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. 11
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, logic of law, 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 you can do 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. etc. 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 (shuffle) the initial arrangement of square pieces. 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 interconnected 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 posterity. 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 “logicism,” 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 distinguished 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.
References
1. Vasiliev N. A. Imaginary logic. Selected works. - M.: Science. 1989; - pp. 94-123.
2. Kulik B.A. Basic principles of common sense philosophy (cognitive aspect) // Artificial Intelligence News, 1996, No. 3, p. 7-92.
3. Kulik B.A. Logical foundations of common sense / Edited by D.A. Pospelov. - St. Petersburg, Polytechnic, 1997. 131 p.
4. Kulik B.A. The logic of common sense. - Common Sense, 1997, No. 1(5), p. 44 - 48.
5. Styazhkin N.I. Formation of mathematical logic. M.: Nauka, 1967.
6. Soloviev A. Discrete mathematics without formulas. 2001//http://soloviev.nevod.ru/2001/dm/index.html
XI REGIONAL SCIENTIFIC AND PRACTICAL CONFERENCE “KOLMOGOROV READINGS”
Section "Mathematics"
Subject
"Solving Logical Problems"
Municipal budgetary general education
school No. 2 st. Arkhonskaya,
7th grade.
Scientific supervisor
mathematics teacher MBOU secondary school No. 2 st. Arkhonskaya
Trimasova N.I.
"Solving Logical Problems"
7th grade
secondary educational institution
school No. 2, st. Arkhonskaya.
Annotation
This work discusses different ways to solve logical problems and a variety of techniques. Each of them has its own area of application. In addition, in the work you can get acquainted with the basic concepts of the direction of “mathematics without formulas” - mathematical logic, and learn about the creators of this science. You can also see the results of the diagnostic “solving logical problems among middle-level students.”
Content
1.Introduction_______________________________________________________________ 4
2. The founders of the science of “logic”___________________________ 6
3.How to learn to solve logical problems?______________________ _8
4. Types and methods of solving logical problems______________________ 9
4.1 Problems of the “Who’s Who?” type 9
a) Graph method___________________________________________ 9
b) Tabular method__________________________________________ 11
4.2 Tactical tasks_______________________________________________ 13
a) method of reasoning_______________________________________________ 13
4.3 Problems of finding the intersection or union of sets_________________________________________________ 14
a) Euler circles_____________________________________________ 14
Letter puzzles and star problems__________________ 16
4.5 Truth problems_____________________________________________ 17
4.6 “Hat” type problems_____________________________________________ 18
5. Practical part__________________________________________________________ 19
5.1 Study of the level of logical thinking of middle-level students__________________________________________________________ 19
6. Conclusion_________________________________________________________ 23
7. Literature_________________________________________________________ 24
"Solving Logical Problems"
Krutogolova Diana Alexandrovna
7th grade
Municipal budgetary general education
secondary educational institution
school No. 2, st. Arkhonskaya.
1. Introduction
The development of creative activity, initiative, curiosity, and ingenuity is facilitated by solving non-standard problems.Despite the fact that the school mathematics course contains a large number of interesting problems, many useful problems are not covered. These tasks include logical tasks.
Solving logic problems is very exciting. There seems to be no mathematics in them - no numbers, no functions, no triangles, no vectors, but there are only liars and wise men, truth and lies. At the same time, the spirit of mathematics is felt most clearly in them - half of the solution to any mathematical problem (and sometimes much more than half) is to properly understand the condition, to unravel all the connections between the participating objects.
A mathematical problem invariably helps to develop correct mathematical concepts, to better understand various aspects of the relationships in the surrounding life, and makes it possible to apply the theoretical principles being studied. At the same time, problem solving contributes to the development of logical thinking.
While preparing this work, I settarget - develop your ability to reason and draw correct conclusions. Only solving a difficult, non-standard problem brings the joy of victory. When solving logical problems, you have the opportunity to think about an unusual condition and reason. This arouses and maintains my interest in mathematics. A logical decision is the best way to unleash your creativity.
Relevance. Nowadays, very often a person’s success depends on his ability to think clearly, reason logically and clearly express his thoughts.
Tasks: 1) familiarization with the concepts of “logic” and “mathematical logic”; 2) study of basic methods for solving logical problems; 3) conducting diagnostics to identify the level of logical thinking of students in grades 5-8.
Research methods: collection, study, generalization of experimental and theoretical material
2. The founders of the science of “logic”
Logic is one of the most ancient sciences. It is currently not possible to establish exactly who, when and where first turned to those aspects of thinking that constitute the subject of logic. Some of the origins of logical teaching can be found in India, at the end of the 2nd millennium BC. e. However, if we talk about the emergence of logic as a science, that is, about a more or less systematized body of knowledge, then it would be fair to consider the great civilization of Ancient Greece as the birthplace of logic. It was here in the V-IV centuries BC. e. During the period of rapid development of democracy and the associated unprecedented revival of socio-political life, the foundations of this science were laid by the works of Democritus, Socrates and Plato.
The founder of logic as a science is the ancient Greek philosopher and scientist Aristotle (384-322 BC). He first developed the theory of deduction, that is, the theory of logical inference. It was he who drew attention to the fact that in reasoning we deduce others from some statements, based not on the specific content of the statements, but on a certain relationship between their forms and structures.
Even then, schools were created in Ancient Greece in which people learned to debate. The students of these schools learned the art of searching for the truth and convincing other people that they were right. They learned to select the necessary ones from a variety of facts, build chains of reasoning that connect individual facts with each other, and draw the right conclusions.
Already from these times, it was generally accepted that logic is a science about thinking, and not about objects of objective truth.
