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Basic approaches to the problem of cognition

Epistemology is a branch of philosophy that studies the nature of knowledge, the ways, sources and methods of knowledge, as well as the relationship between knowledge and reality.

There are two main approaches to the problem of cognition.

1. Epistemological optimism, whose supporters recognize that the world is knowable regardless of whether we can currently explain some phenomena or not.

This position is adhered to by all materialists and some consistent idealists, although their methods of cognition are different.

The basis of cognition is the ability of consciousness to reproduce (reflect) to a certain degree of completeness and accuracy an object existing outside it.

The main premises of the theory of knowledge of dialectical materialism are the following:

1) the source of our knowledge is outside of us, it is objective in relation to us;

2) there is no fundamental difference between “phenomenon” and “thing in itself”, but there is a difference between what is known and what is not yet known;

3) cognition is a continuous process of deepening and even changing our knowledge based on the transformation of reality.

2. Epistemological pessimism. Its essence is doubt in the possibility of knowability of the world.

Types of epistemological pessimism:

1) skepticism - a direction that questions the possibility of knowing objective reality (Diogenes, Sextus Empiricus). Philosophical skepticism turns doubt into a principle of knowledge (David Hume);

2) agnosticism - a movement that denies the possibility of reliable knowledge of the essence of the world (I. Kant). The source of knowledge is the external world, the essence of which is unknowable. Any object is a “thing in itself”. We cognize only phenomena with the help of innate a priori forms (space, time, categories of reason), and we organize our experience of sensation.

At the turn of the 19th and 20th centuries, a type of agnosticism was formed - conventionalism. This is the concept that scientific theories and concepts are not a reflection of the objective world, but the product of agreement between scientists.

Human cognition

Cognition is the interaction of a subject and an object with the active role of the subject itself, resulting in some kind of knowledge.

The subject of cognition can be an individual, a collective, a class, or society as a whole.

The object of knowledge can be the entire objective reality, and the subject of cognition can be only its part or area directly included in the process of cognition itself.

Cognition is a specific type of human spiritual activity, the process of comprehending the surrounding world. It develops and improves in close connection with social practice.

Cognition is a movement, a transition from ignorance to knowledge, from less knowledge to more knowledge.

In cognitive activity, the concept of truth is central. Truth is the correspondence of our thoughts to objective reality. A lie is a discrepancy between our thoughts and reality. Establishing the truth is the act of transition from ignorance to knowledge, in a particular case - from misconception to knowledge. Knowledge is a thought that corresponds to objective reality and adequately reflects it. A misconception is an idea that does not correspond to reality, a false idea. This is ignorance, presented, accepted as knowledge; a false idea presented or accepted as true.

A socially significant process of cognition is formed from millions of cognitive efforts of individuals. The process of transforming individual knowledge into universally significant knowledge, recognized by society as the cultural heritage of humanity, is subject to complex sociocultural patterns. The integration of individual knowledge into the commonwealth is carried out through communication between people, critical assimilation and recognition of this knowledge by society. The transfer and transmission of knowledge from generation to generation and the exchange of knowledge between contemporaries are possible thanks to the materialization of subjective images and their expression in language. Thus, cognition is a socio-historical, cumulative process of obtaining and improving knowledge about the world in which a person lives.

Structure and forms of knowledge

The general direction of the process of cognition is expressed in the formula: “From living contemplation to abstract thinking and from it to practice.”

In the process of cognition, stages are distinguished.

1. Sensory cognition is based on sensory sensations that reflect reality. Through feelings a person contacts the outside world. The main forms of sensory cognition include: sensation, perception and representation. Sensation is an elementary subjective image of objective reality. A specific feature of sensations is their homogeneity. Any sensation provides information only about one qualitative aspect of an object.

A person is able to significantly develop the subtlety and acuity of feelings and sensations.

Perception is a holistic reflection, an image of objects and events in the surrounding world.

An idea is a sensory recollection of an object that does not currently affect a person, but once acted on his senses. Because of this, the image of an object in the imagination, on the one hand, is of a poorer character than in sensations and perceptions, and on the other hand, the purposeful nature of human cognition is more strongly manifested in it.

2. Rational knowledge is based on logical thinking, which is carried out in three forms: concepts, judgments, and inferences.

A concept is an elementary form of thought in which objects are reflected in their general and essential properties and features. Concepts are objective in content and source. Specific abstract concepts are identified that differ in degrees of generality.

Judgments reflect connections and relationships between things and their properties and operate with concepts; judgments deny or affirm something.

Inference is a process as a result of which a new judgment is obtained from several judgments with logical necessity.

3. Intuitive knowledge is based on the fact that a sudden decision, the truth, independently comes to a person on an unconscious level, without preliminary logical proof.

Features of everyday and scientific knowledge

Knowledge differs in its depth, level of professionalism, use of sources and means. Everyday and scientific knowledge are distinguished. The former are not the result of professional activity and, in principle, are inherent to one degree or another in any individual. The second type of knowledge arises as a result of deeply specialized activities that require professional training, called scientific knowledge.

Cognition also differs in its subject matter. Knowledge of nature leads to the development of physics, chemistry, geology, etc., which together constitute natural science. Knowledge of man and society determines the formation of humanitarian and social disciplines. There is also artistic and religious knowledge.

Scientific knowledge as a professional type of social activity is carried out according to certain scientific canons accepted by the scientific community. It uses special research methods and also evaluates the quality of the knowledge obtained based on accepted scientific criteria. The process of scientific knowledge includes a number of mutually organized elements: object, subject, knowledge as a result and research method.

The subject of knowledge is the one who realizes it, that is, a creative person who forms new knowledge. The object of knowledge is a fragment of reality that is the focus of the researcher’s attention. The object is mediated by the subject of cognition. If the object of science can exist independently of the cognitive goals and consciousness of the scientist, then this cannot be said about the object of knowledge. The subject of knowledge is a certain vision and understanding of the object of study from a certain point of view, in a given theoretical-cognitive perspective.

The cognizing subject is not a passive contemplative being, mechanically reflecting nature, but an active, creative personality. In order to get an answer to the questions posed by scientists about the essence of the object being studied, the cognizing subject has to influence nature and invent complex research methods.

