Inference schemes in logic examples. Deductive reasoning (propositional logic). The study of forms of thought

Well, we got to the most important thing. The main task of logic is the analysis of reasoning, and reasoning is made up of sentences and words, or, in other words, of judgments and concepts. Therefore, we began our acquaintance with logic by considering those simple elements from which complex mental structures are formed. Now you can get acquainted with these structures themselves.

Inference is a form of thinking in which a new judgment is obtained from one or more judgments on the basis of certain rules.

Our reasoning in Everyday life or in the professional sphere - these are inferences or chains of inferences. Inference is a means of extracting new knowledge from existing knowledge. The knowledge that we receive as a result of direct contact with environment, is very small - it does not greatly exceed the knowledge of animals. But on this small foundation, man has erected a colossal edifice that includes knowledge about stars and galaxies, about the structure of the atom and elementary particles, about the laws that govern heredity, about ancient civilizations, about vanished languages and the depths of the ocean. All this knowledge is obtained thanks to the ability of a person to build conclusions.

Sometimes the human mind is defined as the ability to draw conclusions, draw conclusions. Maybe the mind is not only in this, but, undoubtedly, the ability to draw conclusions and draw conclusions from the available information is one of its most important aspects. You look at the thermometer hanging outside the window in the morning and see that the mercury in it has dropped to -70°C. Here is everything you have. But from here you conclude that it is cold outside. You haven’t been outside yet, you haven’t felt the bites of the wind on your skin, but you already know that it’s cold there. Where did you get this knowledge? It gave you an inference. You can draw another conclusion: when going out, you need to dress warmly. You foresee the effect the frost will have on you. Foresight is also a conclusion. An intelligent person is one who is able to extract maximum new information from existing knowledge, to foresee the course of events and the consequences of his actions. Sherlock Holmes and his friend Dr. Watson often walk together, see and hear the same thing, but Holmes is able to extract much more from this than Watson, and therefore seems to us smarter and more insightful than his friend.

Any conclusion consists of two parts: those judgments from which we proceed, on which we rely in the conclusion, are called its premises, the new judgment, which we extract from the premises, is called the conclusion. All reasoning is divided into two large groups - deductive and inductive.

Such inferences are called deductive, in which the conclusion from the premises follows with necessity, i.e. If the premises of the inference are true, then the conclusion must be true. For example, if we know that all Gascons are French and d'Artagnan is a Gascon, then from this we can conclude that d'Artagnan is French. And this conclusion will certainly be true.

We will talk about inductive reasoning separately later (in the section "Induction"), and now we will get acquainted with some simple and most commonly used deductive reasoning. We intuitively use them in everyday reasoning, but we often make mistakes, because we do not realize what they are.

1) Along the walls of the square bastion, the commandant placed 16 sentries, 5 people on each side, as shown in the figure:

After some time, the colonel came, expressed dissatisfaction with the arrangement of sentries and rearranged them so that there were 6 people on each side. However, after that, the general appeared. He also expressed dissatisfaction and rearranged the sentries in such a way that there were 7 of them on each side.

How did the colonel arrange the sentries? How did the general arrange them? The total number of sentries remains the same.

Immediate inferences

Direct inferences are called inferences from one premise, which is a simple judgment.

The transformation consists in inserting two negations into our premise, one before the connective and the other before the predicate, and thus we obtain a new judgment. It is customary to depict inferences as follows: first, the premise (or premises) is written, a line is drawn under it, denoting the word “therefore”, and under the line the conclusion is written. Let our premise be a universally affirmative judgment, then the transformation looks like this:

All S's are P's

No S is a non-P

For example, the proposition "All metals are electrically conductive" becomes the proposition "No metal is non-conductive".

If we take a general negative judgment as a premise, then the transformation will look like this:

No S is P

All S is non-P

For example, the proposition "No swindler is an honest person" becomes the proposition "All swindlers are dishonest people". When we insert “not” before the link here, then two “not” are obtained in front of it. We eliminate them, relying on the principle: a double negative is equivalent to an affirmation.

Of course, the conclusion in such inferences gives very little new in comparison with the premise. This is quite natural, since, in fact, we only give the same judgment a different linguistic form. This is not so much a logical as a grammar game. However, a transformation of this kind is capable of making explicit some shades of the meaning of the original judgment that were hidden in the original formulation. We often use the transformation of judgments in everyday life when we want to express our thoughts more clearly and distinctly. This is part of our language ability.

Another type of direct inference is conversion. In inversion, the inference is obtained by putting the predicate of the premise in the place of the subject, and the subject of the premise in the place of the predicate. The general circulation scheme looks like this:

For example, from the proposition "Birds are vertebrates" we obtain the conclusion "Vertebrates are birds" by inversion. In order to actually carry out the conversion, we must not just swap the subject and the predicate, but make the object represented by the predicate of the sending the object of our thought, i.e. turn it into the subject of a new judgment. Sometimes, for example, the inversion is made as follows: from the proposition "All fish breathe with gills" they get the conclusion "All fish breathe with gills." There is no logical conversion operation here! We just swapped subject and verb. In order to obtain the reversal of the original judgment, we must make the "gill-breathers" the subject of our thought, and say of them: "The gill-breathers are fish."

In the premise, the subject is preceded by the word (quantifier): "all" or "some". The question arises: what should we put before the predicate of the premise when we make it the subject of the conclusion, “all” or “some”? "All gill-breathers" or only "some gill-breathers" eat fish? Trying to answer this question, we begin to think about the meaning of the concept of "breathing with gills", we remember, and who else, besides fish, could breathe with gills, perhaps frogs or some newts? You don't need all this! Logic is a formal science and is not at all obliged to know what frogs or fish are doing, just as mathematics, adding 2 and 3, is not at all interested in what you count - rubles, dollars or bricks. Logic sets formal rules that do not depend on the content of our concepts and judgments. In this case, the rule is this: if the premise is an affirmative judgment, then when referring to the predicate put the word "some"; if the premise is a negative proposition, then the word "all" is placed before the predicate. Our premise "All fish breathe with gills" is an affirmative proposition, so we can conclude from it "Some fish breathe with gills." But from the negative premise "No elephant lives in the Arctic" we can draw a general conclusion "Everyone living in the Arctic is not an elephant."

2) Three travelers wandered into an inn, ate well, paid the hostess 30 rubles. and move on. Some time after they left, the hostess discovered that she had taken too much from the travelers. Being an honest woman, she kept 25 rubles for herself, and 5 rubles. gave to the boy, telling him to catch up with the travelers and give them the money. The boy ran fast and soon caught up with the travelers. How do they divide 5 rubles. for three persons? Each of them took 1 ruble, and 2 rub. left the boy as a reward for speed.

Thus, they paid 10 rubles for lunch, but 1 rub. received back, therefore, they paid: 9x3 = 27 rubles. Yes 2 rub. left with the boy: 27 + 2 = 29 rubles. But in the beginning it was 30 rubles! Where did the 1 ruble go?

3) Once upon a time there were two shepherds, Ivan and Peter, they grazed sheep. And somehow Ivan says: “Listen, give me one sheep, then I will have 3 times more sheep than you!”. “No,” Peter answers, “you’d better give me one sheep, then we will have them equally!”

How many sheep did Ivan have and how many did Peter have?

