Alexander Leonidovich Nikiforov
Logics
Logics
Foreword
The biological species to which we belong is called "homo sapiens" - "reasonable man" in Latin. Among our ancestors was a "handy man" who began to make the first stone tools; "straightened man", Pithecanthropus; Neanderthal, immediate ancestor modern man. Our distinctive feature compared with animals and predecessors is the presence of reason. But what is "mind"? It turns out that it is still extremely difficult to answer this question. We know much more about the structure of matter, about stars and galaxies, about plants and animals, than about the most important property person.
But there is one side of the mind - the most important, perhaps - which has been studied quite well. This is thinking, i.e. the ability to navigate in the world around, to reason, to build explanations of certain phenomena, to make predictions. The science of thinking is called logic.
“When I consider,” wrote the English economist, philosopher, and logician John Stuart Mill a hundred years ago, “how simple the theory of inference is, how little time is enough to acquire a complete knowledge of its principles and rules, and even considerable experience in their application, I do not I find no excuse for those who, desiring to engage successfully in any mental work, miss this study. Logic is the great pursuer of obscure and confused thinking; it dispels the fog that hides our ignorance from us and makes us think we understand the subject when we do not. I am convinced that nothing in modern education is more useful for developing precise thinkers who remain true to the meaning of words and sentences and are constantly on guard against terms vague and ambiguous, like logic.
Yes, the mind must be able to use, it must and can be developed. And this is taught by the only science - logic. Therefore, acquaintance with it is necessary for any person. An elephant is stronger than a man, a horse is faster, an eagle sees farther and better. But the mind allows a person to immeasurably surpass them all in strength, and in speed, and in vigilance.
Chapter 1
The subject and meaning of logic
What does logic study
Of course, the subject of any science becomes quite clear only in the process of getting to know the science itself, therefore all preliminary definitions of this subject remain incomplete and not very clear. However, you have to start somewhere.
Logic is the science of the forms and laws of cognitive thinking.
Logic studies thinking, but not all, but only those mental processes that are aimed at discovering and substantiating truth, at solving a certain problem, at finding ways to overcome certain difficulties that confront us both in professional activity and in Everyday life. Sometimes in the morning, looking at yourself in the mirror, you may think: “Whoa, what a disgusting face I have today!” Logic hardly touches such thought processes; their study is a matter of psychology or even psychiatry. Logic is primarily interested in cognitive processes. So what is “cognitive thinking”? Try to answer a few simple questions.
I have two coins in my pocket, which add up to 15 kopecks, but one of these coins is not a nickel. What are these coins? You start thinking: 7 and 8 kopecks? There are no such coins. Maybe 2, 3 and 10 kopecks? But then it turns out three coins, and there are only two of them. Finally, the thought comes: one of them is not a piglet, but the other one can be a piglet! Answer: 10 and 5 kopecks.
The night watchman died not at night, but during the day. Will he get a pension? You start to remember what you know about pension legislation, then you suddenly catch on: but he died, why would he need a pension? Answer: No, they won't.
Can a decent man marry his widow's sister? The word "decent" grabs your attention and you wonder: is it ethical to marry your wife's sister? But soon the thought comes: the wife is already a widow, which means that the person in question is dead! It is clear that he can no longer marry anyone, and the words "decent man" simply divert our attention from this decisive circumstance.
In all these cases, the search for a solution comes down to putting forward some assumption and deriving consequences from it. If the consequences do not agree with the conditions of the problem, we discard the original assumption and put forward a new one, and so on. Despite the simplicity of the given examples, the search for their solution reproduces the main features of any cognitive process: a problem - a hypothetical solution - checking the solution and discarding it, if it does not agree with the conditions of the problem and with the truths known to us - putting forward a new solution ... Logic just studies those forms in which cognitive thinking proceeds, and those general principles which it must obey in order to achieve its goal.
Logic is only interested in the form of our thoughts, not their content. The content of our thoughts is infinitely diverse: we can think and talk about volcanoes and stars, about love and entrepreneurship, about sausage, which is always in short supply, and about electrons, from which you don’t know where to go. However, all this diversity fits into a relatively small number of forms. These are the forms that logic studies. Figuratively speaking, the logician is interested in vessels - bottles, buckets, barrels, and not what is poured into them. Whatever you pour into a bottle - narzan, beer, wine - it will remain a bottle and you can study it just like a bottle: explore it geometric shape, volume, wall thickness, etc. In the same way, mental forms, regardless of their content, have certain properties that are studied by logic.
In this respect, logic is similar to grammar, which we studied at school. Grammar also explores and describes the forms of linguistic expressions, abstracting from their content. To illustrate this circumstance, the famous Soviet linguist L.V. Shcherba gave an example of the following sentence invented by him: “The glitched kuzdra shteko boked up the bokra and tails the bokra.” We can’t say anything about the content of this sentence, but knowledge of grammar allows us to assert that the word “kuzdra” is the subject here, “budlanula” is the predicate, “bokra” is the object, etc. We can talk about the gender, number, case of our nouns without having the slightest idea of what the corresponding words mean. Analogous knowledge about the forms of thought is given to us by logic.
Truth and correctness
We have already mentioned the word "truth" several times. It's time to explain it. Questions about what is truth, whether it is possible to get it and how, philosophy explores. Here we confine ourselves to a brief explanation sufficient for the purposes of logic.
