Reviews of the book "Amazing Logic" by Dmitry Gusev. It is necessary to select a variety of parcels

Dmitry Alekseevich Gusev

Amazing logic

Dmitry Alekseevich Gusev

Amazing logic

Preface

Have you ever read popular science or educational book with the feeling that you don’t quite understand what it says? If yes, then, most likely, based on the “author’s presumption of innocence,” you blame yourself for this: for your insufficient high level education, narrow-mindedness, lack of necessary abilities. However, it would be more correct to proceed from the “presumption of your own innocence,” since if you carefully read, but do not understand the text addressed to you (according to the book annotation), then it is not you who are to blame, but the author. After all, he undertook to write a book for you, and not for himself or narrow circle their colleagues. However, for an intelligible and intelligible presentation of the material, he did not have enough logical culture .

What is logical culture? This is knowledge and compliance with the basic principles and requirements correct construction and expression of thoughts both verbally and in writing. The absence of such a culture leads to various logical errors, which clog not only scientific, but also everyday thinking, prevent us from thinking, communicating, understanding each other and ourselves. The ambiguity and uncertainty of thinking, its inconsistency and confusion, inconsistency and groundlessness are a direct result of the lack of the proper level of logical culture.

Thinking that meets the requirements of logic is like a transparent stream: through the waters of which every pebble and grain of sand at the bottom is visible. Thinking built on violations of logical laws is like a muddy stream: nothing is visible in it. True, some say that it is more convenient to “fish” in troubled waters, that is, to construct such statements and create such texts - complex and obscure for the addressee - in which external thoughtfulness and scientificity mask internal inconsistency and sometimes vacuity. It is unlikely that a conscientious person can be a supporter of such “fishing”.

I undertook to write a book not for myself, but for the reader who is beginning to master logic “from scratch.” How far I succeeded is for the reader to judge.

The book consists of five chapters and one hundred entertaining tasks. The first three chapters are devoted to the forms of thinking in which the entire world of our thoughts, infinite in content, is expressed: concept, judgment and inference. In the fourth chapter we're talking about about the basic laws of logic and their common violations. The fifth chapter is devoted to the conditions and methods of discussion. The examples given in the book are intended to show that logic is not old, dry and lifeless wisdom, but an eternally young, useful and even interesting science, which may well help a person in life.

One hundred entertaining logical problems, completing the book, differ both in the type of their construction and in the level of complexity. For their the right decision a non-standard approach is required and creative work thoughts. The tasks are aimed at developing thinking, memory, attention and imagination; they will help you spend your leisure time interestingly and usefully. Not required to solve problems theoretical knowledge According to logic, life experience and ingenuity are enough, that is, intuitive logic, which all people possess to a greater or lesser extent, regardless of gender, age and level of education. Answers and comments are provided for all problems. However, don’t rush to look into them, try to “rack your brain” and cope with them without any hints - to experience the joy of solving it yourself.

Introduction

Logics is the science of the forms and laws of correct thinking. It appeared around the 4th century BC. e. V Ancient Greece. Its creator is considered to be the famous ancient Greek philosopher and scientist Aristotle. As you can see, logic is approximately 2.5 thousand years old. However, she still retains her practical significance. Many sciences and arts Ancient world are forever a thing of the past and represent for us only “museum” value, interesting solely as ancient monuments, but some of them have survived centuries, and at present we continue to use them. These include Euclid's geometry (which is what we study at school) and Aristotle's logic, also called traditional logic. In the 19th century, symbolic (or mathematical) logic appeared and began to develop rapidly. In traditional logic, to study correct thinking, we use natural language(the one we speak, write, read), and in symbolic logic - artificial language, or a symbolic language similar to the language of mathematics. Symbolic logic is a rather specific and difficult science; it can be considered as a branch of mathematics and computer science. Aristotelian logic, on the contrary, being broader, is a kind of universal science: its mastery is equally useful and even necessary for every person, regardless of which areas of knowledge and subjects are closer to him - social and humanitarian, natural and mathematical or technical . Therefore, our book is devoted to Aristotelian, or traditional, logic.

So why do we need logic, what role does it play in our lives? Logic helps us construct our thoughts correctly and express them correctly, convince other people and better understand our interlocutor, explain and defend our point of view, and avoid errors in reasoning.

Logical culture is knowledge and observance of the basic principles and requirements for the correct construction and expression of thoughts both in oral and written speech. The absence of such a culture leads to numerous and varied logical errors that clog not only scientific, but also everyday thinking, preventing us from thinking, communicating, understanding each other and ourselves. The ambiguity and uncertainty of thinking, its inconsistency and confusion, inconsistency and groundlessness are a direct result of the lack of the proper level of logical culture.

Each of us knows well that the content of human thinking is infinitely diverse, because you can think (think) about anything, for example, about the structure of the world and the origin of life on Earth, about the past of humanity and its future, about books read and films watched, about today's activities and tomorrow's rest... But the most important thing is that our thoughts arise and are built according to the same laws, obey the same principles, fit into the same patterns or forms. Moreover, if the content of our thinking is extremely diverse, then the forms in which this diversity is expressed are very few.

Let's give a simple example. Let's look at three statements: All crucian carp are fish; All triangles are geometric shapes; All chairs are pieces of furniture. Despite different content, these statements have something in common, something that unites them. What is this? They are united by form. Although they differ in content, they are similar in form, because each of the three statements is built according to form All A's are B's Where A And IN - any objects. It is clear that the statement itself All A's are B's devoid of any content. This statement is a pure form that can be filled with any content, for example: All pines are trees; All cities are settlements; All schools are educational establishments; All tigers are predators and so on.

Another example. Let’s take three statements with different contents: If autumn comes, the leaves fall; If tomorrow it will rain, then there will be puddles on the street; If a substance is a metal, then it is electrically conductive. Although different in content, these statements are similar to each other in that they are constructed according to the same form: If A, then B. It is clear that many meaningful statements can be selected for this form, for example: If you don't prepare for test work, then you can get a two; If the runway is covered with ice, planes will not be able to take off; If a word is at the beginning of a sentence, then it must be written with capital letters and so on.

Logic is not interested in the content of thinking (other sciences deal with this), it studies only the forms of thinking; she's not interested in What we think, otherwise How we think, that's why it is often called formal logic . For example, if the content of the statement All mosquitoes are insects is normal, and the statement All Cheburashkas are aliens - absurd, then for logic these two statements are equivalent, since it deals with forms of thinking, and the form of these statements is the same: All A's are B's.

As we see, form of thinking is a way of expressing thoughts, or a scheme for constructing them. There are three forms of thinking: concept, judgment and inference.

Concept is a form of thinking that designates an object or a feature of an object. Examples of concepts: pencil, plant, heavenly body, chemical element, courage, stupidity, negligence.

Judgment- this is a form of thinking that consists of concepts related to each other, and affirms or denies something. Examples of judgments: All planets are celestial bodies, Some schoolchildren are poor students, All triangles are not squares.

Inference is a form of thinking in which a new judgment (conclusion) follows from two or more initial judgments (premises).

In logic, it is customary to place the premises and the conclusion under each other and to separate the conclusion from the premises (in the book this is done using the => sign).

Examples of inferences:

All planets are moving.

Jupiter is a planet.

=> Jupiter is moving.

Iron is electrically conductive.

Copper is electrically conductive.

Mercury is electrically conductive.

Iron, copper, mercury are metals.

=> All metals are electrically conductive.

The entire endless world of our thoughts is expressed in concepts, judgments and conclusions. About these three forms of thinking and we'll talk on the pages of the book.