The ancient Greek mathematician Euclid (330-275 BC) was the first to attempt to organize the extensive information on geometry that had accumulated by that time. He laid the foundation for the understanding of geometry as an axiomatic theory, and of all mathematics as a set of axiomatic theories.
Over the course of many centuries, various philosophers and entire philosophical schools supplemented, improved and changed Aristotle's logic. This was the first, pre-mathematical, stage in the development of formal logic. The second stage is associated with the use of mathematical methods in logic, which was started by the German philosopher and mathematician G. W. Leibniz (1646-1716). He tried to build a universal language with the help of which disputes between people would be resolved, and then completely “replace all ideas with calculations.”
An important period in the formation of mathematical logic begins with the work of the English mathematician and logician George Boole (1815-1864) “Mathematical Analysis of Logic” (1847) and “Investigations into the Laws of Thought” (1854). He applied to logic the methods of contemporary algebra - the language of symbols and formulas, the composition and solution of equations. He created a kind of algebra - the algebra of logic. During this period, it took shape as propositional algebra and was significantly developed in the works of the Scottish logician A. de Morgan (1806-1871), the English one - W. Jevons (1835-1882), the American one - C. Pierce and others. The creation of the algebra of logic was the final link in the development of formal logic.
A significant impetus to a new period in the development of mathematical logic was given by the creation in the first half of the 19th century by the great Russian mathematician N. I. Lobachevsky (1792-1856) and independently by the Hungarian mathematician J. Bolyai (1802-1860) of non-Euclidean geometry. In addition, the creation of the analysis of infinitesimals led to the need to substantiate the concept of number as a fundamental concept of all mathematics. The paradoxes discovered at the end of the 19th century in set theory completed the picture: they clearly showed that the difficulties of substantiating mathematics were difficulties of a logical and methodological nature. Thus, mathematical logic was faced with problems that did not arise before Aristotle’s logic. In the development of mathematical logic, three directions in the substantiation of mathematics were formed, in which the creators tried in different ways to overcome the difficulties that arose.
3. How to learn to solve logical problems?
Many people only think what they think.
They find the thought process unpleasant:
this requires skill and a certain amount of effort,
Why bother when you can do it without it.
Ogden Nash
Logical ornon-numeric problems constitute a broad class of non-standard problems. This includes, first of all, word problems in which it is necessary to recognize objects or arrange them in a certain order according to existing properties. In this case, part of the statements of the problem conditions can appear with a different truth value (be true or false).
Text logic problems can be divided into the following types:
all statements are true;
not all statements are true;
problems about truth-tellers and liars.
It is advisable to practice solving each type of problem gradually, step by step.
So, we will learn how logic problems can be solved in different ways. It turns out there are several such techniques, they are varied and each of them has its own area of application. After getting acquainted in detail, we will figure out in what cases it is more convenient to use one or another method.
4. Types and methods of solving logical problems
4.1 Problems of the “Who is who?” type
Problems like “Who is who?” very diverse in complexity, content and ability to solve. They are certainly of interest.
a) Graph method
One way is to solve using graphs. A graph is several points, some of which are connected to each other by segments or arrows (in this case, the graph is called oriented). Let us need to establish a correspondence between two types of objects (sets). Dots denote elements of sets, and the correspondence between them - segments. The dashed line will merge two elements that do not correspond to each other.
Problem 1 . Three friends Belova, Krasnova and Chernova met. One of them was wearing a black dress, the other was wearing a red dress, and the third was wearing a white dress. A girl in a white dress says to Chernova: “We need to exchange dresses, otherwise the color of our dresses does not match our surnames.” Who was wearing what dress?
Solution. Solving the problem is simple if you consider that:
Each element of one set necessarily corresponds to an element of another set, but only one
If an element of each set is connected to all elements (except one) of another set by dashed segments, then it is connected to the latter by a solid segment.
Instead of solid line segments, you can use colored ones, in which case the solution is more colorful,
Let's denote the girls' surnames in the picture with the letters B, Ch, K, and connect the letter B and the white dress with a dotted line, which will mean: “Belova is not in a white dress.” Next we get three more dotted lines corresponding to the minuses in the table. A white dress can only be worn by Krasnova - we will connect the letter K and the white dress with a solid line, which will mean “Krasnova in a white dress,” etc.
In the same way, you can find correspondence between three sets.
Task 2. Three friends met in a cafe: the sculptor Belov, the violinist Chernov and the artist Ryzhov. “It’s wonderful that one of us has white hair, another has black, and the third has red hair, but none of our hair color matches our surname,” the black-haired man noted. “You’re right,” said Belov. What color is the artist's hair?
Solution. First, all conditions are plotted on the diagram. The solution boils down to finding three solid triangles with vertices in different sets (Fig. 2.).
Belov Chernov Ryzhov
sculptor violinist artist
white black red
The artist is black-haired
When solving, we can get triangles of three types:
a) all sides are continuous segments (solution to the problem);
b) one side is a solid segment, and the others are dashed;
c) all sides are dashed segments.
Thus, it is impossible to obtain a triangle in which two sides are solid segments and the third is a dashed segment.
Task 3. Who's where?
Oak,maple, pine, birch, stump!
Hiding behind them, they lurk
Beaver, hare, squirrel, lynx, deer.
Who's where? Try to figure it out."
Where is the lynx, neither hare nor beaver
Neither on the left nor on the right - it’s clear.
ANDnext to the squirrel - that’s cunning -
Don't look for them in vain either.
There is no lynx next to the deer.
And there is no hare on the right and on the left.
And the squirrel on the right is where the deer is!
Now start your search with confidence.
And wants to give you advice
A tall stump covered with moss:
- Who's where? Find the right trail
A squirrel and a deer will help.
Solution. Let's find the answer using graphs, denoting each animal with a dot and its placement with arrows. All that remains is to count the arrows (Fig.)
