The Problem of Forming and Developing Students’ Logical Thinking in the Context of Subject Specialization in Secondary School

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Table 2. Logical Laws

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In the process of solving such problems students demonstrate the ability to think sequentially, to refute wrong conclusions, to select relevant and valid arguments. These capabilities require further involving, improving and developing both actual mathematical skills and social experience of students. To achieve this goal, the so-called problem of “knights and liars”, the task to establish a correspondence between objects, “the problem to the truth” (when it is required to establish the truth or falsity of one or more statements) can be used.

For example, firstly, one can offer the students the following problems.

Objective 1. Imagine that you are in the room that has two doors. One door points the way to freedom, and the other – to an imminent death. There is one guard near each of the doors. One of the guards tells only the truth, the other only lies. Guards can answer only “yes” or “no”. You don’t know where the door to freedom is and which of the guards stands at that door. However, you know that the guards are aware of the point, who of them is the liar and who tells the truth. You can ask only one question and only to one of the guards in order to find the correct door which leads to freedom. What question should you ask?

Objective 2 (“Three Friends”). Three school graduate friends Igor, Ivan and Oleg were going to enter colleges in different cities of Ukraine: Donetsk, Kiev and Kharkov. A graduate of the class with advanced study of mathematics tried to enter a college in Donetsk, but failed. A graduate of the class with advanced study of chemistry entered Kharkov University. Oleg confirmed his deep knowledge of jurisprudence during the interview, which he had received in the class of the corresponding profile and entered the faculty of law. Ivan refused to accept the parents’ advice to study in Kharkiv. In which city did the boys continue their studies?

In the process of solving these problems it is advisable to emphasize that students have used the logical laws intuitively, but these laws can be represented verbally (Table 2) or with the help of symbols.

However, it should be pointed out that intuitive knowledge may not always be sufficient to reach a conclusion on the validity of reasoning. For example, the students may be asked to determine if the conclusion below is valid (objective 3).

Objective 3. Determine if the conclusion below are true: “If a quadrangle happens to be a rhombus, then its diagonals are mutually perpendicular. If the diagonals of a quadrangle are not mutually perpendicular, then the quadrangle cannot be a square. If a quadrilateral is a square, then it can be fitted into a circle. What is not true means that a given quadrangle cannot be fitted into a circle or that its diagonals are not mutually perpendicular. As a result, the quadrangle is neither a rhombus, nor a square”.

Trying to give a reasonable answer to this question may cause some difficulties. It proves the need for instruction in the field of mathematics that allows to “calculate reasoning”, without being bound to the content. Students are encouraged to seek historical information related to the inception and development of mathematical logic, outstanding mathematicians that have contributed to this field of science and it’s current and relevant problems.

Next, we provide a list of key concepts that, according to the program in mathematics for schools and classes with advanced study, the students should be able to define and describe:

• statements and logical operations applicable to them;

• predicates, and logical operations applicable to them;

• equivalent statements and equivalent predicates;

• truth functions;

• domain and range of the truth predicates;

• quantifiers, both existential and universal;

• formulas of Boolean Algebra;

• formulas of predicate calculus.

Let’s now discuss distinct features that characterize how these concepts may be introduced.

Introducing the concept of statement, it is advisable to carry out the following concrete inductive scheme (in the process of practicing exercises), pointing out the characteristic features of this concept, giving examples and counter examples, practicing exercises with the purpose of leading the object to the concept and making conclusions of belonging of the object to this concept.

Introducing the concept of a “truth function” is possible through the exercises such as the ones we list below.

Exercise 1. The set of true statements is pointed by symbol T (from English word, True), and the set of false statements is pointed by symbol F (from English word, False). 1) Give the examples of the elements of the sets T and F; 2) which of the claims , is true?

If the statement A is true, it is pointed as or . And in this case it is said that A has the meaning of truth 1. If the statement A is False then it is said that A has the meaning of truth 0. In other words, a function can be defined with the range of meanings {0, 1} on a set of given statements. This function is called the function of truth.

Exercise 2. Determine the truth value of statements:

1) the equality is true if and only if ;

2) the equality is true for all

3) the equality is false for any value;

4) the inequality is proved for any values ;

5) the equality is true for all .

