Research work in mathematics "solving logical problems". Abstract research work in mathematics: Topic: "Method of mathematical induction" - the work of my students Municipal budgetary educational institution

This section of our website presents research paper topics on logic in the form of logical problems, sophisms and paradoxes in mathematics, interesting games on logic and logical thinking. The work supervisor should directly guide and assist the student in his research.

The topics presented below for research and design work on logic are suitable for children who love to think logically, solve non-standard problems and examples, explore paradoxes and mathematical problems, and play non-standard logic games.

In the list below, you can select a logic project topic for any grade in a secondary school, from elementary school to high school. To help you correctly design a mathematics project on logic and logical thinking, you can use the developed requirements for the design of work.

The following topics for logic research projects are not final and may be modified due to the requirements set before the project.

Topics of research papers on logic:

Sample topics for research papers on logic for students:

Interesting logic in mathematics.
Algebra logic
Logic and us
Logics. Laws of logic
Logic box. A collection of entertaining logic problems.
Logical tasks with numbers.
Logic problems
Logic problems "Funny arithmetic"
Logical problems in mathematics.
Logical problems for determining the number of geometric shapes.
Logical tasks for the development of thinking
Logical problems in mathematics lessons.
Logic games
Logical paradoxes
Mathematical logic.
Methods for solving logical problems and methods for composing them.
Simulation of logic problems
Educational presentation "Fundamentals of Logic".
Basic types of logical problems and methods for solving them.
In the footsteps of Sherlock Holmes, or Methods for solving logical problems.
Application of graph theory in solving logical problems.
Problems of four colors.
Solving logical problems
Solving logical problems using the graph method.
Solving logical problems in different ways.
Solving logic problems using graphs
Solving logical problems using diagrams and tables.
Solving logical problems.
Syllogisms. Logical paradoxes.

Logic project topics

Sample topics for logic projects for students:
Sophistry
Sophistry around us
Sophisms and paradoxes
Methods for composing and methods for solving logical problems.
Learning to solve logical problems
Algebra of logic and logical foundations of a computer.
Types of tasks for logical thinking.
Two ways to solve logical problems.
Logic and mathematics.
Logic as a science
Logic riddles.

Attention students! Coursework is completed independently in strict accordance with the chosen topic. Duplicate topics are not allowed! You are kindly requested to inform the teacher about the chosen topic in any convenient way, either individually or in a list indicating your full name, group number and title of the course work.

Sample topics for coursework in the discipline
"Mathematical Logic"

1. The resolution method and its application in propositional algebra and predicate algebra.

2. Axiomatic systems.

3. Minimal and shortest CNFs and DNFs.

4. Application of methods of mathematical logic in the theory of formal languages.

5. Formal grammars as logical calculi.

6. Methods for solving text logic problems.

7. Logic programming systems.

8. Logic game.

9. Undecidability of first-order logic.

10. Non-standard models of arithmetic.

11. Diagonalization method in mathematical logic.

12. Turing machines and Church's thesis.

13. Computability on the abacus and recursive functions.

14. Representability of recursive functions and negative results of mathematical logic.

15. Solvability of addition arithmetic.

16. Second-order logic and definability in arithmetic.

17. The method of ultraproducts in model theory.

18. Gödel’s theorem on the incompleteness of formal arithmetic.

19. Solvable and undecidable axiomatic theories.

20. Craig's interpolation lemma and its applications.

21. The simplest information converters.

22. Switching circuits.

24. Contact structures.

25. Application of Boolean functions to relay contact circuits.

26. Application of Boolean functions in the theory of pattern recognition.

27. Mathematical logic and artificial intelligence systems.

The course work must consist of 2 parts: the theoretical content of the topic and a set of problems on the topic (at least 10) with solutions. It is also allowed to write a term paper of a research type, replacing the second part (solving problems) with an independent development (for example, a working algorithm, program, sample, etc.) created on the basis of the theoretical material discussed in the first part of the work.

1) Barwise J. (ed.) Reference book on mathematical logic. - M.: Nauka, 1982.

2) Brothers of programming languages. - M.: Nauka, 1975.

3) Boulos J., computability and logic. - M.: Mir, 1994.

4) Hindikin logic in problems. - M., 1972.

5), Palyutin logic. - M.: Nauka, 1979.

6) Ershov solvability and constructive models. - M.: Nauka, 1980.