The conclusions from one premise are simple. Somewhat more complex are the conclusions from two premises. Among them, one of the most common is a simple categorical syllogism. It was discovered in our everyday reasoning and described by Aristotle, and to a large extent that is why he is considered the creator of logic as a science. Here is an example of a simple categorical syllogism:

All people are mortal.

Socrates is a man.

Socrates is dead.

Here we already see two premises: "All people are mortal" and "Socrates is a man." From these two judgments we derive a new judgment, "Socrates of death." If you pay attention to your reasoning, you will very soon find that you often use this method of inference.

The concepts that make up the premises and conclusion of a syllogism are called its terms. There are only three terms in a syllogism.

The lesser term of the syllogism is the subject of the conclusion. It is denoted by the letter "S", as a subject in the structure of a simple proposition. But here this letter denotes a lesser term, which in the premise can also occur in the place of the predicate. In our example, the lesser term would be Socrates.

The big term of a syllogism is the inference predicate. It is denoted by the letter "P", as a predicate in the structure of a simple proposition, but here this letter denotes a larger term, which in the premise can also stand in the place of the subject. In our example, the big term will be the concept of "mortal".

Finally, the middle term of a syllogism is a concept that is included in both premises, but is absent in the conclusion. It is denoted by the letter "M". In our example, the middle term is the concept of "people". (The words "people" and "man" express the same concept, the difference between them is only grammatical, do not pay attention to it.)

A syllogism is a conclusion that speaks about the ratio of the volumes of the concepts included in it. The first premise says that the class of people is included in the class of mortal beings; the second premise says that Socrates belongs to the class of people; Based on these two relationships, we conclude that Socrates is included in the class of mortal beings.

We often build our reasoning in the form of a simple categorical syllogism, relying on our intuition. But we often get it wrong. Logic establishes some simple rules that help to avoid mistakes and incorrect conclusions.

For example, a syllogism should have only three terms. If a fourth term appears, the syllogism breaks down: we cannot find a middle term and draw a conclusion. You are given, say, the following parcels:

All artists are selfish.

Oleg Tabakov is talented.

There are four terms here. Which one is considered average? Which one is smaller or larger? They are simply two unrelated judgments from which no new knowledge can be extracted. The error associated with the violation of this rule is called “quadrupling terms”. It seems that this mistake is difficult to make. However, it is quite common and is due to the ambiguity of the words of our everyday language. The same word in one premise can be used in one sense, and in another premise - in a different sense and thus express two different concepts. It turns out four terms, although there are only three words. For example:

Movement is eternal.

Going to college is movement.

Going to college forever.

Here the word "motion" in one premise is used to express the philosophical concept of motion as a universal property of the material world, and in another premise it expresses the everyday, everyday concept of motion. Therefore, a ridiculous conclusion is obtained.

The coat is warm.

"Shuba" is a Russian word.

Some Russian words are warm.

Here the quotation marks show us that the word "fur coat" is used in different senses in the first and second premises. However, in oral speech, this difference may go unnoticed. The examples given are simple and transparent, but in many cases the quadrupling of terms is more subtle and not easy to recognize.

Another rule says: no conclusion can be drawn from two negative premises. For example:

The bright red flowers are odorless.

This flower is odorless.

Can we conclude that this flower is bright red? No, it can be any color.

The other rules of the syllogism are just as simple. Now take a look at the following four syllogisms and try to understand how they differ from each other.

All fish swim.

Pikes are fish.

Pikes swim.

Every person has two legs.

Pinocchio has two legs.

Pinocchio is a man.

You may notice that the middle term in these examples is in different places in the premises. In the first example, the middle term "fish" in the first premise is in the place of the subject, and in the second - in the place of the predicate. In the second, the middle term "has two legs" in both premises stands in place of the predicate. In the third, the middle term "birds" in both premises stands in the place of the subject. Finally, in the fourth example, the middle term "parallelogram" in the first premise is in place of the predicate, and in the second, in the place of the subject. All these are different ways of reasoning, built in the form of a simple categorical syllogism. They are called figures of the syllogism. In other words: the figures of a syllogism are its varieties, which differ from each other in the location of the middle term in the premises. There are only four figures. Here is their schematic representation:

Substituting different concepts for the letters "S", "P" and "M", we will get reasoning that looks like one of the figures of the syllogism.

However, in everyday speech we rarely use extended syllogisms, because our language is a great lazybones! He almost never fully says everything that we want to say (although sometimes he blurts out things that would be better kept silent). Pay attention to your speech, to the speech of your friends and acquaintances, and you will easily see how much we do not agree on, it is understood how easy it is to make a mistake when conjecturing the speech of the interlocutor. For example, two friends are talking:

- Well, how did your quarrel with your wife end yesterday?

“Oh, I made her kneel before me.

– That's how! And what did she say?

“Get out from under the bed, you vile coward!”

This is how we shorten our syllogisms, without explicitly expressing all its premises or conclusions in the hope that the interlocutor himself will think of the missing link and understand us. This is quite natural. It's hard to talk to a person who tends to say out loud even the most obvious things. He is reminiscent of Colonel Friedrich Kraus von Zillergut from the novel by J. Hasek "The Adventures of the Good Soldier Schweik", who loved to explain and explain everything and, as a result, earned the fame of the greatest donkey and bore. It is unlikely that you will endure such reasoning for a long time, for example: “The road, on both sides of which ditches stretch, is called a highway. Yes, gentlemen. Do you know what a ditch is? A ditch is a depression dug by a significant number of workers. Yes, sir. Digging ditches with pickaxes. Do you know what a pick is?"

A syllogism in which one of the parts, the premise or conclusion, is omitted and only implied is called an enthymeme. In everyday life, we use abbreviated syllogisms - enthymemes. This is quite natural, but it also causes many errors in our reasoning. When the syllogism is presented in full, the error is easy to notice. But if some part of it is omitted, implied, then it is precisely in it that the error can be hidden - either the implied part is false, or forms an incorrect syllogism. Suppose I arrogantly declare:

"This man is stupid because he doesn't know logic!" This is an enthymeme.

Restore the implied premise and write down the complete syllogism:

Any person who does not know logic is stupid.

This man does not know logic.

This person is stupid.

It immediately becomes clear that the implied and restored premise is false: not every person who does not know logic is stupid. Many people who have never studied logic have nonetheless a sharp and penetrating mind. Conversely, some people spend their whole lives occupied with logic, while remaining very narrow-minded personalities. Logic helps our reason, but still, you need to have reason - just like you need to have legs so that crutches help you.

4) There was a theft and three suspects were detained. One of them is a thief who constantly lies; the other is an accomplice and lies only occasionally; the third is an honest person who never lies. The inquiry began with questions about the profession of each of the detainees. The investigator received such answers.

Shchukin: I am a painter, Karasev is a piano tuner, and Okunev is a designer.

Karasev: I am a doctor, Okunev is an insurance agent. As for Shchukin, if you ask him, he will answer that he is a house painter.

Okunev: Karasev is a piano tuner, Shchukin is a designer, and I am an insurance agent.

Based on these answers, the investigator guessed who was who. Guess you too!

If you went to school, then, apparently, you remember a simple reasoning scheme that looks like: “If a, then b; if in, then with; therefore, if a, then c. For example, in arithmetic this reasoning is represented by the principle: if two quantities are separately equal to a third, then they are equal to each other. This kind of reasoning is called conditional syllogism: here both the premises and the conclusion are conditional propositions. Here is an example of a conditional syllogism taken from the story of V. Bilibin, a Russian writer of the early 20th century:

“If the Sun did not exist in the world, then we would have to constantly burn candles and kerosene.