A thought is called true if it corresponds to its subject, i.e. represents an object, a situation, a state of affairs as they exist in reality, by themselves. If the thought does not correspond to its subject, distorts it, it is called false. For example, the idea that the Russian composer A.P. Borodin was a chemist, is true, since Borodin really owns a number of works and discoveries in the field of chemistry. However, the idea that bananas grow on an apple tree will be false, because it gives a distorted idea of \u200b\u200bthe apple tree.
The logical correctness of reasoning is its compliance with the rules, laws of logic. If you rely on true data and reason correctly, then you will always get a true conclusion. This logic guarantees. Unfortunately, it is possible to reason correctly, but at the same time proceed from false premises. In this case, you can come to any conclusion - both true and false. As the saying goes, everything follows from a lie. For example, if you accepted the premise "All tigers eat grass", then from it you can make both a true conclusion: "Some herbivores are striped", and a false one - "Some herbivores eat tigers". It is important to keep in mind the following: logic cannot tell whether certain premises are true - this is the task of specific sciences and everyday practice - but it helps us to make our reasoning correct. If you rely on lies, your reasoning can take you anywhere. If you rely on the truth, right reasoning will only lead you to the truth.
Thinking and language
Cognitive thinking, studied by logic, is always expressed in language, therefore logic considers thought in its linguistic expression. Sometimes we will just talk about words and sentences, referring to their mental content. Of course, there are thoughts that it can be difficult for us to express in language - everyone has come across this: sometimes a schoolboy or student in an exam seems to understand everything, but he cannot say anything. Logic cannot deal with such thoughts, cannot study and analyze them, they remain "inside" the individual consciousness. It is assumed that the cognitive thinking of all people is approximately the same and does not depend on their nationality, social status and cultural differences. Natural languages spoken by humans different peoples, differ quite significantly, which we are disappointed to see when we begin to study foreign languages. Logic studies the forms of thought, not language, therefore its laws and principles are valid for all thinking, regardless of the language in which it is presented. Take, for example, the following two language expressions:
"Snow is white", "Der Schnee ist weiss".
They consist of different signs and generally differ significantly from one another. However, both express the same thought, which in Russian is conveyed by the sentence "Snow is white." Thought is something that is common to all these three completely different linguistic expressions. This idea is studied by logic. Nevertheless, the fundamental characteristics of language also influence our thought, so logic often has to take into account the peculiarities of the linguistic expression of thought.
Stages of development of logic
Logic is one of ancient sciences developed into an independent discipline. Its creator is ancient Greek philosopher and the scientist Aristotle (384-322 BC).
Aristotle's appeal to a systematic exposition of logical knowledge and the development of logic was largely due to social need. A citizen of the ancient polis actively participated in the life of his small state. He regularly attended people's meetings, where issues of war and peace, urban construction, trade, etc. were discussed, annually participated in the election of government officials, and during his life he himself could repeatedly occupy certain positions. For the administration of public posts in ancient Greece, no payment was supposed, so few sought to occupy them. Sometimes citizens were forced to draw lots to appoint someone to command the army, oversee markets or public water pipes. The ancient Greeks sued a lot and with pleasure, and the Greek court did not know the prosecutors and lawyers, the plaintiff and the defendant themselves had to speak before the judges and the public present. The outcome of the case often depended on the ability to clearly state the essence of the case and convincingly substantiate one's claims. In short, a good citizen needed a certain amount of education in order to fulfill his civic duties and exercise his civil rights.
That is why in ancient Greece there was universal education and teachers were highly valued, they were called sophists (from the Greek word "sophia" - wisdom), which originally meant "sage" or "teacher of wisdom". They taught children to read and write, the basics of science, brought up the ability to speak clearly and convincingly, to defend their opinion in disputes. Over time, however, the sophists began to invent tricks aimed at fooling the interlocutor, presenting the enemy in a ridiculous or stupid way, instilling distrust of the truth, and, on the contrary, presenting a lie in snow-white robes of truth. Such unscrupulous tricks and tricks were called "sophisms", and the word "sophist" acquired the negative connotation that he still has. Here are examples of some sophisms preserved in the writings of ancient authors:
“What you didn’t lose, you have. You didn't lose your horns. Therefore, you are horny."
“The one sitting stood up. Who stood up, he stands. Therefore, the seated person is standing.
From the time of Antiquity, the following story has come down to us. The famous sophist Protagoras had a student named Euathlus, who studied law. The teacher and the student entered into an agreement, according to which Euathlus had to pay the tuition only after he won his first trial. However, after completing his studies, Euathlus was in no hurry to appear in court. The teacher's patience ran out, and he sued his student.
“In any case, Euathlus will have to pay me,” Protagoras reasoned. He will either win this trial or lose it. If he wins, he will pay by virtue of our contract with him; if he loses, he will pay by virtue of the verdict of the court.
“Nothing of the kind,” Euathlus reasoned in turn. - Indeed, I either win the process or lose it. If I win, the court decision will release me from payment; if I lose, I will not pay by virtue of our agreement.”
The student, as you can see, was worthy of his teacher! Perplexed by this turn of the matter, Protagoras devoted a special essay to this dispute with Euathlus, “Litigation for Payment,” which, unfortunately, has not come down to us. And if you are publicly asked, for example, such questions (having agreed in advance that you will answer only “yes” or “no”):
“Have you stopped beating your mother?”, “Have you stopped drinking cognac in the morning?” etc. Here you can fall into a slight tetanus: say at least “yes”, at least “no” - you still find yourself in a stupid position.