7), Taitslin theory // Uspekhi Mat. Nauk, 1965, 20, No. 4, p. 37-108.

8) Igoshin - workshop on mathematical logic. - M.: Education, 1986.

9) Igoshin logic and theory of algorithms. - Saratov: Publishing house Sarat. University, 1991.

10) In Ts., using Turbo Prolog. - M.: Mir, 1993.

11) introduction to metamathematics. - M., 1957.

12) athematic logic. - M.: Mir, 1973.

13) ogics in problem solving. - M.: Nauka, 1990.

14) Kolmogorov logic: a textbook for universities math. specialties /, - M.: Publishing house URSS, 2004. - 238 p.

15) story with knots / Transl. from English - M., 1973.

16) ogic game / Trans. from English - M., 1991.

17), Maksimov on set theory, mathematical logic and theory of algorithms. - 4th ed. - M., 2001.

18), Sukacheva logic. Lecture course. Practical problem book and solutions: Study guide. 3rd ed., rev. - St. Petersburg.

19) Publishing house "Lan", 2008. - 288 p.

20) Lyskova in computer science / , . - M.: Laboratory of Basic Knowledge, 2001. - 160 p.

21) Mathematical logic / Under the general editorship and others - Minsk: Higher School, 1991.

22) introduction to mathematical logic. - M.: Nauka, 1984.

23) Moshchensky on mathematical logic. - Minsk, 1973.

24) Nikolskaya with mathematical logic. - M.: Moscow Psychological and Social Institute: Flint, 1998. - 128 p.

25) Nikolskaya logic. - M., 1981.

26) Novikov mathematical logic. - M.: Nauka, 1973.

27) Rabin theory. In the book: Reference book on mathematical logic, part 3. Recursion theory. - M.: Nauka, 1982. - p. 77-111.

28) Tey A., Gribomon P. et al. Logical approach to artificial intelligence. T. 1. - M.: Mir, 1990.

29) Tey A., Gribomon P. et al. Logical approach to artificial intelligence. T. 2. - M.: Mir, 1998.

30) Chen Ch., Li R. Mathematical logic and automatic proof of theorems. - M.: Nauka, 1983.

31) introduction to mathematical logic. - M.: Mir, 1960.

32) Shabunin logic. Propositional logic and predicate logic: textbook /, rep. ed. ; Chuvash state University named after . - Cheboksary: ​​Chuvash Publishing House. University, 2003. - 56 p.

Municipal educational budgetary institution -

Secondary school No. 51

Orenburg.

Project on:

mathematic teacher

Egorcheva Victoria Andreevna

2017

Hypothesis : If graph theory is brought closer to practice, then the most beneficial results can be obtained.

Target: Get acquainted with the concept of graphs and learn how to apply them in solving various problems.

Tasks:

1) Expand knowledge about methods of constructing graphs.

2) Identify types of problems whose solution requires the use of graph theory.

3) Explore the use of graphs in mathematics.

“Euler calculated, without any visible effort, how a person breathes or how an eagle soars above the earth.”

Dominic Arago.

I. Introduction. p.

II . Main part.

1. The concept of a graph. Problem about the Königsberg bridges. p.

2. Properties of graphs. p.

3. Problems using graph theory. p.

Sh. Conclusion.

The meaning of graphs. p.

IV. Bibliography. p.

I . INTRODUCTION

Graph theory is a relatively young science. “Graphs” has the root of the Greek word “grapho,” which means “I write.” The same root is in the words “graph”, “biography”.

In my work, I look at how graph theory is used in various areas of people's lives. Every math teacher and almost every student knows how difficult it is to solve geometric problems, as well as algebra word problems. Having explored the possibility of using graph theory in a school mathematics course, I came to the conclusion that this theory greatly simplifies understanding and solving problems.

II . MAIN PART.

1. The concept of a graph.

The first work on graph theory belongs to Leonhard Euler. It appeared in 1736 in publications of the St. Petersburg Academy of Sciences and began with a consideration of the problem of the Königsberg bridges.

You probably know that there is such a city as Kaliningrad; it used to be called Koenigsberg. The Pregolya River flows through the city. It is divided into two branches and goes around the island. In the 17th century there were seven bridges in the city, arranged as shown in the picture.