If you had to constantly burn candles and kerosene, then the officials would not have enough of their salaries and they would take bribes. Therefore, officials do not take bribes because the sun exists in the world.

Even more common are reasonings in which one premise is a conditional proposition, the second premise and conclusion are simple categorical propositions. Such an argument is called a conditionally categorical syllogism. For example, when you feel unwell, the first thing you do is put yourself a thermometer. And when you come to the clinic, then again, you first put a thermometer. We proceed from the premise: "If a person has a fever, then the person is sick." If you really have a fever, then you are recognized as sick, released from work or school, your family members walk around you on tiptoe and try to give you tea with raspberries. At the same time, we argue as follows:

If a person has a fever, then the person is sick.

This person has a fever. Therefore, this person is sick. Let us present our reasoning in symbolic form. Let us denote the judgment “The person has a fever” by the letter A, the judgment “The person is sick” by the letter B. Then our reasoning will take the form:

(the arrow "->" is read as "if ... then"). We remember that the first part of the conditional premise is called the basis, the second - the consequence. The second premise of our reasoning asserts that the reason takes place, hence we conclude that the consequence must also take place. An argument of this form is called the affirmative mode of a conditionally categorical syllogism (or modus ponens, to use Latin): here we pass from the statement of the foundation to the statement of the consequence of the conditional premise.

However, with the same conditional premise, the reasoning can proceed differently. They put you a thermometer, but the temperature was normal. From this they conclude that you are not sick, you are not released from work, you are not given tea. The reasoning looks like:

With the same conditional premise, one can move towards a conclusion, affirming or denying its consequence. Thus, a conditionally categorical syllogism has only four modes:

The first and last are called "correct" modes: they provide valid inference; the second and third are "wrong" modes: they do not give a reliable conclusion - it is impossible to reason like that, it will lead to an error, which is easy to see.

You have not been found to have a fever, but each of us knows that this does not mean at all that you are not sick: many diseases are not accompanied by fever. Therefore, the conclusion that a person is not sick may be erroneous. In the third mode, from the fact that a person is sick, we conclude that he must have a fever. For the same reasons, this conclusion may be erroneous. Finally, the fourth mode tells us that if a person is not sick, then he does not have a temperature. This conclusion is quite reliable: if you are healthy, then your temperature is normal.

Thus, if you build your reasoning according to the first and last modes, you are reasoning correctly; if you build your reasoning according to the second or third mode, you risk making a mistake.

5) “Come here,” I once said to three students. - Here I have 5 hats: 3 white and 2 black. Close your eyes and I will put a hat on each of you. When you open your eyes, you can see what color hats your comrades are wearing. You will not be able to see your own hat, and you will not see what hats I have left. Anyone who guesses what color the hat is on will immediately receive a credit in logic.

After a while, without exchanging a single word, the students shouted: "I'm wearing a white hat!" I had to put all three of them off. Would you guess?

For example, you wake up in the morning and, while still in bed, you begin to reason: “This afternoon I can go on a date or to class. I'll go on a date. Therefore, I won’t go to class.” Here, the first premise of your argument is the disjunctive proposition "I can go on a date (A) or go to class (B)", symbolically: A v B. The second premise asserts one of the possibilities indicated in the disjunctive premise: "I will go on a date » (A). The conclusion denies the second possibility: "Therefore, I will not go to class" (Not-B). It is clear that you can argue in a slightly different way: “No, I will not go on a date. Therefore, I will go to class." Symbolically, these two modes of reasoning can be represented as follows:

They are called modes of divisive-categorical syllogism. The first mode is called affirmative-denying, the second - denying-asserting. Both modes can lead to both correct and erroneous conclusions. In order not to make mistakes in reasoning that has the form of a divisive-categorical syllogism, it is necessary to fulfill the requirement for a dividing premise. In the affirmative-denying mode, the dividing premise must be strictly divisive, i.e. the alternatives must be mutually exclusive. If this requirement is not met, the conclusion may be erroneous. For example, you meet a friend walking with a lady, and you think: "This lady is his mother or wife." It turns out that the lady is his wife. “Yeah,” you conclude, “it means she’s not his mother.” This is an affirmative-denying mode, and its divisive premise is strictly divisive. The conclusion is quite reliable.

But here is another case. You see your friend, with a haggard look, wandering down the street. “He is sick or poor,” you think. It turns out that your friend has long and terminally ill. “So he is not poor,” you conclude. Alas, the divisive premise is not strictly divisive: sickness and poverty are by no means mutually exclusive, especially in our time. The conclusion may be wrong.

For the negating-affirming mode, the requirement is as follows: the dividing premise must be exhaustive, i.e. should cover all the possibilities that exist in this area of reasoning. Otherwise, the output may be incorrect.

The logical structure of this particular mode often underlies many detective stories and real investigative practice. A crime has been committed, and the investigator outlines the circle of possible participants in the crime. His further work or plot development is that he checks the suspects and weeds them out one by one: this one was sick, that one was in prison at the time of the crime, that one was seen by several people in another place, etc. Who remains - that and the criminal. This is the denying-affirming mode: the crime could have been committed by A or B; A could not have committed the crime, so B did.

It is good if all possible participants in the crime are listed in the separating premise. And if not? They condemn B, and after a while it turns out that the investigation has lost sight of a certain C, who is the real criminal: not all possibilities were taken into account in the dividing premise of the reasoning. The investigator made a mistake, the court could make a mistake. Therefore, we first need to prove that the distributive premise is exhaustive, and only then draw a conclusion. Then it will be quite reliable.

Of course, in everyday life and in professional activity we are not limited to those simple conclusions with which we have become acquainted. We can connect and combine them in a wide variety of ways, for example, in one reasoning, we can combine conditionally categorical and divisive-categorical syllogisms, then we get what is called a dilemma:

If you go right, you will lose your horse. If you go to the left, you will lose your head. But you have to go right or left. You will have to lose a horse or a head.

But complex combinations of inferences can be decomposed into their simple forms and thus the correctness of our reasoning can be tested.

6) Once three peasants came to an inn. They asked the hostess to cook them a pot of potatoes, and they themselves fell asleep. The hostess boiled the potatoes and put the pot on the table.

One peasant woke up, counted the number of potatoes and ate exactly 1/3 of it. After that, he went back to sleep. Another peasant woke up, counted the potatoes and, thinking that no one had eaten yet, ate exactly 1/3 of it. And also lay down to sleep. Finally, the third peasant woke up, counted the number of potatoes and, thinking that no one had eaten yet, ate exactly 1/3 of it. Then his comrades woke up. We looked into the pot, and there were only 8 potatoes left.

The question is: how many potatoes did the hostess cook in total? How many pieces did each peasant eat? How much more should each peasant eat to get everyone equally?

7) Once upon a time there was one farmer, and he had 17 bases and 3 sons. Dying, he bequeathed to divide the donkeys between his sons in this way: 1/2 - to the eldest son; 1/3 - middle and 1/9 - junior. The brothers rushed to divide the inheritance, but something didn’t work out in any way: they couldn’t chop the donkey into pieces! They called the judge for help, but he could not come up with anything. Someone advised the brothers to seek help from a wise old man who lives in a neighboring village. He arrived, divided the donkeys between the brothers as his father had bequeathed, and left, accompanied by thanks.