You feel that you are being cheated a little, but it's hard to say what's the matter. Sophistry began to interfere with the social life of the Greeks, just as, indeed, it interferes with us now. Aristotle systematized the logical knowledge known by that time, adding a lot of new things to them, and wrote several works in which he presented logic as a means of protecting the truth and exposing sophistry and lies. In this capacity, she has been serving people for more than two millennia. A significant contribution to the development of logic was made by the medieval scholastics, and the Latin terminology introduced by them is still preserved.
In the middle of the XIX century. logic has experienced its own scientific revolution: mathematical (symbolic) logic arose and began to develop rapidly, using mathematical tools and methods to analyze reasoning. It was she who laid theoretical basis subsequent development of programming languages for computer technology. Aristotelian logic has since become known as traditional.
In Russia, until 1917, logic was taught in the upper grades of the gymnasium, and Russian logicians made a significant contribution to the development of this science. However, shortly after October revolution logic - for a variety of reasons - disappears in Russia. Only in the late 40s. it returns again, first to legal institutes and universities, then to pedagogical and other universities, and in last years Logic is starting to infiltrate schools again.
Why is it important to get familiar with logic?
In our pragmatic age, when faced with something new, people first ask:
"Why do I need it?" Alas, simple curiosity is gradually disappearing, and the eternal pursuit of a career, success, just for a piece of bread leaves almost no time and energy for activities that do not bring immediate benefits. Therefore - why? Why should I read this book? Perhaps the following considerations will seem to you worthy of attention.
First of all, familiarity with logic teaches us to think accurately and express our thoughts clearly. Many people are not able to connect two words at all. Others speak, but so incoherently and vaguely that you will not understand anything. Logic contributes to the formation of coherent and clear speech.
Logic brings up the ability to justify your ideas and decisions and to convince other people. If you are able to substantiate your thought, the solution of a particular issue, then your speech will be not only clear, but also convincing. Whatever type of activity you are engaged in, often it is - necessary condition her success.
More importantly, familiarity with logic gradually forms the habit of analyzing one's own and others' reasoning. Logic also equips us with the means to detect, pinpoint and eliminate the error of reasoning. It helps us cope with demagogy and sophistry, saves us from that earthy innocence that easily pushes us into the arms of sweet-talking crooks. For example, they turn to you with the following reasoning: “I am a person, and you are not me, therefore, you are not a person.” And even if you feel that something is wrong here, can you adequately object? Unlikely. At best, grumble something like "You're a fool yourself!" and walk away with a sense of intellectual humiliation. Familiarity with logic will enable you to determine what kind of reasoning it is, what requirements it must satisfy, and which of these requirements is violated here. Having pointed out all this, you will shame the demagogue or the ignoramus, and he will already move away from you, sprinkling ashes on his head.
Logic will teach you to argue. Both in everyday life and in professional activities, we often have to engage in polemics on various issues. We, as a rule, do not know how to argue, and our collisions most often end in a squabble, shouting, and even a fight. Having become acquainted with logic, you will learn how to correctly defend your opinion, refute the erroneous belief of your opponent, find compromises, and expose unscrupulous tricks and tricks.
And yet the most important thing is that logic develops the habit of thinking. Modern life forces a person to know a lot, so the systems of school and higher education are built in such a way as to put as much information into the student's head as possible. But they, as a rule, do not teach to think, do not strive to develop this precious human ability. Therefore, many do not like and do not know how to think. Instead of thinking and finding our own solution to certain problems, we willingly rely on the opinion of some television broadcaster, friends or acquaintances. Of course, it is difficult to think, intense thinking takes as much strength as a miner or a hammerer spends. But after all, it is necessary to think if you do not want to live your whole life as a doll, which is pulled by the strings by clever manipulators. And when thinking becomes a habit, it becomes pleasurable. So an athlete, crunching his spine, sweating, with groans develops his muscles. But then, what delight the play of these muscles gives him, when every cell of the body sings about the joy of bodily existence!
It is for this purpose that a variety of tasks are included in the book. They are simple, but still make you think a little. Think! But this is not a textbook on logic. For a deeper acquaintance with this science, you need to refer to the specialized literature.
Chapter 2
Word and concept
Our everyday and professional conversations, speeches, disputes consist of words and sentences. We will assume that you know how to distinguish a word or phrase from a sentence, and we will first analyze words - it is from them, like bricks, that the “buildings” of our reasoning are built.
Among the words we use, names are the most important, and it is they that make up most words. Well, of course! Words are needed first of all in order to somehow designate, name the objects, phenomena, events around us. We are well acquainted with the class of proper names, say, "Petya", "Ivan Kuzmich", "Adelaida Mitrofanovna", "Alexander Pushkin", "Moscow", etc. But the words “house”, “birch”, “red” are also designations, i.e. names of certain objects or properties! Even when we meet an unfamiliar object, we immediately give it a name - “an unfamiliar object”; and if we come across a thing that we do not know what to call, we still call it - "a thing without a name." In short, all objects, properties, relations of the world around us, any object of our attention can be called by some name.
A name is a language expression denoting a single object, a collection of objects, a property or a relationship.