They say that one day a city resident asked his friend if he could walk across all the bridges so as to visit each of them only once and return to the place where the walk began. Many townspeople became interested in this problem, but no one could come up with a solution. This issue has attracted the attention of scientists from many countries. The famous mathematician Leonhard Euler managed to solve the problem. Leonhard Euler, a native of Basel, was born on April 15, 1707. Euler's scientific achievements are enormous. He influenced the development of almost all branches of mathematics and mechanics, both in the field of fundamental research and in their applications. Leonhard Euler not only solved this specific problem, but also came up with a general method for solving these problems. Euler did the following: he “compressed” the land into points, and “stretched” the bridges into lines. The result is the figure shown in the figure.

Such a figure, consisting of points and lines connecting these points, is calledcount. Points A, B, C, D are called the vertices of the graph, and the lines that connect the vertices are called the edges of the graph. In a drawing of vertices B, C, D 3 ribs come out, and from the top A - 5 ribs. Vertices from which an odd number of edges emerge are calledodd vertices, and vertices from which an even number of edges emerge areeven.

2. Properties of the graph.

While solving the problem about the Königsberg bridges, Euler established, in particular, the properties of the graph:

1. If all the vertices of the graph are even, then you can draw a graph with one stroke (that is, without lifting the pencil from the paper and without drawing twice along the same line). In this case, the movement can start from any vertex and end at the same vertex.

2. A graph with two odd vertices can also be drawn with one stroke. The movement must begin from any odd vertex and end at another odd vertex.

3. A graph with more than two odd vertices cannot be drawn with one stroke.

4.The number of odd vertices in a graph is always even.

5. If a graph has odd vertices, then the smallest number of strokes that can be used to draw the graph will be equal to half the number of odd vertices of this graph.

For example, if a figure has four odd numbers, then it can be drawn with at least two strokes.

In the problem of the seven bridges of Königsberg, all four vertices of the corresponding graph are odd, i.e. You cannot cross all the bridges once and end the journey where it started.

3. Solving problems using graphs.

1. Tasks on drawing figures with one stroke.

Attempting to draw each of the following shapes with one stroke of the pen will result in different results.

If there are no odd points in the figure, then it can always be drawn with one stroke of the pen, no matter where you start drawing. These are figures 1 and 5.

If a figure has only one pair of odd points, then such a figure can be drawn with one stroke, starting drawing at one of the odd points (it doesn’t matter which). It is easy to understand that the drawing should end at the second odd point. These are figures 2, 3, 6. In figure 6, for example, drawing must begin either from point A or from point B.

If a figure has more than one pair of odd points, then it cannot be drawn with one stroke at all. These are figures 4 and 7, containing two pairs of odd points. What has been said is enough to accurately recognize which figures cannot be drawn with one stroke and which ones can be drawn, as well as from what point the drawing should begin.

I propose to draw the following figures in one stroke.

2. Solving logical problems.

TASK No. 1.

There are 6 participants in the table tennis class championship: Andrey, Boris, Victor, Galina, Dmitry and Elena. The championship is held in a round robin system - each participant plays each of the others once. To date, some games have already been played: Andrey played with Boris, Galina, Elena; Boris - with Andrey, Galina; Victor - with Galina, Dmitry, Elena; Galina - with Andrey, Victor and Boris. How many games have been played so far and how many are left?

SOLUTION:

Let's build a graph as shown in the figure.

7 games played.

In this figure, the graph has 8 edges, so there are 8 games left to play.

TASK #2

In the courtyard, which is surrounded by a high fence, there are three houses: red, yellow and blue. The fence has three gates: red, yellow and blue. From the red house, draw a path to the red gate, from the yellow house to the yellow gate, from the blue house to the blue one so that these paths do not intersect.

SOLUTION:

The solution to the problem is shown in the figure.

3. Solving word problems.

To solve problems using the graph method, you need to know the following algorithm:

1.What process are we talking about in the problem?2.What quantities characterize this process?3.What is the relationship between these quantities?4.How many different processes are described in the problem?5.Is there a connection between the elements?

Answering these questions, we analyze the condition of the problem and write it down schematically.

For example . The bus traveled for 2 hours at a speed of 45 km/h and for 3 hours at a speed of 60 km/h. How far did the bus travel during these 5 hours?

S
¹=90 km V ¹=45 km/h t ¹=2h

S=VT

S ²=180 km V ²=60 km/h t ²=3 h

S ¹ + S ² = 90 + 180

Solution:

1)45x 2 = 90 (km) - the bus traveled in 2 hours.