In addition to forms of thinking, logic also deals with the laws of thinking. Laws of thinking- these are such objective (i.e., existing in themselves and independent of our desires and preferences) principles or rules of thinking, the observance of which always leads reasoning (regardless of its content) to true conclusions, provided that the initial judgments are true. There are four basic laws of thinking (or laws of logic): the law of identity, the law of contradiction, the law of the excluded middle and the law of sufficient reason. Each of them will be considered in detail after studying the forms of thinking. Violation of these laws leads to various logical errors, usually to false conclusions. Sometimes the laws of logic are violated involuntarily, out of ignorance, but sometimes this is done deliberately, in order to confuse the interlocutor and prove to him some false idea. Such deliberate violations of logical laws for the outwardly correct proof of false thoughts are called sophistry .

Common sense and life experience alone are often enough to solve any problems. For example, anyone unfamiliar with logic can find a catch in the following reasoning:

Movement is eternal.

=> Going to school forever.

A false conclusion is obtained due to the use of the word movement V different meanings: in the first judgment it is used in a broad, philosophical sense, and in the second – in a narrow, mechanical sense. However, finding errors in reasoning is not always easy. Consider this example:

All my friends speak English.

The current president of America speaks English.

=> The current President of America is my friend.

It is clear that there is something wrong with this reasoning. But what exactly? Anyone who is familiar with logic will say that in this case an error was made, which is called “non-distribution of the middle term in a simple syllogism.” Don't let this unfamiliar thing scare you at first glance. complex expression: in the process of further reading the book, you will be convinced that there is nothing complicated, much less incomprehensible, here.

Or this example:

All cities above the Arctic Circle have white nights.

St. Petersburg does not lie beyond the Arctic Circle.

=> There are no white nights in St. Petersburg.

As we see, a false conclusion follows from two true judgments. There is also an error in this reasoning. It is unlikely that a person unfamiliar with logic will be able to immediately find it. And anyone who has a logical culture will immediately establish the reason: “an extension of a larger term in a simple syllogism.” Don't be alarmed: we'll soon find out what it is.

So, common sense and life experience are usually enough to navigate various difficult situations. But if to our common sense And life experience add a logical culture, then we will only benefit from this. Of course, logic will not solve all problems, but it can certainly help in life.

Concept

The definition should not be only negative.

Extraction square root- This mathematical operation, which is neither multiplication, nor division, nor exponentiation.

Man is neither a bird nor a fish.

Judgment

Any proposition is true or false. 4. Judgments can be simple or complex. Complex judgments consist of simple,... As we see, judgment is more complex shape

Inference

thinking versus concept. It is not surprising, therefore, that the judgment...

Inference Let's draw a conclusion (What is an inference)

is a form of thinking in which from two or more judgments, called premises, a new judgment, called a conclusion, follows. For example:

All living organisms feed on moisture.

All plants are living organisms.

=> All plants feed on moisture.

In the above example, the first two judgments are premises, and the third is a conclusion. The premises must be true propositions and must be related to each other. If at least one of the premises is false, then the conclusion is false:

All birds are mammals.

All sparrows are birds.

=> All sparrows are mammals.

As we can see, in the above example, the falsity of the first premise leads to a false conclusion, despite the fact that the second premise is true. If the premises are not related to each other, then it is impossible to draw a conclusion from them. For example, no conclusion follows from the following two premises:

All pines are trees.

Let us pay attention to the fact that inferences consist of judgments, and judgments consist of concepts, that is, one form of thinking is included in another as an integral part.

All inferences are divided into direct and indirect. IN immediate

In inferences, the conclusion is drawn from one premise. For example:

All flowers are plants.

=> Some plants are flowers.

=> It is not true that some flowers are not plants.

It is not difficult to guess that direct inferences are operations of transformation of simple judgments already known to us and conclusions about the truth of simple judgments using a logical square. The first given example of direct inference is the transformation of a simple judgment by inversion, and in the second example by a logical square from the truth of a judgment of the form A a conclusion is drawn about the falsity of a judgment of the form ABOUT.

All inferences are divided into direct and indirect. indirect In inferences, a conclusion is drawn from several premises. For example:

All fish are living beings.

All crucian carp are fish.

=> All crucian carp are living beings.

Indirect inferences are divided into three types: deductive, inductive and analogical inferences.

Deductive inferences (deduction) (from lat. deductio“derivation”) are inferences in which a conclusion is drawn from a general rule for a particular case (from a general rule one derives special case). For example:

All stars emit energy.

The sun is a star.

=> The sun emits energy.

As we can see, the first premise is a general rule, from which (using the second premise) a special case follows in the form of a conclusion: if all stars emit energy, then the Sun also emits it, because it is a star.

In deduction, reasoning proceeds from the general to the particular, from the greater to the lesser, knowledge is narrowed, due to which deductive conclusions are reliable, that is, accurate, obligatory, necessary. Let's look again at the example given. Could any other conclusion follow from two given premises than the one that follows from them? Could not. The following conclusion is the only possible one in this case. Let us depict the relationships between the concepts that made up our conclusion using Euler circles. Scope of three concepts: stars (3); bodies that emit energy(T) and Sun(C) will be located schematically in the following way(Fig. 33).

If the scope of the concept stars included in the scope of the concept bodies that emit energy and the scope of the concept Sun included in the scope of the concept stars, then the scope of the concept Sun is automatically included in the scope of the concept bodies that emit energy due to which the deductive conclusion is reliable.

The undoubted advantage of deduction lies in the reliability of its conclusions. Let's remember the famous literary hero Sherlock Holmes used the deductive method to solve crimes. This means that he structured his reasoning in such a way as to deduce the particular from the general. In one work, explaining to Dr. Watson the essence of his deductive method, he gives the following example. Scotland Yard detectives found a smoked cigar near the murdered Colonel Ashby and decided that the colonel had smoked it before his death. However, Sherlock Holmes irrefutably proves that the colonel could not smoke this cigar, because he wore a large, bushy mustache, and the cigar was smoked to the end, that is, if Colonel Ashby smoked it, he would certainly have set his mustache on fire. Therefore, another person smoked the cigar.

In this reasoning, the conclusion looks convincing precisely because it is deductive - from the general rule: Anyone with a big, bushy mustache can't smoke a cigar all the way through, a special case is displayed: Colonel Ashby couldn't finish smoking his cigar because he had such a mustache. Let us bring the considered reasoning to the one accepted in logic standard form recording inferences in the form of premises and conclusion:

Anyone with a big, bushy mustache can't finish a cigar.

Colonel Ashby wore a large, bushy mustache.

=> Colonel Ashby could not smoke the cigar completely.

Inductive inference (induction) (from lat. inductio“guidance”) are inferences in which a general rule is derived from several particular cases. For example:

Jupiter is moving.

Mars is moving.

Venus is moving.

Jupiter, Mars, Venus are planets.

=> All planets are moving.

The first three premises represent special cases, the fourth premise brings them under one class of objects, unites them, and the conclusion speaks about all objects of this class, i.e., a certain general rule is formulated (following from three special cases).

It is easy to see that inductive inferences are based on the principle opposite construction deductive reasoning. In induction, reasoning proceeds from the particular to the general, from the lesser to the greater, knowledge expands, due to which inductive conclusions (unlike deductive ones) are not reliable, but probabilistic. In the example of induction discussed above, a feature found in some objects of a certain group is transferred to all objects of this group, a generalization is made, which is almost always fraught with error: it is quite possible that there are some exceptions in the group, and even if many objects from a certain group characterized by some feature, this does not mean that all objects of this group are characterized by such a feature. The probabilistic nature of the conclusions is, of course, a disadvantage of induction. However, its undoubted advantage and advantageous difference from deduction, which is narrowing knowledge, is that induction is expanding knowledge that can lead to something new, while deduction is the analysis of the old and already known.

Inferences by analogy(analogy) (from Greek. analogia“correspondence”) are inferences in which, based on the similarity of objects (objects) in some characteristics, a conclusion is made about their similarity in other characteristics. For example:

Planet Earth is located in solar system, it has atmosphere, water and life.

The planet Mars is located in the solar system, it has an atmosphere and water.

=> There is probably life on Mars.