How did the sage manage to fulfill his father's will?

Induction

Where do the premises of deductive inferences come from? What gives us reason to believe they are true? Of course, sometimes they can be deducted from more general propositions and thereby justify their truth. However, sooner or later we will reach such judgments for the justification of which there are no more general premises, therefore, their truth cannot be substantiated deductively. In such cases, we resort to the help of induction.

Inductive inferences are called inferences that expand our knowledge and give not a reliable, but only a probable conclusion. The premises of inductive reasoning only to some extent confirm or make the conclusion probable, but by no means ensure its reliability. The most typical inductive conclusion is a conclusion from particular cases to a general statement.

In everyday life, we draw such conclusions at every step. When you walk into a government office and bribe first one official and then another, you think to yourself, “All the officials here are bribe-takers!” Or a girl, having met one young man and became disillusioned with him, then meeting another, perhaps not so young, and again experiencing disappointment, sometimes comes to the conclusion:

"All men are scoundrels!"

Distinguish between popular and scientific induction. With popular induction, we hasten to generalize, relying on the first special cases that come across. Our examples just demonstrate this kind of induction. The reliability of the conclusion with popular induction is very low, it is very easy to make a mistake here, which we usually do.

If we consciously strive to increase the reliability of the inductive conclusion and take certain measures for this, then such induction is called scientific. In particular, it is desirable to investigate as many representatives of the class of objects to which the generalization refers as possible. Further, the facts studied should be as diverse as possible. Finally, these facts must be typical of the given class of phenomena. If these conditions are met, the reliability of inductive inference increases significantly. So, if you wanted to make your conclusion about the officials of this institution more reliable, you should not be limited to one or two officials you met, but to get acquainted with a large number of them, moreover, belonging to different levels of the bureaucratic hierarchy. Numerous examples of such conclusions can be found in sociology: in trying to ensure the validity of their statements, the sociologist, in fact, takes care of observing the rules of scientific induction.

However, it should be remembered that even if these rules are observed, we can come to erroneous conclusions. The frequent mistakes of the same sociologists clearly demonstrate this. But here is an example invented by physicists, illustrating how things are in natural science: “It is dangerous to eat cucumbers - all bodily ailments and human misfortunes in general are associated with them. Almost all people suffering from chronic diseases ate cucumbers. 99.9% of all people who died of cancer ate cucumbers during their lifetime. 99.7% of all victims of car and plane crashes ate cucumbers in the two weeks preceding the fatal accident. 93.1% of all juvenile delinquents come from families where cucumbers were constantly consumed.” This example shows how easy it is to fit an erroneous hypothesis with statistical data and pass off stupidity as scientific truth.

It must always be remembered that no matter how well founded an inductive inference, no matter how numerous the evidence in its favor, from a logical point of view, it always remains problematic. Therefore, any going beyond the limits of existing knowledge, any attempt to obtain new knowledge is associated with a risk - with the risk of making a mistake. But precisely because of this, the history of human knowledge is not a dull sequence of unchanging successes, but a dramatic adventure in which victories are replaced by defeats, ups and downs, successes into disappointments. It is the risk that makes the scientific game so exciting and reckless.

1) This task is solved simply: you need to rearrange sentries from the middle of the bastion to its corners, as shown in the following figures:

2) Unfortunately, this is a simple and impudent deception. Travelers really paid 27 rubles. But that's all, no 30 rubles. not anymore! Of these 27 rubles. the hostess took 25 rubles. and 2 rubles. left with the boy. On what basis to these 27 rubles. I add 2 more rubles.? Where did I get them from? Where are they? Both the hostess's money and the boy's money have already been counted - they are in the paid 27 rubles. I invented these 2 rubles to mislead you.

3) To solve this problem, simple arithmetic operations are sufficient. If Ivan gives 1 sheep to Peter, then they will have equal number of sheep. This allows us to make an equality: Peter's sheep + 1 = Ivan's sheep - 1. From this we easily conclude that Ivan has 2 more sheep. More in the same vein. Answer: Peter had 3 sheep, Ivan had 5.

4) Don't know where to start. But there is one clue that helps unwind the ball. Karasev said: "If you ask Shchukin about his profession, he will answer that he is a painter." And Shchukin really said that he was a house painter! This means that Karasev told at least one truth, therefore, he cannot be a thief who always lies. Maybe Karasev is an accomplice who sometimes tells the truth and sometimes lies? Then Shchukin and Okunev must be a thief and an honest man, and their answers must be completely different from one another, since one of them always tells the truth, and the other constantly lies. No, this does not work: the answers of Shchukin and Okunev coincide in one point. Therefore, only Karasev can be an honest person and everything he said is true. Okunev's answers in one point coincide with Karasev's answers, therefore, Okunev is an accomplice in the crime. And of course, Shchukin cannot be anything but a thief.

5) Let's designate the students with the letters A, B, C and put ourselves in the place of A. He argues as follows: “I see two white caps in front of me. So I'm wearing a white or black hat. If I am wearing a black hat, then B sees a black and white hat in front of him. But B also argues: “If I had a black hat on, then C would see two black hats in front of him and would immediately guess that he himself was wearing a white hat. But C is silent, which means I have a white cap on. Thus, - continues to argue A, - if I had a black hat on, then B would already have guessed that he himself should be wearing a white hat. But B is silent. So he doesn't see the black cap on me. Therefore, I have a white hat on! So each of them reasoned, and since all the students thought equally quickly, they solved the problem at the same time.

6) The logic of the reasoning leading to the decision is important here. We need to move from the end to the beginning. At the end, 8 potatoes remained, which is equal to 2/3 of the amount that the third peasant found in the iron. So, in total, he found 12 pieces. But this is equal to 2/3 of the amount that the second peasant found. So there were 18 pieces. Again, this is equal to 2/3 of the amount of potatoes that the first farmer discovered. Consequently, the first one found 27 potatoes in a cast iron pot. So many potatoes cooked by the hostess. The first one ate 9 pieces and cannot claim anything else. The second ate 6 pieces, and he is still entitled to 3 potatoes. The third ate only 4 pieces and should get 5 more potatoes.

7) This task is difficult, I'm afraid not everyone coped with it. Indeed, 17 is not divisible either in half, or in three parts, or in nine parts. But you remember: the wise man came, he came on a donkey! By adding his donkey to the donkeys of his brothers, he got 18 donkeys. Half, i.e. 9 donkeys, he gave to his older brother; the third part, 6 donkeys, he gave to the middle brother and the ninth part - two donkeys - he gave to the younger one. So: 9 + 6 + 2 = 17. After that, he got on his donkey and left.

The properties of the basic concepts are revealed in axioms- proposals accepted without proof.

For example, in school geometry there are axioms: “a straight line can be drawn through any two points and only one” or “a straight line divides a plane into two half-planes.”

The system of axioms of any mathematical theory, revealing the properties of the basic concepts, gives their definitions. Such definitions are called axiomatic.

Proved properties of concepts are called theorems, consequences signs, formulas, rules.

Prove the theorem AIN- it means to set in a logical way that whenever the property is executed A, the property will be executed IN.

Proof in mathematics, a finite sequence of sentences of a given theory is called, each of which is either an axiom or is derived from one or more sentences of this sequence according to the rules of inference.

The proof is based on reasoning - a logical operation, as a result of which one or more sentences related in meaning result in a sentence containing new knowledge.

