Practical tasks on the mathematical logic of statements and operations on them. Elements of propositional logic

Plan

Statements with external negation.

conjunctive statements.

disjunctive statements.

Strictly disjunctive statements.

Statements about equivalence.

Implicit statements.

Statements with external negation.

A statement with external negation is a statement (judgment) in which the absence of a certain situation is affirmed. It is most often expressed in a sentence that begins with the phrase “it is not true that...” or “it is wrong that...”. External negation is indicated by the symbol “ù”, called the negation sign. This sign is determined by the following truth table:

In statements with external negation, the situation in A is denied. For example, if A: “The Volga flows into the Black Sea,” then ùA: “It is not true that the Volga flows into the Black Sea.”

conjunctive statements.

Conjunctive statements are those in which the simultaneous existence of two situations is affirmed. Conjunctive statements are formed from two statements using the unions “and”, “a”, “but”. Form of conjunctive utterance: (A&B). Each of statements A and B can take both the value "true" and the value "false". These values ​​are denoted for brevity by the letters i, l. The truth table for conjunctive statements is as follows:

In conjunctive statements, it is stated that the situation described in A and in B takes place simultaneously. Examples of conjunctive statements: “Earth is a planet, and the Moon is a satellite”; “Petrov mastered logic well, but Sidorov mastered logic badly”; “It's dark outside and the lights are on in the auditorium”; “Petrov gave the official a bribe in cash, and Sidorov gave him a bottle.”

disjunctive statements.

Disjunctive statements are statements that assert the existence of at least one of the two situations described in A and B. A disjunction is denoted by the symbol V and is expressed in natural language by the union “or”.

The tabular definition of the disjunction sign is as follows:

An example of a disjunctive statement: "Roman Sergeevich Ivanov is a teacher, or Roman Sergeevich Ivanov is a graduate student."

Strictly disjunctive statements.

Strictly disjunctive statements are statements that assert the existence of exactly one of the two situations described in A and B. Such statements are most often carried out by means of sentences with the union “or ..., or ...” (“either ..., or ...”). Strict disjunction is denoted by the symbol V* (read “either... or...”).

The tabular definition of the strict disjunction sign is as follows:

An example of a strictly disjunctive statement: "Either it's sunny outside, or it's raining."

Among the possible truth values ​​of a linguistic variable Truth two values ​​attract particular attention, namely the empty set and unit interval, which correspond to the smallest and largest elements (with respect to inclusion) of the lattice of fuzzy interval subsets. The importance of these truth values ​​is due to the fact that they can be interpreted as truth values undefined and unknown respectively. For convenience, we will denote these truth values ​​by the symbols and , understanding that and are determined by the expressions

Values unknown and undefined, interpreted as degrees of membership, are also used in the representation of fuzzy sets of type 1. In this case, there are three possibilities for expressing the degree of membership of a point in : 1) a number from the interval ; 2) ( undefined); 3) (unknown).

Let's consider a simple example. Let

Take a fuzzy subset of the set of the form

In this case, the degree of membership of the element in the set is unknown, and the membership degree is undefined. In a more general case, it may be

where it is understood that the degree of membership of an element in a set is partially unknown, and the member is interpreted as follows:

. (6.56)

It is important to clearly understand the difference between and. When we say that the degree of membership of a point in a set is , we mean that the membership function not defined at point . Suppose, for example, that is the set of real numbers, and is a function defined on the set of integers, and , if - even, and , if - odd. Then the degree of membership of the number in the set is , and not 0. On the other hand, if it were defined on the set of real numbers and if and only if is an even number, then the degree of membership of the number in the set would be equal to 0.

Since we can calculate the truth values ​​of propositions and, or and not given the linguistic truth values ​​of statements and , it is easy to calculate the values ​​, , , when . Suppose, for example, that

, (6.57)

. (6.58)

Applying the principle of generalization, as in (6.25), we obtain

, (6.59)

After simplification (6.59) reduces to the expression

. (6.61)

In other words, the truth value of the statement and, where , is a fuzzy subset of the interval , whose degree of membership of the point is equal to (membership function ) on the interval .

Rice. 6.4. The conjunction and disjunction of the truth values ​​of a statement with the truth value is unknown ().