As we can see, two objects are compared (planet Earth and planet Mars), which are similar to each other in some significant, important features (being in the solar system, having an atmosphere and water). Based on this similarity, it is concluded that perhaps these objects are similar to each other in other ways: if there is life on Earth, and Mars is in many ways similar to Earth, then the presence of life on Mars is not excluded. The conclusions of analogy, like the conclusions of induction, are probabilistic.

When all propositions are simple (Categorical syllogism)

All deductive reasoning is called syllogisms(from Greek syllogismos –"counting, summing up, drawing conclusions"). There are several types of syllogisms. The first of them is called simple, or categorical, because all the judgments included in it (two premises and a conclusion) are simple, or categorical. These are judgments of the types already known to us A, I, E, O.

Consider an example of a simple syllogism:

All flowers (M)- these are plants (R).

All roses (S)- this is flowers (M).

=> All roses (S)- these are plants (R).

Both premises and conclusion are simple judgments in this syllogism, and both premises and conclusion are judgments of the form A(general affirmative). Let us pay attention to the conclusion presented by the judgment All roses are plants. In this conclusion, the subject is the term roses, and the predicate is the term plants. The subject of the inference is present in the second premise of the syllogism, and the predicate of the inference is in the first. Also in both premises the term is repeated flowers, which, as is easy to see, is connecting: it is thanks to it that the terms that are not connected, separated in premises plants And roses can be linked in the output. Thus, the structure of a syllogism includes two premises and one conclusion, which consist of three ( in various ways located) terms.

The subject of the conclusion is located in the second premise of the syllogism and is called smaller term of the syllogism(the second premise is also called less).

The predicate of inference is located in the first premise of the syllogism and is called major term of the syllogism(the first premise is also called greater). The inference predicate is usually in scope big concept than the subject of inference (in the given example, the concept roses And plants are in relation to generic subordination), due to which the predicate of inference is called bigger term, and the subject of the output is smaller .

A term that is repeated in two premises and connects a subject with a predicate (minor and major terms) is called middle term of the syllogism and is designated Latin letter M(from lat. medium –"average").

The three terms of a syllogism can be arranged in different ways. The relative arrangement of terms relative to each other is called figure of a simple syllogism. There are four such figures, i.e. all possible options relative position The terms in the syllogism are limited to four combinations. Let's look at them.

First figure of the syllogism- this is an arrangement of its terms in which the first premise begins with the middle term, and the second ends with the middle term. For example:

All gases (M)- these are chemical elements (R).

Helium (S)- it's gas (M).

=> Helium (S)is a chemical element (R).

Considering that in the first premise the middle term is associated with the predicate, in the second premise the subject is associated with the middle term, and in the conclusion the subject is associated with the predicate, we will draw up a diagram of the arrangement and connection of terms in the given example (Fig. 34).

Straight lines in the diagram (except for the one that separates the premises from the conclusion) show the relationship between the terms in the premises and in the conclusion. Since the role of the middle term is to connect the greater and lesser terms of the syllogism, in the diagram the middle term in the first premise is connected by a line to the middle term in the second premise. The diagram shows exactly how the middle term connects the other terms of the syllogism in its first figure. Additionally, the relationships between the three terms can be depicted using Euler circles. In this case, the following diagram will be obtained (Fig. 35).

Second figure of the syllogism- this is an arrangement of its terms in which both the first and second premises end with the middle term. For example:

All fish (R)breathe with gills (M).

All whales (S)do not breathe with gills (M).

=> All whales (S)not fish (R).

Schemes of the relative arrangement of terms and relations between them in the second figure of the syllogism look as shown in Fig. 36.

The third figure of the syllogism- this is an arrangement of its terms in which both the first and second premises begin with the middle term. For example:

All tigers (M)- these are mammals (R).

All tigers (M)- these are predators (S).

=> Some predators (S)- these are mammals (R).

Schemes of the relative arrangement of terms and relations between them in the third figure of the syllogism are shown in Fig. 37.

The fourth figure of the syllogism- this is an arrangement of its terms in which the first premise ends with the middle term, and the second begins with it. For example:

All squares (R)- these are rectangles (M).

All rectangles (M)- these are not triangles (S).

=> All triangles (S)- these are not squares (R).

Schemes of the relative arrangement of terms and relations between them in the fourth figure of the syllogism are shown in Fig. 38.

Note that the relationships between the terms of the syllogism in all figures may be different.

Any simple syllogism consists of three propositions (two premises and a conclusion). Each of them is simple and belongs to one of four types ( A, I, E, O). The set of simple propositions included in a syllogism is called mode of simple syllogism. For example:

All celestial bodies move.

All planets are celestial bodies.

=> All planets are moving.

In this syllogism the first premise is a simple proposition of the form A(generally affirmative), the second premise is also a simple proposition of the form A, and the conclusion in this case is a simple judgment of the form A. Therefore, the considered syllogism has the mode AAA or Barbara. The last Latin word does not mean anything and is not translated in any way - it is simply a combination of letters, selected in such a way that it contains three letters A, symbolizing the mode of syllogism AAA. Latin “words” to denote modes of a simple syllogism were invented in the Middle Ages.

The following example is a syllogism with mode EAE, or cesare:

All magazines are periodicals.

All books are not periodicals.

=> All books are not magazines.

And one more example. This syllogism has the mode A.A.I. or darapti.

All carbons are simple bodies.

All carbons are electrically conductive.

=> Some electrical conductors are simple bodies.

Total modes in all four figures (i.e. possible combinations simple propositions in a syllogism) – 256. Each figure has 64 modes. However, of these 256 modes, only 19 give reliable conclusions, the rest lead to probabilistic conclusions. If we take into account that one of the main signs of deduction (and therefore of syllogism) is the reliability of its conclusions, then it becomes clear why these 19 modes are called correct, and the rest - incorrect.

Our task is to be able to determine the figure and mode of any simple syllogism. For example, you need to establish the figure and mode of the syllogism:

All substances are made up of atoms.

All liquids are substances.

=> All liquids are made of atoms.

First of all, you need to find the subject and predicate of the conclusion, that is, the minor and major terms of the syllogism. Next, you should establish the location of the minor term in the second premise and the larger one in the first. After this, you can determine the middle term and schematically depict the arrangement of all terms in the syllogism (Fig. 39).

All substances (M)consist of atoms (R).

All liquids (S)- these are substances (M).

=> All liquids (S)consist of atoms (R).

As you can see, the syllogism under consideration is built on the first figure. Now we need to find its mode. To do this, you need to find out what type of simple judgments the first and second premises and conclusion belong to. In our example, both premises and conclusion are judgments of the form A(generally affirmative), i.e. the mode of a given syllogism – AAA, or b a rb a r a . So, the proposed syllogism has the first figure and mode AAA.

Going to school forever (General rules of syllogism)

The rules of syllogism are divided into general and specific.

The general rules apply to all simple syllogisms, regardless of the figure by which they are constructed. Private the rules apply only to each figure of the syllogism and are therefore often called figure rules. Let's consider general rules syllogism.

A syllogism must have only three terms. Let us turn to the already mentioned syllogism, in which this rule is violated.

Movement is eternal.

Going to school is movement.

=> Going to school forever.

Both premises of this syllogism are true propositions, but a false conclusion follows from them, because the rule in question is violated. Word movement used in two premises in two different meanings: movement as universal world change and movement like mechanical movement bodies from point to point. It turns out that there are three terms in the syllogism: movement, going to school, eternity, but meanings (since one of the terms is used in two different meanings) four, i.e. the extra meaning seems to imply an extra term. In other words, in the given example of a syllogism there were not three, but four (in meaning) terms. An error that occurs when the above rule is violated is called quadrupling terms .

The middle term must be distributed in at least one of the premises. The distribution of terms in simple judgments was discussed in the previous chapter. Let us recall that the easiest way to establish the distribution of terms in simple judgments is with the help of circular diagrams: we need to depict the relations between the terms of the judgment with Euler circles, while full circle in the diagram will be denoted by a distributed term (+), and an incomplete term will be denoted by an undistributed term (–). Let's look at an example of a syllogism.

