The functional nature of the predicate entails the introduction of another concept - quantitor. (Quantum - from Lat. "How many") quantifier operations can be considered as a generalization of conjunction and disjunction operations in the case of finite and endless areas.
Quantitor community (Everything, every, everyone, anyone (all - "all")). The corresponding verbal expression sounds like this:
"For any x p (x) truly." The entry of the variable in the formula may be associated if the variable is located either immediately after the sign of the quantifier or in the range of the quantifier, after which the variable is worth. All other entries are free, the transition from P (x) x (px) or (px) is called the binding of the variable x or hanging the quantifier to the variable x (or to the predicate P) or the quantity of the variable x. The variable on which the quantifier is hung, called connectedunrelated quantization variable called free.
For example, the variable x in the predicate p (x) is called free (x - any of the M), in the statement of P (x), the variable X is called the associated variable.
Fair equivofinity P (x 1) p (x 2) ... P (x n),
P (x) - predicate defined on the set M \u003d (x 1, x 2 ... x 4)
Quantitor existence(EXIST - "exist"). The verbal expression corresponding to it sounds like this: "There is an X, at which P (x) is true." The expression XR (X) is no longer dependent on x, the variable x is associated with a quantitor.
Fair equivalent:
xp (x) \u003d p (x 1) p (x 2) ... p (x n), where
P (x) - predicate defined on the set M \u003d (x 1, x 2 ... x n).
Quantitor community and quantator of existence are called dual, sometimes the designation of Quantitor is used! - "There is, and moreover, only one."
It is clear that the statement Xp (x) is true only in the only case when p (x) is identically true predicate, and the statement is falsely only when P (x) is identically false predicate.
Quantum operations apply to multi-family predicates. The use of a quantifier operation to a predicate P (x, y) in the variable x puts in accordance with a two-bed predicate P (x, y), a single predicate XP (x, y) or xp (x, y), depending on y and independent of x.
You can apply quantify operations for both variables to a two-bed predicate. Then we get eight statements:
1. p (x, y); 2. p (x, y);
3. p (x, y); 4. p (x, y);
5. P (x, y); 6. p (x, y);
7. p (x, y); 8. P (x, y)
Example 3.Consider possible options for hanging quantifiers to predicate P (x, y) – “x. divided by y.", Defined on the set of natural numbers (without zero) N.. Give the verbal formulations of the received statements and determine their truth.
Operation of testing quantifiers leads to the following formulas:
Statements "For any two natural numbers there is a division of one to another" (or 1) All natural numbers are divided into any natural number; 2) any natural number is a divider for any natural number) false;
Statements "There are such two natural numbers that the first is divided into second" (1. "There is such a natural number X, which is divided into some number Y"; 2. "There is such a natural number Y, which is a divider of some kind of natural X ") True;
Statement "There is a natural number that is divided into any natural", false;
Saying "For every natural number there is such a natural, which is divided into the first" (or for every natural number there is its divime), true;
A statement "For any natural X, there is such a natural number Y, which is divided" (or "for any natural number there is a divider"), true;
Statement "There is a natural number that is a divider of every natural number", true (such a divider is one).
In the general case, the change in the order of the quantifier changes the meaning of the statement and its logical value, i.e. For example, the statements p (x, y) and p (x, y) are different.
Let predicate p (x, y) mean that X is a mother for Y, then P (x, y) means that every person has a mother - a true statement. P (x, y) means that there is a mother of all people. The truth of this statement depends on the set of values \u200b\u200bthat can take Y: if it is a lot of brothers and sisters, then it is true, otherwise it is false. Thus, the permutation of quantifiers of universality and existence can change the meaning itself and the value of the expression.
a) replace the initial sign (or) to the opposite
b) put a sign in front of the rest of the predicate
Consider several proposals with a variable:
-
« 
- simple natural number "; The area of \u200b\u200bpermissible values \u200b\u200bof this predicate - a plurality of natural numbers;
-
« 
- an integer "; The area of \u200b\u200bpermissible values \u200b\u200bof this predicate - a plurality of integers;
-
« 
- equilateral ";
-
« 
»
- "Student 
got an estimate 
»
-
« 
divided by 3 "
Definition. If the proposal with variables with any replacement of variables by permissible values \u200b\u200bturns into a statement, then such a proposal is called a predicate.