As an example, consider the reasoning of a schoolboy who needs to establish the ratio "less than" between the numbers 7 and 8. The student says: "7< 8, потому что при счете 7 называют раньше, чем 8».

Let us find out on what facts the conclusion obtained in this reasoning is based.

There are two such facts: First: if the number A when counting, they call before the number b, That a< b. Second: 7 is called earlier than 8 when counting.

The first sentence is general in nature, since it contains a general quantifier - it is called a general premise. The second sentence concerns the specific numbers 7 and 8 - it is called a private premise. A new fact is obtained from two premises: 7< 8, его называют заключением.

There is a certain connection between the premises and the conclusion, thanks to which they constitute an argument.

Reasoning, between the premises and the conclusion of which there is a relation of consequence, is called deductive.

In logic, instead of the term "reasoning", the word "inference" is more often used.

inference It is a way of obtaining new knowledge on the basis of some existing one.

An inference consists of premises and a conclusion.

Parcels- is containing the original knowledge.

Conclusion- this is a statement containing new knowledge obtained from the original.

As a rule, the conclusion is separated from the premises with the help of the words "therefore", "means". Inference with parcels R 1, R 2, …, рn and conclusion R we will write in the form: or (R 1, R 2, …, рn) R.

Examples inferences: a) Number a =b. Number b = c. Therefore, the number a = s.

b) If the numerator is less than the denominator, then the fraction is proper. In fraction numerator less than denominator (5<6) . Therefore, the fraction - correct.

c) When it rains, there are clouds in the sky. There are clouds in the sky, so it's raining.

Inferences can be right or wrong.

The inference is called correct if the formula corresponding to its structure and representing the conjunction of the premises, connected with the conclusion by the sign of the implication, is identically true.

For that to determine whether the conclusion is correct, proceed as follows:

1) formalize all premises and conclusion;

2) write down a formula representing the conjunction of premises connected by an implication sign with the conclusion;

3) make up a truth table for this formula;

4) if the formula is identically true, then the conclusion is correct, if not, then the conclusion is incorrect.

In logic, it is believed that the correctness of an inference is determined by its form and does not depend on the specific content of the statements included in it. And in logic, such rules are proposed, observing which, one can build deductive conclusions. These rules are called inference rules or schemes of deductive reasoning.

There are many rules, but the following are the most commonly used:

1. - conclusion rule;

2. - the rule of negation;

3. - the rule of syllogism.

Let's bring example inference made by rule conclusions:"If the entry of a number X ends with a number 5, that number X divided by 15. Writing a number 135 ends with a number 5 . Therefore, the number 135 divided by 5 ».

As a general premise in this conclusion, the statement “if Oh), That B(x)", Where Oh) is a "record of a number X ends with a number 5 ", A B(x)- "number X divided by 5 ". A private premise is a statement that results from the condition of a general premise when
x = 135(those. A(135)). The conclusion is a statement derived from B(x) at x = 135(those. B(135)).

Let's bring an example of a conclusion made according to the rule negations:"If the entry of a number X ends with a number 5, that number X divided by 5 . Number 177 not divisible by 5 . Therefore, it does not end with a number 5 ».

We see that in this conclusion the general premise is the same as in the previous one, and the private one is a negation of the statement "the number 177 divided by 5 » (i.e.). The conclusion is the negation of the sentence "Recording the number 177 ends with a number 5 » (i.e.).

And finally, consider example of an inference based on syllogism rule: "If the number X multiple 12, then it is a multiple 6. If number X multiple 6 , then it is a multiple 3 . Therefore, if the number X multiple 12, then it is a multiple 3 ».

There are two premises in this conclusion: “if Oh), That B(x)" and if B(x), That C(x)”, where A (x) - “number X multiple 12 », B(x)- "number X multiple 6 " And C(x)- "number X multiple 3 ". The conclusion is the statement "if Oh), That C(x)».

Let's check if the following conclusions are correct:

1) If a quadrilateral is a rhombus, then its diagonals are mutually perpendicular. ABCD- rhombus. Therefore, its diagonals are mutually perpendicular.

2) If the number is divisible by 4 , then it is divisible by 2 . Number 22 divided by 2 . Therefore, it is divided into 4.

3) All trees are plants. Pine is a tree. So pine is a plant.

4) All students of this class went to the theatre. Petya was not in the theatre. Therefore, Petya is not a student of this class.

5) If the numerator of a fraction is less than the denominator, then the fraction is correct. If the fraction is correct, then it is less than 1. Therefore, if the numerator of the fraction is less than the denominator, then the fraction is less than 1.

Solution: 1) To resolve the issue of the correctness of the conclusion, we will identify its logical form. Let us introduce the notation: C(x)- quadrilateral X- rhombus, B(x)- in a quadrilateral X diagonals are mutually perpendicular. Then the first message can be written as:
C(x) B(x), second - C(a), and the conclusion B(a).

Thus, the form of this inference is as follows: . It is built according to the rule of conclusion. Therefore, this reasoning is correct.

2) Let's introduce the notation: Oh)- "number X divided by 4 », B(x)- "number X divided by 2 ". Then we write the first message: Oh)B(x), second B(a), and the conclusion is A(a). The conclusion will take the form: .

There is no such logical form among the known ones. It is easy to see that both premises are true and the conclusion is false.

This means that this reasoning is wrong.

3) Let us introduce the notation. Let Oh)- "If X tree", B(x) - « X plant". Then the messages will look like: Oh)B(x), A(a), and the conclusion B(a). Our conclusion is built in the form: - Conclusion rules.

So our reasoning is correct.

4) Let Oh) - « X- Students in our class B(x)- “students X went to the theatre." Then the messages will be as follows: Oh)B(x),, and the conclusion.

This conclusion is built according to the rule of negation:

- means it is correct.

5) Let's reveal the logical form of the conclusion. Let A(x) -"numerator of a fraction X less than the denominator. B (x) - "fraction X- correct. C(x)- "fraction X less 1 ". Then the messages will look like: Oh)B(x), B(x) C(x), and the conclusion Oh)C(x).

Our conclusion will be of the following logical form: - the rule of syllogism.

So this conclusion is correct.

In logic, various methods of checking the correctness of inferences are considered, among which analysis of the correctness of inferences using Euler circles. It is carried out as follows: the conclusion is written in the set-theoretic language; depict the parcels on the circles of Euler, considering them to be true; they look to see if the conclusion is always true. If so, then the conclusion is said to be correct. If a drawing is possible from which it is clear that the conclusion is false, then the conclusion is said to be wrong.

Table 9

Verbal formulation of the sentence

Recording in set-theoretic language

Image on Euler circles

Anything A There is IN

Some A There is IN

Some A do not eat IN

none A do not eat IN

A There is A
A do not eat A

Let us show that the inference made according to the rule of conclusion is deductive. Let us first write this rule in set-theoretic language.

Package Oh)B(x) can be written in the form TATV, Where TA And TV- truth sets of propositional forms Oh) And B(x).

private package A(a) means that ATA, and the conclusion B(a) shows that ATV.

The whole inference, built according to the rule of conclusion, will be written in the set-theoretic language as follows: .

Having depicted on the Euler circles the sets TA And TV and denoting the element ATA, we will see that ATV(Fig. 58). Means, AT aT.

Rice. 58.

Examples.

1. Is the conclusion correct “If the entry of a number ends with a number 5, then the number is divisible by 5. Number 125 divided by 5. Therefore, writing a number 125 ends with a number 5 »?