All cats (TO)- these are living beings (J. s).

Socrates (WITH)- This is also a living being.

=> Socrates is a cat.

A false conclusion follows from two true premises. Let us depict the relations between the terms in the premises of the syllogism using Euler circles and establish the distribution of these terms (Fig. 40).

As we can see, the middle term ( living things) in this case is not distributed in any of the premises, but according to the rule it must be distributed in at least one. An error that occurs when the rule in question is violated is called - undistribution of the middle term in each premise .

A term that was not distributed in the premise cannot be distributed in the conclusion. Let's look at the following example:

All apples (I)– edible items (S.p.).

All pears (G)- these are not apples.

=> All pears are inedible items.

The premises of a syllogism are true propositions, but the conclusion is false. As in the previous case, let us depict the relations between the terms in the premises and the conclusion of the syllogism using Euler circles and establish the distribution of these terms (Fig. 41).

In this case, the predicate of inference, or larger term of the syllogism ( edible items), in the first premise it is undistributed (–), and in the conclusion it is distributed (+), which is prohibited by the rule in question. An error that occurs when it is violated is called extension of a larger term. Let us remember that a term is distributed when we are talking about all the objects included in it, and undistributed when we are talking about some of the objects included in it, which is why the error is called extension of the term.

A syllogism should not have two negative premises. At least one of the premises of the syllogism must be positive (both premises can be positive). If two premises in a syllogism are negative, then the conclusion from them either cannot be drawn at all, or, if it is possible to draw it, it will be false or, at least, unreliable, probabilistic. For example:

Snipers cannot have poor eyesight.

All my friends are not snipers.

=> All my friends have poor eyesight.

Both premises in a syllogism are negative judgments, and, despite their truth, a false conclusion follows from them. The error that occurs in this case is called two negative premises.

There should not be two partial premises in a syllogism.

Some schoolchildren are first graders.

Some schoolchildren are tenth graders.

It is necessary to select a variety of parcels.

3. It is necessary to draw a conclusion only on the basis of significant features. If,... These are the basic rules of incomplete induction. Now let's look at its most common mistakes. Speaking of...

Basic laws of logic

Logic of the discussion

Conclusion

We got acquainted with the main sections of logic - the science of the forms and laws of correct thinking. Surprising as it may seem at first glance, any person knows logic, regardless of whether he has studied it or not. Everyone has had to deal with such widespread expressions in life as: This reasoning is logical; It is illogical. There is no logic in their actions; Where is the logic here? etc. When they talk about something logical or illogical, we, as a rule, guess what they are talking about, even if we are completely unfamiliar with Aristotelian logic. This indicates that all people, regardless of gender, age, nationality,, social environment historical era

and other factors, one way or another use logic in thinking and speech.

Practical logic is often called intuitive logic. It is formed spontaneously in the process of life experience around the age of 6–7 years. Any person not familiar with the laws of logic will notice the logical incorrectness and even the absurdity of the statement I'm going in new trousers, and you're going to the gymnasium. I'm walking in trousers, and you're walking in shorts; I'm going to the gymnasium, and you're going to the lyceum. Anyone who has studied logic knows that the first statement violates the logical law of identity, since it confuses two different situations: walking in some clothes and going somewhere. It turns out that even before becoming familiar with the law of identity, we already practically use it, we know about it, only implicitly, intuitively.

In the same way, it is unlikely that a person of sound mind will not notice a logical error in the statement Driver P. grossly violated the rules of the garage cooperative: when leaving the territory, he did not take verbal permission in writing. Not everyone will be able to qualify this error as the result of a violation of the logical law of contradiction. However, even without knowing about this law, people successfully use it in practice. And finally, everyone is familiar with the situation when we tell our interlocutor (or he tells us) something like the following: Why should I trust you? How will you prove this? Justify! In this case, what is happening is nothing more than a practical and (intuitive) use of the law of sufficient reason, which, most likely, those people who have not specifically studied logic are not aware of. However, this does not at all prevent them from unknowingly using this law.

So, practically we use logic long before we begin to study it theoretically. The same thing happens with our native language: we begin to use it at 2.5–3 years of age, and learn it only from school age. Why do we study? native language at school, if we already know it well? In order to own it even better. It’s the same with logic: using it intuitively and every day, we can get acquainted with it as a science, study it - in order to master it much better and more effectively. When we study logic, our intuition is supplemented and reinforced, honed and systematized, improved and enriched with theoretical knowledge, which raises us to a new, higher level of intellectual life.

Entertaining tasks

The proposed tasks vary significantly both in the type of their construction and in the level of complexity. One of them…

Problem conditions

1. Each of 10 bags contains 10 coins. Each coin weighs 10 grams. But in one bag all the coins are counterfeit - not 10, but 11 grams each... 2. The labels on all three iron cans of cookies are mixed up: “Oatmeal cookies”, “Shortbread cookies” and “Chocolate...

- How old is your father? - they ask the boy.

“The same as me,” he replies calmly.

- How is this possible?

– It’s very simple: my father became my father only when I was born, because before I was born he was not my father, which means that my father is the same age as me.

Is this reasoning correct? If not, what mistake was made in it?

77. There are 24 kilograms of nails in a bag. How can you measure 9 kilograms of nails on a cup scale without weights?

78. Peter lied from Monday to Wednesday and told the truth on other days, and Ivan lied from Thursday to Saturday and told the truth on other days. One day they said the same thing: “Yesterday was one of the days when I lie.” What day was yesterday?

79. A three-digit number was written down in numbers and then in words. It turned out that all the numbers in this number are different and increase from left to right, and all the words begin with the same letter. What number is this?

80. In an equation made up of matches:

Х I I I = V I I–V I,

a mistake was made. How should one match be rearranged for the equality to be true?

81. How many times will it increase three digit number, if the same number is added to it?

82. If there were no time, there would not be a single day. If there were not a single day, it would always be night. But if it was always night, then there would be time. Therefore, if there were no time, there would be time. What is the reason for this misunderstanding?

83. There are 12 apples in each of the two baskets. Nastya took several apples from the first basket, and Masha took from the second as much as was left in the first. How many apples are left in the two baskets together?

84. One farmer has 8 pigs: 3 pink, 4 brown and 1 black. How many pigs can say that in this small herd there is at least one other pig of the same color as her own?

85. The only son of the shoemaker's father is a carpenter. How does a shoemaker relate to a carpenter?

86. If 1 worker can build a house in 5 days, then 5 workers can build it in 1 day. Therefore, if 1 ship crosses Atlantic Ocean in 5 days, then 5 ships will cross it in 1 day. Is this statement true? If not, what is the mistake made in it?

87. Returning from school, Petya and Sasha went into a store, where they saw large scales.

“Let’s weigh our portfolios,” Petya suggested.

The scales showed that Petya’s briefcase weighed 2 kilograms, and the weight of Sasha’s briefcase turned out to be 3 kilograms. When the boys weighed the two briefcases together, the scales showed 6 kilograms.

- How so? – Petya was surprised. – After all, 2 plus 3 does not equal 6.

– Don’t you see? – Sasha answered him. – The arrow on the scale has moved.

What is the actual weight of the portfolios?

88. How to place 6 circles on a plane so that you get 3 rows of 3 circles in each row?

89. After seven washes, the length, width and height of the bar of soap were halved. How many washes will the remaining piece last?

90. How to cut 1/2 m from a piece of material 2/3 m long without any help measuring instruments?

91. It is often said that one must be born a composer (or an artist, or a writer, or a scientist). Is this true? Do you really have to be born a composer (artist, writer, scientist)?

92. You don't have to have eyes to see. Without the right eye we see. We also see it without the left one. And since we have no other eyes besides the left and right eyes, it turns out that not a single eye is necessary for vision. Is this statement true? If not, what mistake was made in it?