Lynx Hare
Squirrel Hare Beaver Deer Squirrel Lynx
Deer Oak Maple Pine Birch Stump
beaver
b) Tabular method
The second way to solve logical problems - using tables - is also simple and intuitive, but it can only be used when it is necessary to establish a correspondence between two sets. It is more convenient when sets have five or six elements.
Task 4. One day, seven married couples gathered at a family celebration. The men's surnames: Vladimirov, Fedorov, Nazarov, Viktorov, Stepanov, Matveev and Tarasov. The women's names are: Tonya, Lyusya, Lena, Sveta, Masha, Olya and Galya.
Solution. When solving the problem, we know that each man has one last name and one wife.
Rule 1: Each row and column of the table can contain only one matching sign (for example, “+”).
Rule 2: If in a row (or column) all the “places”, except one, are occupied by an elementary prohibition (a discrepancy sign, for example “-”), then you need to put a “+” sign on the free space; if there is already a “+” sign in a row (or column), then the remaining places should be occupied by a “-” sign.
Having drawn a table, you need to place known prohibitions in it based on the conditions of the problem. Having filled out the table according to the conditions of the problem, we immediately obtain solutions: (Fig. 3).
Tonya
Lucy
Lena
Sveta
Masha
Olya
Galya
Vladimirov
Fedorov
Nazarov
Viktorov
Stepanov
Matveev
Tarasov
4.2 Tactical tasks
Solving tactical and set-theoretic problems involves drawing up a plan of action that leads to the correct answer. The difficulty is that the choice must be made from a very large number of options, i.e. these possibilities are not known, they need to be invented.
a) Problems of moving or correctly placing pieces can be solved in two ways: practical (actions in moving pieces, selecting) and mental (thinking about a move, predicting the result, guessing a solution -method of reasoning ).
In the method of reasoning, the following help when solving: diagrams, drawings, short notes, the ability to select information, the ability to use the enumeration rule.
This method is usually used to solve simple logical problems.
Problem 5 . Lena, Olya, Tanya took part in the 100 m race. Lena ran 2 seconds earlier than Olya, Olya ran 1 second later than Tanya. Who came earlier: Tanya or Lena and by how many seconds?
Solution. Let's make a diagram:
Lena Olya Tanya
Answer. Earlier, Lena arrived at 1st.
Let's consider a simple problem.
Problem 6 . Remembering the autumn cross, Squirrels argue for two hours:
The hare won the race.Athe second was a fox!
- No, says another squirrel,
- you to mejokes
The first one, I remember, was an elk!
- “I,” said the important owl,
- I won’t get involved in someone else’s dispute.
But in each of your words
There is one error.
The squirrels snorted angrily.
It became unpleasant for them.
After weighing everything, you decide
Who was first, who was second.
Solution.
Hare - 1 2
Fox - 2
Moose - 1
If we assume that the correct statement is the hare came 1, then the fox 2 is then not true, i.e. in the second group of statements, both options remain incorrect, but this contradicts the condition. Answer: Elk - 1, Fox - 2, Hare - 3.
4.3 Problems of finding the intersection or union of sets (Eulerian circles)
Another type of problem is one in which it is necessary to find some intersection of sets or their union, observing the conditions of the problem.
Let's solve problem 7:
Of the 52 schoolchildren, 23 collect badges, 35 collect stamps, and 16 collect both badges and stamps. The rest are not interested in collecting. How many schoolchildren are not interested in collecting?
Solution. The conditions of this problem are not so easy to understand. If you add 23 and 35, you get more than 52. This is explained by the fact that we counted some schoolchildren twice here, namely those who collect both badges and stamps.To make the discussion easier, let's use Euler circles
There is a big circle in the picturedenotes the 52 students in question; circle 3 depicts schoolchildren collecting badges, and circle M depicts schoolchildren collecting stamps.
The large circle is divided by circles 3 and M into several areas. The intersection of circles 3 and M corresponds to schoolchildren collecting both badges and stamps (Fig.). The part of circle 3 that does not belong to circle M corresponds to schoolchildren who collect only badges, and the part of circle M that does not belong to circle 3 corresponds to schoolchildren who collect only stamps. The free part of the large circle represents schoolchildren who are not interested in collecting.
We will sequentially fill out our diagram, entering the corresponding number in each area. According to the condition, both badges and stamps are collected by 16 people, so we will write the number 16 at the intersection of circles 3 and M (Fig.).
Since 23 schoolchildren collect badges, and 16 schoolchildren collect both badges and stamps, then only 23 - 16 = 7 people collect badges. In the same way, only stamps are collected by 35 - 16 = 19 people. Let's write numbers 7 and 19 in the corresponding areas of the diagram.
From the picture it is clear how many people are involved in collecting. To find out thisyou need to add the numbers 7, 9 and 16. We get 42 people. This means that 52 - 42 = 10 schoolchildren remain not interested in collecting. This is the answer to the problem; it can be entered into the free field of the large circle.
Euler's method is indispensable for solving some problems, and also greatly simplifies reasoning.
4.4 Letter puzzles and problems with asterisks
Letter puzzles and examples with asterisks are solved by selecting and considering various options.
Such problems vary in complexity and solution scheme. Let's look at one such example.
Problem 8 Solve a number puzzle
CIS
KSI
ISK
Solution. Amount AND+ C
(in the tens place) ends in C, but I ≠ 0 (see the units place). This means I = 9 and 1 ten in the units place is remembered. Now it is easy to find K in the hundreds place: K = 4. For C there is only one possibility left: C = 5.
4.5 Truth problems
We will call problems in which it is necessary to establish the truth or falsity of statements truth problems.