2)60x 3 = 180 (km) - the bus traveled in 3 hours.

3)90 + 180 = 270 (km) - the bus traveled in 5 hours.

Answer: 270 km.

III . CONCLUSION.

As a result of working on the project, I learned that Leonhard Euler was the founder of graph theory and solved problems using graph theory. I concluded for myself that graph theory is used in various areas of modern mathematics and its numerous applications. There is no doubt about the usefulness of introducing us students to the basic concepts of graph theory. Solving many mathematical problems becomes easier if you can use graphs. Data presentation V the form of a graph gives them clarity. Many proofs are also simplified and become more convincing if you use graphs. This especially applies to such areas of mathematics as mathematical logic and combinatorics.

Thus, the study of this topic has great general educational, general cultural and general mathematical significance. In everyday life, graphic illustrations, geometric representations and other visual techniques and methods are increasingly used. For this purpose, it is useful to introduce the study of elements of graph theory in primary and secondary schools, at least in extracurricular activities, since this topic is not included in the mathematics curriculum.

V . BIBLIOGRAPHY:

2008

Review.

The project on the theme “Graphs around us” was completed by Nikita Zaytsev, a student of class 7 “A” at Municipal Educational Institution No. 3, Krasny Kut.

A distinctive feature of Nikita Zaitsev’s work is its relevance, practical orientation, depth of coverage of the topic, and the possibility of using it in the future.

The work is creative, in the form of an information project. The student chose this topic to show the relationship of graph theory with practice using the example of a school bus route, to show that graph theory is used in various areas of modern mathematics and its numerous applications, especially in economics, mathematical logic, and combinatorics. He showed that solving problems is greatly simplified if it is possible to use graphs; presenting data in the form of a graph gives them clarity; many proofs are also simplified and become convincing.

The work addresses issues such as:

1. The concept of a graph. Problem about the Königsberg bridges.

2. Properties of graphs.

3. Problems using graph theory.

4. The meaning of graphs.

5. School bus route option.

When performing his work, N. Zaitsev used:

1. Alkhova Z.N., Makeeva A.V. "Extracurricular work in mathematics."

2. Magazine “Mathematics at school”. Appendix “First of September” No. 13

2008

3. Ya.I.Perelman “Entertaining tasks and experiments.” - Moscow: Education, 2000.

The work was done competently, the material meets the requirements of this topic, the corresponding drawings are attached.

Introduction. 3

1. Mathematical logic (meaningless logic) and “common sense” logic 4

2. Mathematical judgments and inferences. 6

3. Mathematical logic and “Common sense” in the 21st century. eleven

4. Unnatural logic in the foundations of mathematics. 12

Conclusion. 17

References… 18

The expansion of the area of ​​logical interests is associated with general trends in the development of scientific knowledge. Thus, the emergence of mathematical logic in the middle of the 19th century was the result of centuries-old aspirations of mathematicians and logicians to build a universal symbolic language, free from the “shortcomings” of natural language (primarily its polysemy, i.e. polysemy).

The further development of logic is associated with the combined use of classical and mathematical logic in applied fields. Non-classical logics (deontic, relevant, legal logic, decision-making logic, etc.) often deal with the uncertainty and fuzziness of the objects under study, with the nonlinear nature of their development. Thus, when analyzing rather complex problems in artificial intelligence systems, the problem of synergy between different types of reasoning when solving the same problem arises. Prospects for the development of logic in line with convergence with computer science are associated with the creation of a certain hierarchy of possible reasoning models, including reasoning in natural language, plausible reasoning and formalized deductive conclusions. This can be solved using classical, mathematical and non-classical logic. Thus, we are not talking about different “logics”, but about different degrees of formalization of thinking and the “dimension” of logical meanings (two-valued, multi-valued, etc. logic).

Identification of the main directions of modern logic:

1. general or classical logic;

2. symbolic or mathematical logic;

3. non-classical logic.

Mathematical logic is a rather vague concept, due to the fact that there are also infinitely many mathematical logics. Here we will discuss some of them, paying more tribute to tradition than to common sense. Because, quite possibly, this is common sense... Logical?