Philosophy of scientific knowledge

The theory of scientific knowledge (epistemology) is one of the areas of philosophical knowledge.

Science is a field of human activity, the essence of which is to obtain knowledge about natural and social phenomena, as well as about man himself.

The driving forces of scientific knowledge are:

1) practical need for knowledge. Most sciences grew out of these needs, although some of them, especially in such areas as mathematics, theoretical physics, cosmology, were born not under the direct influence of practical need, but from the internal logic of the development of knowledge, from contradictions in this knowledge itself;

2) curiosity of scientists. The task of a scientist is to ask nature questions through experiments and get answers to them. An incurious scientist is not a scientist;

3) the intellectual pleasure that a person experiences when discovering something that no one knew before (in the educational process, intellectual pleasure is also present as the student discovering new knowledge “for himself”).

The means of scientific knowledge are:

1) the mind, logical thinking of a scientist, his intellectual and heuristic (creative) abilities;

2) sense organs, in unity with the data of which mental activity is carried out;

3) instruments (appeared since the 17th century), which provide more accurate information about the properties of things.

A device is like one or another organ of the human body that has gone beyond its natural boundaries. The human body distinguishes degrees of temperature, mass, illumination, current, etc., but thermometers, scales, galvanometers, etc. do this much more accurately. With the invention of instruments, human cognitive capabilities have expanded incredibly; Research became available not only at the level of short-range action, but also long-range action (phenomena in the microcosm, astrophysical processes in space). Science begins with measurement. Therefore, the scientist’s motto is: “Measure what can be measured, and find a way to measure what cannot yet be measured.”

Practice and its functions in the process of cognition

Practice and knowledge are closely related to each other: practice has a cognitive side, knowledge has a practical side. As a source of knowledge, practice provides initial information that is generalized and processed by thinking. Theory, in turn, is a generalization of practice. In practice and through practice, the subject learns the laws of reality; without practice there is no knowledge of the essence of objects.

Practice is also the driving force of knowledge. Impulses emanate from it, largely determining the emergence of a new meaning and its transformation.

Practice determines the transition from the sensory reflection of objects to their rational reflection, from one research method to another, from one thinking to another, from empirical thinking to theoretical thinking.

The purpose of knowledge is to achieve true meaning.

Practice is a specific method of development in which the result of an activity is adequate to its purpose.

Practice is a set of all types of socially significant, transformative activities of people, the basis of which is production activity. This is the form in which the interaction of object and subject, society and nature is realized.

The importance of practice for the cognitive process, for the development and development of scientific and other forms of knowledge has been emphasized by many philosophers of different directions.

The main functions of practice in the process of cognition:

1) practice is a source of knowledge because all knowledge is caused in life mainly by its needs;

2) practice acts as the basis of knowledge, its driving force. It permeates all aspects, moments of knowledge from its beginning to its end;

3) practice is directly the goal of knowledge, for it exists not for the sake of simple curiosity, but in order to direct them to correspond to images, to one degree or another regulate the activities of people;

4) practice is the decisive criterion, that is, it allows one to separate true knowledge from misconceptions.
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Perhaps this is the most famous work of Lord Bertrand Arthur William Russell (1872–1970), who left a bright mark on English and world philosophy, logic, sociology, and political life. Following G. Frege, he, together with A. Whitehead, attempted a logical substantiation of mathematics (see Principles of Mathematics). B. Russell is the founder of English neorealism, as a type of neopositivism. B. Russell did not recognize either materialism or religion. Bertrand Russell is very widely cited, and when I came across no less than 10 references in the books I read, I decided it was time bite into in this considerable work...

Bertrand Russell. Human knowledge, its spheres and boundaries. – Kyiv: Nika-Center, 2001. – 560 p. (The book was first published in English in 1948)

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The medieval Christian cosmos is fashioned from certain elements of poetic fantasy that paganism retained to the end. Both the scientific and poetic elements of the medieval cosmos were expressed in Dante's Paradise. It was precisely this picture of the universe that the pioneers of the new astronomy opposed. It is interesting to compare the noise created around Copernicus with the almost complete oblivion that befell Aristarchus.

The theory of the Sun and planets as a complete system was practically completed by Newton. Contrary to Aristotle and medieval philosophers, she showed that the Sun, not the Earth, is the center of the solar system; that the celestial bodies, left to themselves, would move in straight lines and not in circles; that in fact they move not in straight lines or in circles, but in ellipses, and that no external action is necessary to maintain their movement. But Newton said nothing scientific about the origin of the solar system.

General relativity holds that the universe is finite in size - not in the sense that it has an edge beyond which there is something that is no longer part of the universe, but that it is a sphere having three dimensions in which the straightest possible lines return over time to their starting point, as on the surface of the Earth. The theory stipulates that the universe must be either contracting or expanding; it uses observed facts about nebulae to decide the question in favor of expansion. According to Eddington, the universe doubles in size every 1,300 million years or so. If this is so, then the universe was once very small, but will eventually become quite large (by the time the book was written - 1948 - the Big Bang concept had not yet become dominant).

Galileo introduced two principles that contributed to the possibility of mathematical physics: the law of inertia and the law of parallelogram. Aristotle thought that the planets needed gods to move them in their orbits, and that movements on earth could begin independently in animals. Movements in matter, according to this view, can only be explained by immaterial causes. The law of inertia changed this view and made it possible to calculate the movements of matter through the laws of dynamics alone. Newton's law of parallelogram concerns what happens to a body when two forces act on it at once.

From the time of Newton until the end of the 19th century, the progress of physics did not provide any essentially new principles. The first revolutionary news was Planck's introduction of the quantum constant h in 1900. Newton's view concerned the apparatus of dynamics and had, as he pointed out, empirical grounds for his preference. If the water in a bucket rotates, it rises up the sides of the bucket, and if the bucket rotates while the water is at rest, the surface of the water remains flat. We can therefore distinguish between the rotation of water and the rotation of a bucket, which we could not do if the rotation were relative. Einstein showed how Newton's conclusion could be avoided and space-time position could be made purely relative.