We see that logic greatly expands our ordinary understanding of what a name is. In everyday life, we consider as names only proper names denoting individual, one-of-a-kind objects. But the designations of groups of objects, relations, properties are also naturally considered as their names! Of course, this may seem somewhat unusual, and if the question "What is your name?", You will answer:
"My name is human," the questioner may be a little surprised. However, science often departs from ordinary common sense, penetrating into the essence of things. So it is here: logic highlights the most important function of our words - to designate something, and from this point of view, almost all of them turn out to be names.
concept
Language is used to express thoughts. Names not only designate certain objects, but also express this or that thought. This thought (more precisely, the form of thought) is called a concept.
A concept is a form of thought expressed by a name.
When we name a separate object or a group of objects, for example, "the current president of Russia" or "chicken", we simultaneously think about some features, properties of these objects, about their appearance, origin, relation to other objects, etc. The first of the above names denotes a specific person, but at the same time expresses the idea of \u200b\u200ba state structure modern Russia and chapter Russian state. The second name denotes an extensive class of objects and at the same time expresses the idea that these are birds, and domestic ones, that they carry eggs, cannot fly, etc. These thoughts, expressed by names, are concepts.
Every concept is expressed in a name, and every name expresses a concept. Therefore, in what follows we will often not distinguish between them. However, it should be remembered that a name is an expression of language, and a concept is a thought.
There is no rigid connection between names and concepts: the same concept can be expressed by different names, and the same name can express different concepts. This is manifested in synonymy, when two words express the same concept - for example, "brave" and "brave", and in homonymy, when one word is used to express two different concepts - for example, "bow" from which they shoot, and "onion" that is eaten. It is in the absence of a rigid connection between names and concepts that the source of misunderstanding between people, errors in reasoning, demagogy and sophistry lies. The writer V. Nabokov said that a German linguist translated Pushkin's "At Lukomorye ..." with the turnover "On the shore of the onion sea ..." He confused weapons with a plant!
1) You are the pilot of an airplane flying from New York to Moscow. During the flight, a terrorist is announced on board, demanding to land the plane in Lisbon. The crew doesn't know what to do. Passengers are in a panic. How old is an airplane pilot?
2) A small child is walking along the road and laments: “I have a mother and a father, but it’s a pity, I’m not their son!” Who is the child?
Each concept or name associated with it has scope and content.
For example, in the concept of "tree" the following properties are conceived: to be a plant, to have roots, a trunk, a crown. These properties form the content of the concept of "tree".
The scope of a concept is the set of those objects, each of which has properties included in the content of the concept.
For example, the scope of the concept of "tree" is formed by all those objects that have the properties listed above, i.e. all trees growing on earth. The scope of the concept of "table" will include all existing tables, in the scope of the concept of "students" - all people who study somewhere, etc. It should be borne in mind that the content of a concept is a set of properties, and the scope of a concept is a set of objects that have these properties.
Anagrams are words in which letters are rearranged, for example, "table" - "post". If the letters are rearranged according to some system, then a cipher is obtained. It is not always easy to guess which word was originally. Try it!
3) Shiamna, teevr, fekri, ezezhol, labosak, diropom, soratak, two, rmeo, nalep, dokilork, vtekots, surtal, korayezh, kasim, lukab, trix.
Concept classification
Concepts are divided into three groups according to their size.
Concepts, the volume of which includes only one subject, are called singular.
They are expressed, as a rule, by proper names or equivalent expressions, for example: “Moon”, “Paris”, “Napoleon”, “author of the Iliad”, “first cosmonaut”, etc.
General concepts are called, the scope of which includes more than one subject, i.e. two or more. For example, "river", "mountain", "planet", "house", "dog", etc.
And finally, concepts will be empty, the scope of which does not include a single real object.
Such concepts have content, i.e. they think of a set of some properties, but in the surrounding world there are no objects that would have these properties, for example: “mermaid”, “centaur”, “Baba Yaga”, “a man who lived 1000 years”, etc. In the concept of "mermaid" the following properties are conceived: a woman to the waist, a fish below the waist, lives in the water, comes ashore at night. But there are no creatures that have these properties. Therefore, the concept of "mermaid" is empty in scope. The use of empty concepts requires special care. If, say, the concept of "Mr. D's bank account" is empty in scope, and you argue as if it had scope, you might be making a mistake.
They say that a certain doctor from Toulouse, wanting to have some fun, placed the following announcement in the local newspaper: “In connection with my departure abroad, I am selling the house and all property, including the rarest historical relic, namely the skull of Voltaire the child.” And what do you think? Within one week, the editorial office of the newspaper received more than 100 requests from citizens about the price of this precious relic!
They say that 30 years ago, young guides of the St. Petersburg "Hermitage" had fun in a similar way. They led a group of innocent visitors into the hall where the skulls of human ancestors were displayed, and with serious look they began to say: “You see the skull of Peter I at the age of 5, but his skull at the age of 20, but this is how he became at the age of 50!” Some visitors noted with interest the changes that the emperor's skull underwent over time.
Specific concepts refer to objects, things, persons that have an independent existence.
Abstract concepts refer to the properties or relations of i objects.
The difference between concrete and abstract concepts is as follows. For example, a house White House, a horse is an independently existing objects and concepts about them are concrete. But “whiteness” or “horseness” are properties of objects that exist only in connection with objects, therefore the corresponding concepts are abstract. The concepts of “higher”, “big”, “redness”, “goodwill”, etc. will also be abstract.