We introduce the concept of predicate on concrete inductive basis, proposing to determine which of the items 1) - 7) are statements:

1) ; 2) ; 3) ; 4) ; 5) 6) ; 7)

We define characteristic features of sentences which are not statements. Thus, the truth value of 3) and 4) depends on the variable x. Item 6) is true for all a and b. Item 7) is true for all We emphasize that if a sentence has a variable, it is necessary to define the domain of this variable. Another specific feature of items 3, 4, 6 and 7 is that if a variable takes a value, then the sentence becomes a statement.

Together with pupils we list characteristic properties of predicates: 1) expression should contain a variable; 2) the domain of the variable must be defined; 3) if a variable takes a value, then an expression becomes a statement. We give the following examples of predicates: 1) on the set of triangles x: the square of one side of the triangle equals the sum of squares of the other two sides; 2) on the set of integers: ; 3) on the set of natural numbers: .

After summing up, we point out that arbitrary numerical equality or inequality is the example of statement, arbitrary equation or inequality with variables is the example of predicate.

Pupils are offered to choose predicates. If the sentence is not a predicate, it should be completed to become a predicate:

• (this is not a predicate because properties 2 and 3 are not fulfilled);

• (this is not a predicate because the domain is not defined);

• let’s say M is the set of students, “x practice sports” (predicate);

• let’s say M is the set of students, “y come to me” (is not a predicate because it is not a statement if y takes a value from M);

• let M be the set of students, “x and y are sisters” (predicate).

On the basis of the given examples pupils get acquainted with single-variable and two-variable predicates. Special attention should be concentrated on the fact that if one variable of a two-variable predicate takes a value from the domain, we obtain a single-variable predicate. A statement can be considered as a 0-variable predicate.

The following questions could be useful for the generalization of the concept of the domain: 1) Let’s consider number expression: . Is it an example of a proposition? Prove your argument. 2) Give an example of single-variable, two-variable, three-variable, four-variable and six-variable predicates. Which of them induce the given proposition when all variable of a predicate take values from the set of natural numbers?

The concept of equivalent predicates is useful to introduce using the technique of analogues with preliminary actualization of students’ knowledge about equivalent equations or inequalities. For example, we could assign exercises (5 and 6).

Еxercise 5.

Choose the equation, which is the consequence of equation

Еxercise 6.

Choose the equation, which is equivalent to the equation

It’s useful to remind students of definitions of concepts “equation consequence”, “equivalent equations” in the process of solving these problems. It’s also recommended to remind them which elimination of equations leads to the consequence of equations and which ones to equivalent equations. On this basis, while using analogy, one can introduce the meaning of predicate consequence concept and equivalent predicates. While using the method of comparison and analogy it is possible to give the meaning of operations with predicates.

The method of analogy is beneficial to use for introducing operations with statements. For realizing the meaning of logical operations of conjunction, disjunction, implications, equality of statements, students should be supplied with truth tables of logical operations completed in such a way (Table 3- Table 5).

However, not all the new concepts of the considered theme are to be introduced on concrete-inductive basis. Experience shows that definitions of operations with statements and operations with predicates are better to be introduced following a deductive scheme. This scheme is recommended when universal quantifier and existential quantifier are introduced.

Table 3. Truth Table for Negation

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Table 4. Truth Table for Conjunction

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Table 5. Truth Table for Disjunction

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Teachers should pay special attention for reviling the logical structure of mathematical statements and theorems. “Elements of mathematical logic” for pupils who study advanced mathematics in this context is very significant. It should be noted that mathematical facts can be formulated as either categorical or implication statements. Students are proposed to provide examples of theorems formulated with either categorical or implication statements. Then students are recommended to re-formulate theorems formulated with categorical statements using implication statements and vice versa.

To it is necessary to remember while formalizing the formulae of theorem that four types of categorical statements should be chosen (affirmative, partial affirmative, negative, partial negative. It is suggested to use Table 6 for the demonstration of examples of matching symbols and means of formulization. It is advisable to start with examples followed by illustrations using the Euler-Venn diagrams while working with Table 6. The generalized formulation of a certain kind of categorical statements should be considered then and the way they are formalized.