General relativity contains in its equations what is called the “cosmic constant,” which determines the size of the universe at any time. According to this theory, the universe is finite, but limitless, like the surface of a sphere in three-dimensional space. All this implies non-Euclidean geometry and may seem mysterious to those whose imagination is connected with Euclidean geometry (for more details, see). The size of the universe is measured at between 6,000 and 60,000 million light years, but the size of the universe doubles approximately every 1,300 million years. All this, however, can be doubted.

Quantum equations differ from the equations of classical physics in a very important respect, namely, that they are “nonlinear.” This means that if you have discovered the effect of only one cause, and then the effect of only another cause, then you cannot find the effect of both of them by adding two separately determined effects. It turns out a very strange result.

The theory of relativity and experiments have shown that mass is not constant, as previously thought, but increases with rapid movement; if a particle could move at the speed of light, its mass would become infinitely large. Quantum theory made an even greater attack on the concept of “mass”. It now appears that wherever energy is lost by radiation, there is also a corresponding loss of mass. The Sun is believed to be losing mass at a rate of four million tons per second.

CHAPTER 4. BIOLOGICAL EVOLUTION. It has turned out to be much more difficult for humanity to take a scientific point of view in relation to life than in relation to celestial bodies. If what the Bible says is taken literally, then the world was created in 4004 BC. The brevity of the time allowed by the book of Genesis was at first the most serious obstacle to scientific geology. All previous battles between science and theology in this area have faded in the face of the great battle over the question of evolution, which began with the publication of Darwin's On the Origin of Species in 1859, and which has not yet ended in America (since the book was written, the situation in the United States has probably , has only gotten worse; see, for example, Less than Half of Americans Believe in Darwin's Theory).

Thanks to Mendel's theory, the process of inheritance became more or less clear. According to this theory, in the egg and in the sperm there is a certain, but very small number of “genes” that carry hereditary traits (for more details, see). The doctrine of evolution now enjoys general acceptance. But the special driving force assumed by Darwin, namely the struggle for existence and survival of the fittest, is not as popular among biologists now as it was fifty years ago. Darwin's theory was an extension of the economic principle of laisser-faire to life in general; Now that this type of economics, like its corresponding kind of politics, has fallen out of fashion, people prefer other ways of explaining biological changes.

There is no reason to assume that living matter is governed by different laws than non-living matter, and there is good reason to think that everything in the behavior of living matter can theoretically be explained in terms of physics and chemistry (this approach is called reductionism; see its criticism).

CHAPTER 5. PHYSIOLOGY OF SENSATION AND VILLATION. From the point of view of orthodox psychology, there are two boundaries between the mental and physical worlds, namely sensation and volition. “Sensation” can be defined as the first mental effect of a physical cause, “volition” - as the last mental cause of a physical action.

The problem of the relationship between consciousness and matter, which belongs to the field of philosophy, concerns the transition from phenomena in the brain to sensation and from volition to other phenomena in the brain. This is, therefore, a double problem: how does matter affect consciousness in sensation, and how does consciousness affect matter in volition?

There are two types of nerve fibers, some that conduct stimulation to the brain and others that conduct impulses from it. The first are related to the physiology of sensation.

Can the process in the brain that connects the arrival of sensory stimulation with the departure of impulses to the muscles be fully expressed in physical terms? Or is it necessary to resort to “psychic” mediators - such as sensation, reflection and volition?

There are reflexes in which the response is automatic and not controlled by the will. Conditioned reflexes are sufficient to explain most human behavior; whether there is a residue in it which cannot be thus explained is a question which at present remains open.

CHAPTER 6. SCIENCE OF SPIRIT. Psychology as a science was damaged by its association with philosophy. The distinction between spirit and matter, which was not sharply drawn by the Pre-Socratics, received special significance in Plato. Gradually the distinction between soul and body, which at first was an obscure metaphysical subtlety, became part of the generally accepted world view, and few metaphysicians in our time dare to doubt it. The Cartesians reinforced the absoluteness of this distinction by denying all interaction between thought and matter. But their dualism was followed by Leibniz's monadology, according to which all substances are souls. In France in the 18th century, materialists appeared who denied the soul and argued for the existence of only material substance. Among the great philosophers, Hume alone denied all substance in general and thereby pointed the way for modern debates about the difference between the mental and the physical.

Psychology can be defined as the science of such phenomena that, by their very nature, can only be observed by the person experiencing them. There is often, however, such a close resemblance between the simultaneous perceptions of different people that insignificant differences can be ignored for many purposes; in such cases we say that all these people perceive the same phenomenon, and we attribute such a phenomenon to the public world, but not to the personal one. Such phenomena are the data of physics, while phenomena that do not have such a social character are (as I believe) data of psychology.

This definition faces serious objections from psychologists who believe that "introspection" is not a true scientific method and that nothing can be scientifically known except what is obtained from public data. “Social” data are those that cause the same sensations in all persons perceiving them. It is difficult to draw a definite line between public and personal data. I come to the conclusion that there is knowledge of personal data and that there is no reason to deny the existence of a science about it.

Are there any causal laws that operate only in consciousness? If such laws exist, then psychology is an autonomous science. For example, psychoanalysis strives to uncover purely mental causal laws. But I do not know of a single psychoanalytic law that would claim to predict what will always happen under such and such circumstances. Although at present it is difficult to give any significant examples of truly precise mental causal laws, it still seems absolutely certain, on the basis of ordinary common sense, that such laws exist.

PART TWO. LANGUAGE

CHAPTER 1. LANGUAGE USE. Language primarily serves as a means of making statements and conveying information, but this is only one and perhaps not its most basic function. Language can be used to express emotions or to influence the behavior of others. Each of these functions; can be accomplished, although with less success, with the help of pre-verbal means.

Language has two primary functions: the function of expression and the function of communication. In ordinary speech both elements are usually present. Communication is not just about conveying information; it must include orders and questions. Language has two interrelated virtues: the first is that it is social, and the second is that it is a means for society to express “thoughts” that would otherwise remain private.

There are two other very important uses of language: it enables us to conduct our affairs with the outside world by means of signs (symbols) that have (1) a certain degree of constancy in time and (2) a significant degree of discreteness in space. Each of these virtues is more evident in writing than in speaking.