Problem 9 . Three friends Kolya, Oleg and Petya were playing in the yard, and one of them accidentally broke the window glass with a ball. Kolya said: “It wasn’t me who broke the glass.” Oleg said: “Petya broke the glass.” It was later discovered that one of these statements was true and the other was false. Which boy broke the glass?
Solution. Let's assume that Oleg told the truth, then Kolya also told the truth, and this contradicts the conditions of the problem. Consequently, Oleg told a lie, and Kolya told the truth. From their statements it follows that Oleg broke the glass.
Problem 10 Four students - Vitya, Petya, Yura and Sergei - took four first places at the Mathematical Olympiad. When asked what places they took, the following answers were given:
a) Petya - second, Vitya - third;
b) Sergey - second, Petya - first;
c) Yura - second, Vitya - fourth.
Indicate who took what place if only one part of each answer is correct.
Solution. Suppose that the statement “Peter - II” is true, then both statements of the second person are incorrect, and this contradicts the conditions of the problem.
Suppose that the statement “Sergey - II” is true, then both statements of the first person are incorrect, and this contradicts the conditions of the problem.
Suppose that the statement "Jura - II" is true, then the first statement of the first person is false, and the second is true. And the first statement of the second person is incorrect, but the second is correct.
Answer: first place - Petya, second place - Yura, third place - Vitya, fourth place Sergey.
4.6 “Hats” type problems
The most famous problem is about wise men who need to determine the color of the hat on their head. To solve such a problem, you need to restore the chain of logical reasoning.
Problem 11 . “What color are the berets?”
Three friends, Anya, Shura and Sonya, sat in the amphitheater one after another without birets. Sonya and Shura cannot look back. Shura sees only the head of Sonya sitting below her, and Anya sees the heads of both friends. From a box containing 2 white and 3 black berets (all three friends know about this), they took three out and put them on their heads, not to mention what color the beret was; two berets remained in the box. When Anya was asked about the color of the beret they put on her, she was unable to answer. Shura heard Anya’s answer and said that she also could not determine the color of her beret. Based on the answers of her friends, can Sonya determine the color of her beret?
Solution. You can reason this way. From Anya’s answers, both girlfriends concluded that they both could not have two white berets on their heads. (Otherwise Anya would have immediately said that she had a black beret on her head). They have either two black ones, or white and black. However, if Sonya had a white beret on her head, then Shura also said that she did not know which beret she had on her head, then, therefore, Sonya had a black beret on her head.
5. Practical part
Study of the level of logical thinking of middle school students.
In the practical part of the research work, I selected logical problems like:Who is who?
The tasks corresponded to the level of knowledge of the 5th and 6th, 7th and 8th grades, respectively. The students solved these problems, and I analyzed the results. Let us consider the results obtained in more detail.
The following tasks were proposed for grades 5 and 6:
Problem 1. Remembering the autumn cross, Squirrels argue for two hours:
The hare won the race.Athe second was a fox!
- No, says another squirrel,
- you to mejokesthrow these away. The hare was second, of course
The first one, I remember, was an elk!
- “I,” said the important owl,
- I won’t get involved in someone else’s dispute.
But in each of your words
There is one error.
The squirrels snorted angrily.
It became unpleasant for them.
After weighing everything, you decide
Who was first, who was second.
Task 2. Three friends of Belova, Krasnova and Chernova met. One of them was wearing a black dress, the other was wearing a red dress, and the third was wearing a white dress. A girl in a white dress says to Chernova: “We need to exchange dresses, otherwise the color of our dresses does not match our surnames.” Who was wearing what dress?
Among students in grades 5 and 6, there were 25 people with proposed tasks like “Who is who?” 11 people completed it, including 5 girls and 6 boys. The results of solving logical problems by students in grades 5 and 6 are presented in the figure:
The figure shows that 44% successfully solved both “Who is who?” problems. Almost all students coped with the first task; the second task, using graphs or tables, caused difficulties for the children.
To summarize, we can conclude that, in general, 5th and 6th grade students cope with simpler tasks, but if a little more elements are added in reasoning, then not all of them cope with such tasks.
The following tasks were proposed for 7th and 8th grades:
Problem 1. Lena, Olya, Tanya took part in the 100 m race. Lena ran 2 seconds earlier than Olya, Olya ran 1 second later than Tanya. Who came earlier: Tanya or Lena and by how many seconds?
Problem 2. Three friends met in a cafe: the sculptor Belov, the violinist Chernov and the artist Ryzhov. “It’s wonderful that one of us has white hair, another has black, and the third has red hair, but none of our hair color matches our surname,” the black-haired man noted. “You’re right,” said Belov. What color is the artist's hair?
Problem 3. Once upon a time, seven married couples gathered at a family holiday. The men's surnames: Vladimirov, Fedorov, Nazarov, Viktorov, Stepanov, Matveev and Tarasov. The women's names are: Tonya, Lyusya, Lena, Sveta, Masha, Olya and Galya.At the evening, Vladimirov danced with Lena and Sveta, Nazarov - with Masha and Sveta, Tarasov - with Lena and Olya, Viktorov - with Lena, Stepanov - with Sveta, Matveev - with Olya. Then they started playing cards. First, Viktorov and Vladimirov played with Olya and Galya, then Stepanov and Nazarov replaced the men, and the women continued the game. And finally, Stepanov and Nazarov played one game with Tonya and Lena.
Try to determine who is married to whom if it is known that at the evening not a single man danced with his wife and not a single married couple sat down at the same time at the table during the game.
In the 7th and 8th grades among 33 people with all the problems like “Who is who?” 18 people completed it, including 8 girls and 10 boys.
The results of solving logical problems by students of the 7th and 8th grades are presented in the figure:
The figure shows that 55% of students coped with all the tasks, 91% completed the first task, 67% successfully solved the second task, and the last task turned out to be the most difficult for the children and only 58% completed it.