,
,
,
- Predicates from one variable (single predicates). Predicates from two variables: 
,
- Double predicates. Spellings - zero predicates.
Quantitor community.
Definition. Symbol 
called quantor community.
read: for any 
, for each 
, for all 
.
Let be 
- Single predicate.
read: for any 
- Truth.
Example.
- "All natural numbers are simple" - a false statement.
- "All integers are even numbers" - a false statement.
- "All students received an assessment 
"- Single predicate. Covered quantifier to two predicate, got a single predicate. Similarly 
-n-local predicate, then 
- (n - 1) -fest predicate. 
- (N-2) is a predicate.
In Russian, quantifier community falls.
Quantitor existence.
Definition.Symbol 
called quantator existence.
reading: exists 
, there is 
There is found 
.
Expression 
where 
- Single predicate, read: exists 
, for which 
true.
Example.
- "There are simple natural numbers." (and)
- "There are integer numbers." (and).
- "There is a student who received an assessment 
"- Single predicate.
If there is 1 quantitor to hang on the N-local predicate, then we get (n-1) a predicate, if you can hang a tanker, then we get a zero predicate, i.e. Statement.
If you inspire the quantifier of one species, then the order of padding of the quantifiers is indifferent. And if different quantifiers are hung on the predicate, then the order of hanging quantifiers cannot be changed.
Building a denial of statements containing quantifiers. Laws de Morgana.
De Morgana law.
When building a denial of statement containing a quantifier community, this quantifier community is replaced by a quantifier of existence, and the predicate is replaced by its denial.
De Morgana law.
When building a denial of statements containing a quantifier of existence, a quantifier of existence should be replaced by a community quantifier, and predicate 
- His denial. Similarly, the denial of statements containing several quantifiers are built: a community quantifier is replaced by a quantifier of existence, a quantifier of existence is a quantifier of community, the predicate is replaced by its denial.
P. Elements of set theories (intuitive theory of sets). Numeric sets. Many valid numbers.
Description of the set: Under the word, the set is understood as a combination of objects, which is considered as one. Instead of the word "set" sometimes they say "Aggregate", "Class".
Definition. The object included in the set is called it element.
Record 
indicates that 
is an element of set 
. Record 
indicates that 
not the element of the set 
. About any object can be said, it is an element of a set or not. We write this statement using logical symbols:
There is no object that simultaneously belongs to the set and does not belong, that is,
The set cannot contain the same elements, i.e. If from the set containing the element 
Delete item 
, then a set that does not contain element 
.
Definition.Two sets 
and 
called equal if they contain some of the same items.
Logic and argument: EDUCATION. Handbook for universities. Ruzavin Georgy Ivanovich
4.2. Quantitor
4.2. Quantitor
The significant difference between the logic of predicates from the logic of statements is also the fact that the first introduces the quantitative characteristic of statements or, as they say in logic, quantifies them. Already in the traditional logic, judgments were classified not only in quality, but in quantity, i.e. General judgments were different from private and isolated. But no theory of communication between them was not. Modern logic considers the quantitative characteristics of statements in the special theory of quantification, which constitutes an integral part of the calculation of predicates.
For quantification (quantitative characteristics) of statements This theory introduces two main quantifiers: a community quantifier, which we denote the symbol (x), and a quantifier of the existence indicated by the symbol (EX). They are placed immediately before statements or formulas that are related to. In the case when quantifiers have a wider area of \u200b\u200baction, brackets are put before the corresponding formula.
Quantitor community shows that the predicate marked with a specific symbol belongs to all objects of this class or the union of reasoning.
So, judgment: "All material bodies have a mass" can be translated into a symbolic language like this:
where x - denotes the material body:
M - Mass;
(x) - quantifier community.
Similar to this, the approval about the existence of extrasensory phenomena can be expressed through a quantifier of existence:
where through x indicated phenomena:
E - inherent in such phenomena, the property of extrasensity;
(EX) - quantifier existence.