Logics. Tutorial Gusev Dmitry Alekseevich
Introduction, Or what is logic and why is it needed?
Starting to get acquainted with any science, we first of all answer the question of what it studies, what it is devoted to, what it does. Logic is the science of thinking. But psychology, pedagogy, and many other sciences are also involved in thinking. This means that logic does not deal with all the questions and problems associated with thinking, not with all its areas or aspects, but only with some of them. What interests logic in thinking?
Each of us is well aware 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 mankind and its future, about books read and films watched, about today's studies and tomorrow's rest, etc., etc.
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 schemes or forms. Moreover, if the content of our thinking, as already mentioned, is infinitely diverse, then there are very few forms in which this diversity is expressed.
Let's take a simple example to illustrate this idea. Consider three completely different statements in content:
1. All carp are fish;
2. All triangles are geometric figures;
3. All chairs are pieces of furniture.
Despite the different content, these three statements have something in common, something unites them. What? They are united not by content, but by form. Differing in content, they are similar in form: after all, each of these three statements is built according to a scheme or form - "All A's are B", where A and B are any objects. It is clear that the statement itself "All A's are B" devoid of any content (What exactly is it talking about? Nothing!). This statement is a pure form, which, as you can guess, can be filled with any content, for example: All pines are trees; All cities are towns; All schools are educational institutions; All tigers are predators etc.
Let's take another example. Let's take three different statements:
1. If autumn comes, then the leaves fall;
2. If it rains tomorrow, then there will be puddles on the street;
3. If the substance is a metal, then it is electrically conductive.
Being unlike each other in content, these three statements are similar to each other in that they are built according to the same form: "If A, then B". It is clear that a huge number of different meaningful statements can be selected for this form, for example: If not prepared for control work, then you can get a deuce; If the runway is covered with ice, the planes cannot take off; If the word is at the beginning of a sentence, then it must be capitalized. etc.
So, we noticed that in terms of content our thinking is infinitely diverse, but all this diversity fits into just a few forms. So logic is not interested in the content of thinking (other sciences are concerned with it), it studies only the forms of thinking, it is not interested in what what we think, but as we think, therefore it is also often called formal logic. So, for example, if the content of the statement All mosquitoes are insects is normal, understandable, meaningful, and the statement All Cheburashki are aliens is meaningless, absurd, absurd, then for logic these two statements are equivalent: after all, it deals with forms of thinking, and the form of these two statements was the same - "All A's are B".
Thus, form of thinking is the way in which we express our thoughts, or the scheme by which they are built. There are three forms of thinking.
1. concept is a form of thinking that denotes some object or feature of an object (examples of concepts: pencil, plant, celestial body, chemical element, courage, stupidity, negligence etc.).
2. 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 losers; All triangles are not squares etc.).
3. inference is a form of thinking in which a new judgment or conclusion follows from two or more initial judgments. 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 whole endless world of our thoughts is expressed in concepts, judgments and conclusions. We will talk about these three forms of thinking in detail on other pages of the book.
In addition to forms of thought, logic also deals with laws of thought, that is, such rules, the observance of which always leads reasoning, regardless of its content, to true conclusions and protects against false ones (provided that the original judgments are true). There are four basic laws of thought (or laws of logic). Here we will only list (let's name) them, and we will consider each of them in detail after we consider all forms of thinking.
1. The law of identity.
2. The law of contradiction.
3. Law of the excluded middle.
4. The law of sufficient reason.
Violation of these laws leads to various logical errors, as a rule, to false conclusions. Sometimes these laws are violated involuntarily, not on purpose, out of ignorance. The resulting errors are called paralogisms. However, sometimes this is done deliberately, in order to confuse the interlocutor, confuse him and prove to him some false thought. Such deliberate violations of logical laws in order to prove outwardly correct false thoughts are called sophistry, which will be discussed ahead.
So, Logic is the science of the forms and laws of correct thinking.
Logic appeared around the 5th century. BC e. in Ancient Greece. The famous ancient Greek philosopher and scientist Aristotle (384–322 BC) is considered its creator. As you can see, logic is 2.5 thousand years old, but it still retains its practical significance. Many of the sciences and arts of the Ancient World have gone forever into the past and are for us only a “museum” value, they are of interest to us only as ancient monuments. But some few creations of the ancients survived the ages, and we continue to use them at the present time. These include the geometry of Euclid (we study it in school) and the logic of Aristotle, which is also often called traditional logic.
In the XIX century appeared and began to develop rapidly symbolic or mathematical or modern logics, which is based on ideas put forward long before the 19th century. by the German mathematician and philosopher Gottfried Leibniz (1646–1716), about making a complete transition to an ideal (i.e., completely freed from content) logical form using a universal symbolic language similar to the language of algebra. Leibniz spoke of the possibility of presenting a proof as a mathematical calculation. The Irish logician and mathematician George Boole (1815–1864) interpreted inference as the result of solving logical equalities, as a result of which the theory of inference took the form of a kind of algebra, differing from ordinary algebra only in the absence of numerical coefficients and degrees. Thus, one of the main differences between symbolic logic and traditional logic is that in the latter, when describing correct thinking, ordinary, or natural language is used; and symbolic logic investigates the same subject ( right thinking) by building artificial, special, formalized languages, or, as they are also called, calculus.