Analyzing the results obtained, in general we can say that students in the 7th and 8th grades coped better with solving logical problems. Pupils of the 5th and 6th grades showed worse results, perhaps the reason for this is that solving this type of problem requires a good knowledge of mathematics; pupils of the 5th grade do not yet have experience in solving such problems.
I also conducted social. survey among students in grades 5-8. I asked everyone the question: “Which problems are easier to solve: mathematical or logical? 15 people took part in the survey. 10 people answered - mathematical, 3-logical, 2 - they can’t solve anything. The survey results are shown in the figure:
The figure shows that mathematical problems are easier to solve for 67% of respondents, logical problems for 20%, and 13% will not be able to solve any problem.
6.Conclusion
In this work you got acquainted with logical problems. With what logic is. We have brought to your attention various logical tasks that help develop logical and imaginative thinking.
Any normal child has a desire for knowledge, a desire to test himself. Most often, schoolchildren’s abilities remain undiscovered for themselves, they are not confident in their abilities, and are indifferent to mathematics.
For such students, I propose using logical tasks. These tasks can be considered in club and elective classes.
They must be accessible, awaken intelligence, capture their attention, surprise, awaken them to active imagination and independent decisions.
I also believe that logic helps us cope with any difficulties in our lives, and everything we do should be logically understood and structured.
We encounter logic and logical problems not only in school in mathematics lessons, but also in other subjects.
7. Literature
Dorofeev G.V. Mathematics 6th grade.-Enlightenment,: 2013.
Matveeva G. Logical problems // Mathematics. - 1999. No. 25. - P. 4-8.
Orlova E. Solution methods logical problems and number problems //
Mathematics. - 1999. No. 26. - P. 27-29.
4. Sharygin I.F. , Shevkin E.A. Tasks for ingenuity.-Moscow,: Education, 1996.-65 p.
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. Course of lectures. 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.
MINISTRY OF EDUCATION AND SCIENCE OF THE REPUBLIC OF BURYATIA
MUNICIPAL BUDGET EDUCATIONAL INSTITUTION
"MALOKUDARINSKAYA SECONDARY SCHOOL"
RESEARCH WORK
Topic: “Logical tasks
Completed the job:
Igumnov Matvey, 3rd grade student
MBOU "Malokudarinskaya secondary school"
Head: Serebrennikova M.D.
1. INTRODUCTION …………………………………………………………..3-4
2. MAIN PART
What is logic………………………………………………………. …5
Types of logical problems………………………………………………………6
Solving a logical problem…………………………………………………….10
Practical part …………………………………………………….. 10-12
3. CONCLUSION……………………………………………………… 14
4. LIST OF REFERENCES AND INTERNET SOURCES………. 15
5.APPLICATIONS
Introduction
The development of creative activity, initiative, curiosity, and ingenuity is facilitated by solving non-standard and logical problems.
Solving logic problems is very exciting. There seems to be no mathematics in them - there are no numbers, no geometric figures, but there are only liars and wise men, truth and lies. At the same time, the spirit of mathematics is felt most clearly in them - half of the solution to any mathematical problem (and sometimes much more than half) is to properly understand the condition, to unravel all the connections between the objects of the problem.
While preparing this work, I set target- develop your ability to reason and draw correct conclusions. Only solving a difficult, non-standard problem brings the joy of victory. When solving logical problems, you have the opportunity to think about an unusual condition and reason. This arouses and maintains my interest in mathematics. Relevance. Nowadays, very often a person’s success depends on his ability to think clearly, reason logically and clearly express his thoughts.
Purpose of the study: can a logic problem have multiple correct answers?
Tasks: 1) familiarization with the concepts of “logic” and types of logical problems; 2) solving a logical problem, determining the dependence of the change in the answer of the problem on the size of the nuts
Research methods: collection, study of material, comparison, analysis
Hypothesis If we change the size of the nuts, will the answer to the problem change?
Field of study: logical problem.
What is logic?
The following definitions of logic can be found in the scientific literature:
Logic is the science of acceptable methods of reasoning.
Logic is the science of the forms, methods and laws of intellectual cognitive activity, formalized using logical language.
Logic is the science of correct thinking.
Logic is one of the most ancient sciences. Some of the origins of logical teaching can be found in India, at the end of the 2nd millennium BC. The founder of logic as a science is the ancient Greek philosopher and scientist Aristotle. It was he who drew attention to the fact that in reasoning we deduce others from some statements, based not on the specific content of the statements, but on a certain relationship between their forms and structures.
How to learn to solve logical problems? Logical or non-numeric problems constitute a broad class of non-standard problems. This includes, first of all, word problems in which it is necessary to recognize objects or arrange them in a certain order according to existing properties. In this case, part of the statements of the problem conditions can appear with a different truth value (be true or false). So, we will learn how logic problems can be solved in different ways. It turns out there are several such techniques, they are varied and each of them has its own area of application.
Types of logic problems
1"Who is who?"
2 Tactical tasks Solving tactical and set-theoretic problems involves drawing up a plan of action that leads to the correct answer. The difficulty is that the choice must be made from a very large number of options, i.e. these possibilities are not known, they need to be invented.
3 Problems on finding the intersection or union of sets
4 Letter and number puzzles and star problems
Letter puzzles and examples with asterisks are solved by selecting and considering various options.
5 Tasks that require establishing the truth or falsity of statements
6 “Hats” type problems
The most famous problem is about wise men who need to determine the color of the hat on their head. To solve such a problem, you need to restore the chain of logical reasoning.
SOLVING A LOGICAL PROBLEM
There are many types of nuts. Let's find out whether the answer to this problem depends on the size of the nuts?
Let's look at some of them.