With the help of a community quantitor, empirical and theoretical laws can be expressed, generalizations about the relationship between phenomena, universal hypotheses and other general statements. For example, the law of thermal expansion of bodies symbolically can be represented as a formula:
(x) (T (x)? P (x)),
where (x) is a community quantator;
T (x) - body temperature;
P (x) - its expansion;
Sign of implication.
The quantifier of existence applies only to a certain part of objects from this area of \u200b\u200breasoning. Therefore, for example, it is used for a symbolic recording of statistical laws, which argue that the property or relation is applied only to characterize a specific part of the objects under study.
The introduction of quantifiers makes it possible first of all to turn predicates into certain statements. Predicates themselves are neither true nor false. They become such if, instead of variables, specific statements are substituted, or if they are associated with quantifiers, quantify. On this basis, the separation of variables into associated and free is introduced.
Changes are called variables that are subject to the signs of quantifiers of community or existence. For example, formulas (x) a (x) and (x) (p (x)? Q (x)) contain a variable x. In the first formula, the quantifier of community is directly in front of the predicate A (X), Vacifora - the quantifier extends its effect on the variables included in the previous and subsequent members of the implication. Similarly, this quantifier of existence may relate to both a separate predicate and their combination, formed by the logical operations of denial, conjunction, disjunction, etc.
The free variable does not fall under the signs of quantifiers, so it characterizes the predicate or a propositional function, and not a statement.
With the help of a combination of quantifiers, you can express in the symbolic language of logic quite complex proposals of the natural language. At the same time, the statements, where we are talking about the existence of objects that satisfy a certain condition are introduced using a quantifier of existence. For example, the approval about the existence of radioactive elements is recorded using the formula:
where R denotes the property of radioactivity.
The statement that there is a danger to smoking to get sick with cancer, can be expressed as follows: (ex) (K (x)? P (x)), where it is the property "to be smoking", and R - "get sick." With the famous reservations, the same could be expressed "by means of a quantifier of generality: (x) (K (x)? P (x)). But the statement that any smoke can get cancer would be incorrect, and therefore it is best to record it with the help of a quantitor of existence, not a community.
Community quantifier is used for statements, in which it is argued that a specific predicate A satisfies any object from the region of its values. In science, as already mentioned, a community quantator is used to express universal assertions, which are verbally presented with the help of such phrases as "for any", "every", "any", "any", etc. By denial of the quantifier, generality can be expressed by general-negative statements, which in the natural language are introduced by the words "no", "none", "no one", etc.
Of course, with a translation on the symbolic language of statements of the natural language, there are certain difficulties, but at the same time the necessary accuracy and unambiguousness of the expression is achieved. It is impossible, however, it is impossible to think that the formal language is richer than the natural language, on which it is not only sense, but also different shades. It may therefore talk only about a more accurate representation of the expressions of the natural language as a universal means of expressing thoughts and exchange them in the process of communication.
Most often, quantifiers of community and existence are encountered together. For example, to express symbolically assertion: "For each valid number x, there is such a number that X will be less from", we denote the predicate "be less" with the symbol<, известным из математики, и тогда утверждение можно представить формулой: (х) (Еу) < (х, у). Или в более привычной форме: (х) (Еу) (х < у). Это утверждение является истинным высказыванием, поскольку для любого действительного числа х всегда существует другое действительное число, которое будет больше него. Но если мы переставим в нем кванторы, т.е. запишем его в форме: (Еу) (х) (х < у), тогда высказывание станет ложным, ибо в переводе на обычный язык оно означает, что существует число у, которое будет больше любого действительного числа, т.е. существует наибольшее действительное число.
From the very determination of quantifiers of generality and existence it immediately follows that there is a certain relationship between them, which is usually expressed using the following laws.
1. The laws of permutation of quantifiers:
(x) (y) a ~ (y) (x) A;
(EX) (EU) A ~ (EU) (EX) A;
(EX) (y) A ~ (y) (EX) A;
2. The laws of denial of quantifiers:
¬ (x) A ~ (EX) ¬ A;
¬ (EX) A ~ (x) ¬ A;
3. The laws of mutual dissipability of quantifiers:
(x) A ~ ¬ (EX) ¬ A;
(EX) A ~ ¬ (x) ¬ A.