CHAPTER 2. VISUAL DEFINITION can be defined as “the process by which a person, by any means, to the exclusion of the use of other words, learns to understand a word.” There are two stages in the process of mastering a foreign language: the first is when you understand it only through translation into your language, and the second is when you can already “think” in a foreign language. Knowledge of a language has two aspects: passive - when you understand what you hear, active - when you can speak yourself. The passive side of visual definition is a well-known act of association, or conditioned reflex. If a certain stimulus A produces a certain response R in a child and is frequently associated with the word B, then in time it will happen that B will produce the response R or some part of it. As soon as this happens, word B will acquire “meaning” for the child: it will already “mean” A.

The active side of language learning requires other abilities. For every child it is a discovery that there are words, that is, sounds with meaning. Learning to pronounce words is a rewarding game for a child, especially because this game gives him the opportunity to communicate his desires more definitely than through shouts and gestures. It is thanks to this pleasure that the child does the mental work and muscular movements that are necessary to learn to speak.

CHAPTER 3. PROPER NAMES. There is a traditional distinction between "proper" names and "class" names; this distinction is explained by the fact that proper names refer to only one object, while class names refer to all objects of a certain kind, no matter how numerous they may be. Thus, “Napoleon” is a proper name, and “man” is a class name.

CHAPTER 4. EGOCENTRIC WORDS. I call “egocentric words” those words whose meaning changes with changes in the speaker and his position in time and space. The four basic words of this kind are “I”, “this”, “here” and “now”.

CHAPTER 5. DELAYED REACTIONS: COGNITION AND FAITH. Let's say that you are going to take a train trip tomorrow, and today you are looking for your train in the train schedule; you do not at this moment intend to use the knowledge you have acquired in any way, but when the time comes, you will act accordingly. Cognition, in the sense in which it is not merely the recording of actual sense impressions, consists chiefly of preparations for such delayed reactions. Such preparations can in all cases be called "faith" and are called "knowledge" only when they promise successful reactions or at least turn out to be connected with the facts relating to them in such a way that they can be distinguished from preparations that could be would be called "mistakes".

Another example is the difficulty that uneducated people have with hypotheses. If you tell them, “Let's assume such and such and see what follows from this assumption,” then such people will either tend to believe your assumption, or they will think that you are simply wasting your time. Therefore, reductio ad absurdum is an incomprehensible form of argumentation for those not familiar with logic or mathematics; if a hypothesis is proven to be false, they are unable to conditionally accept the hypothesis.

CHAPTER 6. PROPOSALS. Words that denote objects can be called “indicative” words. Among these words I include not only names, but also words denoting qualities, such as “white,” “hard,” “warm,” as well as words denoting perceived relationships, such as “before,” “above,” “ V". If the only purpose of language was to describe sensory facts, then we would be content with indicative words alone. But such words are not sufficient to express doubt, desire or disbelief. They are also not sufficient to express logical connections, for example: “If this is so, then I will eat my hat” or: “If Wilson had been more tactful, then America would have joined the League of Nations.”

CHAPTER 7. RELATIONSHIP OF IDEAS AND BELIEFS TO THE EXTERNAL. The relation of an idea or image to something external consists of a belief, which, when identified, can be expressed in the words: “This has a prototype.” In the absence of such faith, even in the presence of a real prototype, there is no relation to the external. Then it is a case of pure imagination.

CHAPTER 8. TRUTH AND ITS ELEMENTARY FORMS. In order to define "true" and "false", we must go beyond sentences and consider what they "express" and what they "express". A sentence has a property that I will call “sense (meaning).” What distinguishes truth from falsehood must be sought not in the sentences themselves, but in their meanings. Some sentences, which at first glance seem quite well constructed, are in fact absurd in the sense that they have no meaning (meaning). For example, “Necessity is the mother of invention” and “Continuous procrastination steals time.”

What an asserted proposition expresses is belief; that which makes it true or false is a fact, which is generally distinct from belief. Truth and lies are related to the attitude towards the external; this means that no analysis of a proposition or belief will tell whether it is true or false.

A sentence of the form “This is A” is said to be “true” when it is caused by what “A” stands for. We can say, moreover, that a sentence of the form “it was A” or “That will be A” is “true” if the sentence “This is A” was or will be true in the sense indicated. This applies to all sentences which state what is, was, or will be a fact of perception, and also to those in which we correctly infer from the perception its ordinary concomitants by means of the animal faculty of inference. One important point that can be made about our definition of “meaning” and “truth” is that both depend on an understanding of the concept of “cause.”

CHAPTER 9. LOGICAL WORDS AND LIES. We examine propositions of those kinds that can be proven or disproved when the relevant observational evidence is known. When it comes to such propositions, we must no longer consider the relation of belief or propositions to something which is in general neither belief nor proposition; instead we must consider only the syntactic relations between sentences in virtue of which the certain or probable truth or falsehood of a certain sentence follows from the truth or falsehood of certain other sentences.

In such inferences there are certain words, of which one or more always take part in the inference, and which I will call “logical” words. These words are of two kinds, which may be called "conjunctions" and "common words" respectively, although not quite in the usual grammatical sense. Examples of conjunctions are: “not”, “or”, “if - then”. Examples of general words are “all” and “some”.

With the help of conjunctions we can draw various simple conclusions. If "P" is true, then "not - P" is false, if "P" is false, then "not - P" is true. If "P" is true, then "P or q" is true; if "q" is true, then "P or q" is true. If "P" is true and "q" is true, then "P and q" are true. And so on. I will call sentences containing conjunctions “molecular” sentences; in this case, the connected “P” and “q” are understood as “atoms”. Given the truth or falsity of atomic sentences, the truth or falsity of each molecular sentence composed of these atomic sentences follows the syntactic rules and does not require a new observation of the facts. We are truly in the realm of logic here.

When an indicative sentence is expressed, we are dealing with three points: firstly, in the cases considered, there is a cognitive attitude of the affirmer - belief, disbelief and hesitation; secondly, there is a content denoted by the sentence, and thirdly, there is a fact (or facts) in virtue of which the sentence is true or false, which I call a “verifier fact” or “falsifying fact ( falsifier)" sentences.