Mathematical logic teaches you to reason logically no more than any other branch of mathematics. This is due to the fact that the “logicality” of reasoning in logic is determined by logic itself and can be used correctly only in logic itself. In life, when thinking logically, as a rule, we use different logics and different methods of logical reasoning, shamelessly mixing deduction with induction... Moreover, in life we ​​build our reasoning based on contradictory premises, for example, “Don’t put off until tomorrow what can be done today" and "You'll make people laugh in a hurry." It often happens that a logical conclusion we don’t like leads to a revision of the initial premises (axioms).

Perhaps the time has come to say about logic, perhaps the most important thing: classical logic is not concerned with meaning. Neither healthy nor any other! To study common sense, by the way, there is psychiatry. But in psychiatry, logic is rather harmful.

Of course, when we differentiate logic from sense, we mean first of all classical logic and the everyday understanding of common sense. There are no forbidden directions in mathematics, therefore the study of meaning by logic, and vice versa, in various forms is present in a number of modern branches of logical science.

(The last sentence worked out well, although I won’t attempt to define the term “logical science” even approximately). Meaning, or semantics if you will, is dealt with, for example, by model theory. And in general, the term semantics is often replaced by the term interpretation. And if we agree with philosophers that the interpretation (display!) of an object is its comprehension in some given aspect, then the borderline spheres of mathematics, which can be used to attack the meaning in logic, become incomprehensible!

In practical terms, theoretical programming is forced to take an interest in semantics. And in it, in addition to just semantics, there is also operational, and denotational, and procedural, etc. and so on. semantics...

Let us just mention the apotheosis - THE THEORY OF CATEGORIES, which brought semantics to a formal, obscure syntax, where the meaning is already so simple - laid out on shelves that it is completely impossible for a mere mortal to get to the bottom of it... This is for the elite.

So what does logic do? At least in its most classic part? Logic does only what it does. (And she defines this extremely strictly). The main thing in logic is to strictly define it! Set the axiomatics. And then the logical conclusions should be (!) largely automatic...

Reasoning about these conclusions is another matter! But these arguments are already beyond the bounds of logic! Therefore, they require a strict mathematical sense!

It may seem that this is a simple verbal balancing act. NO! As an example of a certain logical (axiomatic) system, let's take the well-known game 15. Let's set (mix) the initial arrangement of square chips. Then the game (logical conclusion!), and specifically the movement of chips to an empty space, can be handled by some mechanical device, and you can patiently watch and rejoice when, as a result of possible movements, a sequence from 1 to 15 is formed in the box. But no one forbids control mechanical device and prompt it, BASED ON COMMON SENSE, with the correct movements of the chips in order to speed up the process. Or maybe even prove, using for logical reasoning, for example, such a branch of mathematics as COMBINATORICS, that with a given initial arrangement of chips it is impossible to obtain the required final combination at all!

There is no more common sense in that part of logic that is called LOGICAL ALGEBRA. Here LOGICAL OPERATIONS are introduced and their properties are defined. As practice has shown, in some cases the laws of this algebra may correspond to the logic of life, but in others they do not. Because of such inconstancy, the laws of logic cannot be considered laws from the point of view of the practice of life. Their knowledge and mechanical use can not only help, but also harm. Especially psychologists and lawyers. The situation is complicated by the fact that, along with the laws of algebra of logic, which sometimes correspond or do not correspond to life reasoning, there are logical laws that some logicians categorically do not recognize. This applies primarily to the so-called laws of the EXCLUSIVE THIRD and CONTRADICTION.

2. Mathematical judgments and inferences

In thinking, concepts do not appear separately; they are connected with each other in a certain way. The form of connection of concepts with each other is a judgment. In each judgment, some connection or some relationship between concepts is established, and this thereby affirms the existence of a connection or relationship between the objects covered by the corresponding concepts. If judgments correctly reflect these objectively existing dependencies between things, then we call such judgments true, otherwise the judgments will be false. So, for example, the proposition “every rhombus is a parallelogram” is a true proposition; the proposition “every parallelogram is a rhombus” is a false proposition.

Thus, a judgment is a form of thinking that reflects the presence or absence of the object itself (the presence or absence of any of its features and connections).

To think means to make judgments. With the help of judgments, thought and concept receive their further development.

