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Lecture notes, cheat sheets
Логика. Умозаключение. Общая характеристика дедуктивных умозаключений (конспект лекций)
Directory / Lecture notes, cheat sheets Table of contents (expand) LECTURE No. 15. Inference. General characteristics of deductive reasoning 1. The concept of inference Inference - this is a form of abstract thinking, through which new information is derived from previously available information. In this case, the sense organs are not involved, that is, the entire process of inference takes place at the level of thinking and is independent of the information received at the moment from the outside. Visually, the conclusion is reflected in the form of a column in which there are at least three elements. Two of them are premises, the third is called the conclusion. Parcels and conclusions are usually separated from each other by a horizontal line. The conclusion is always written below, the premises - above. Both the premises and the conclusion are judgments. Moreover, these judgments can be both true and false. For example: All mammals are animals. All cats are mammals. All cats are animals. This conclusion is true. Inference has a number of advantages before the forms of sensory knowledge and experimental research. Since the process of inference takes place only in the realm of thinking, it does not affect real objects. This is a very important property, since often the researcher does not have the opportunity to get a real object for observation or experiments due to its high cost, size or remoteness. Some items at the moment can generally be considered inaccessible for direct research. For example, space objects can be attributed to such a group of objects. As is known, human exploration of even the closest planets to the Earth is problematic. Another advantage of inferences is that they provide reliable information about the object under study. For example, it was through inference that D. I. Mendeleev created his own periodic system of chemical elements. In the field of astronomy, the position of the planets is often determined without any visible contact, based only on the information already available about the regularities in the position of celestial bodies. Inference flaw one can say that conclusions are often characterized by abstractness and do not reflect many of the specific properties of the subject. This does not apply, for example, to the above-mentioned periodic table of chemical elements. It is proved that with its help, elements and their properties were discovered, which at that time were not yet known to scientists. However, this is not the case in all cases. For example, when determining the position of a planet by astronomers, its properties are reflected only approximately. Also, it is often impossible to speak about the correctness of the conclusion until it has passed the test in practice. Inferences can be true and probabilistic. The former reliably reflect the real state of affairs, the latter are of an uncertain nature. The types of inference are: induction, deduction and conclusion by analogy. Inference - this is primarily the derivation of consequences, it is applied everywhere. Every person in his life, regardless of profession, made conclusions and received consequences from these conclusions. And here the question of the truth of such consequences arises. A person who is not familiar with logic uses it at a philistine level. That is, he judges things, draws conclusions, draws conclusions based on what he has accumulated in the process of life. Despite the fact that almost every person is trained in the basics of logic at school, learns from their parents, the philistine level of knowledge cannot be considered sufficient. Of course, in most situations this level is enough, but there is a percentage of cases when logical preparation is simply not enough, although it is in such situations that it is most needed. As you know, there is such a type of crime as fraud. Most often, scammers use simple and proven schemes, but a certain percentage of them are engaged in highly skilled deception. Such criminals know logic almost perfectly and, in addition, have abilities in the field of psychology. Therefore, it often costs them nothing to deceive a person who is not prepared. All this speaks of the need to study logic as a science. Inference is a very common logical operation. As a general rule, in order to obtain a true judgment, the premises must also be true. However, this rule does not apply to evidence to the contrary. In this case, knowingly false premises are deliberately taken, which are necessary in order to determine the necessary object through their negation. In other words, false premisses are discarded in the process of deriving a consequence. 2. Deductive reasoning Like much in classical logic, the theory of deduction owes its appearance to the ancient Greek philosopher Aristotle. He developed most of the issues related to this kind of reasoning. According to the works of Aristotle deduction is the transition in the process of inference from the general to the particular. In other words, deduction is the gradual concretization of a more abstract concept. It goes through several steps, each time deriving a consequence from several premises. It must be said that true knowledge must be obtained through the process of deductive reasoning. This goal can be achieved only if the necessary conditions and rules are met. There are two types of inference rules: direct inference rules and indirect inference rules. Direct inference means obtaining a conclusion from two premises that will be true if the rules of direct inference are followed. Thus, the premises must be true and the rules for obtaining consequences must be observed. Subject to these rules, one can speak of the correctness of thinking regarding the subject taken. This means that in order to obtain a true judgment, new knowledge, it is not necessary to have all the information. Part of the information can be recreated in a logical way and fixed. Consolidation is necessary, because without it the process of obtaining new information becomes meaningless. It is not possible to transfer such information or use it in any other way. Naturally, such consolidation occurs through the language (spoken, written, programming language, etc.). Consolidation in logic occurs primarily with the help of symbols. For example, these can be conjunction symbols, disjunctions, implications, literal expressions, brackets, etc. The following types of inferences are deductive: conclusions of logical connections and subject-predicate conclusions. Also deductive inferences are direct. They are made from one premise and are called transformation, inversion and opposition to the predicate, the conclusions on the logical square are considered separately. Such conclusions are derived from categorical judgments. Let's consider these conclusions. The transformation has a scheme: S is P S is not non-R. This diagram shows that there is only one package. This is a categorical judgment. The transformation is characterized by the fact that when the quality of the premise changes in the process of inference, its quantity does not change, and the predicate of the consequence negates the predicate of the premise. There are two ways of transformation - double negation and replacement of a negation in a predicate by a negation in a connective. The first case is shown in the diagram above. In the second, the transformation is reflected in the scheme as S is not-P - S is not P. Depending on the