CHAPTER 10. GENERAL COGNITION. By "general cognition" I mean the knowledge of the truth or falsity of sentences containing the word "all" or the word "some" or the logical equivalents of these words. One might think that the word "some" means less generality than the word "all", but this would be a mistake. This is clear from the fact that the negation of a sentence with the word “some” is a sentence with the word “all”, and vice versa. The negation of the sentence: “Some people are immortal” is the sentence: “All people are mortal,” and the negation of the sentence: “All people are mortal” is the sentence: “Some people are immortal.” From this it is clear how difficult it is to refute sentences with the word “some” and, accordingly, to prove sentences with the word “all”.

CHAPTER 11. FACT, FAITH, TRUTH AND KNOWLEDGE. A fact, in my understanding of this term, can only be defined visually. I call everything that exists in the universe a “fact.” The sun is a fact; Caesar's crossing of the Rubicon was a fact; If I have a toothache, then my toothache is a fact. Most facts do not depend on our will, which is why they are called “harsh”, “stubborn”, “irremovable”.

Our entire cognitive life is, from a biological point of view, part of the process of adaptation to facts. This process takes place, to a greater or lesser extent, in all forms of life, but is called "cognitive" only when it reaches a certain level of development. Since there is no sharp boundary between the lowest animal and the most eminent philosopher, it is clear that we cannot say exactly at what point we pass from the sphere of simple animal behavior into a sphere that deserves by its dignity the name “cognition.”

Faith is manifested in the affirmation of a proposition. Sniffing the air, you exclaim: “God! There's a fire in the house! Or, when a picnic is starting, you say: “Look at the clouds. It will be raining". I am inclined to think that sometimes a purely bodily state may deserve the name of "faith." For example, if you walk into your room in the dark and someone has placed a chair in an unusual place, you may bump into the chair because your body believed there was no chair in that place.

Truth is a property of faith and, as a derivative, a property of sentences expressing faith. Truth consists in a certain relationship between a belief and one or more facts other than the belief itself. When this relationship is absent, the belief turns out to be false. We need a description of the fact or facts that, if they actually exist, make the belief true. I call such a fact or facts a “fact verifier” of faith.

Knowledge consists, first, of certain facts of fact and certain principles of inference, neither of which needs extraneous evidence, and, secondly, of everything that can be asserted by the application of principles of inference to facts. According to tradition, it is believed that factual data is supplied by perception and memory, and the principles of inference are the principles of deductive and inductive logic.

There is much that is unsatisfactory in this traditional doctrine. First, this doctrine does not provide a meaningful definition of “knowledge.” Secondly, it is very difficult to say what the facts of perception are. Third, deduction turned out to be much less powerful than previously thought; it does not give new knowledge, except new forms of words for the establishment of truths, in a sense already known. Fourthly, methods of inference which may be called in the broadest sense "inductive" have never been satisfactorily formulated.

PART THREE. SCIENCE AND PERCEPTION

CHAPTER 1. KNOWLEDGE OF FACTS AND KNOWLEDGE OF LAWS. When we examine our belief in evidence, we find that sometimes it is based directly on perception or memory, and at other times on inference. The same external stimulus entering the brains of two people with different experiences will produce different results, and only what is common in these different results can be used to make inferences about external causes. There is no reason to believe that our sensations have external causes.

CHAPTER 2. SOLIPSISM. The doctrine called "solipsism" is usually defined as the belief that there is only one self. We can distinguish two forms of solipsism. Dogmatic solipsism says: “There is nothing but the data of experience,” and skeptical says: “It is not known that anything else exists except the data of experience.” Solipsism can be more or less radical; when it becomes more radical, it becomes both more logical and at the same time more implausible.

The Buddha was pleased that he could think while the tigers roared around him; but, if he were a consistent solipsist, he would believe that the roaring of the tigers stopped as soon as he stopped noticing it. When it comes to memories, the results of this theory are extremely strange. The things I remember at one moment turn out to be completely different from the things I remember at another moment, but the radical solipsist must admit only those things that I remember now.

CHAPTER 3. PROBABLE CONCLUSIONS OF ORDINARY COMMON SENSE. A “probable” conclusion is one in which the premises are true and the construction is correct, but the conclusion is nevertheless not certain, but only more or less probable. In the practice of science, two types of conclusions are used: purely mathematical conclusions and conclusions that can be called “substantial”. The derivation from Kepler's laws of the law of gravitation as applied to the planets is mathematical, and the derivation of Kepler's laws from the noted apparent motions of the planets is substantive, since Kepler's laws are not the only hypotheses logically consistent with observed facts.

Pre-scientific knowledge is expressed in the conclusions of ordinary common sense. We must not forget the difference between inference, as it is understood in logic, and that inference which can be called “animal”. By “animal inference” I mean what happens when some event A is the cause of belief B without any conscious intervention.

If in the life of a given organism A was often accompanied by B, then A will be simultaneously or in quick succession accompanied by the “idea” of B, that is, an impulse to actions that could be stimulated by B. If A and B are emotionally interesting for the organism, then even one case of them connection may be sufficient to form a habit; if not, many cases may be needed. The connection between the number 54 and the multiplication of 6 by 9 is of negligible emotional interest for most children; hence the difficulty of learning the multiplication table.

Another source of knowledge is verbal evidence, which turns out to be very important, precisely in that it helps to learn to distinguish the public world of feelings from the personal world of thought, which is already well established when scientific thinking begins. One day I was giving a lecture to a large audience when a cat snuck into the room and lay down at my feet. The behavior of the audience convinced me that this was not my hallucination.

CHAPTER 4. PHYSICS AND EXPERIMENT. From the earliest times there have been two types of theories of perception: one is empirical and the other is idealistic.