Pine nuts are considered the smallest. Moreover, their sizes depend on the type. The nuts of European cedar, Siberian dwarf cedar and Korean cedar differ in size. Among them, the smallest are dwarf pine nuts. Their length is 5 mm.
Conclusion: There are many types of nuts. They have different sizes: in diameter. Therefore, we substitute nuts of different sizes into the problem.
PRACTICAL PART
Practical work.
Job No. 1. Practical work with walnuts.
Tools and materials: ruler, chalk, colored measures, 10 pieces of walnuts.
Preparatory work. We cut out measurements from colored cardboard: 3 measurements from green cardboard, 2 cm long and 2 cm wide, for the first row and 5 measurements from yellow cardboard, 1 cm long and 2 cm wide, for the second row.
Description of work. Mark a point on the table with chalk. We put a nut on it. Place a 2 cm measure and a second nut, a 2 cm measure and a third nut, a 2 cm measure and a fourth nut. With chalk we mark the beginning and end of the length of the first row. The beginning of the second row is clearly marked with chalk under the beginning
first and put a nut, a 1 cm measure and a second nut, a 1 cm measure and a third, a measure and a fourth, a measure and a fifth, a measure and a sixth. We mark the end of the length of the second row with chalk. Compare the lengths of the rows.
Answer: the second row is longer.
2. Practical work with pine nuts. (See job description #1.)
Answer: the second row is longer.
3. Practical work with hazelnuts (hazelnuts).
(See job description #1.)
Answer:
the second row is longer.
4. Practical work with peanuts. (Fig.4)
(See job description #1.)
Answer: :
the second row is longer.
Conclusion: the answer to the problem does not change depending on the size of these nuts.
All nuts more than 5 mm.
DRAWINGS
Let's check this in the drawings using scale.
Scale
1. The ratio of the length of lines on a map or drawing to the actual length.
.
CONCLUSION
My hypothesis was confirmed: when the size of the nuts changes, the answer to the problem changes
Conclusion: For nuts up to 5 mm in size, the first row is longer.
When the nut size is 5 mm, the length of the rows is the same.
For nuts larger than 5 mm, the second row is longer.
Practical significance. The solutions proposed in the work are very simple; any student can use them. I showed them to my friends. Many students became interested in this task. Now, when solving logical problems, everyone will think about its answer.
Prospects: I really enjoyed experimenting with nuts, arranging them, looking for the answer. I shared all my findings with friends and classmates. Logical problems interested me: in the future I want to try to create my own problem that is just as interesting, with different answer options.
I tried changing the problem condition. I took meters for the spaces between the nuts. Substituting nuts of different sizes, I got the same answer: the first row is longer. Why is this so? I started measuring everything again: everything was the same. If I increased the intervals by 100 times, then the size of the nuts should also be increased by 100 times. Now I realized that I don’t have such a large nut of 50 cm or more. All nuts are less than 50 cm. According to my conclusion, for the lengths to be equal, the nut must be 50 cm, and if it is more than 50 cm, then the second row will be longer. This means that my conclusion is also suitable for this task.
6.Conclusion
In this work you got acquainted with logical problems. Various options for solving a logical problem were offered to your attention.
Any normal child has a desire for knowledge, a desire to test himself. Most often, schoolchildren’s abilities remain undiscovered for themselves, they are not confident in their abilities, and are indifferent to mathematics.
For such students, I propose using logical tasks.
They must be accessible, awaken intelligence, capture their attention, surprise, awaken them to active imagination and independent decisions.
I also believe that logic helps us cope with any difficulties in our lives, and everything we do should be logically understood and structured.
Literature
1. Ozhegov S.I. and Shvedova N.Yu. Explanatory dictionary of the Russian language: 80,000 words and phraseological expressions / Russian Academy of Sciences. Institute of Russian Language named after V.V. Vinogradov. - 4th ed., supplemented. – M.: Azbukovnik, 1999. – 944 pp.
2. Encyclopedia for children. Biology. Volume 2. “Avanta+”, M. Aksenov, S. Ismailova,
M.: “Avanta+”, 1995
3. I explore the world: Det.Entsik.: Plants / Comp. L.A. Bagrova; Khud.A.V.Kardashuk, O.M.Voitenko;
Under general ed. O.G. Hinn. – M.: AST Publishing House LLC, 2000. – 512 p.
4. Encyclopedia of living nature. - M.: AST-PRESS, 2000. - 328 p.
5. Rick Morris. Secrets of living nature (translation from English by A.M. Golov), M.: “Rosman”, 1996.
6. David Burney. Large illustrated encyclopedia of living nature (translation from English) M.: “Swallowtail”, 2006
MINISTRY OF EDUCATION
REPUBLIC OF BELARUS
Minsk region Borisov district
State educational institution
"Loshnitsa district gymnasium"
Research work
in mathematics
Karpovich Anna Igorevna, 11th grade student,
Melech Alexey Vladimirovich, 9th grade student,
Demidchik Artyom Alekseevich, 9th grade student
Supervisor:
Yakimenko Ivan Viktorovich, mathematics teacher
Loshnitsa, 2006-2008
Introduction 3
Relevance of the selected topic 3
Literature review on topic 4
Formation of concepts 4
Level of development of the problem 4
Object of study 5
Subject of study 5
Setting goals 5
Setting goals 5
Main part 6
Empirical basis of the study 6
Description of research paths and methods 6
1. Study of bibliography 6
2. Trial and error 6
3. Variation 7
Research results 8
Reliability of the results obtained 8
Conclusion 9
Summing up. Conclusions 9
Practical significance of the results obtained 9
Scientific novelty of the results obtained 9
Applications 10
Appendix 1. Classification of logic games 10
Appendix 2. Rules of the game “Dozen” 10
Appendix 3. Rules of the game “Devil's Dozen” 10
Appendix 4. Classification of figures in the game “Dozen” 11
Appendix 5. Additional pieces of the game “Dozen” 12
Appendix 6. Figures of the game “Devil's Dozen” 17
Literature 18
Introduction
Relevance of the selected topic
Not a single child, from first-grader to graduate, has ever refused to just play, especially instead of or during a lesson.You don’t need any special equipment for this, just a notebook sheet and a pen. School games are easy to play, always have an ending, and guarantee all three outcomes: win, lose, draw.
