rational or there are no rational numbers that are real.”

Example 7. Create a truth table for the propositional logic formula , which in the table can be designated f .

Solution. We begin compiling a truth table by recording values ​​(“true” or “false”) for single statements (atoms) p , q And r. All possible values ​​are written in eight rows of the table. Further, when determining the values ​​of the implication operation and moving to the right in the table, we remember that the value is equal to “false” when “false” follows from “true”.

p	q	r					f
AND	AND	AND	AND	AND	AND	AND	AND
AND	AND	L	AND	AND	AND	L	AND
AND	L	AND	AND	L	L	L	L
AND	L	L	AND	L	L	AND	AND
L	AND	AND	L	AND	L	AND	AND
L	AND	L	L	AND	L	AND	L
L	L	AND	AND	AND	AND	AND	AND
L	L	L	AND	AND	AND	L	AND

Note that no atom has the form ~ A , (A ∧ B) , (A ∨ B) , (A → B) , (A ↔ B) . Complex formulas have this type.

The number of parentheses in propositional logic formulas can be reduced if we accept that

1) in a complex formula we will omit the outer pair of brackets;

2) let’s arrange the signs of logical operations “in order of precedence”:

↔, →, ∨, ∧, ~ .

In this list, the ↔ sign has the largest scope and the ~ sign has the smallest scope. The scope of an operation sign refers to those parts of the formula of propositional logic to which the occurrence of this sign in question is applied (on which it acts). Thus, it is possible to omit in any formula those pairs of parentheses that can be restored, taking into account the “order of precedence”. And when restoring parentheses, first all parentheses related to all occurrences of the sign ~ are placed (we move from left to right), then to all occurrences of the sign ∧, and so on.

Example 8. Restore the parentheses in the propositional logic formula B ↔ ~ C ∨ D ∧ A .

Solution. The brackets are restored step by step as follows:

B ↔ (~ C) ∨ D ∧ A

B ↔ (~ C) ∨ (D ∧ A)

B ↔ ((~ C) ∨ (D ∧ A))

(B ↔ ((~ C) ∨ (D ∧ A)))

Not every propositional logic formula can be written without parentheses. For example, in formulas A → (B → C) and ~( A → B) further exclusion of brackets is not possible.

Tautologies and contradictions

Logical tautologies (or simply tautologies) are formulas of propositional logic such that if letters are arbitrarily replaced by statements (true or false), the result will always be a true statement.

Since the truth or falsity of complex statements depends only on the meanings, and not on the content of the statements, each of which corresponds to a certain letter, then checking whether a given statement is a tautology can be done in the following way. In the expression under study, the values ​​1 and 0 (respectively “true” and “false”) are substituted for the letters in all possible ways, and the logical values ​​of the expressions are calculated using logical operations. If all these values ​​are equal to 1, then the expression under study is a tautology, and if at least one substitution gives 0, then it is not a tautology.

Thus, a propositional logic formula that takes the value “true” for any distribution of the values ​​of the atoms included in this formula is called identical to the true formula or tautology .

The opposite meaning is a logical contradiction. If all the values ​​of the statements are equal to 0, then the expression is a logical contradiction.

Thus, a propositional logic formula that takes the value “false” for any distribution of the values ​​of the atoms included in this formula is called identically false formula or contradiction .

In addition to tautologies and logical contradictions, there are formulas of propositional logic that are neither tautologies nor contradictions.

Example 9. Construct a truth table for a propositional logic formula and determine whether it is a tautology, a contradiction, or neither.

Solution. Let's create a truth table:

AND	AND	AND	AND	AND
AND	L	L	L	AND
L	AND	L	AND	AND
L	L	L	L	AND

In the meanings of the implication we do not find a line in which “true” implies “false”. All values ​​of the original statement are equal to "true". Consequently, this formula of propositional logic is a tautology.