Here everywhere and denotes any formula of the object (subject) language. The meaning of the denying quantifier is obvious: if it is incorrect that for any x takes place a, then there are such x, for which and does not have the place. It also follows that if: any x is inherent, then there is no such x that it would be inherent in it, which is symbolically represented in the first law of mutual prosecution.
In addition to we know the logical operations for predicates, two new things are introduced: the operation of hanging quantifiers of existence and community.
"for all h." (for anyone h., for each h.) Called quantitor community And denotes x.
The statement "exists h." (for some h., at least for one x, There is such h.) Called quantitor existence And denotes x.
Saying "There is one and only one h."(For the sole value h.) Called quantitor uniqueness : ! x.
For example: "All shrubs are plants." This statement contains a community quantator ("all"). Statement "There are numbers, multiple 5 »Contains a quantifier of existence (" exist ").
In order to obtain a statement from a multi-seat predicate, it is necessary to tie each variable by quantifiers. For example, if a P (x; y) - Double predicate, then (xx) (oy) p (x; y) - Saying.
If not each variable is associated with a quantitor, then no statement is obtained, but a predicate depending on the variable that is not associated with a quantitor. So, if before the predicate P (x; y) Set quantitor y, I get predicate (oy) p (x; y)variable x.
Thusill which of the following offers are statements, and which predicates: a) there is such x, what x + y \u003d 2;
b) for any h. and w. There is equality x + y \u003d u + x.
Decision: I will identify the logical structure of these proposals.
a) the offer "there is such x, what x + y \u003d 2»Can be written as (xr) x + y \u003d 2. Since only the variable x is associated with the quantor, then the proposal under consideration with two variables is a predicate.
b) offer "for any h. and w. occurs x + y \u003d u + x»Can be written as : (xr) (ur) x + y \u003d u + x,where both variables are associated. Consequently, this offer is a statement.
If any subject variable in the formula is not related to quantitor, then it is called free variable.
For example: (x) Hu \u003d Wow. Here is variables w.not connected by any quantitor, so it is free. The truth of this statement does not depend on it.
Quantitor (x) (x) Called dualeach other.
The quantizerns of the same name can be changed in places that does not affect the truth of the statement.
For example: (y) (x) x + y \u003d 5.it the statement has the same meaning, that I. (x) (y) x + y \u003d 5.
For variepete quantifiers, the change in order can lead to a change in the truth of the statement.
For example: (x) (y) x<у . For any number h. There is a greater number w. - True statement.
Change the quantifairs in some places: (x) (y) x
In connection with the introduction of quantifiers, it is necessary to take into account the following:
1. The formula of the predicate logic cannot contain the same substantive variable, which would be connected in one part of the formula and freely into another.
2. The same variable can not be in the area of \u200b\u200bdual friend of quantifiers.
Violation of these conditions is called collision of variables.
How is the value of the truth of saying with Quantitor?
To prove the approval with a community quantifier It is necessary to make sure that when substituting each of the values h. in predicate P (x) The latter turns into a true statement. If the set X is of course, this can be done by extinguishing all cases; If the set x is infinite, then it is necessary to conduct reasoning in general.
Statement (x) p (x)false if you can specify this value butH.in which P (x) appeals to false statement P (a). Therefore, to refute the statement with a quantator of community It is enough to give an example.
Statement (x) p (x) True, if you can specify this value butH.in which P (x) refers to true statement R (a). Therefore, to make sure the truth of saying with Quantitor existence , just give an example and thus prove.
In order to make sure with Quantitor existence (x) p (x), need to make sure of each P (H.), P (H.), …, P (H.). If the set H. Of course, this can be done by busting. If the set H.infinitely, it is necessary to conduct reasoning in general.
Examples.
1. Find the value of the truth "Bondage 1, 2, 3, 4 there is a simple number. "
Decision: The statement contains a quantifier of existence and therefore can be represented as a disjunction of statements: " 1 - a simple number "or" 2 - a simple number "or" 3 - a simple number "or" 4 - Prime number". To prove the truth of the disjunction, the truth of at least one statement, for example, " 3 - Easy ", which is truly true. Consequently, true and initial statement.