However, most of the games that schoolchildren play have long been known, and therefore studied and uninteresting. For example, two strong players will never lose to each other at tic-tac-toe. This “game vacuum” inevitably leads to a search for novelty in one of the following directions:
- in game rules ry (“Tic Tac Toe” up to five),
- in the size of the playing field(dimensionless “Angles”),
- in the number of players(crossover "Battleship").
In this regard, we consider it relevant to invent, test and explore new games for schoolchildren.
The relevance of the research topic is confirmed by the unflagging interest in charades, rebuses, and puzzles, which serve as a testing ground for schoolchildren to test their capabilities in solving problems and tasks of any complexity. In other words, by developing logic, we learn to survive.
Gottfried-Wilhelm Leibniz noted in a letter to his colleague: “...even games, both those requiring dexterity and those based on chance, provide enormous material for scientific studies. Moreover, the most ordinary children’s fun could attract the attention of the greatest mathematician.”(, pp. 19-20).
And finally, we were haunted by the laurels of Erne Rubik, the inventor of the most famous (and most commercial!) puzzle - the Rubik's cube.
During the previous year, we created the game “Dozen” (see. Appendix 2). Work on the game continued this year with the goal of refinement, research of game combinations and development of new game options.
Literature review on the topic
Formation of concepts
Logics. 1. The science of the laws of thinking and its forms. 2. Course of reasoning, conclusions. 3. Reasonableness, internal regularity.(, p.167)Game. Doing something that serves for entertainment, relaxation, or participating in competitions in something.(, p.127)
Even at the first comparison, the inconsistency of these two concepts is striking, and even the phrase "logical games" generally seems like verbal nonsense.
Based on the above definitions, a logic game can be considered as activity for entertainment and development of thinking.
The following terms will be used in this work:
"Paper Game" is a game for two or more players that uses a piece of paper and a pen.
Under "computer game" we will understand a paper or other logic game for which a computer version exists or can be created.
Term "inventory game" is understood as a game that requires additional, specially made equipment.
"Math Game"- a game that requires mathematical knowledge from various branches of algebra or geometry.
"Winning Strategy" is interpreted in the usual sense, that is, as a way of playing a game that inevitably leads to victory.
"Game Outcome"- end of the game. There are three possible game outcomes: victory, defeat, draw.
Degree of development of the problem
Studying the literature on the issue under study, we noted that, upon coming to the attention of mathematicians, any fact, dependence, phenomenon is immediately measured, calculated, classified, and so on."The Queen Problem"(, p. 100) is described in detail in theory and for n=8 it demonstrably has 92 solutions (ibid.).
Ancient math fun "Bashe's Game", "Jianshizi" And "Nim" are generally called games, “the theory of which has been developed with exhaustive completeness” (p. 59).
However, in the sources studied, there was not even a mention of such a famous game as "Dots".
The widespread problem of filling a chess field with the move of a chess knight (, p. 104) is considered for both the nxn field and the mxn field. However, in the literature the problem has only one variation for a truncated 9x9 field without corners (, p. 20), which means it may have other, unexplored initial conditions.
The question of whether there are solutions for "Magic squares" of any size still remains open (, p.25, , p.89).
Thus, the study in the literature of logical games, ingenuity tasks, gaming and entertaining tasks does not exhaust the entire variety of conditions and solutions, and therefore the degree of development of the problem can be defined as insufficient.
Object of study
The object of the study is educational And creative interests of students 8-11 grades.Subject of research
The subject of the study is a game created by the authors "Dozen" and its sequel - the game "The Devil's Dozen."Setting goals
The purpose of this study is development, testing and study of new logic games.Setting goals
Realization of this goal requires solving the following specific tasks:
Study literature on a topic of interest.
Classify winning outcomes of the game (pieces).
Improve and expand your own game.
Clarify the relevance and demand of the created games.
Formulate recommendations for creating games.
Main part
Empirical basis of the study
The empirical basis of our research is the results after testing the game "Dozen".This also includes numerous handwritten versions of the game itself, tested by the authors and respondents, and a mini-tournament held as part of the week of exact sciences.
Description of research paths and methods
During the work, the following methods were used:1. Studying the bibliography
At this stage, when studying the literature on the issue of interest (mainly books on entertaining mathematics), we looked for logic games and classified them according to certain criteria (see Appendix 3).It turned out that none of the games are specific, i.e. cannot refer to only one species.
For example, the game "Pentamino"(, p. 13) consists of using any pentomino figures (a flat figure made up of five equal squares) to form a large figure - a square, rectangle, etc. We draw pentominoes on checkered paper - a paper game, cut them out of cardboard - an inventory game. But we are more familiar with this game as a continuation of the computer game. "Tetris" – "Pentix".
In addition, we were once again convinced that all games are educational to one degree or another and develop the thinking abilities of players.
2. Trial and error
Briefly describe the rules of the game "Dozen" Whoever gets one of the pre-agreed pieces first wins (see Appendices 2,4,5).At first glance, with such rules the game cannot have a draw, because only one player makes the final move, and it is simply impossible not to draw at least one piece with such variety. However, both players should have an equal chance, so let's let them make an equal number of moves and then they can "both win".