Traditional and symbolic logic are not, as it may seem, different sciences, but represent two successive periods in the development of one and the same science: the main content of traditional logic entered symbolic logic, was refined and expanded in it, although much was rethought.
Now let's answer the question why we need logic, what role it plays in our life. Logic helps us build our thoughts correctly and express them correctly, convince other people and understand them better, explain and defend our point of view, and avoid errors in reasoning. Of course, it is quite possible to do without logic: one common sense and life experience is often enough to solve any problems. For example, any person who is not familiar with logic can find a catch in the following reasoning:
Movement is eternal.
Going to school is movement.
Therefore, going to school is forever.
Everyone will notice that a false conclusion is obtained due to the use of the word "motion" in different senses (in the first original judgment it is used in a broad, philosophical sense, and in the second in a narrow, mechanical sense). However, it is not always easy to find an error in reasoning. Consider this example:
All my friends speak English.
The current president of America also knows English.
Therefore, the current President of America is my friend.
Any person will see that there is some catch in this reasoning, something in it is not right or wrong. But what? Those who are not familiar with logic, most likely, will not be able to determine exactly what mistake is made here. Anyone who is familiar with logic will immediately say that in this case a mistake was made - "non-distribution of the middle term in a simple syllogism." Or this example:
In all cities beyond the Arctic Circle there are white nights.
Petersburg is not located beyond the Arctic Circle.
Consequently, there are no white nights in St. Petersburg.
As we see, a false conclusion follows from two true judgments. It is clear that in this reasoning, too, something is wrong, there is some error. But what? It is unlikely that a person not familiar with logic will be able to immediately find it. And the one who owns a logical culture will immediately establish given error- "an extension of a larger term in a simple syllogism."
After reading this book, you will learn not only how logical laws are violated in such reasoning, but also a lot of other interesting and useful information.
So, common sense and life experience, as a rule, is enough to navigate in various difficult situations. But if to our common sense and add a logical culture to life experience, then we will not lose at all from this, but even, on the contrary, will win. Of course, logic will never solve all problems, but it can certainly help in life.
Common sense is often called practical, or intuitive logic. It is formed spontaneously in the process of life experience, approximately by the age of 6–7, that is, by school age or even earlier, and we all own it. For example, the very word "logics", most likely, was familiar to you long before you started reading this book. In life, we often come across expressions such as "logical reasoning", "illogical act", "iron logic" etc. Even if we have never studied logic, we still fully understand what is at stake when talking about logic, logical or illogical.
Consider this example: any person who is not familiar with logic will notice the logical incorrectness and even the absurdity of the statement: I go in new trousers, and you go to the gymnasium. And everyone will say that the following statement would be correct and meaningful: I go in trousers and you go in shorts or: I go to the gymnasium, and you go to the lyceum. When we study logic, we learn that in the given example the logical law of identity is violated, since two different (unequal or non-identical) situations are mixed in it: to walk in some clothes and to go somewhere. It turns out that even before we get acquainted with the law of identity, we already practically use it, we know about it, only implicitly, intuitively. Similarly, the law of identity is violated in the statement: Today we will dig a trench from this pillar until lunch. Even if a person does not know anything about the law of identity and its various and numerous violations, he, nevertheless, will definitely pay attention to the fact that there is some kind of logical error in this statement (even though he could not determine which one ).
In the same way, any person, most likely, cannot fail to notice some logical violation in the following statements: He did not take oral permission to writing; Let's go tomorrow evening at dawn; She was a young girl of old age etc. Not everyone can qualify this error as a violation of the logical law of contradiction. However, even if we do not know anything about this law, we feel, or feel it is violated.
Finally, in everyday life, each of us often hears and uses expressions such as: Why should I trust you? How will you prove it? On what basis? Justify! Motivate! etc. When we say this, we use the logical law of sufficient reason. Anyone who has not studied logic is most likely not familiar with this law and has not heard anything about it. However, as we see, ignorance of this logical law does not prevent us from using it practically or intuitively.
These examples testify in favor of the fact that all people own logic, regardless of whether they studied it or not. Thus, we practically use logic long before we begin to study it theoretically. The question arises: why do we need to study logic if we already own it?
Answering this question, it can be noted that the same thing happens with our native language: in practice, we begin to use it at the age of 2.5–3 years of our life, and we begin to study it only from school age. Why do we study our native language at school, if we know it so well long before it? At 2.5–3 years old, we use the language intuitively, or unconsciously: practically owning it, we know nothing not only about declensions and conjugations, but also about words and letters, and even about the very fact that in life we constantly we use language. We learn about all this only when we begin to study it at school (or senior preschool) age, as a result of which our intuitive use of the language gradually turns into a conscious one - we begin to master it much better.
So it is with logic: knowing it intuitively and using it practically every day, we study it as a science in order to turn the spontaneous use of logic into a conscious one, to master it even better and use it more effectively.
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Introduction, or What is logic and why is it needed?
Starting to get acquainted with any science, we first of all answer the question of what it studies, what it is devoted to, what it does. Logic is the science of thinking. But psychology, pedagogy, and many other sciences are also involved in thinking. This means that logic does not deal with all the questions and problems associated with thinking, not with all ᴇᴦο areas or aspects, but only with some of them. What interests logic in thinking?