Solution: This conclusion is made according to the scheme , which corresponds . There is no such scheme among the known to us. Let's find out if it is a rule of deductive reasoning?

Let's use the Euler circles. In set-theoretic language

The resulting rule can be written as follows:

. Let us represent on the Euler circles the sets TA And TV and denote the element A from many TV.

It turns out that it can be contained in the set TA, or maybe not belong to him (Fig. 59). In logic, it is believed that such a scheme is not a rule of deductive reasoning, since it does not guarantee the truth of the conclusion.

This conclusion is not correct, since it is made according to a scheme that does not guarantee the truth of the reasoning.

Rice. 59.

b) All verbs answer the question "what to do?" or “what to do?”. The word "cornflower" does not answer any of these questions. Therefore, "cornflower" is not a verb.

Solution: a) Let us write this conclusion in the set-theoretic language. Denote by A- a lot of students of the pedagogical faculty, through IN- many students who are teachers, through WITH- many students over 20 years old.

Then the conclusion will take the form: .

If you depict these sets on circles, then 2 cases are possible:

1) sets A, B, C intersect;

2) set IN intersects with many WITH And A, and the set A intersects IN, but does not intersect with WITH.

b) Denote by A many verbs, and IN many words that answer the question "what to do?" or “what to do?”.

Then the conclusion can be written as follows:

Let's look at a few examples.

Example 1 The student is asked to explain why the number 23 can be represented as the sum 20 + 3. He argues: “The number 23 is two-digit. Any two-digit number can be represented as a sum of bit terms. Therefore, 23 = 20 + 3."

The first and second sentences in this inference of the premise, and one of a general nature is the statement “any two-digit number can be represented as a sum of bit terms”, and the other is private, it characterizes only the number 23 - it is two-digit. The conclusion - this sentence that comes after the word "therefore" - is also private, since it deals with the specific number 23.

Inferences that are commonly used in proving theorems are based on the concept of logical consequence. Moreover, from the definition of logical consequence it follows that for all values of propositional variables for which the original statements (premises) are true, the conclusion of the theorem is also true. Such inferences are deductive.

In the example discussed above, the above inference is deductive.

Example 2 One of the methods for introducing younger students to the commutative property of multiplication is as follows. Using various visual aids, students, together with the teacher, establish that, for example, 6 3 = 36, 52 = 25. Then, based on the obtained equalities, they conclude: for all natural numbers a And b true equality ab=ba.

In this conclusion, the premises are the first two equalities. They state that such a property holds for concrete natural numbers. The conclusion in this example is a general statement - the commutative property of multiplication of natural numbers.

In this conclusion, the premises of a particular nature show that some natural numbers have the property that the product does not change from a permutation of factors. And on this basis, it was concluded that all natural numbers have this property. Such reasoning is called incomplete induction.

those. for some natural numbers it can be argued that the sum is less than their product. So, based on the fact that some numbers have this property, we can conclude that all natural numbers have this property:

This example is an example of reasoning by analogy.

Under analogy understand a conclusion in which, based on the similarity of two objects in some features and in the presence of an additional feature, one of them concludes that the other object has the same feature.

The conclusion by analogy is in the nature of an assumption, a hypothesis and therefore needs either proof or refutation.

CONCLUSION - THE THIRD FORM OF THINKING

What is an inference?

inference- this is the third (after the concept and judgment) form of thinking, in which one, two, or several judgments, called premises, follow a new judgment, called the conclusion, or conclusion.

In logic, it is customary to place the premises and the output one under the other and to separate the premises from the output with a line:

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 the premises, and the third is the conclusion. It is clear that the premises must be true judgments and must be connected with 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 you 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 connected with each other, then it is impossible to draw a conclusion from them.

For example, no conclusion follows from the following two premises:

All planets are celestial bodies.

All pines are trees.

Let us pay attention to the fact that inferences consist of judgments, and judgments - of concepts, i.e. one form of thought enters into another as an integral part.

All inferences are divided into direct and indirect. IN immediate inferences, the conclusion is made from one premises.

For example:

All flowers are plants.

Some plants are flowers.

Another example:

It is true that all flowers are plants.

It is not true that some flowers are not plants.

It is not difficult to guess that direct inferences are operations for transforming simple judgments and conclusions about the truth of simple judgments in a logical square. The first example of direct inference given above is a transformation of a simple proposition by inversion, and in the second example, by the logical square, from the truth of a proposition of type A, a conclusion is drawn about the falsity of a proposition of type O.

IN mediated inferences, the conclusion is drawn from several premises.

For example:

All fish are living beings.

All carp are fish.

All carp are living beings.

Since direct inferences are various logical operations with judgments, then under inferences are meant, first of all, indirect inferences. In the future, we will talk about them.

Indirect inferences are divided into three types. They are deductive, inductive and reasoning by analogy.

deductive reasoning, or deduction - these are inferences in which a conclusion is drawn from a general rule for a particular case (a special case is derived from a general rule).

For example:

All stars radiate energy.

The sun is a star.

The sun radiates energy.

As you can see, the first premise is a general rule, from which (with the help of the second premise) a special case follows in the form of a conclusion: if all stars radiate energy, then the Sun also radiates it, because it is a star. In deduction, reasoning goes from the general to the particular, from the greater to the lesser, knowledge is narrowed, due to which the deductive conclusions are reliable, i.e. accurate, obligatory, necessary, etc. Let's look again at the example above. Could any other conclusion follow from these two premises than the one that follows from them? Could not! The following conclusion is the only one possible in this case. Let us depict the relationship between the concepts of which our conclusion consisted, Euler circles. Volumes of three concepts: stars; body, radiating energy; Sun schematically arranged as follows.

If the scope of the concept stars included in the concept body, radiating energy, and the scope of the concept Sun included in the concept stars, then the scope of the concept Sun automatically included in the scope of the concept bodies that radiate energy, which makes the deductive inference valid.

However, he (Sherlock Holmes) irrefutably proves that Colonel Morin could not smoke this cigar, because he wore a large, lush mustache, and the cigar was smoked to the end, i.e. if Morin had smoked it, he would certainly have set his mustache on fire. Therefore, the cigar was smoked by another person. In this reasoning, the conclusion looks convincing precisely because it is deductive: from the general rule ( Anyone with a big, bushy mustache can't finish a cigar.) a special case is displayed ( Colonel Morin could not finish his cigar because he wore such a mustache).

Inductive reasoning, or induction - these are inferences in which a general rule is deduced from several special cases (several special cases lead to a general rule).

For example:

Jupiter is moving.

Mars is moving.

Venus is moving.

Jupiter, Mars, Venus are planets.

All planets are moving.

As you can see, the first three premises are special cases, the fourth premise brings them under one class of objects, combines them, and the output refers to all objects of this class, i.e. some general rule is formulated (following from three particular cases). In induction, reasoning goes from the particular to the general, from less to more, knowledge expands, due to which inductive conclusions (unlike deductive ones) are not reliable, but probabilistic. The probabilistic nature of the conclusions is, of course, a disadvantage of induction. However, its undoubted advantage and advantageous difference from deduction, which is a narrowing knowledge, is that induction is an expanding knowledge that can lead to a new one, while deduction is an analysis of the old and already known.

Inference by analogy, or analogy- these are conclusions in which, on the basis of the similarity of objects (objects) in some features, a conclusion is made about their similarity, and in other features, a conclusion is made about their similarity in other features.