Similarly, we find that the truth value of the statement or expressed as

. (6.62)

It should be noted that expressions (6.61) and (6.62) are easy to obtain using the graphical procedure described above (see (6.38) and below). An example illustrating this is shown in Fig. 6.4.

Turning to the case, we find

(6.63)

and similarly for .

It is instructive to follow what happens to the above relations when we apply them to a particular case of two-valued logic, i.e. to the case when the universal set has the form

or in the more familiar form

where means true, a - false. Since there is , we can identify the truth value unknown with meaning true or false, i.e.

The resulting logic has four truth values ​​, , and and is a generalization of two-valued logic in the sense of Remark 6.5.

Since the universal set of truth values ​​consists of only two elements, it is advisable to construct truth tables for the operations , and in this four-valued logic directly, i.e. without using the general formulas (6.25), (6.29) and (6.31). Thus, applying the principle of generalization to the operation , we immediately obtain

whence it necessarily follows that

Along the way, we arrive at the usual definition of the connective ⟹ in two-valued logic in the form of the following truth table:

As the example above shows, the notion of a truth value unknown in combination with the principle of generalization helps to understand some of the concepts and relationships of ordinary two-valued and three-valued logics. These logics, of course, can be considered as degenerate cases of fuzzy logic, in which the truth value unknown is the entire unit interval, not the set 0 + 1.

Here: 1 - true, 0 - false.

• 1. X: triangle ABC - acute. X: It is not true that triangle ABC is acute. It's like: X: triangle ABC - right or obtuse
• 2. A: Ivanova M. In the exam in mathematics, she received 4. : It is not true that Ivanova M. in mathematics received 4.

Definition: The disjunction of a statement A and B is a statement AB that is true provided that at least one of the statements A or B is true.

It is read "A or B".

Truth table for AB

Example: 1. This time the defendant appeared and the trial took place. - truth

2. In right triangle the sum of any two angles is greater than or equal to the third angle and the hypotenuse is less than the leg. - False

Definition: An implication of statements A and B is a statement AB that is false only if A is true and B is false.

It is read: "If A, then B."

truth table

Example: 1. If I pass the test, I will go to the cinema.

2. If the triangle is isosceles, then the angles at its base are equal. Definition: An equivalent of statements A and B is a statement AB that is true if and only if A and B have the same truth (i.e. either both are true or both are false).

They read: “A if and only if B” or “A is necessary and sufficient for B”

truth table

The second task, solved by means of propositional algebra, is to determine the truth of a particular proposition on the basis of compiling its formula (formalization process) and compiling a truth table.

Example: If Saratov is located on the banks of the Neva, then polar bears live in Africa.

A: Saratov is located on the banks of the Neva River;

Q: Polar bears live in Africa

Definition: A formula that is true, regardless of what values ​​its propositional variables take, is called a tautology or an identically true formula.

Definition: Formulas F 1 and F 2 are called equivalent if their equivalent is a tautology.

Definition: If the formulas F 1 and F 2 are equivalent, then the sentences Р 1 and Р 2 that initiate these formulas are called equivalent in propositional logic.

The basic, most common equivalences are called the laws of logic. We list some of them:

• 1. X X - the law of identity
• 2. X L - the law of contradiction
• 3. XI - the law of exclusion of the third
• 4. X - the law of double negation

• 5. laws of commutativity
• 6. X (Y Z) (X Y) Z associativity law

X (Y Z) (X Y) Z distributive law

7. De Morgan's laws

8. laws of articulation of a variable with a constant

Using the laws of logic, you can transform formulas.

4. Of the many formulas that are equivalent to each other, consider two. They are perfect conjunctive normal form (CKNF) and perfect disjunctive normal form (PDNF). They are built for a given formula based on its truth table.

Building SDNF:

• -- rows corresponding to the truth values ​​(1) of the given formula are selected;
• -- for each selected line, we compose a conjunction of variables or their negations so that the sets of values ​​of the variables presented in the line correspond to the true values ​​of the conjunction (for this, you need to take the variables that in this line took the values ​​false (0) with a negation sign, and the variables , taking values ​​of truth (1) without negation);
• -- a disjunction of the resulting conjunctions is made.

It follows from the algorithm that for any formula it is possible to compose an SDNF, and, moreover, the only one, if the formula is not identically false, i.e. accepting only false values.