93. The parrot lived less than 100 years and can only answer yes and no questions. How many questions should he be asked to find out his age?

94. How many cubes are shown in Fig. 51?

95. Three calves - how many legs?

96. One man who was in captivity says the following: “My dungeon was in the upper part of the castle. After many days of effort, I managed to break out one of the bars in the narrow window. It was possible to crawl into the resulting hole, but the distance to the ground was too great to simply jump down. In the corner of the dungeon I found a rope forgotten by someone. However, it turned out to be too short to go down. Then I remembered how one wise man lengthened a blanket that was too short for him by cutting off part of it from the bottom and sewing it on top. So I hastened to divide the rope in half and tie the two pieces together again. Then it became long enough, and I safely went down it.” How did the narrator manage to do this?

97. The interlocutor asks you to think of any three-digit number, and then asks you to write down its digits in reverse order to make another three-digit number. For example, 528–825, 439–934, etc. Next, he asks from more subtract the smaller one and tell him the last digit of the difference. After this he names the difference. How he does it?

98. Seven walked and found seven rubles. If not seven, but three had gone, would they have found much?

99. Divide the drawing, consisting of seven circles, into seven parts with three straight lines so that each part contains one circle (Fig. 52).

100. The globe was pulled together with a hoop along the equator. Then the length of the hoop was increased by 10 meters. At the same time, a small gap formed between the surface of the globe and the hoop. Will a person be able to crawl through this gap? The length of the earth's equator is approximately 40,000 kilometers.

Answers and comments

1. You need to take out one coin from the first bag, two from the second, three from the third, etc. (all 10 coins from the tenth bag). Next comes one... 2. You need to take cookies from a jar labeled “Oatmeal cookies” (you can use any other one). Since the can is labeled...

81. 1001 times. In order to establish this, you need to divide the six-digit number obtained by duplicating a three-digit number by this three-digit number. The result is 1001 (see also problem 51).

82. The error of this reasoning lies in the statement that if there were no time, then there would not be a single day, which means it would always be night. Just the opposite - if there was no time, then there could not be a single day and not a single night, because the concept of night (like the concept of day) refers specifically to time (both day and night are certain time intervals).

83. Let's take the number of apples that Nastya took from the first basket as X, then in the first basket there are (12 – X) apples. That’s exactly how many apples Masha took from the second basket. This means there is left in the second basket

(12 – (12 – X)) apples.

Left in two baskets together

(12 – X) + 12 – (12 – X) = 12 – X + 12–12 + X = 12.

There are 12 apples left in the two baskets together.

84. No pig can say this, because pigs, as you know, do not speak. This not very serious problem is based on the ambiguity of the question: “How many pigs can say...?” The word “say” in this question can be taken literally - to speak articulately human speech, and can also be perceived in figurative meaning- someone speaks on behalf of or for those who themselves cannot (do not know how) to speak.

85. A shoemaker and a carpenter are one person. This can be easily verified by compiling simple diagram:

86. The reasoning is incorrect. The mistake is mixing the two various situations in the same words. When workers build a house, their efforts add up, so the work goes faster and is completed in less time. short term. When ships cross the Atlantic Ocean, their “efforts” do not add up: each ship crosses the ocean alone anyway, and therefore the time spent crossing the ocean does not decrease as the number of ships increases.

87. The arrow on the scales was shifted not to the right of zero, but to the left, i.e. the scales showed 1 kilogram less. This means that Petya’s briefcase weighs 3 kilograms, and Sasha’s - 4 kilograms. Together their briefcases weigh 7 kilograms. When the boys weighed them, the scales showed 1 kilogram less, i.e. 6 kilograms.

88. At first sight, In a similar way You can place only 9 circles, but the condition does not say that the rows of circles must be horizontal or vertical. They can be anything. You can arrange the circles different ways(Fig. 63).

89. It may seem that the remaining piece is enough for seven washes. However, it is not. If the length, width and height of a bar of soap are halved, then its volume is reduced not by 2 times, but by 8 times:

If after seven washes the volume of a bar of soap has decreased by 8 times, it means that the remaining bar is enough for just one wash (Fig. 64).

90. A 2/3 meter piece of material must be folded in half. The resulting fold line will divide it into two equal parts of 1/3 meter each. Then you need to fold it in half again. The resulting fold lines will divide the piece of material into four equal parts of 1/6 meter each. Three such parts are 3/6 of a meter, or the desired 1/2 of a meter (Fig. 65).

91. Of course, a composer, as well as an artist, writer or scientist, must be born, because if a person is not born, then he will not be able to compose music, draw pictures, write novels or do scientific discoveries. This joke problem is based on the ambiguity of the question: “Do you really have to be born?” This question can be taken literally: is it necessary to be born in order to engage in any type of activity; and this question can be understood in figuratively: is the talent of a composer (artist, writer, scientist) innate, given by nature, or is it acquired during life through hard work?

92. The reasoning is, of course, incorrect. Its external correctness is based on the almost imperceptible exclusion of one more option, which also needed to be considered in this argument. This is an option when no eye can see. It was he who was missed: “We see without the right eye, without the left one too, which means that eyes are not necessary for vision.” The correct statement should be: “Without the right eye we see, without the left we also see, but without the two together we do not see, which means we see either with one eye, or with the other, or with both eyes together, but we cannot see without eyes, which, thus essential for vision.”

93. At first glance, it may seem that you can ask a parrot up to 99 questions. In reality, you can get by with a much smaller number of questions. Let’s ask him this way: “Are you over 50 years old?” If he answers yes, then his age is from 51 to 99 years; if he answers “no,” then he is from 1 to 50 years old. The number of options for his age after the first question is halved. Next similar question: “Are you over (you can ask, less than) 25 years old?”, “Are you over (less than) 75 years old?” (depending on the answer to the first question) reduces the number of options by 4 times, etc. As a result, the parrot needs to be asked only 7 questions.

94. This drawing can be seen in different ways. Take a closer look at it, and you will notice how the image will turn over in one direction or the other, as if shimmering before your eyes. In one case we see six cubes - three on top, two in the middle and one on the bottom, and in the other case we see one cube in the middle of the picture. Thus, there are seven cubes in total in the figure.

95. You can rub the calf as long as you like, but no matter how many calfs there are, it will still have four legs. This joke task is based on the fact that the numeral “three” has a homonym - a verb in the imperative mood.

96. The narrator divided the rope not across, as most likely it might seem, but along, making two ropes out of it same length. When he tied the two pieces together, the rope became twice as long as it was originally.

97. When subtracting smaller number From the larger one, one regularity applies: the sum of all digits of the difference will always be equal to 18 (regardless of the original numbers). In addition, the second digit of the difference will always be 9. Thus, knowing the last digit of the difference (or the first), you can accurately determine the entire difference.

98. If not seven, but three had gone, they would still have found the same seven rubles.

99. See fig. 66.

100. At first glance, it may seem that the gap will be so small (after all, 10 meters is almost nothing compared to 40,000 kilometers) that not only a person, but even a cat will not be able to fit into it. In fact, the size of the gap will be approximately 1.6 meters, i.e. a person will not only be able to crawl into it, but even walk through it (maybe by slightly tilting his head).

As you know, the circumference of a circle is 2ϖ R, Where R – its radius. This means that the radius of the circle is equal to L/2ϖ, where L – circumference. Thus, the length of a circle and its radius are in a relationship of direct proportionality, but the radius is less than the length.

Increasing the length of the equatorial hoop is an increase in the circumference. Using the above formula, it is easy to establish the increase in its radius, which will be the size of the gap formed between the hoop and the surface globe. By making simple calculations, you will see that if the length of the equatorial hoop increases by just 1 meter, its radius increases by approximately 16 centimeters. A cat can crawl into such a gap. Increasing the length of the hoop by 10 meters (as in the problem statement) increases the gap by approximately 1.6 meters, and a person can pass through it. If the length of the equatorial hoop increases by 100 meters, then the gap will be approximately 16 meters. A five-story building could easily fit into such a gap.