Each of us is well aware 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 mankind and the future, about books read and films watched, about today's studies and tomorrow's rest, etc. etc. 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 schemes or forms. Moreover, if the content of our thinking, as already mentioned, is infinitely diverse, then there are very few forms in which this diversity is expressed.
Let's take a simple example to illustrate this idea.
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Consider three completely different statements in terms of content. 1. All carp are fish; 2. All triangles are geometric figures; 3. All chairs are pieces of furniture. Despite the different content, these three statements have something in common, something unites them. What? They are united not by content, but by form. Differing in content, they are similar in form, because each of these three statements is built according to a scheme or form - “All A is B”, where A and B are some objects. It is clear that the very statement “All A is B” is devoid of any content (What exactly is it talking about? Nothing!). This statement is a pure form, which, as you can guess, can be filled with any content, for example˸ All pines are trees; All cities are towns; All schools are educational institutions; All tigers are predators etc. etc.
Let's take another example.
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Let's take three statements with different content˸ 1. If autumn comes, then the leaves fall; 2. If it rains tomorrow, then there will be puddles on the street; 3. If the substance is a metal, then it is electrically conductive. Being dissimilar to each other in content, these three statements are similar to each other in that they are built according to the same form˸ “If A, then B”. It is clear that a huge number of different meaningful statements can be selected for this form, for example ˸ If you do not prepare for the test, you can get a deuce; If the runway is covered with ice, the planes cannot take off; If the word is at the beginning of a sentence, then ᴇᴦο must be capitalized etc. etc.
So, we noticed that in terms of content our thinking is infinitely diverse, but all this diversity fits into just a few forms. So logic is not interested in the content of thinking (other sciences are concerned with it), it studies only the forms of thinking, it is not interested in what what we think, but as we think, therefore it is also often called formal logic. So, for example, if the content of the statement All mosquitoes are insects is normal, understandable, meaningful, and the statement All Cheburashki are aliens is meaningless, absurd, absurd, then for logic these two statements are equivalent, because it deals with forms of thinking, and the form of these two statements was the same - “All A is B”.
Introduction, or What is logic and why is it needed? - concept and types. Classification and features of the category "Introduction, or What is logic and why is it needed?" 2015, 2017-2018.
The first year student was constantly impressed by the wisdom and wit of the undergraduate philosophy students he met. One day he plucked up the courage to ask one of them, “How is it that all undergraduate philosophy students are so smart?” The philosopher replied: “There is no mystery in this. It’s just that we all studied logic.” "Seriously?" the freshman asked. “Is that all it takes? So if I study logic, I will become very smart?” "Of course," replied the philosopher. “Really, it’s too late to sign up for the course right now. However, I'll tell you what, you can use my old textbook on logic and study it yourself. Here, I have it with me. I'll give it to you for twenty dollars." “Oh, thank you,” the freshman was delighted. The deal was completed, and the freshman gleefully set out to improve his IQ with his textbook under his arm. Later he ran into the philosopher again. "Hey!" he shouted. “That textbook on logic that you sold me for twenty dollars - I saw it in a bookstore for ten. Everything you told me about how logic would make me smarter. Now I see - you just wanted to rob me! "You see," replied the philosopher. "It's already starting to work." Get to know the reasons why philosophy and logic should be on your list of required subjects that you need to learn.
Symbolic logic is fun
Learning basic symbolic logic is like learning a new language, but its vocabulary is very small and there are very few grammar rules. You can learn to do all sorts of things with the new symbols: you will use them to analyze the logic of ordinary sentences, test arguments for validity, and create proofs for complex arguments whose validity is not obvious. The exercises to help you become more proficient in this section of logic look like jigsaw puzzles, so if you like Sudoku, you'll probably love logic too.
Knowing Whether an Argument Is Valid or Not is a Valuable Skill
Logic is essentially the study of the validity of an argument. All people use reasoning to draw conclusions that are useful to them. If your car won't start, you conclude that the battery may not be working. So you are testing the battery. If it works, you conclude that the problem must be something else, most likely the starter. So you check the starter. Etc. This reasoning example is simple, however sometimes the reasoning chains can get quite complex and confusing. If you practice generating good arguments and spotting bad ones, you can develop a skill that will be useful in almost every area of life. This will help you find the truth and recognize lies.
Good logic is an effective persuasion tool
The art of persuasion is called rhetoric. Rhetoric, like logic, used to be an important part of the liberal arts curriculum. Unfortunately, neither logic nor rhetoric are required subjects anymore. But at the same time, rhetoric includes almost all forms of persuasion, with the exception of bribes, extortion, and physical violence. It includes, for example, an appeal to emotions, provocative images or clever wordplay. There is no doubt that these approaches can be persuasive, but it is also worth paying attention to good persuasive reasoning. No one is saying that good arguments can always beat smart rhetoric: people aren't Vulcans like Mr. Spock, after all. However, in the long run, good arguments will always lead you to success.
Learning Logic Will Help You Spot Fallacies
Erroneous thinking abounds in modern culture. Politicians, experts, advertisers and official representatives appeal to the opinion of the majority, reject a view of something just because they do not like the person who expresses this opinion, and so on. Becoming familiar with fallacious thinking and fallacies of this kind will help you become a more critical reader, listener, and thinker.