Table 6. Four Types of Categorical Statements

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It should be recalled that the record of theorems in the form of implicative statements has the structure in which is the condition of the theorem, is a conclusion. Explanatory part of the theorem contains a list of sets M, on which the predicates are specified. The explanatory part of the theorem is often omitted for shortening formulations. Students are suggested to create all possible types of implicative statements and matching theorems on the basis of predicates (Table 7).

Table 7. Four types of theorems

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Students can be suggested to prove the equivalence of the direct theorem, and the opposite of the inverse one. They can do it in the following way. Since -is a direct theorem, then . The system of exercises directed on pupils’ mastering of mathematical statement formulation should provide performing direct and opposite action. Taking this into consideration, it is useful to complete the following exercises together with pupils.

Exercise 1. Write down a statement with a help of quantifiers and specify their negations.

• Every single number is an odd number.

• All natural numbers are whole numbers, but not all whole numbers are natural ones.

• A set N (natural numbers) contains the smallest number.

• A set R (real numbers) contains neither the smallest nor the largest number.

• For the preassigned arbitrary triangle there is a matching triangle equal to it in the specified location on a given ray.

• For the arbitrary straight line, there are points lying and not lying on it.

• For any two points on the straight line, there are points lying between them.

Exercise 2. “Move” symbolic record into the native language, determine the validity of each of the statements:

Exercise 3. Write down stereometry axioms and some theorems with the help of quantifiers and symbols.

Axiom 1. There are points lying in the given plane, and points not lying there.

Axiom 2. Through 3 arbitrary points that are not lying on the same line, we can draw only one plane.

Axiom 3. If two points of a line lie in a plane, then every point of this line lies in this plane.

Theorem 1. One and only one plain can be drawn through two intersecting lines.

Theorem 2. Only one plain can be drawn via a line and a point not lying on it.

Theorem 3. The diagonals of a rhombus intersect at right angles.

Theorem 4. A line parallel to one of two given parallel lines is parallel to other given line.

3.3. Considering the Applied Aspect of the “Elements of Mathematical Logic”

As it was mentioned in the mathematics curriculum for advanced classes, different approaches should be applied for the mathematical training of students: they should be prepared for obligatory mastering of specific concepts and facts of mathematical curriculum (the theoretical aspects), as well as for using the skills for modeling of real processes, and solving applied tasks. Students should be oriented on systematic use of ideas and methods of mathematics in natural sciences. The applied aspect of the “Elements of mathematical logic” can be realized through the analysis and synthesis of relay-contact or combined circuits. Students are suggested with a problem situation.

Problem situation. At the entrance of a three-story house, there is only one lamp. It is necessary to create the electrical circuit of the house in the way by using one’s own key: a lamp on each floor could be switched on and switched off at the entrance regardless of the position of switchers on other floors.

It is possible to solve the given problem by using elements of the theory of relay contact circuits. It is necessary to point out the idea of relay contact circuit of connecting and disconnecting contact, of function of conductivity matching the relay contacting circuit among the ideas which deserve to be considered.

The consideration of sequent and parallel connections of conductors makes it possible to establish some specific analogy with calculus of statements, to be more exact: the consequence circuit realizes the operation of conjunction, and the parallel - the disjunction operation (Figure 1).

Figure 1. Sequent and parallel circuit

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The main task of the relay contact circuit theory is the analysis (building on the given circuit with its matching formula and the function of conductivity) and the synthesis (building of the circuit according to the formula or the function of conductivity). For consolidation of basic ideas the following exercises can be suggested.

Exercise 1. Create the function of conductivity for the ladder circuit (relay contact circuit) given on Figure 2.

The solution can be built this way: the contact A is connected successively with a group of contacts (they are limited by a dotted line), which, in its turn, is the parallel connection or . For this reason, the matching formula looks like . It can be simplified by using equal logical formulas .

Figure 2. To the exercise 1

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Now, it is necessary to build the truth table for the last formula to find the matching function of conductivity.

Exercise 2. Simplify relay contact circuit shown on Figure 3.

Figure 3. To the exercise 2

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