Compilation of SKNF is carried out according to the following algorithm:

• -- select those rows of the table in which the formula evaluates to false (0);
• -- from the variables in each such line, make a disjunction, which should take the values ​​- false (0). To do this, all variables must enter it with the value false, therefore those that are true (1) must be replaced by their negation;
• - make a conjunction from the obtained disjunctions.

Obviously, any formula that is not a tautology has SKNF.

SDNF and SKNF are used to obtain consequences from this formula.

Example: Make a truth table of SDNF and SKNF for the formula: .

Truth table of SDNF and SKNF

5. Consider the propositional form "The river flows into the Black Sea." It contains one variable and can be represented as "River x flows into the Black Sea".

Depending on the values ​​of the variable X, the sentence is either true or false, i.e. the mapping of a set of rivers onto a two-element set is specified. Let's denote this mapping, then:

Thus, we have a function, all values ​​of which belong to the set.

Definition: A function whose values ​​all belong to a set is called a predicate.

Letters denoting predicates are called predicate symbols.

Predicates can be set:

a) a propositional formula,

b) formula, i.e. specifying the interpretation of the predicate symbol,

c) a table.

1) P - "flow into the Black Sea."

This formula means that "River a flows into the Black Sea."

• 2) The predicate P is given by the propositional formula: “to be prime number on the set of the first 15 natural numbers.
• 3) In tabular form, the predicate has the form:

The scope of predicates can be any set.

If the predicate loses its meaning for any set of input variables, then it is generally assumed that this set corresponds to the value of A.

If the predicate contains one variable, then it is called unary, two variables - two-place, n variables - n-place predicate.

To translate texts into the language of predicates and determine their truth, it is necessary to introduce logical operations on predicators and quantifiers.

Operations are performed on predicates in the same way: negations, conjunctions, disjunctions, implications, equivalences.

Definition: A subset of the set M on which the predicate P is given, consisting of those and only those elements of M that correspond to the value AND of the predicate P, is called the truth set of the predicate P.

The truth set is denoted.

Definition: The negation of a predicate P is a predicate that is false for those sets of variable values ​​that make P true, and true for those sets of variable values ​​that make P false.

Negative is indicated.

Be a student of ABC.

Do not be a student of ABiK.

If, then the set, where M is the set on which the predicates P and Q are given.

Definition: A conjunction of predicates is a predicate that is true for those and only those values ​​of the variables included in it that make both predicates also true.

be a football player

To be a student

: be a footballer and be a student.

Definition: A disjunction of predicates is a predicate that is false for those sets of variables that make both predicates false

be even natural number

be an odd natural number

: be a natural number.

Definition: An implication of predicates is a predicate that is false for those and only those sets of variables included in it that turn into a true predicate, and - into a false one.

Designated:

Be a prime number on the set N

be an odd number

False for and true for other natural numbers.

Definition: A predicate equivalence is a predicate that becomes true if both predicates are true or both are false.

Designated:

- “win”, i.e. x wins y

Better to know chess history, x knows better than y

means that x beats y in chess if and only if he knows the theory better.

Definition: The predicate follows from the predicate if the implication is true for any of the values ​​of the variables included in it.

The following are designated: .

To be a student

go to college

There are 2 ways to turn a predicate into a statement:

1) giving a variable a specific value

; x - student

Ivanov is a student.

2) Hanging quantifiers - any, any, each

There is, there is.

The notation where has the property P means that every object x has the property P. Or in other words, "all x have the property P."

The notation means that there is an object x that has the property P.

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

The logic of propositions is abstracted from the meaningful load of propositions and studies their truth value, that is, whether the proposition is true or false.

The figure 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 be branded a liar a priori. In the natural layered world on the subject of "truth" or "falsehood" is evaluated only separately taken statements . And later in this lesson, you will be introduced to the opportunity to evaluate many statements on this subject (and then look at the correct answers). Including complex statements in which simpler ones are interconnected by signs of logical operations. But first let us consider these operations on propositions themselves.

Propositional logic is used in computer science and programming in the form of declaring logical variables and assigning them the 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, that boolean variable is often given a name, such as "flag" and the "flag" is implied when that 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 a lot of logical variables, professionals are required to come up with names of logical variables that have the form of statements and a semantic load that distinguishes them from other logical variables and is understandable to other professionals who will read the text of this program.