Logic is a fundamental discipline
Logic is fundamental to every realm that uses arguments. She has particularly close ties to mathematics, computer science, and philosophy. Both Aristotelian and modern symbolic logic are both impressive archives of knowledge containing very important intellectual achievements.
Clear thinking makes you a better person
Dubious methods of persuasion, such as "criticizing" a candidate's views while demonstrating their not the best image, are very often used, in particular in election campaigns. There is no doubt that they are effective in some situations, but that should not be a reason to choose them over good and clear reasoning. On the contrary, that is why now logical thinking people need more than ever before.
Why do we need Logic? Why do we need mathematics - one of the most obvious manifestations of logic? All this is necessary to help a person solve problems and, as expected, based on this knowledge, we will be able to better understand life.
Why do you think a graduate of, say, a textile technical school can figure it out without any problems and after that work quite qualified, say, at a power plant, if he was practically not taught the features of this specialty? Why sometimes strangers understand each other perfectly, and sometimes they “fight” over this for years? Why do women misunderstand men and men misunderstand women?
In all these cases, the reason is the same - their Logic: if people have the same way of reasoning, they will easily understand each other (just like two graduates of completely different technical universities), and if the way of reasoning of one differs from the way of reasoning of another - complete mutual understanding impossible.
So Logic can be called a way of reasoning. There is standard logic, female logic, associative logic, and so on. There are many ways of reasoning, and, for example, you, the Reader, personally use your way of reasoning, your Logic.
During the search for Truths, we were most interested in such a way of reasoning, which would UNMISTAKELY predict the future in any life situation. We were interested in that method of reasoning that would work without a single error, without a single exception, and, at the same time, would be simple enough to be used in everyday life when communicating with people and predicting their behavior and actions. And we found this method! It is set forth in this book and every chapter written here is “soaked through and through”.
And now - short review what we have discarded as "methods of reasoning having exceptions" i.e. we will show you some examples of Truths that are not Truths.
Here is an example of mathematical logic: 0.5+0.5=1. In ordinary language, if you add two halves, for example, of the same substance, you get one whole. Indeed, if you add half a bucket of rice to the same amount, you get exactly a bucket of rice. However, if we take two halves of an apple and attach one to the other, we will not get a whole apple, we get all the same two halves of it. Those. this kind of Logic is the Truth, but not the Truth.
Yu.V. Ivlev in the textbook for universities "Logic" gives an example of research (in order to find out whether the ability to think logically is inherent in people), during which peasants were asked: "Does the third statement (and called them) follow from two statements (about the presence in the neighboring district center post office)? Do you know what the peasant answered? “What I don’t know, I don’t know. I've never been there". Further, the author concludes: “The logical culture of a modern literate person is higher than the logical culture of the peasants that were discussed. It even seems strange to us not to understand such simple reasoning.” Do you know what is the most amazing thing? The peasant answered the question from the position of CONFIDENCE! He was offered to draw a conclusion from the words he had said (they said that there is a post office in every district center) and asked: “Is there a post office in the neighboring district center?” And the peasant's Real Intention to make mistakes (unlike logicians) turned out to be equal to zero! He did not want to look at this situation from any other point of view, except from the point of view of CONFIDENCE!
In general, the fact that standard Logic studies judgments, statements, assertions is very indicative - i.e. all those things that we have designated by the word "Declared". But it would be much easier to compare statements with facts - it's much faster and this method is really error-free!
In conclusion, we can say that the representatives of standard Logic, in addition to other Real Intentions, have the following: "To get confused in words and try to confuse others."
There are also examples of “everyday” logic (I don’t want to capitalize this word!) such as “jealous - it means love”, “the longer the thread - the lazier the seamstress”, etc. etc. - the whole darkness of proverbs, sayings and aphorisms. And some people build their lives (it would be funny if it weren't sad!) on the basis of statements, among which are full of lies and truths. Yes, full of truths and lies! If you are overcome by doubts - take any collection of such statements and try to find Truths there (rules that have no exceptions)!
There is also a very "wonderful" pattern of reasoning among believers, which is built on quotations from sacred books, there are other ways of reasoning - however, they are also based on someone else's information, i.e. on VERA.
Let's define LOGIC.
LOGIC is a way of reasoning that includes certain rules and principles applied to find the best way actions and to solve problems that arise in the course of life.
Ask people around why they do something, how they understand this or that word, and you can get to the bottom of the rules that people use in life and that make up their Logic. The results that you get will be able to stun anyone, it will be something like the logic of one of the interviewed girls: “I am looking for a smart (!) man, so first of all I look at his shoes - if he has them polished, then (!) he is good to himself (!) and will be good to his wife (!) - so he is a suitable candidate for me. It is not surprising that this girl, instead of mutual understanding, will find only "polished shoes" in marriage! By the way, what did you find in your marriage as a result of using your Logic?
There is Logic, using which you will never get into trouble. Here is its definition.
ERROR-FREE LOGIC - logic based on Truths.
The same girl, using ERROR-FREE LOGIC, would not associate her life with a man whose most important Real Intentions for him are different from the Most Important Real Intentions for her, because. she would understand that the life of spouses whose Real Intentions are different is a life of conflicts, quarrels and resentments. Here's an example of Infallible Logic!
Only with the help of such logic it is possible to SOLVE (!) any problem in this universe!
The TRUTH in this book are just such rules, and they are right in front of you! Can someone stop you from using them?
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