Svintsov V.I. Logics. Elementary course For humanitarian specialties. – M.: Skorina, 1998. P. 68.

Leonhard Euler - famous mathematician XVIII century

Concepts modern natural science. Ed. V. N. Lavrinenko and V. P. Ratnikova. – M.: UNITY, 1997. P. 264.

Svintsov V.I. Logics. Elementary course for humanities majors. – M.: Skorina, 1998. P. 60–61.

Svintsov V.I. Logics. Elementary course for humanities majors. – M.: Skorina, 1998. P. 144.

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Dmitry Alekseevich Gusev

Amazing logic

Preface

Have you ever read a non-fiction or educational book and felt like you didn't quite understand what it said? If so, then, most likely, based on the “presumption of innocence of the author,” you blame yourself for this: for your insufficiently high level of education, narrow-mindedness, lack of necessary abilities. However, it would be more correct to proceed from the “presumption of your own innocence,” since if you carefully read, but do not understand the text addressed to you (according to the book annotation), then it is not you who are to blame, but the author. After all, he undertook to write a book for you, and not for himself or a narrow circle of his colleagues. However, for an intelligible and intelligible presentation of the material, he did not have enough logical culture.

What is logical culture? This is knowledge and observance of the basic principles and requirements of the correct construction and expression of thoughts both in oral and written speech. The absence of such a culture leads to a variety of logical errors that clog not only scientific, but also everyday thinking, preventing us from thinking, communicating, understanding each other and ourselves. The ambiguity and uncertainty of thinking, its inconsistency and confusion, inconsistency and groundlessness are a direct result of the lack of the proper level of logical culture.

I undertook to write a book not for myself, but for the reader who is beginning to master logic “from scratch.” How far I succeeded is for the reader to judge.

The book consists of five chapters and one hundred entertaining problems. The first three chapters are devoted to the forms of thinking in which the entire world of our thoughts, infinite in content, is expressed: concept, judgment and inference. In the fourth the head goes we are talking about the basic laws of logic and their common violations. The fifth chapter is devoted to the conditions and methods of discussion. The examples given in the book are intended to show that logic is not old, dry and lifeless wisdom, but an eternally young, useful and even interesting science, which may well help a person in life.

One hundred entertaining logic problems that complete the book differ both in the type of their construction and in the level of complexity. To solve them correctly requires a non-standard approach and creative work of thought. The tasks are aimed at developing thinking, memory, attention and imagination; they will help you spend your leisure time interestingly and usefully. To solve problems, theoretical knowledge of logic is not required; life experience and ingenuity are enough, that is, intuitive logic, which all people possess to a greater or lesser extent, regardless of gender, age and level of education. Answers and comments are provided for all problems. However, don’t rush to look into them, try to “rack your brain” and cope with them without any hints - to experience the joy of solving it yourself.

Introduction

Logics is the science of the forms and laws of correct thinking. It appeared around the 4th century BC. e. in Ancient Greece. Its creator is considered to be the famous ancient Greek philosopher and scientist Aristotle. As you can see, logic is approximately 2.5 thousand years old. However, it still retains its practical significance. Many sciences and arts of the Ancient world are forever a thing of the past and are of only “museum” value to us, interesting solely as ancient monuments, but some of them have survived centuries, and at present we continue to use them. These include Euclid's geometry (which is what we study at school) and Aristotle's logic, also called traditional logic. In the 19th century, symbolic (or mathematical) logic appeared and began to develop rapidly. In traditional logic, natural language (the one we speak, write, read) is used to study correct thinking, and in symbolic logic an artificial language, or the language of symbols, similar to the language of mathematics, is used. Symbolic logic is a rather specific and difficult science; it can be considered as a branch of mathematics and computer science. Aristotelian logic, on the contrary, being broader, is a kind of universal science: its mastery is equally useful and even necessary for every person, regardless of which areas of knowledge and subjects are closer to him - social, humanitarian, natural, mathematical or technical . Therefore, our book is devoted to Aristotelian, or traditional, logic.

Let's give a simple example. Let's look at three statements: All crucian carp are fish; All triangles are geometric shapes; All chairs are pieces of furniture. Despite their different content, these statements have something in common, something that unites them. What is this? They are united by form. Although they differ in content, they are similar in form, because each of the three statements is built according to form All A's are B's Where A And IN - any objects. It is clear that the statement itself All A's are B's devoid of any content. This statement is a pure form that can be filled with any content, for example: All pines are trees; All cities are populated areas; All schools are educational institutions; All tigers are predators and so on.

Another example. Let’s take three statements with different contents: If autumn comes, the leaves fall; If it rains tomorrow, there will be puddles on the street; If a substance is a metal, then it is electrically conductive. Although different in content, these statements are similar to each other in that they are constructed according to the same form: If A, then B. It is clear that many meaningful statements can be selected for this form, for example: If you don’t prepare for the test, you can get a bad mark; If the runway is covered with ice, planes will not be able to take off; If a word appears at the beginning of a sentence, it must be capitalized and so on.

Logic is not interested in the content of thinking (other sciences deal with this), it studies only the forms of thinking; she's not interested in What we think, otherwise How we think, that's why it is often called formal logic. For example, if the content of the statement All mosquitoes are insects is normal, and the statement All Cheburashkas are aliens - absurd, then for logic these two statements are equivalent, since it deals with forms of thinking, and the form of these statements is the same: All A's are B's.

As we see, form of thinking is a way of expressing thoughts, or a scheme for constructing them. There are three forms of thinking: concept, judgment and inference.

Logic is not taught in school. Nevertheless, we use its laws from childhood: we learn to think and make decisions, comprehend what is happening, comprehend various sciences and, most importantly, we communicate with other people - we explain our position, object, argue, convince...

Modern smart developed person simply must own logical thinking– it organizes acquired knowledge, gives clarity of speech, makes convincing arguments and allows you to achieve victory in discussions.

The book “Amazing Logic” requires a certain tension mental strength and can serve as a kind of verification of basic logical abilities person. At the same time, it allows you to develop personal intelligence and creative search skills. non-standard solutions. In a word, it teaches you to think.

The original logic problems given at the end of the publication also serve testing and developmental purposes.

The book is addressed primarily to high school students and students who are interested in logic and want to actively use its laws to achieve personal success.

Preface

logical culture

I undertook to write a book not for myself, but for the reader who is beginning to master logic “from scratch.” How far I succeeded is for the reader to judge.

The book consists of five chapters and one hundred entertaining problems. The first three chapters are devoted to the forms of thinking in which the entire world of our thoughts, infinite in content, is expressed: concept, judgment and inference. The fourth chapter deals with the basic laws of logic and their common violations. The fifth chapter is devoted to the conditions and methods of discussion. The examples given in the book are intended to show that logic is not old, dry and lifeless wisdom, but an eternally young, useful and even interesting science, which may well help a person in life.

Introduction

is the science of the forms and laws of correct thinking. It appeared around the 4th century BC. e. in Ancient Greece. Its creator is considered to be the famous ancient Greek philosopher and scientist Aristotle. As you can see, logic is approximately 2.5 thousand years old. However, it still retains its practical significance. Many sciences and arts of the Ancient world are forever a thing of the past and are of only “museum” value to us, interesting solely as ancient monuments, but some of them have survived centuries, and at present we continue to use them. These include Euclid's geometry (which is what we study at school) and Aristotle's logic, also called

traditional logic

In the 19th century, symbolic (or mathematical) logic appeared and began to develop rapidly. In traditional logic, natural language (the one we speak, write, read) is used to study correct thinking, and in symbolic logic an artificial language, or the language of symbols, similar to the language of mathematics, is used. Symbolic logic is a rather specific and difficult science; it can be considered as a branch of mathematics and computer science. Aristotelian logic, on the contrary, being broader, is a kind of universal science: its mastery is equally useful and even necessary for every person, regardless of which areas of knowledge and subjects are closer to him - social, humanitarian, natural, mathematical or technical . Therefore, our book is devoted to Aristotelian, or traditional, logic.