Let us remember that the game got its name from the number of risks that make up the winning figure.
The development of the topic was computer interpretation. The game has three electronic versions: one in MicroSoft Word and two in MicroSoft Excel. In order to play "A Dozen", you need to customize the Office interface, for which it is convenient to create a new work panel.
3. Variation
The variation method consists of playing out (going through, thinking through) different options for a situation. Variation is the work of logical thinking. In our case it is:Formulation of the easiest and most quickly remembered rules of the game,
Determination of optimal field sizes,
Increasing the number of possible figures.
Trying to put ourselves in the place of a leader or an outsider, we looked for ways out of the current position on the field. The most important thing in this work was the search for possible winning strategy, because if such a thing is found, after some time our game will become just as hackneyed as the others.
The playing field is a set of risks:
Horizontal – 6x7=42,
Vertical – 6x7=42,
Diagonal – 2x36=72,
Total – 2x42+72=156.
An elementary calculation - 156:12 = 13 shows that 13 figures can be constructed on the field at the same time, consisting of the required 12 marks. The multiplicity of the total number of risks to the number 13 became the first clue to changing the rules of the game.
^ General directions The following rule changes were varied:
prohibition on drawing a second diagonal (considerably speeds up the game and provides additional opportunities for a draw);
prohibition on using other people's risks (makes the game too “transparent” for the opponent);
resizing a field (increasing had a negative effect; when decreasing, some basic figures are lost);
addition to the basic set of winning pieces (asymmetric, non-convex polygons, open figures);
increase in the number of marks in basic figures .
Research results
It was the last two directions in variation that gave the most encouraging results. Firstly, the variety of resulting figures was so great that a special classification had to be invented for them (see. Appendix 4). Moreover, most of the figures obtained according to the rules of the game are non-convex axisymmetric polygons.Secondly, moving to asymmetrical figures, we felt urgent need add another risk to the figures! With the addition of the 13th mark it became difficult to achieve symmetry. This made the game even more exciting. The name of the new game appeared by itself: "The Devil's Dozen"».
Research into a modernized game will likely lead to significant rule changes. For example, if you allow different pieces on the field, in one game you can “earn” as many points as the winning piece contains risks. For the pieces different shapes(see Classification) you can also enter bonus points, etc.
Reliability of the results obtained
The reliability of the research results is ensured by:
practical confirmation of the main provisions of the study (the created game is a huge scope for research for schoolchildren of any age);
careful processing of data obtained during the study (when changing the rules of the game, all general directions of changes in game outcomes and winning strategy are considered).
Conclusion
Summing up. Conclusions
Game "Dozen"can be used in the study of mathematics at all levels of education.
Game "The Devil's Dozen""is a continuation, logical development of the game "Dozen».
"The Devil's Dozen"» fully meets the requirements set in goal setting.
The topic requires development in the form of a study of logic games.
Practical significance of the results obtained
The modernized game has practical valueHow educational tool For:
Mathematicians (development of logical thinking, familiarity with geometric figures).
Computer scientists (familiarity with MicroSoft Office programs, mouse skills, working with the Office clipboard).
Primary and secondary schoolchildren (modernization of games as part of research work).
Players of any age (competitions, tournaments).
Scientific novelty of the results obtained
The original game "12" and the modernized game "13", according to the author, manager and respondents, have no analogues and are the intellectual property of their developers.Applications
Appendix 1. Classification of logic games
Inventory
Paper
Educational (mathematical)
Linguistic
Computer
Appendix 2. Rules of the game “Dozen”
The game "Dozen" ("Twelve") is intended for schoolchildren aged 6-16 years.The player’s task is to draw a pre-agreed figure consisting of 12 lines before the opponent. To obtain a piece, you can use both your own and the risks drawn by your opponent.
Appendix 3. Rules of the game “Devil's Dozen”
The game "Devil's Dozen" ("Thirteen") is intended for schoolchildren aged 10-17 years.The playing field is a 6x6 square. Two people are playing. A move is considered to be drawing one of 4 lines: the horizontal side of the cell, the vertical side of the cell, or any diagonal of the cell. A move can only be made from an already drawn risk. Diagonal marks may intersect.
The player’s task is to draw a pre-agreed figure consisting of 13 lines before the opponent. To obtain a piece, you can use both your own and the risks drawn by your opponent.
A bonus is considered to be the receipt of a new figure (by mutual agreement of the players).
Appendix 4. Classification of figures in the game “Dozen”
By symmetry:1) axial symmetry:
side symmetry (the axis of symmetry runs along the side of the cell);
diagonal symmetry (the axis of symmetry runs along the diagonal of the cell);
secondary (the axis of symmetry passes inside the cell).
3) universal symmetry (lateral, diagonal and central at the same time);
4) asymmetry.
By convexity:
convex;
non-convex.
geometric shapes;
animate objects;
inanimate objects.
Appendix 5. Additional pieces of the game “Dozen”
heart
shorts
wolf
boomerang
butterfly
swift
Appendix 6. Figures of the game “Devil's Dozen”
snake
wolf
hedgehog
airplane
Literature
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Bakhankov A.E.; Explanatory dictionary of the Russian language. Mn.: NGO “Bel. assoc. “Competition”, 2006. – 416 p.
Bondareva L.A. etc.; tasks with an asterisk. Mn.: NGO “Bel. assoc. “Competition”, 2006. – 159 p.
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Kordemsky B.A.; Essays on mathematical problems for ingenuity. M.: “Uchpedgiz”, 1958. – 116 p.
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Lehman Johannes; editor E.K. Vakulina; 2x2 = joke. M.: “Enlightenment” 1974. – 192 p.
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Repkin V.V.; Educational dictionary of the Russian language. M.: Infoline, 1999. – 656 pp.: ill.
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