So, a logical variable with the name "UserRegistered" (or its English equivalent) can be declared, having the form of a statement, which can be assigned the logical value "true" if the conditions are met that the data for registration is sent by the user and this data is recognized by the program as valid. In further calculations, the values ​​of the variables may change depending on what logical value ("true" or "false") the "UserLogged in" variable has. In other cases, a variable, for example, with the name "More than Three Days Until Day", can be assigned the value "True" up to a certain block of calculations, and during the further execution of the program this value can be saved or changed to "false" and the course of further execution depends on the value of this variable programs.

If the program uses several logical variables whose names have the form of propositions, and more complex propositions are built from them, then it is much easier to develop a program if, before developing it, all operations from propositions are written in the form of formulas used in propositional logic than we do in the course of this lesson and let's do it.

Logical operations on statements

For mathematical statements, one can always choose between two different alternatives "true" and "false", but for statements made in "verbal" language, the concepts of "true" and "false" are somewhat more vague. However, for example, such verbal forms as "Go home" and "Is it raining?" are not utterances. Therefore, it is clear that utterances 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 by the values ​​"true" and "false".

Propositions, on the other hand, can be viewed as a quantity that can take on two values: "true" and "false".

For example, 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 - "false", the third - "true", and the fourth - "false". Such an interpretation of propositions is the subject of propositional algebra. We will denote statements in capital Latin letters A, B, ..., and their values, that is, true and false, respectively And and L. In ordinary speech, connections are used between the statements "and", "or" and others.

These connections make it possible, by combining various statements, to form new statements - complex statements . For example, a bunch of "and". Let the statements be given: π greater 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". The statement "if π irrational, then π ² is also irrational" is obtained by linking two statements with the link "if - then". Finally, we can get a new - complex statement - from any statement - negating the original statement.

Considering propositions as quantities taking on the values And and L, we 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 - is the formation of a new statement, which we will denote A ∧ B and which is true if and only if A and B true. In ordinary speech, this operation corresponds to the connection of statements with a bunch of "and".

Truth table for conjunction:

2 . The 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 the connection of statements with a bunch of "or". However, here we have a non-separative "or", which is understood in the sense of "either-or" when A and B both cannot be true. In the definition of propositional logic A ∨ B true if only one of the statements is true, and if both statements are true A and B.

Truth table for disjunction:

3 . The third logical operation on statements A and B, expressed as A → B; the resulting statement is false if and only if A true, and B false. A called parcel , B - consequence , and the statement A → B - following , also called an implication. In ordinary speech, this operation corresponds to the link "if - then": "if A, then B". But in the definition of propositional logic, this proposition is always true, regardless of whether the proposition is true or false B. This circumstance can be briefly formulated as follows: "anything you like follows from the false." In turn, if A true, and B false, then the whole statement A → B false. It will be true if and only if A, and B true. Briefly, this can be formulated as follows: "false cannot follow from the true."

Truth table to follow (implication):

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

Truth table for negation:

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

Truth table for equivalence:

Most programming languages ​​have special symbols for logical values ​​of propositions, they are written in almost all languages ​​as true (true) and false (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 ones. Elementary statements are considered as whole, not decomposable into parts.

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

For logical operations are true laws of the algebra of logic, which can be used to simplify boolean expressions. At the same time, it should be noted that in the logic of propositions they are abstracted from the semantic content of the proposition and are limited 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 value of the statement in the first brackets is "true", the value 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), so the logical value of this entire statement is "true".

2) The meaning of the statement in brackets is "false". This statement is preceded by a logical negation operation, so the logical value of this whole 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". The statements are connected by the logical operation "OR" and none of the statements has the value "true". Therefore, the logical meaning of this whole statement is "false".

4) The meaning of the statement in brackets is "false". This statement is preceded by a logical negation operation. Therefore, the logical meaning of the whole given statement is "true".

5) In the first brackets, the statement in the inner brackets is negated. This statement in parentheses evaluates to "false", so its negation will evaluate to the logical value "true". The statement in the second brackets has the value "false". These two statements are connected by the logical operation "AND", that is, "true AND false" is obtained. Therefore, the logical meaning of the whole given 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 whole 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" is obtained. Therefore, the logical meaning of the whole given statement is "true".

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

1) "User not registered";

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

3) "The user is registered when and only when the data sent by the user is found to be 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- "Data sent by the user is valid", logical operation: .