Amazing logic

Logic is not taught in school. Nevertheless, we have been using its laws since childhood: we learn to think and make decisions, comprehend what is happening, comprehend various sciences and, most importantly, communicate with other people - we explain our position, object, argue, convince...

A modern intelligent, developed person simply must master logical thinking - it organizes acquired knowledge, gives clarity to speech, makes convincing arguments and allows you to achieve victory in discussions.

The book “Amazing Logic” requires a certain amount of mental effort and can serve as a kind of test of a person’s basic logical abilities. At the same time, it allows you to develop personal intellectual data and creative skills in finding non-standard solutions. In a word, it teaches you to think.

The original logic problems given at the end of the publication also serve testing and developmental purposes.

The book is addressed primarily to high school students and students who are interested in logic and want to actively use its laws to achieve personal success.

Dmitry Alekseevich Gusev

Amazing logic

Preface

I undertook to write a book not for myself, but for the reader who is beginning to master logic “from scratch.” How far I succeeded is for the reader to judge.

The book consists of five chapters and one hundred entertaining problems. The first three chapters are devoted to the forms of thinking in which the entire world of our thoughts, infinite in content, is expressed: concept, judgment and inference. The fourth chapter deals with the basic laws of logic and their common violations. The fifth chapter is devoted to the conditions and methods of discussion. The examples given in the book are intended to show that logic is not old, dry and lifeless wisdom, but an eternally young, useful and even interesting science, which may well help a person in life.

Introduction

Logic is not interested in the content of thinking (other sciences deal with this), it studies only the forms of thinking; she's not interested in What we think, otherwise How we think, that's why it is often called formal logic. For example, if the content of the statement All mosquitoes are insects is normal, and the statement All Cheburashkas are aliens - absurd, then for logic these two statements are equivalent, since it deals with forms of thinking, and the form of these statements is the same: All A's are B's.

As we see, form of thinking is a way of expressing thoughts, or a scheme for constructing them. There are three forms of thinking: concept, judgment and inference.

Concept is a form of thinking that designates an object or a feature of an object. Examples of concepts: pencil, plant, celestial body, chemical element, courage, stupidity, carelessness.

Judgment- this is a form of thinking that consists of concepts related to each other and affirms or denies something. Examples of judgments: All planets are celestial bodies, Some schoolchildren are poor students, All triangles are not squares.

Inference is a form of thinking in which a new judgment (conclusion) follows from two or more initial judgments (premises).

In logic, it is customary to place the premises and the conclusion under each other and to separate the conclusion from the premises (in the book this is done using the => sign).

Examples of inferences:

All planets are moving.

Jupiter is a planet.

=> Jupiter is moving.

Iron is electrically conductive.

Copper is electrically conductive.

Mercury is electrically conductive.

Iron, copper, mercury are metals.

=> All metals are electrically conductive.

The entire endless world of our thoughts is expressed in concepts, judgments and conclusions. These three forms of thinking will be discussed on the pages of the book.

Common sense and life experience alone are often enough to solve any problems. For example, anyone unfamiliar with logic can find a catch in the following reasoning:

Movement is eternal.

Going to school is movement.

=> Going to school forever.

A false conclusion is obtained due to the use of the word movement in different meanings: in the first judgment it is used in a broad, philosophical sense, and in the second - in a narrow, mechanical sense. However, finding errors in reasoning is not always easy. Consider this example:

All my friends speak English.

The current president of America speaks English.

=> The current President of America is my friend.

It is clear that there is something wrong with this reasoning. But what exactly? Anyone who is familiar with logic will say that in this case an error was made, which is called “non-distribution of the middle term in a simple syllogism.” Don’t let this unfamiliar and, at first glance, complex expression scare you: as you continue reading the book, you will be convinced that there is nothing complicated, much less incomprehensible, here.

Or this example:

All cities above the Arctic Circle have white nights.

St. Petersburg does not lie beyond the Arctic Circle.

=> There are no white nights in St. Petersburg.

Philosophers and medieval European logicians. Founder logic in ancient Greek philosophy... - turns into law. Not marvelous, that the supreme law of the universe is... from the directions of modern mathematical logic - logic predicates. An important step V...

Logics (20)

Abstract >> Logic

1. Brief essay stories logic Main directions in history logic Initially logics originated and developed in the depths of philosophy... the study of the phenomena of the micro- and macroworld came to amazing, a paradoxical discovery: the passage of time depends...

Stages of development logic as sciences and the main directions of modern symbolic logic

Law >> Logic

Deep: “In the very idea of non-uniqueness logic of course there is nothing amazing. In fact, with what... there is no reason. Amazing, on the contrary, it would be if logics was the only one...

The problem began in "Science" logic" G.W.F. Hegel.

Coursework >> Philosophy

On understanding the method of scientific presentation, engaged in presentation logic, giving away all this comprehension in mental definitions...which is not. But the irony of history amazing- the idea was declared the basis... written. 8) What should it be logics. Logics, as a developmental science...

I undertook to write a book not for myself, but for the reader who is beginning to master logic “from scratch.” How far I succeeded is for the reader to judge.

The book consists of five chapters and one hundred entertaining problems. The first three chapters are devoted to the forms of thinking in which the entire world of our thoughts, infinite in content, is expressed: concept, judgment and inference. The fourth chapter deals with the basic laws of logic and their common violations. The fifth chapter is devoted to the conditions and methods of discussion. The examples given in the book are intended to show that logic is not old, dry and lifeless wisdom, but an eternally young, useful and even interesting science, which may well help a person in life.

To solve problems, theoretical knowledge of logic is not required; life experience and ingenuity are enough, that is, intuitive logic, which all people possess to a greater or lesser extent, regardless of gender, age and level of education. Answers and comments are provided for all problems. However, don’t rush to look into them, try to “rack your brain” and cope with them without any hints - to experience the joy of solving it yourself.

Introduction

Logic is not interested in the content of thinking (other sciences deal with this), it studies only the forms of thinking; she's not interested in What we think, otherwise How we think, that's why it is often called formal logic. For example, if the content of the statement All mosquitoes are insects is normal, and the statement All Cheburashkas are aliens - absurd, then for logic these two statements are equivalent, since it deals with forms of thinking, and the form of these statements is the same: All A's are B's.

As we see, form of thinking is a way of expressing thoughts, or a scheme for constructing them. There are three forms of thinking: concept, judgment and inference.

Concept is a form of thinking that designates an object or a feature of an object. Examples of concepts: pencil, plant, celestial body, chemical element, courage, stupidity, carelessness.

Inference is a form of thinking in which a new judgment (conclusion) follows from two or more initial judgments (premises).

In logic, it is customary to place the premises and the conclusion under each other and to separate the conclusion from the premises (in the book this is done using the => sign).

Examples of inferences:

All planets are moving.

Jupiter is a planet.

=> Jupiter is moving.

Iron is electrically conductive.

Copper is electrically conductive.

Mercury is electrically conductive.

Iron, copper, mercury are metals.

=> All metals are electrically conductive.

The entire endless world of our thoughts is expressed in concepts, judgments and conclusions. These three forms of thinking will be discussed on the pages of the book.

Common sense and life experience alone are often enough to solve any problems. For example, anyone unfamiliar with logic can find a catch in the following reasoning:

Movement is eternal.

Going to school is movement.

=> Going to school forever.

All my friends speak English.

The current president of America speaks English.

=> The current President of America is my friend.

It is clear that there is something wrong with this reasoning. But what exactly? Anyone who is familiar with logic will say that in this case an error was made, which is called “non-distribution of the middle term in a simple syllogism.” Don’t let this unfamiliar and, at first glance, complex expression scare you: as you continue reading the book, you will be convinced that there is nothing complicated, much less incomprehensible, here.