For example:

Planet Earth is located in the solar system, it has an 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 you can see, two objects are compared (compared) (the planet Earth and the planet Mars), which are similar to each other in some essential, 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.

In this lesson, we finally move on to the topic that is the core of any reasoning and any logical system - inferences. In the fourth lesson, we said that reasoning is a set of judgments or statements. Obviously, such a definition is not complete, because it does not say anything about why some different statements suddenly appeared side by side. If we give a more precise definition, then reasoning is the process of substantiating a statement with the help of its consistent conclusion from other statements. This conclusion is most often carried out in the form of inferences.

inference- this is a direct transition from one or more statements A 1, A 2, ..., A n to the statement B. A 1, A 2, ..., A n are called premises. There can be one parcel, there can be two, three, four, in principle - as many as you like. The parcels contain information known to us. B is the conclusion. In conclusion, there is already new information that we have extracted from the parcels with the help of special procedures. This new information was already contained in the parcels, but in a hidden form. So the task of the inference is to make this hidden explicit. In addition, sometimes the premises are called arguments, and the conclusion is called the thesis, and the conclusion itself in this case is called justification. The difference between inference and justification is that in the first case, we do not know what conclusion we will come to, and in the second case, we already know the thesis, we just want to establish its connection with the premises-arguments.

As an illustration of the conclusion, we can take Hercule Poirot's reasoning from Agatha Christie's "Murder on the Orient Express":

But I felt that he was rebuilding on the go. Suppose he wanted to say, "Didn't they burn her?" Therefore, McQueen knew both about the note and that it was burned, or, in other words, he was the killer or an accomplice of the killer.

Above the line are the premises, below the line is the conclusion, and the line itself denotes the relation of logical consequence.

Criteria for the truth of inferences

As well as for judgments, for inferences there are certain conditions for their truth. When determining whether a conclusion is true or false, two aspects must be taken into account. First aspect is the truth of the premises. If at least one of the premises is false, then the conclusion drawn will also be false. Since the conclusion is the information that was hidden in the premises and which we simply brought to light, it is impossible to accidentally obtain the correct conclusion from incorrect premises. It can be compared to trying to make a carrot steak. Perhaps carrots can be given the color and shape of a steak, but the inside will still be carrots, not meat. No culinary operations will transform one into the other.

Second aspect- this is the correctness of the conclusion itself from the point of view of its logical form. The fact is that the truth of the premises is an important but not sufficient condition for the conclusion to be correct. It is not uncommon for the premises to be true but the conclusion to be false. As an example of an incorrect inference with the truth of the premises, one can cite the conclusion of the dove from Carroll's Alice in Wonderland. Dovewing accuses Alice of not being a snake. Here's how she comes to this conclusion:

Snakes eat eggs.
Girls eat eggs.
So girls are snakes.

While the premises are correct, the conclusion is absurd. The conclusion as a whole is wrong. To avoid such errors, logicians have identified such inferences, the logical forms of which, if the premises are true, guarantee the truth of the conclusion. They are called correct inferences. Thus, in order for the conclusion to be made correctly, it is necessary to monitor the truth of the premises and the correctness of the very form of the conclusion.

We will consider various forms of correct reasoning using the example of syllogistic. In this lesson, we will analyze the simplest one-terminal conclusions. In the next lesson - more complex conclusions: syllogisms, enthymemes, multi-premise conclusions.

To make it easier to remember exactly what types of inferences are possible between categorical attributive statements, logicians have come up with a special logical square depicting the relationship between them. Therefore, some one-term inferences are also called logical square inferences. Let's look at this square:

Let's start with subordination relations. We already encountered them in the fourth lesson, when we considered the truth conditions for particular affirmative and particular negative statements. We said that from the statement "All S are P" it would be logical to deduce the statement "Some S are P", and from the statement "No S are P" - "Some S are not P". Thus, the following types of inferences are possible:

All S's are P's
Some S's are P's
All birds have a beak. Therefore, some birds have beaks.
No S is P
Some S's are not P's
No goose wants to be caught and roasted. Consequently, some geese do not want to be caught and roasted.

In addition, according to the rule of contraposition, two more correct conclusions can be deduced from the subordination relations. The rule of contraposition is a logical law that says: if statement A implies statement B, then the statement “it is not true that B” will follow the statement “it is not true that A”. You can try to test this law with a truth table. So, the following conclusions on contraposition will also be true:

It is not true that all S are P
It is not true that some cars do not have wheels. Therefore, it is not true that all cars do not have wheels.
It is not true that all S are not P
It is not true that some wines are not spirits. Thus, it is not true that all wines are non-alcoholic beverages.

Contrarality relation(opposites) means that statements like "All S are P" and "No S is P" cannot both be true, but they can be both false. This is clearly seen from the truth table for categorical attributive statements, which we built in the last lesson. From this we can deduce the so-called law of counter-contradiction: It is not true that all S are P and at the same time none of S is P.

According to the law of contradiction, the following types of inferences will be true:

All S's are P's
All apples are fruits. Therefore, it is not true that no apple is a fruit.
No S is P
It is not true that all S are P
Not a single whale can fly. Therefore, it is not true that all whales can fly.

Subcontrary relations(subopposites) mean that statements like "Some S are P" and "Some S are not P" cannot both be false, although they can be both true. On this basis, the law of subcontrary excluded middle can be formulated: Some S is not P or Some S is P.

According to this law, the following conclusions will be correct:
It is not true that some S are P
Some S's are not P's
It is not true that certain foods are good for health. Therefore, some foods are not healthy.
It is not true that some S are not P
Some S's are P's
It is not true that some of the students in our class are not Losers. Thus, some students from our class are losers.

Relations of contradiction(contradictions) say that the statements contained in them cannot be both true and false. On the basis of these relations, two laws of contradiction and two laws of the excluded middle can be formulated. First law of contradiction: It is not true that all S are P and some S are not P. Second law of contradiction: It is not true that no S is P and some S are P. First law of excluded middle: All S are P or some S are not is P. Second Law of the Excluded Middle: No S is P or some S is P.

The following types of conclusions are based on these laws:

All S's are P's
It is not true that some S are not P
All children need to be taken care of. Therefore, it is not true that some children do not need care.
Some S's are not P's
It is not true that all S are P
Some books are not boring. Therefore, it is not true that all books are boring.
It is not true that all S are P
Some S's are not P's
It is not true that all employees of our firm work hard. Thus, some employees of our firm do not work hard.
It is not true that some S are not P
All S's are P's
It is not true that some zebras do not have stripes on their skin. Hence, all zebras have stripes on their skin.
No S is P
It is not true that some S are P
Not a single painting in this room belongs to the 20th century. Therefore, it is not true that some of the paintings in this room belong to the 20th century.
Some S's are P's
It is not true that no S is a P
Some students go in for sports. Thus, it is not true that no student goes in for sports.
It is not true that no S is a P
Some S's are P's
It is not true that no scientist is interested in art. Hence, some scientists are interested in art.
It is not true that some S are P
No S is P
It is not true that some cats smoke cigars. Thus, no cat smokes cigars.

As you most likely noticed in all these inferences, the statements above the line and below the line carry the same information, just presented in a different form. An important detail is that the meaning of some of these statements is perceived easily and intuitively, while the meaning of others is obscure, and sometimes you have to rack your brains over them. For example, the meaning of affirmative statements is easier to perceive than the meaning of negative statements, the meaning of statements with one negative is more understandable than the meaning of statements with two negatives. Thus, the main purpose of reasoning on the logical square is to bring difficult-to-perceive, incomprehensible statements to the most simple and clear form.