Since every concept reflects a certain class of objects, phenomena or relationships between them, any judgment can be considered as the inclusion or non-inclusion (partial or complete) of one concept in the class of another concept. For example, the proposition “every square is a rhombus” indicates that the concept “square” is included in the concept “rhombus”; the proposition “intersecting lines are not parallel” indicates that intersecting lines do not belong to the set of lines called parallel.

A judgment has its own linguistic shell - a sentence, but not every sentence is a judgment.

A characteristic feature of a judgment is the obligatory presence of truth or falsity in the sentence expressing it.

For example, the sentence “triangle ABC is isosceles” expresses some judgment; the sentence “Will ABC be isosceles?” does not express judgment.

Each science essentially represents a certain system of judgments about the objects that are the subject of its study. Each of the judgments is formalized in the form of a certain proposal, expressed in terms and symbols inherent in this science. Mathematics also represents a certain system of judgments expressed in mathematical sentences through mathematical or logical terms or their corresponding symbols. Mathematical terms (or symbols) denote those concepts that make up the content of a mathematical theory, logical terms (or symbols) denote logical operations with the help of which other mathematical propositions are constructed from some mathematical propositions, from some judgments other judgments are formed, the entirety of which constitutes mathematics as a science.

Generally speaking, judgments are formed in thinking in two main ways: directly and indirectly. In the first case, the result of perception is expressed with the help of a judgment, for example, “this figure is a circle.” In the second case, judgment arises as a result of a special mental activity called inference. For example, “the set of given points on a plane is such that their distance from one point is the same; This means that this figure is a circle.”

In the process of this mental activity, a transition is usually made from one or more interconnected judgments to a new judgment, which contains new knowledge about the object of study. This transition is inference, which represents the highest form of thinking.

So, inference is the process of obtaining a new conclusion from one or more given judgments. For example, the diagonal of a parallelogram divides it into two congruent triangles (first proposition).

The sum of the interior angles of a triangle is 2d (second proposition).

The sum of the interior angles of a parallelogram is equal to 4d (new conclusion).

The cognitive value of mathematical inferences is extremely great. They expand the boundaries of our knowledge about objects and phenomena of the real world due to the fact that most mathematical propositions are a conclusion from a relatively small number of basic judgments, which are obtained, as a rule, through direct experience and which reflect our simplest and most general knowledge about its objects.

Inference differs (as a form of thinking) from concepts and judgments in that it is a logical operation on individual thoughts.

Not every combination of judgments among themselves constitutes a conclusion: there must be a certain logical connection between the judgments, reflecting the objective connection that exists in reality.

For example, one cannot draw a conclusion from the propositions “the sum of the interior angles of a triangle is 2d” and “2*2=4”.

It is clear what importance the ability to correctly construct various mathematical sentences or draw conclusions in the process of reasoning has in the system of our mathematical knowledge. Spoken language is poorly suited for expressing certain judgments, much less for identifying the logical structure of reasoning. Therefore, it is natural that there was a need to improve the language used in the reasoning process. Mathematical (or rather, symbolic) language turned out to be the most suitable for this. The special field of science that emerged in the 19th century, mathematical logic, not only completely solved the problem of creating a theory of mathematical proof, but also had a great influence on the development of mathematics as a whole.

Formal logic (which arose in ancient times in the works of Aristotle) ​​is not identified with mathematical logic (which arose in the 19th century in the works of the English mathematician J. Boole). The subject of formal logic is the study of the laws of the relationship of judgments and concepts in inferences and rules of evidence. Mathematical logic differs from formal logic in that, based on the basic laws of formal logic, it explores the patterns of logical processes based on the use of mathematical methods: “The logical connections that exist between judgments, concepts, etc., are expressed in formulas, the interpretation of which is free from ambiguities that could easily arise from verbal expression. Thus, mathematical logic is characterized by formalization of logical operations, more complete abstraction from the specific content of sentences (expressing any judgment).

Let us illustrate this with one example. Consider the following inference: “If all plants are red and all dogs are plants, then all dogs are red.”

Each of the judgments used here and the judgment that we received as a result of restrained inference seems to be patent nonsense. However, from the point of view of mathematical logic, we are dealing here with a true sentence, since in mathematical logic the truth or falsity of a conclusion depends only on the truth or falsity of its constituent premises, and not on their specific content. Therefore, if one of the basic concepts of formal logic is a judgment, then the analogous concept of mathematical logic is the concept of a statement-statement, for which it only makes sense to say whether it is true or false. One should not think that every statement is characterized by a lack of “common sense” in its content. It’s just that the meaningful part of the sentence that makes up this or that statement fades into the background in mathematical logic and is unimportant for the logical construction or analysis of this or that conclusion. (Although, of course, it is essential for understanding the content of what is being discussed when considering this issue.)