Solve propositional logic examples on your own and then look at the solutions

Example 3 Calculate the boolean values ​​of the following statements:

1) ("There are 70 seconds in a minute") OR ("A running clock shows the time");

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

3) ("TV - electrical appliance") and ("Glass - wood");

4) Not((300 > 100) OR ("Thirst can be quenched 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 does not show the time correctly, then you can come to class at the wrong time";

2) "In the mirror you can see your reflection and Paris - the capital of the USA";

Example 5 Determine Boolean Expression

(p → q) ↔ (r → s) ,

p = "278 > 5" ,

q= "Apple = Orange",

p = "0 = 9" ,

s= "The hat covers the head".

Propositional logic formulas

concept logical form complex statement is specified with the help of the concept propositional logic formulas .

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

To denote statements, as in the above 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 henceforth call them elementary formulas or atoms .

To construct propositional logic formulas, in addition to the above letters, the 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 define 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 an enumeration of the rules for the formation of these formulas. According to the definition, every formula of propositional logic is either an atom or is formed from atoms as a result of the successive 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 into the form of verbal propositions the following formulas of propositional logic:

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 no rational numbers that are real."

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

Solution. We begin compiling the truth table by recording the 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, remember that the value is equal to "false" when "true" implies "false".

Note that no atom has the form ~ A , (A ∧ B) , (A ∨ B) , (A → B) , (A ↔ B) . These are complex formulas.

The number of brackets in propositional logic formulas can be reduced by assuming that

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

2) order the signs of logical operations "by seniority":

↔, →, ∨, ∧, ~ .

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

Example 8 Restore parentheses in 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 brackets. For example, in formulas BUT → (B → C) and ~( A → B) no further deletion of brackets is possible.

Tautologies and contradictions

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

Since the truth or falsity of complex statements depends only on the meanings, and not on the content of statements, each of which corresponds to a certain letter, then the test of whether a given statement is a tautology can be substituted 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 using logical operations, the logical values ​​of the expressions are calculated. 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 this 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 identically true formula or tautology .

The opposite meaning is a logical contradiction. If all proposition values ​​are 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 Make a truth table for a propositional logic formula and determine whether it is a tautology, a contradiction, or neither.

Solution. We make a truth table:

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

False and true statements are often used in language practice. The first assessment is perceived as a denial of truth (untruth). In reality, other types of assessment are also used: uncertainty, unprovability (provability), unsolvability. Arguing over for what number x the statement is true, it is necessary to consider the laws of logic.

The emergence of "many-valued logic" led to the use of an unlimited number of truth indicators. The situation with the elements of truth is confusing, complicated, so it is important to clarify it.

Principles of the theory

A true statement is the value of a property (feature), it is always considered for certain action. What is truth? The scheme is as follows: "Proposition X has a truth value Y in the case when proposition Z is true."

Let's look at an example. It is necessary to understand for which of the given statements the statement is true: "Object a has a sign B". This statement is false in that the object has attribute B, and false in that a does not have attribute B. The term "false" in this case is used as an external negation.

Definition of truth

How is a true statement determined? Regardless of the structure of the statement X, only the following definition is allowed: "The statement X is true when there is X, only X."

This definition makes it possible to introduce the term "true" into the language. It defines the act of accepting agreement or utterance with what it says.

Simple sayings

They contain a true statement without a definition. You can limit yourself when saying "Not-X" common definition if this statement is not true. The conjunction "X and Y" is true if both X and Y are true.

Saying example

How to understand for which x the statement is true? To answer this question, we use the expression: "Particle a is located in the region of space b". Consider the following cases for this statement:

• it is impossible to observe the particle;
• particle can be observed.

The second option involves certain possibilities:

• the particle is actually located in a certain region of space;
• it is not in the supposed part of space;
• the particle moves in such a way that it is difficult to determine the area of ​​its location.

In this case, you can use four terms of truth values ​​that correspond to the given possibilities.

For complex structures, it is appropriate to use more terms. This indicates that the truth values ​​are unlimited. For what number the statement is true depends on practical expediency.

Ambiguity principle

In accordance with it, any statement is either false or true, that is, it is characterized by one of two possible truth values ​​- “false” and “true”.

This principle is the basis of classical logic, which is called the two-valued theory. The ambiguity principle was used by Aristotle. This philosopher, arguing over for what number x a statement is true, considered it unsuitable for those statements that concern future random events.