Another type of one-parcel inference is reversal. This is a type of inference in which the subject of the premise coincides with the predicate of the conclusion, and the subject of the conclusion coincides with the predicate of the premise. Roughly speaking, S and P are simply reversed in the conclusion.

Before moving on to inference through inversion, let's build a truth table for statements in which P will take the place of the subject, and S - the place of the predicate.

Compare it with the table we built in the last lesson. The inversion, like other inferences, can only be true when the premise and the conclusion are both true. When comparing two tables, you will see that there are not so many such combinations.

So, there are two types of conversion: pure and restricted. Pure conversion occurs when the quantitative characteristic does not change, that is, if the premise contained the word "all", then the conclusion will also contain the words "all" / "none", if the premise contains the word "some", then the conclusion "some. Accordingly, when handling the restriction, the quantitative characteristic changes: there were “all”, and now there are “some”. For statements like "No S is P" and "Some S are P" the following pure inversion is correct:

No S is P
No P is S
No man can survive without air. Therefore, no living being capable of surviving without air is human.
Some S's are P's
Some P's are S's
Some snakes are venomous. Therefore, some poisonous creatures are snakes.
For propositions like "All S are P" and "No S is P" the inversion with the restriction is true:
All S's are P's
Some P's are S's
All penguins are birds. So some birds are penguins.
No S is P
Some P is not S
No crocodile eats marshmallows. Therefore, some marshmallow-eating creatures are not crocodiles.
Statements like "Some S are not P" are not reversible at all.

Although inversions, like logical square inferences, are single-premise inferences, and we extract all new information from the existing premise in the same way, the premise and conclusion in them can no longer be called simply different formulations of the same information. The information received already refers to another subject, and therefore it no longer seems so trivial.

So, in this lesson, we started to look at the right kinds of inferences. We talked about the simplest single-premise inferences: inferences by a logical square and inferences through reversal. Although these conclusions are quite simple and even trivial in some places, people everywhere make mistakes in them. It's understandable that it's hard to keep all kinds of valid inferences in mind, so when you're doing the exercises or when you're faced with the need to test or make a one-term inference in real life, don't be afraid to use model diagrams and truth tables. They will help you check whether the conclusion is always true when the premises are true, and this is the main thing for a correct conclusion.

Exercise "Pick the key"

In this game, you need to create a key with the correct shape. To do this, set serifs of the desired length (from 1 to 3, 0 - cannot be), and then click the "Try" button. You will be given 2 judgments, how many serifs of the selected length are present in the key (for simplicity, the value is “presence”), and how many of the selected ones are in place (for simplicity, the value is “in place”). Adjust your decision and try until you pick up the key.

Exercises

Make all possible inferences from the following statements on the logical square:

All bears hibernate for the winter.
It is not true that all people are envious.
Not a single gnome reaches a height of two meters.
It is not true that not a single person has been to the North Pole.
Some people have never seen snow.
Some buses run on time.
It is not true that some elephants flew to the moon.
It is not true that some birds do not have wings.

Make appeals with those statements with which it is possible:

Nobody has built a time machine yet.
Some of the waiters are very pushy.
All professionals are experienced in their field.
Some books are not hardcover.

Check if the following conclusions are correct:

Some rabbits don't wear white gloves. Hence, some rabbits wear white gloves.
It is not true that no one has been to the moon. So some people have been on the moon.
All people are mortal. Therefore, all mortals are human.
Some birds cannot fly. Therefore, some creatures that cannot fly are birds.
No lamb has a taste for whiskey. Therefore, no creature that has a taste for whiskey is a lamb.
Some marine animals are mammals. Thus, it is not true that no marine animal is a mammal.

Test your knowledge

If you want to test your knowledge on the topic of this lesson, you can take a short test consisting of several questions. Only 1 option can be correct for each question. After you select one of the options, the system automatically moves on to the next question. The points you receive are affected by the correctness of your answers and the time spent on passing. Please note that the questions are different each time, and the options are shuffled.

Logics. Tutorial Gusev Dmitry Alekseevich

3.2. Types of inference

Inferences, or mediated inferences, are divided into three types. They are deductive, inductive And reasoning by analogy.

deductive reasoning or deduction(from lat. deductio - derivation) - these are inferences in which a conclusion is drawn from a general rule for a particular case (a special case is derived from a general rule).

For example:

All stars radiate energy.

The sun is a star.

The sun radiates energy.

Let's look again at the example above. Could any other conclusion follow from these two 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 relationship between the concepts of which our conclusion consisted of Euler circles. The scope of the three concepts: stars; bodies that radiate energy; Sun schematically arranged as follows:

If the scope of the concept stars included in the concept bodies that radiate energy and the scope of the concept Sun included in the concept stars, then the scope of the concept Sun automatically included in the concept bodies that radiate energy whereby the deductive conclusion is reliable.

The undoubted advantage of deduction, of course, lies in the reliability of its conclusions. Recall that the famous literary hero Sherlock Holmes used the deductive method in solving crimes. This means that he built 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. Near the murdered Colonel Morin, Scotland Yard detectives found a smoked cigar and decided that the colonel had smoked it before his death. However, he (Sherlock Holmes) irrefutably proves that Colonel Morin could not smoke this cigar, because he wore a large, lush mustache, and the cigar was smoked to the end, that is, if Morin smoked it, he would certainly set it on fire your mustache. Therefore, the cigar was smoked by another person. In this reasoning, the conclusion looks convincing precisely because it is deductive: from the general rule ( Anyone with a big, bushy mustache can't finish a cigar.) a special case is displayed ( Colonel Morin could not finish his cigar because he wore such a mustache). Let us bring the considered reasoning to the standard form of writing inferences in the form of premises and conclusions accepted in logic:

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

Colonel Morin wore a large, bushy mustache.

Colonel Morin could not finish his cigar.

Inductive reasoning or induction(from Latin inductio - guidance) - these are inferences in which a general rule is derived from several special cases (several special cases, as it were, lead to a general rule). For example:

Jupiter is moving.

Mars is moving.

Venus is moving.

Jupiter, Mars, Venus are planets.

All planets are moving.

As you can see, the first three premises are special cases, the fourth premise brings them under one class of objects, unites them, and the conclusion refers to 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 reasoning is built on a principle opposite to that of deductive reasoning. In induction, reasoning goes from the particular to the general, from less to more, knowledge expands, due to which inductive conclusions, unlike deductive ones, are not reliable, but probabilistic. In the example of induction considered 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 an error: it is quite possible that there are some exceptions in the group, and even if the set of objects from a certain group is characterized by some attribute, this does not mean with certainty that all objects of this group are characterized by this attribute. The probabilistic nature of the conclusions is, of course, a disadvantage of induction. However, its undoubted advantage and advantageous difference from deduction, which is a narrowing knowledge, is that induction is an expanding knowledge that can lead to a new one, while deduction is an analysis of the old and already known.

Inference by analogy or simply analogy(from the Greek analogia - correspondence) - these are inferences in which, on the basis of the similarity of objects (objects) in some features, a conclusion is made about their similarity in other features. For example:

Planet Earth is located in the solar system, it has an 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.

This text is an introductory piece.

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