It is clear that in mathematics itself meaningful statements are considered. By establishing various connections and relationships between concepts, mathematical judgments affirm or deny any relationships between objects and phenomena of reality.

3. Mathematical logic and “Common sense” in the 21st century.

Logic is not only a purely mathematical, but also a philosophical science. In the 20th century, these two interconnected hypostases of logic turned out to be separated in different directions. On the one hand, logic is understood as the science of the laws of correct thinking, and on the other hand, it is presented as a set of loosely connected artificial languages, which are called formal logical systems.

For many, it is obvious that thinking is a complex process with the help of which everyday, scientific or philosophical problems are solved and brilliant ideas or fatal delusions are born. Language is understood by many simply as a means by which the results of thinking can be transmitted to contemporaries or left to descendants. But, having connected in our consciousness thinking with the concept of “process”, and language with the concept of “means”, we essentially stop noticing the immutable fact that in this case the “means” is not completely subordinated to the “process”, but depending on our purposeful or unconscious choice of certain or verbal cliches has a strong influence on the course and result of the “process” itself. Moreover, there are many cases where such “reverse influence” turns out to be not only an obstacle to correct thinking, but sometimes even its destroyer.

From a philosophical point of view, the task posed within the framework of logical positivism was never completed. In particular, in his later studies, one of the founders of this trend, Ludwig Wittgenstein, came to the conclusion that natural language cannot be reformed in accordance with the program developed by the positivists. Even the language of mathematics as a whole resisted the powerful pressure of “logicalism,” although many terms and structures of the language proposed by the positivists entered some sections of discrete mathematics and significantly supplemented them. The popularity of logical positivism as a philosophical trend in the second half of the 20th century dropped noticeably - many philosophers came to the conclusion that the rejection of many “illogicalities” of natural language, an attempt to squeeze it into the framework of the fundamental principles of logical positivism entails the dehumanization of the process of cognition, and at the same time and the dehumanization of human culture as a whole.

Many reasoning methods used in natural language are often very difficult to map unambiguously into the language of mathematical logic. In some cases, such a mapping leads to a significant distortion of the essence of natural reasoning. And there is reason to believe that these problems are a consequence of the initial methodological position of analytical philosophy and positivism about the illogicality of natural language and the need for its radical reform. The very original methodological setting of positivism also does not stand up to criticism. To accuse spoken language of being illogical is simply absurd. In fact, illogicality does not characterize the language itself, but many users of this language who simply do not know or do not want to use logic and compensate for this flaw with psychological or rhetorical techniques of influencing the public, or in their reasoning they use as logic a system that is called logic only by misunderstanding. At the same time, there are many people whose speech is distinguished by clarity and logic, and these qualities are not determined by knowledge or ignorance of the foundations of mathematical logic.

In the reasoning of those who can be classified as legislators or followers of the formal language of mathematical logic, a kind of “blindness” in relation to elementary logical errors is often revealed. One of the great mathematicians, Henri Poincaré, drew attention to this blindness in the fundamental works of G. Cantor, D. Hilbert, B. Russell, J. Peano and others at the beginning of our century.

One example of such an illogical approach to reasoning is the formulation of the famous Russell paradox, in which two purely heterogeneous concepts “element” and “set” are unreasonably confused. In many modern works on logic and mathematics, in which the influence of Hilbert's program is noticeable, many statements that are clearly absurd from the point of view of natural logic are not explained. The relationship between “element” and “set” is the simplest example of this kind. Many works in this direction claim that a certain set (let's call it A) can be an element of another set (let's call it B).

For example, in a well-known manual on mathematical logic we will find the following phrase: “Sets themselves can be elements of sets, so, for example, the set of all sets of integers has sets as its elements.” Note that this statement is not just a disclaimer. It is contained as a “hidden” axiom in formal set theory, which many experts consider the foundation of modern mathematics, as well as in the formal system that the mathematician K. Gödel built when proving his famous theorem on the incompleteness of formal systems. This theorem refers to a rather narrow class of formal systems (they include formal set theory and formal arithmetic), the logical structure of which clearly does not correspond to the logical structure of natural reasoning and justification.