type of judgment, the transformation can be expressed as follows. All S are P - No S is non-P. No S is P - All S is non-P. Some S are P - Some S are not non-P. Some S's are not P - Some S's are not-P. Treatment - this is a conclusion in which the quality of the premise does not change when the places of the subject and the predicate are changed. That is, in the process of inference, the subject takes the place of the predicate, and the predicate takes the place of the subject. Accordingly, the circulation scheme can be depicted as S is P - P is S. Appeal can be with or without limitation. (it is also called simple or pure). This division is based on a quantitative indicator of the judgment (meaning the equality or inequality of the volumes of S and P). This is expressed in whether the quantified word has changed or not and whether the subject and predicate are distributed. If such a change occurs, then the constraint has been handled. Otherwise, we can speak of pure conversion. Recall that a quantified word is a word - an indicator of quantity. Thus, the words "all", "some", "none" and others are quantified words. Contrasting with a predicate characterized by the fact that the link in the consequence is reversed, the subject contradicts the predicate of the premise, and the predicate is equivalent to the subject of the premise. It must be said that a direct inference with opposition to a predicate cannot be deduced from particular affirmative judgments. Let's give opposition schemes depending on the types of judgments. Some S are not P - Some non-P are S. No S is P - Some non-P are S. All S are P - No P is S. Combining what has been said, we can consider the opposition to the predicate as the product of two immediate inferences at once. The first one is the transformation. Its result is inverted. 3. Conditional and disjunctive inferences Speaking of deductive reasoning, one cannot but pay attention to conditional and disjunctive reasoning. Conditional inferences are called so because they use conditional propositions as premises (if a, then b). Conditional inferences can be reflected in the form of the following diagram. If a, then b. If b, then c. If a, then c. Above is a diagram of inferences, which are a kind of conditional. It is characteristic of such inferences that all of their premises are conditional. Another type of conditional inference is conditional categorical judgments. According to the name in this conclusion, not both premises are conditional propositions, one of them is a simple categorical proposition. It is also necessary to mention modes - varieties of inferences. There are: affirming mode, denying mode and two probabilistic modes (first and second). Approving mode has the widest distribution in thinking. This is due to the fact that it gives a reliable conclusion. Therefore, the rules of various academic disciplines are built mainly on the basis of the affirmative mode. You can display the affirmative mode as a diagram. If a, then b. a. b. Let us give an example of an assertive mode. If the ax falls into the water, it will sink. The ax fell into the water. He will drown. The two true propositions that are the premises of this proposition are transformed in the process of inference into a true proposition. Negative mode expressed in the following way. If a, then b. Non-b. Nope. This judgment is based on the negation of the consequence and the negation of the foundation. Inferences can give not only true, but also indefinite judgments (it is not known whether they are true or false). In this connection it is necessary to speak about probabilistic modes. The first probabilistic mode in the diagram is displayed as follows. If a, then b. b. Probably a. As the name implies, the consequence deduced from the premises with the help of this mode is probable. If a strong wind blows, then the yacht heels to one side. The yacht rolls to one side. Probably a strong wind is blowing. As we see from the statement of the consequence to the statement of the reason it is impossible to draw a true conclusion. The second probabilistic mode in the form of a diagram can be depicted as follows. If a, then b. Nope. Probably not-b. Let's take an example. If a person lies under the sun, he will tan. This man does not lie under the sun. It won't burn. As can be seen from the above example, making a conclusion from the negation of the basis to the negation of the consequence, we will get not a true, but a probabilistic consequence. The formulas of the affirming and denying modes are the laws of logic, while the formulas of probabilistic ones are not. Divisive reasoning are divided into simple disjunctive and divisive-categorical inferences. In the first case, all premises are separating. Accordingly, dividing-categorical judgments have a simple categorical judgment as one of the premises. In this way, inference is considered divisive, all or part of whose premises are disjunctive judgments. The structure of a simple disjunctive inference is reflected as follows. S is A or B or C. And there is A1 or A2. S is A1 or A2 or B or C. An example of such a conclusion is the following. The path can be straight or circular. The roundabout can be with one transfer or with several transfers. The path can be straight or with one transfer, or with several transfers. Separative-categorical inferences can be represented in the form of a diagram. S is A or B. S is A (B). S is not B(A). For example: The shot is accurate and inaccurate. This shot is accurate. This shot is not inaccurate. Here it is necessary to mention conditional-separative inferences. They differ from the above inferences in their premises. One of them is a disjunctive proposition, which is not special, but the second premise of such propositions consists of two or more conditional propositions. A conditional-separative judgment can be either a dilemma or a trilemma. in a dilemma the conditional premise consists of two members. In this case, the separation implies the presence of a choice. In other words, a dilemma is a choice between two options. The dilemma can be simple constructive and complex constructive, as well as simple and complex destructive. The first has two premises, one of which asserts the same outcome of the two proposed situations, the other says that one of these situations is possible. The corollary summarizes the statement of the first premise (the conditional proposition). If you press on a pencil, it will break; if you bend a pencil, it will break. You can press the pencil or bend the pencil. The pencil will break. A complex design dilemma involves a harder choice between alternatives. Trilemma consists of two premises and a consequence and offers a choice of three options or states three facts. If the athlete strikes in time, he will win; if the athlete correctly distributes the forces, then he will win; if the athlete performs the jump cleanly, he will win. The athlete will strike in time or correctly distribute the forces over the distance, or perform the jump cleanly. The athlete will win. There are cases when a conclusion or one of the premises is omitted in conditional, disjunctive or conditionally distributive inferences. Such conclusions are called abbreviated. Author: Shadrin D.A. << Back: logical laws (The concept of logical laws. The law of identity. The law of non-contradiction. The law of excluded middle. Sufficient reason) >> Forward: Syllogism (The concept of syllogism. Simple categorical syllogism. Complex syllogism. Abbreviated syllogism. Abbreviated complex syllogism)
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