We see that physical theories change all the time and that there is no reasonable representative of science who would expect a physical theory to remain unchanged for a hundred years. But because theories change, this change usually provides little new information about the observed phenomena. The practical difference between Einstein's and Newton's theories of gravity is negligible, although the theoretical difference between them is very great. Moreover, in every new theory there are certain parts that are apparently completely reliable, while others remain purely speculative. Einstein's introduction of space-time instead of space and time represents a change in language, the basis for which, like the Copernican change in language, is its simplification. This part of Einstein's theory can be accepted without any hesitation. However, the view that the universe is a three-dimensional sphere and has a finite diameter remains speculative; no one will be surprised if reasons are found that force astronomers to abandon this method of expression.

Our main question is: if physics is true, how can this be established and what, besides physics, must we know in order to deduce it? This problem arises from the physical causation of perception, which makes it plausible to assume that physical objects differ significantly from perception; but if this is really so, how can we infer physical objects from perceptions? Moreover, since perception is regarded as a "mental" event, while its cause is considered "physical", we are faced with the old problem of the relation between spirit and matter. My own opinion is that the "mental" and the "physical" are not as separate from each other as is commonly thought. I would define a "mental" event as one which is known without the aid of inference; therefore, the distinction between “mental” and “physical” refers to the theory of knowledge, and not to metaphysics.

One of the difficulties that led to confusion was the failure to distinguish between perceptual space and physical space. Perceptual space consists of perceptual relations between perceptual parts, whereas physical space consists of inferred relations between inferred physical things. What I see may be outside my perception of my body, but not outside my body as a physical thing.

Perceptions considered in the causal chain arise between events occurring in the centripetal nerves (stimulus) and events in the centrifugal nerves (response), their position in the causal chains being the same as the position of certain events in the brain. Perceptions as a source of knowledge of physical objects can fulfill their purpose only insofar as in the physical world there are separate causal chains, more or less independent of each other. All this is only approximate, and therefore the inference from perceptions to physical objects cannot be completely accurate. Science consists largely of means for overcoming this initial lack of precision, on the assumption that perception gives a first approximation to the truth.

CHAPTER 5. TIME IN EXPERIENCE. There are two sources of our knowledge of time. One of them is the perception of following during one present present, the other is memory. A memory can be perceived and has the quality of being more or less distant, so that all my present memories are arranged in chronological order. But this is subjective time and must be distinguished from historical time. Historical time has a relation of “precedence” to the present, which I know as the experience of change during one present present. In historical time, all my real memories take place now. But, if they are true, they point to events that took place in the historical past. There is no logical reason to believe that memories must be true; From a logical point of view, it can be proven that all my present memories could be exactly the same even if there had never been any historical past. Thus, our knowledge of the past depends on a certain postulate that cannot be revealed by a simple analysis of our present memories.

CHAPTER 6. SPACE IN PSYCHOLOGY. When I have the experience called "seeing a table", the table seen has first of all a position in the space of my instantaneous visual field. Then, through the correlations existing in experience, it obtains a position in space, which embraces all my perceptions. Further, through physical laws, it is correlatively associated with some place in physical space-time, namely with the place occupied by a physical table. Finally, through physiological laws, it refers to another place in physical space-time, namely to the place occupied by my brain as a physical object. If the philosophy of space is to avoid hopeless confusion, it must carefully distinguish between these various correlations. It should be noted that the dual space in which perceptions are contained stands in a relation of very close analogy to the dual time of memories. In subjective time, memories refer to the past; in objective time they take place in the present. Likewise, in subjective space the table I perceive is there, but in physical space it is here.

CHAPTER 7. SPIRIT AND MATTER. I maintain that while mental phenomena and their qualities can be known without inference, physical phenomena are known only in relation to their spatio-temporal structure. The qualities inherent in such phenomena are unknowable - so completely unknowable that we cannot even say whether they are different or not different from the qualities that we know to belong to psychic phenomena.

PART FOUR. SCIENTIFIC CONCEPTS

CHAPTER 1. INTERPRETATION. It often happens that we seem to have sufficient reason to believe in the truth of some formula expressed in mathematical symbols, although we cannot give a clear definition of the ethics of symbols. In other cases it also happens that we can give several different meanings to symbols, each of which makes the formula true. In the first case we do not even have one specific interpretation of our formula, while in the second case we have many interpretations.

As long as we remain in the realm of arithmetic formulas, different interpretations of "number" are equally good. It is only when we begin the empirical use of numbers in enumeration that we find a basis for preferring one interpretation over all others. This situation arises whenever mathematics is applied to empirical material. Let's take geometry for example. If geometry is to be applied to the sensible world, then we must find definitions of points, lines, planes, and so on, in terms of sense data, or we must be able to infer from sense data the existence of non-perceptible entities having the properties that geometry requires. Finding ways or ways to do this or that is a problem in the empirical interpretation of geometry.

CHAPTER 2. MINIMUM DICTIONARIES. Typically, there are several ways in which words used in science can be defined by a small number of terms from among those words. These few terms may have either pictorial or nominal definitions using words that do not belong to the science. I call such a set of initial words the "minimal vocabulary" of a given science if (a) every other word used in the science has a nominal definition by the words of this minimal vocabulary, and (b) none of these initial words has a nominal definition with using other initial words.

Let's take geography as an example. In doing so, I will assume that the geometry vocabulary is already installed; then our first distinctly geographical need is a method of establishing latitude and longitude. Apparently, only two words - "Greenwich" and "North Pole" - are necessary to make geography the science of the surface of the Earth, and not of any other spheroid. It is thanks to the presence of these two words (or two others that serve the same purpose) that geography can tell about the discoveries of travelers. It is these two words that are involved wherever latitude and longitude are mentioned. As this example shows, as science becomes more systematic, it needs less and less of a minimum vocabulary.

CHAPTER 3. STRUCTURE. To identify the structure of an object means to mention its parts and the ways in which they come into relationship. Structure always presupposes relationships: a simple class as such has no structure. Many structures can be built from the members of any given class, just as many different kinds of houses can be built from any given pile of bricks.

CHAPTER 4. STRUCTURE AND MINIMUM DICTIONARIES. Each discovery of a structure allows us to reduce the minimum vocabulary required for a given subject content. Chemistry used to need names for all elements, but now the various elements can be defined in terms of atomic structure using two words: "electron" and "proton."