However, for more than half a century it has been the subject of heated discussion among logicians and philosophers in the context of the general theory of knowledge. With such a broad generalization of this theorem, it turns out that many elementary concepts are fundamentally unknowable. But with a more sober approach, it turns out that Gödel’s theorem only showed the inconsistency of the program of formal justification of mathematics proposed by D. Hilbert and taken up by many mathematicians, logicians and philosophers. The broader methodological aspect of Gödel's theorem can hardly be considered acceptable until the following question is answered: is Hilbert's program for justifying mathematics the only possible one? To understand the ambiguity of the statement “set A is an element of set B,” it is enough to ask a simple question: “What elements are set B formed from in this case?” From the point of view of natural logic, only two mutually exclusive explanations are possible. Explanation one. The elements of the set B are the names of some sets and, in particular, the name or designation of the set A. For example, the set of all even numbers is contained as an element in the set of all names (or designations) of sets separated by some characteristics from the set of all integers. To give a clearer example: the set of all giraffes is contained as an element in the set of all known animal species. In a broader context, the set B can also be formed from conceptual definitions of sets or references to sets. Explanation two. The elements of the set B are the elements of some other sets and, in particular, all the elements of the set A. For example, every even number is an element of the set of all integers, or every giraffe is an element of the set of all animals. But then it turns out that in both cases the expression “set A is an element of set B” does not make sense. In the first case, it turns out that the element of the set B is not the set A itself, but its name (or designation, or reference to it). In this case, an equivalence relation is implicitly established between the set and its designation, which is unacceptable neither from the point of view of ordinary common sense, nor from the point of view of mathematical intuition, which is incompatible with excessive formalism. In the second case, it turns out that set A is included in set B, i.e. is a subset of it, but not an element. Here, too, there is an obvious substitution of concepts, since the relation of inclusion of sets and the relation of membership (being an element of a set) in mathematics have fundamentally different meanings. Russell's famous paradox, which undermined logicians' confidence in the concept of a set, is based on this absurdity - the paradox is based on the ambiguous premise that a set can be an element of another set.

Another possible explanation is possible. Let a set A be defined by a simple enumeration of its elements, for example, A = (a, b). The set B, in turn, is specified by enumerating some sets, for example, B = ((a, b), (a, c)). In this case, it seems obvious that the element of B is not the name of the set A, but the set A itself. But even in this case, the elements of the set A are not elements of the set B, and the set A is here considered as an inseparable collection, which can well be replaced by its name . But if we considered all elements of the sets contained in it to be elements of B, then in this case the set B would be equal to the set (a, b, c), and the set A in this case would not be an element of B, but a subset of it. Thus, it turns out that this version of the explanation, depending on our choice, comes down to the previously listed options. And if no choice is offered, then elementary ambiguity results, which often leads to “inexplicable” paradoxes.

It would be possible not to pay special attention to these terminological nuances if not for one circumstance. It turns out that many of the paradoxes and inconsistencies of modern logic and discrete mathematics are a direct consequence or imitation of this ambiguity.

For example, in modern mathematical reasoning, the concept of “self-applicability” is often used, which underlies Russell’s paradox. In the formulation of this paradox, self-applicability implies the existence of sets that are elements of themselves. This statement immediately leads to a paradox. If we consider the set of all “non-self-applicable” sets, it turns out that it is both “self-applicable” and “non-self-applicable.”

Mathematical logic contributed a lot to the rapid development of information technology in the 20th century, but the concept of “judgment”, which appeared in logic back in the days of Aristotle and on which, as the foundation, rests the logical basis of natural language, fell out of its field of vision. Such an omission did not at all contribute to the development of a logical culture in society and even gave rise to the illusion among many that computers are capable of thinking no worse than humans themselves. Many are not even embarrassed by the fact that against the backdrop of general computerization on the eve of the third millennium, logical absurdities within science itself (not to mention politics, lawmaking and pseudoscience) are even more common than at the end of the 19th century. And in order to understand the essence of these absurdities, there is no need to turn to complex mathematical structures with multi-place relations and recursive functions that are used in mathematical logic. It turns out that to understand and analyze these absurdities, it is quite enough to apply a much simpler mathematical structure of judgment, which not only does not contradict the mathematical foundations of modern logic, but in some way complements and expands them.

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