Or this example:

All cities above the Arctic Circle have white nights.

St. Petersburg does not lie beyond the Arctic Circle.

=> There are no white nights in St. Petersburg.

So, common sense and life experience are usually enough to navigate various difficult situations. But if we add logical culture to our common sense and life experience, then we will only benefit from this. Of course, logic will not solve all problems, but it can certainly help in life.

Concept

Names of things (What is a concept)

In the world around us there is infinite set various objects and properties, and in our consciousness they are reflected in the form of concepts.

Concept is a form of thinking that denotes an object or its property. For example, we call one object mountain, another - celestial body third - plant; one property or sign we call courage, another - cunning. In language, any concept is expressed in a word or phrase, for example: house, autumn leaf, America's first president. Here it may seem that the concept and the word are one and the same thing: for example, the concept Human expressed in a word Human. However, a concept and a word are different things. A concept is a mental designation of an object (a thought about it), and a word is linguistic expression this thought. A concept is a form of thinking, and a word is a form of language. This can best be understood with an example. Concept Human for representatives of all peoples and nationalities - the same thing: a mental reflection or designation of a person, and not a plant, a celestial body, geometric figure or molecules. But the concept Human V different languages will be expressed in completely different words.

Each concept has content and scope.

Contents of the concept- this is the most important sign (or signs) of the object that is designated (expressed) by this concept. For example, to establish the content of the concept Human it is necessary to indicate such a sign that is the most important for a person, which distinguishes him from all other creatures, objects and objects. Such a sign for a person is the presence of reason. Therefore, the content of the concept Human There is only one important sign - the presence of reason. And in the content of the concept man already includes two important signs: the presence of intelligence (this sign is repeated, because any man is a person) and belonging to a certain gender (to one of the halves of humanity; word floor comes from the word half). And if you need to establish the content of the concept Russian man, then three important signs should be indicated: the presence of intelligence, belonging to a certain gender and belonging to a certain nationality. Thus, the content of a concept can include both one sign of an object (or objects) and two or more signs, and their number depends on the object that is denoted by this concept. But why in one case does the content of a concept consist of a single attribute, and in the other - from many attributes? This question is not difficult to answer if you know what the scope of a concept is.

Scope of concept is the number of objects covered by this concept and included in it. For example, the scope of the concept Human much more than the scope of the concept man, because there are fewer men than people in general. And the scope of the concept Russian man much less than the scope of the concept man, because there are much fewer Russian men in the world than all men in general. And finally, the scope of the concept first president of Russia is equal to one because it includes only one person. In the same way, the scope of the concept city very broad, since this concept covers all cities in the world, and the scope of the concept capital less scope of concept city, After all, there are much fewer capitals than cities.

The scope of the concept current capital of Russia is equal to one because it includes only one city.

Let's once again return to the content and scope of the concept and recall the examples given above. What concept - Human or man - more content? Of course, the concept man, because its content includes two characteristics: the presence of intelligence and belonging to a certain gender, and the content of the concept Human includes only one sign: the presence of intelligence. Now let’s answer the question: what concept - Human or man - more in volume? Concept Human more because it covers many more objects than the concept man. Thus, between the volume and content of a concept there is inverse relation: the greater the content of a concept, the smaller its volume, and vice versa. For example, the content of the concept heavenly body is narrow, since it includes only one sign - being outside the Earth, but in scope this concept is very broad, because it covers a huge number of objects: any star, planet, meteorite, comet is a celestial body. And the concept Sun, on the contrary, it is very narrow in volume, since it includes only one object, but very wide, rich in content, which consists of many features: the size of the Sun, its mass, density, chemical composition, temperature, age, etc.

All concepts are divided into several types in terms of volume and content. By volume they are single(the volume includes only one object, for example: The sun, the city of Moscow, the first president of Russia, writer Leo Tolstoy), general(the scope includes many objects, for example: celestial body, city, president, writer) And zero(the scope does not include a single object, for example: Baba Yaga, Koschey the Immortal, Father Frost, perpetual motion machine, Martian inhabitant, that is, the concept exists, but the object it denotes does not exist). In terms of scope, concepts can also be collective(denote objects that consist of, are assembled from some limited set of elements, are divided, fall apart into some component parts, for example: company of soldiers, musical group, Wolf Pack, constellation) And non-collective(denote objects that are not assembled from a limited set of elements, are not divided into any component parts, but are something single, whole, for example: person, plant, star, ocean, pencil).

According to the content of the concepts there are specific(denote an object, for example: table, mountain, tree, planet) And abstract(they do not denote an object, but a sign, a property, for example: courage, stupidity, sloppiness, darkness). According to the content of the concept, there are also positive(denote the presence of something, for example: animal, school, truth, tact) And negative(denote the absence of something, for example: not an animal, not a school, untruth, tactlessness). It is easy to notice that a concept is negative when the word by which it is expressed is used with the particle Not or with prefix without-. However, if the particle Not is part of a word that is not used without it (for example: slob, sloppiness, bad weather, carelessness, ignorance), then the concept expressed by such a word will be positive.

Any concept can be given a logical characterization, that is, it can be analyzed by volume and content. First you need to determine whether it is singular, general or zero, then establish whether it is collective or non-collective, then find out whether it is concrete or abstract, and, finally, answer the question whether it is positive or negative.

For example, the concept Sun - individual (because its volume includes only one object, one celestial body), non-collective (since the Sun does not consist of any parts, is not divided into them), specific (after all, the Sun is an object, not a sign or property), positive (because this concept denotes the presence, not the absence, of an object). Similar plant - this concept is general, non-collective, specific and positive, and the concept constellation Orion - singular, collective, concrete and positive.

A young man and a bad habit
(Definite and indefinite concepts)

The concept is certain when it has clear content and sharp volume. As we already know, the content of a concept is the most important signs of the object that it expresses, and the volume is the number of objects it covers. Thus, a concept has a clear content if it is possible to accurately indicate a set of essential features of the expressed object, as well as accurately establish the boundary between those objects that this concept covers and those that do not belong to its scope. For example, the concept master of Sport is certain. It has a clear content, since one can precisely indicate its most important distinguishing feature - officially possessing the sports rank of master of sports. This concept has a sharp scope - with respect to any person one can say for sure whether he is a master of sports or not, i.e. whether he falls into the scope or does not fall into the scope this concept; in other words, it is possible to draw a line between all masters of sports and all those who are not, to accurately separate some from others.

The concept is uncertain, when it has unclear content and blurred volume. A concept is characterized by unclear content if it is impossible to accurately indicate the most important features the object that it expresses; and the blurred scope of a concept indicates the impossibility of drawing a precise boundary between those objects that are included in the scope of this concept and those that are not included in it. For example, the concept good athlete is uncertain. It has unclear content, since it is impossible to accurately indicate the essential characteristics of a good athlete, because it is impossible to unambiguously answer the question of who should be considered a good athlete. Either this is someone who has a rank of at least master of sports, or someone who has set at least one world record, or a multiple Olympic champion, or someone who considers himself such. It is clear that opinions different people as to who should be classified as good athletes will differ: some will say one thing, others another. Also, this concept has a vague scope - with respect to any person it is impossible to say for sure whether he is a good athlete or not, that is, whether he falls within the scope of this concept or not; in other words, there is no way to draw a line between the many good athletes and all those who are not.

Reviews of the book "Amazing Logic" by Dmitry Gusev. It is necessary to select a variety of parcels

Concept

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Judgment

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Preface

Introduction

Amazing logic

Dmitry Alekseevich Gusev

Amazing logic

Preface

Introduction

Logics (20)

Stages of development logic as sciences and the main directions of modern symbolic logic

The problem began in "Science" logic" G.W.F. Hegel.

Introduction

Concept

Names of things (What is a concept)

A young man and a bad habit (Definite and indefinite concepts)

A young man and a bad habit
(Definite and indefinite concepts)