CHAPTER 6. SPACE IN CLASSICAL PHYSICS. In elementary geometry, straight lines are defined as a whole; their main characteristic is that a straight line is defined if its two points are given. The possibility of considering distance as a straight-line relationship between two points depends on the assumption that there are straight lines. But in modern geometry, adapted to the needs of physics, there are no straight lines in the Euclidean sense, and "distance" is determined by two points only when they are very close to each other. When two points are located far from each other, we must first decide which route we will take from one to the other, and then add up many small segments of this route. The “straightest” line between these two points will be the one in which the sum of the segments is minimal. Instead of straight lines, we should use here "geodesic lines", which are shorter routes from one point to another than any other routes that differ from them. This violates the simplicity of measuring distances, which becomes dependent on physical laws.

CHAPTER 7. SPACE-TIME. Einstein introduced the concept of space-time instead of the concepts of space and time. "Simultaneity" turns out to be a vague concept when it is applied to events occurring in different places. Experiments, especially the Michelson-Morley experiment, lead to the conclusion that the speed of light is constant for all observers, no matter how they move. There is, however, one relation between two events which turns out to be the same for all observers. Previously there were two such relations - distance in space and period of time; now there is only one, called "interval". It is precisely due to the fact that there is only this one relation of interval instead of distance and interval of time that we must instead of two concepts - the concept of space and the concept of time - introduce one concept of space-time.

CHAPTER 8. THE PRINCIPLE OF INDIVIDUATION. How do we determine the difference that makes us distinguish between two objects in the list? Three views have been defended on this issue with some success.

1. What is special is formed through qualities; when all its qualities are listed, it is completely defined. This is Leibniz's view.
2. The special is determined by its spatiotemporal position. This is Thomas Aquinas' view regarding material substances.
3. The numerical difference is finite and indefinable. Such, I think, would be the views of the most modern empiricists, if they took the trouble to have a definite view on this matter.

The second of the three theories mentioned is reducible to either the first or the third, according to how it is interpreted.

CHAPTER 9. CAUSAL LAWS. The practical usefulness of science depends on its ability to foresee the future. A "causal law," as I will use the term, can be defined as the general principle by virtue of which - if there is sufficient evidence about a certain region of space-time - some inference can be drawn about a certain other region of space-time. The conclusion can only be probable, but this probability must be much greater than half if the principle we are interested in deserves the name of “causal law.”

If the law establishes a high degree of probability, it may be almost as satisfactory as if it established certainty. For example, the statistical laws of quantum theory. Such laws, even if we assume that they are completely true, make the events deduced on the basis of them only probable, but this does not prevent them from being considered causal laws, according to the above definition.

Causal laws are of two kinds: those relating to constancy, and those relating to change. The former are often not considered causal, but this is not true. A good example of the law of constancy is the first law of motion. Another example is the law of constancy of matter.

Causal laws concerning change were discovered by Galileo and Newton and formulated in terms of acceleration, that is, a change in speed in magnitude or direction, or both. The greatest triumph of this view was the law of gravitation, according to which every particle of matter produces in every other an acceleration directly proportional to the mass of the attracting particle and inversely proportional to the square of the distance between them. The basic laws of change in modern physics are the laws of quantum theory, which govern the transition of energy from one form to another. An atom can release energy in the form of light, which then moves unchanged until it encounters another atom that can absorb the light energy. Everything we (we think) we know about the physical world depends entirely on the assumption that there are causal laws.

The scientific method consists of inventing hypotheses corresponding to experimental data, which are as simple as is compatible with the requirement of correspondence to experience, and which make it possible to draw conclusions that are then confirmed by observation.

If there is no limit to the complexity of possible laws, then every imaginary course of events will obey laws, and then the assumption of the existence of laws will become a tautology. Let's take, for example, the numbers of all the taxis that I hired during my life, and the times when I hired them. We will get a finite series of integers and a finite number of corresponding times. If n is the number of the taxi that I hired at time t, then in an infinite number of ways it is certainly possible to find a function f such that the formula n = f(t) will be true for all values ​​of n and f that have occurred so far. An infinite number of these formulas will be false for the next taxi I hire, but there will still be an infinite number that will remain true.

The merit of this example for my present purpose lies in its manifest absurdity. In the sense in which we believe in natural laws, we would say that there is no law connecting the n and t of the above formula, and that if any of the proposed formulas turn out to be valid, it will be simply a matter of chance. If we found a formula valid for all cases up to the present, we would not expect it to be valid in the next case. Only a superstitious person, acting under the influence of emotion, will believe in this kind of induction; Monte Carlo players resort to inductions, which, however, no scientist will approve.

PART FIVE. PROBABILITY

CHAPTER 1. TYPES OF PROBABILITY. There have been numerous attempts to create a logic of probability, but fatal objections have been raised against most of them. One of the reasons for the fallacy of these theories was that they did not distinguish - or, rather, deliberately confused - fundamentally different concepts, which in ordinary usage have the same right to be called the word “probability”.

The first very significant fact that we must take into account is the existence of the mathematical theory of probability. There is one very simple concept that satisfies the requirements of the axioms of probability theory. If given a finite class B having n members, and if m number of them are known to belong to some other class A, then we say that if any member of class B is chosen at random, then the chance is that it will belong to to class A, will be equal to the number m/n.

There are, however, two aphorisms which we are all inclined to accept without much examination, but which, if accepted, imply an interpretation of "probability" which does not seem to be reconciled with the above definitions. The first of these aphorisms is Bishop Butler's dictum that “probability is the guide of life.” The second is the position that all our knowledge is only probable, which Reichenbach especially insisted on.

When, as is usually the case, I am not sure what is going to happen, but must act in accordance with some hypothesis, I am usually and quite rightly advised to choose the most probable hypothesis, and always rightly advised to take the degree of probability into account when making a decision.

Probability, which is the guide of life, does not belong to the mathematical form of probability, not only because it does not relate to arbitrary data, but to all data that are relevant to the question from the very beginning, but also because it must take into account something entirely underlying outside the realm of mathematical probability, which can be called “inherent doubt.”