2. We prove that any square is a rectangle.
Decision:The statement contains a community quantator. Therefore, it can be represented in the form of conjunction: "Square - rectangle" and "square - a rectangle" and "square - rectangle", etc. Since all these statements are true, then the conjunction of these statements is true, therefore, a truly and source offer.
3. "Any triangle is a preceded." This is a false statement. To make sure that it is enough to draw a triangle that is not equally chaped.
To build a denial of statements with quantizers It is necessary:
1) a community quantifier to replace the existence quantator, and the quantifier of existence is a community quantifier;
2) Predicate replace it with denial.
Example. We formulate denial for the following statements:
a) all elements of the set Z.even; b) Some verbs answer the question "What to do?".
Decision: a) Replace the quantifier community by quantifier existence, and the statement of its denial: some elements of the set Z.odd.
b) We will replace the quantifier of the existence by a quantifier of community, and its expression is denied: all verbs do not answer the question "what to do?".
In the logic of predicates, two operations are considered, which turn a single predicate into a statement, for this, special words are used that put before predicates. In logic they are called quantifiers.
Two types of quantifiers distinguish:
1. Quantitor community;
2. Quantitor existence.
1. Quantitor community.
Let there be predicate p (x) defined on the set m
Symbol called quantitor Universality(generality). This is an inverted first letter of the English word ALL- all. They read "all", "every", "any", "all sorts". Variable x in predicateP (x) called free (she can be given different values \u200b\u200bfrom M), in sayingsame C called connected Quantitor Universality.
Example №1: P (x) - "A simple number of odd"
Add a quantifier community - "Any simple number x is odd" - a false statement.
Under the expression, the statement is true, when p (x) is true for each element x from the set M and false otherwise. This statement is no longer dependent on x.
2. Quantitor existence.
Let p (x) be predicate defined on the set M. under the expression understand statementwhich is true if there is an element for which P (x) is truly true and false - otherwise. This statement no longer depends on x. The resulting verbal expression sounds like this: "There is an X, at which P (x) is true." Symbol called quantitor existence. In the statement, the variable X is connected by this quantitor (quantifier is brought on it).
(Read: "There is such x from M, at which p from the True")
Under the expression, the statement is understood, which is true if there is an element X € M (at least one), for which P (x) is true, and false otherwise.
Example №2: P (x) "The number X is more than 5"
Any natural number of multiple 5 "
Every natural number of multiple 5 "false statements
All natural numbers are multiple 5 "
There is a natural number of multiple 5
There is a natural number of multipolite 5 True statements
At least one natural number is multiple 5
Quantum operations apply to multi-family predicates. Let, for example, a double predicate P (x, y) is set on the set M. The use of a quantifier operation to a predicate P (x, y) in the variable x puts in accordance with a two-bed predicate P (x, y) a single predicate (or single predicate), depending on the variable Y and independent of the variable x. You can apply quantifying operations for the variable Y, which will already lead to the statements of the following types:
To build denials with quantizers it is necessary:
1) a quantifier of community to replace the quantifier of existence, and the quantifier of existence is on a community quantifier;
2) Predicate replace it with denial.
Thus, Formulas are valid:
Decidence of the proposal to record as, and denial of the sentence - as. Obviously, the proposal is the same meaning, and therefore, the same meaning of truth as the proposal, and the proposal is the same meaning that. In other words, it is equivalent; equivalent.
PRI M E R No. 3. Build a denial of statements "Some two-digit numbers are divided into 12".
Replace the quantifier of existence (it is expressed by the word "some") on the quantifier of the generality "all" and construct a denial of the sentence standing after the word "some" by putting a particle "not" before the verb. We get a statement "All two-digits are not divided into 12".
PRI M E R No. 4. Formulate the denial of statements "In each class, at least one student did not cope with the control work."
This statement contains a community quantifier, expressed with the help of the word "each", and the quantifier of existence, expressed with the help of the words "at least one". According to the rule of constructing of statements with quantities, a quantifier must be replaced with a quantifier of existence, and the quantifier of existence is a quantifier of community and remove the "not" particle from the verb. We get: "There is such a class in which all students coped with the control work."