He established a logical relationship between fatalism and the principle of ambiguity, the predestination of any human action.

In subsequent historical eras, the restrictions that were imposed on this principle, were explained by the fact that it significantly complicates the analysis of statements about planned events, as well as about non-existent (non-observable) objects.

When thinking about which statements are true, it was not always possible to find an unambiguous answer with this method.

The emerging doubts about logical systems were dispelled only after modern logic was developed.

To understand for which of the given numbers the statement is true, two-valued logic is suitable.

The principle of ambiguity

If we reformulate a version of a two-valued statement to reveal the truth, we can turn it into special case polysemy: any statement will have one n truth value if n is either greater than 2 or less than infinity.

Many logical systems based on the principle of ambiguity act as exceptions to additional truth values ​​(above "false" and "true"). Two-valued classical logic characterizes the typical uses of some logical signs: "or", "and", "not".

A multi-valued logic that claims to make them concrete should not contradict the results of a two-valued system.

The belief that the principle of ambiguity always leads to a statement of fatalism and determinism is considered erroneous. Also incorrect is the idea that multiple logic is considered as a necessary means of carrying out indeterministic reasoning, that its acceptance corresponds to the rejection of the use of strict determinism.

Semantics of logical signs

To understand for what number X the statement is true, you can arm yourself with truth tables. Logical semantics is a branch of metalogics that studies the relation to the designated objects, their content of various linguistic expressions.

This problem was considered already in the ancient world, but in the form of a full-fledged independent discipline, it was formulated only at the turn of the 19th-20th centuries. The works of G. Frege, C. Pierce, R. Carnap, S. Kripke made it possible to reveal the essence of this theory, its realism and expediency.

For a long time period, semantic logic relied mainly on the analysis of formalized languages. Only in recent times most of research began to be devoted to natural language.

There are two main areas in this methodology:

• notation theory (reference);
• theory of meaning.

The first involves the study of the relationship of various linguistic expressions to the designated objects. As its main categories, one can imagine: "designation", "name", "model", "interpretation". This theory is the basis for proofs in modern logic.

The theory of meaning is concerned with finding an answer to the question of what is the meaning of a linguistic expression. She explains their identity in meaning.

The theory of meaning plays a significant role in the discussion of semantic paradoxes, in the solution of which any criterion of acceptability is considered important and relevant.

Boolean Equation

This term is used in metalanguage. Under the logical equation, we can represent the record F1=F2, in which F1 and F2 are formulas of the extended language of logical propositions. To solve such an equation means to determine those sets of true values ​​of variables that will be included in one of the formulas F1 or F2, under which the proposed equality will be observed.

The equal sign in mathematics in some situations indicates the equality of the original objects, and in some cases it is used to demonstrate the equality of their values. The entry F1=F2 may indicate that we are talking about the same formula.

In the literature, quite often, formal logic is understood as a synonym for "the language of logical propositions". As " right words» are formulas that serve as semantic units used to build reasoning in informal (philosophical) logic.

The statement acts as a sentence that expresses a specific proposition. In other words, it expresses the idea of ​​the presence of a certain state of affairs.

This fact became the basis of propositional logic. There is a division of statements into simple and complex groups.

When formalizing simple variants of statements, elementary formulas of the zero-order language are used. The description of complex statements is possible only with the use of language formulas.

Logical connectives are necessary to denote unions. When they are applied, simple statements turn into complex types:

• "not",
• "It's not true that..."
• "or".

Conclusion

Formal logic helps to find out for which name a statement is true, involves the construction and analysis of rules for transforming certain expressions that preserve their true meaning regardless of content. As a separate section of philosophical science, it appeared only at the end of the nineteenth century. The second direction is informal logic.

The main task of this science is to systematize the rules that allow deriving new statements based on proven statements.

The foundation of logic is the possibility of obtaining some ideas as a logical consequence of other statements.

Such a fact makes it possible to adequately describe not only a certain problem in mathematical science, but also to transfer logic to artistic creativity.

Logical research presupposes the relationship that exists between premises and the conclusions drawn from them.

It can be attributed to the number of initial, fundamental concepts of modern logic, which is often called the science of "what follows from it."

It is difficult to imagine proving theorems in geometry without such reasoning, explaining physical phenomena, explanation of the mechanisms of reactions in chemistry.