If we assert, as Reichenbach does, that all our knowledge is doubtful, then we cannot determine this doubt mathematically, for in compiling statistics it is already assumed that we know that A is or is not B, that the insured person is dead, or that he is alive. Statistics are built on the structure of the assumed certainty of past cases, and general doubt cannot be only statistical.

I think, therefore, that everything we tend to believe in has some “degree of doubt” or, conversely, some “degree of plausibility.” Sometimes this is due to mathematical probability, and sometimes it is not; it is a broader and more vague concept.

I think that each of the two different concepts has, on the basis of ordinary usage, an equal right to be called "probability." The first of them is mathematical probability, which can be measured numerically and satisfies the requirements of the axioms of probability calculus.

But there is another kind, which I call “degree of plausibility.” This type applies to individual propositions and is always subject to consideration of all relevant evidence. It is applicable even in some such cases in which there is no known evidence. It is this type, and not mathematical probability, that is meant when they say that all our knowledge is only probable and that probability is the guide of life.

CHAPTER 2. CALCULUS OF PROBABILITY. We derive the theory of probability as a branch of pure mathematics from certain axioms, without trying to attribute to them one or another interpretation. Following Johnson and Keynes, we will use the expression p/h to denote the indefinite concept of “the probability of p given h.” When I say that this concept is indeterminate, I mean that it is defined only by axioms or postulates, which must be enumerated. Anything that satisfies the requirements of these axioms is an “interpretation” of the calculus of probability, and one should think that many interpretations are possible here.

Necessary axioms:

1. If p and h are given, then there is only one value of p/h. We can therefore speak of “a given probability p given h.”
2. The possible values ​​of the expression p/h are all real numbers from 0 to 1, including both.
3. If h has a value of p, then p/h=1 (we use "1" to indicate confidence).
4. If h has the value non-p, then p/h=0 (we use “0” to denote impossibility).
5. The probability of p and q given h is the probability of p given h multiplied by the probability of q given p and h, and is also the probability of q given h multiplied by the probability of p given q and h. This axiom is called "conjunctive".
6. The probability of p and q given h is the probability of p given h plus the probability of q given h minus the probability of p and q given h. This is called the "disjunctive" axiom.

It is important to keep in mind that our basic concept p/h is a relation of two sentences (or a conjunction of sentences) and not a property of a single sentence p. This distinguishes probability, as it is in mathematical calculation, from probability, which is guided in practice, since the latter must relate to a proposition taken in itself.

Axiom V is a “conjunctive” axiom. It deals with the probability that each of two events will occur. For example, if I draw two cards from a deck, what is the chance that both will be red? Here "h" represents the given that the deck consists of 26 red and 26 black cards; "p" means "the first card is red" and "q" means "the second card is red." Then (p and q)/h" there is a chance that both cards will be red, "p/h" there is a chance that the first one is red, "q / (p and h)" there is a chance that the second one is red, provided that the first one is red. It is clear that p/h =1/2, q (p and h) =25/51. Obviously, according to the axiom, the chance that both cards will be red is 1/2x25/51.

Axiom VI is a “disjunctive” axiom. In the example above, it gives a chance that at least one of the cards will be red. She says that the chance that at least one will be red is the chance that the first will be red, plus the chance that the second will be red (when it is not given whether the first will be red or not), minus the chance that both will be red. This equals 1/2+1/2 – 1/2x25/51.

From the conjunctive axiom it follows that

This is called the "inverse probability principle." Its usefulness can be illustrated as follows. Let p be some general theory and q be the experimental data relating to p. Then p/h is the probability of theory p with respect to previously known data, q/h is the probability of q with respect to previously known data, and q (p and h) is the probability of q if p is true. Thus, the probability of a theory p after q has been established is obtained by multiplying the former probability of p by the probability of q given p and dividing by the former probability of q. In the most favorable case, the theory p will imply q, so that q/(p and h) =1. In this case

This means that a new given q increases the probability of p in proportion to the previous improbability of q. In other words, if our theory suggests something very unexpected, and that unexpected thing then happens, then this greatly increases the likelihood of our theory.

This principle may be illustrated by the discovery of Neptune, considered as a confirmation of the law of gravitation. Here p is the law of gravity, h is all relevant facts known before the discovery of Neptune, q is the fact that Neptune was discovered in a certain place. Then q/h was the preliminary probability that a hitherto unknown planet would be found in a certain small area of ​​the sky. Let it be equal to m/n. Then, after the discovery of Neptune, the probability of the law of gravity became n/m times greater than before. It is clear that this principle is of great importance in assessing the role of new evidence in favor of the probability of a scientific theory.

There is a very significant proposition, sometimes called Bayes' theorem, which has the following form (for more details, see). Let р 1, р 2, …, р n be n mutually exclusive possibilities, and it is known that one of them is true; let h stand for general data and q for some relevant fact. We want to know the probability of one possibility p, given q, when we know the probability of each p 1 before knowing q, and also the probability of q given p 1 for each r. We have

This sentence allows us to solve, for example, the following problem: given n+1 bags, the first of which contains n black balls and no white ones, the second contains n–1 black balls and one white; The r+1st bag contains n–r black balls and r white balls. One bag is taken, but it is not known which one; m balls are taken out of it, and it turns out that they are all white; What is the probability that bag r was taken? Historically, this problem is important in connection with Laplace's claim to prove induction.

Let us next take Bernoulli's law of large numbers. This law states that if for each number of cases the chance of a certain event occurring is p, then for any two arbitrarily small numbers δ and ε the chance is that, starting with a sufficiently large number of cases, the ratio of cases of the occurrence of an event will always differ from p by more than , than by the value ε, will be less than δ.

Let's explain this using an example of tossing a coin. Let's assume that the front and back sides of the coin are equally likely to fall out. This means that, apparently, after a sufficiently large number of throws, the ratio of the faces thrown will never differ from 1/2 by more than the value ε, no matter how small this value ε; further, no matter how small s is, anywhere after n throws, the chance of such a deviation from 1/2 will be less than δ, unless n big enough.

Since this sentence is of great importance in applications of probability theory, such as statistics, let us try to become more familiar with the exact meaning of what is stated in the above example of tossing a coin. First of all, I argue that, from a certain number of their hits, the percentage of the coin that will land on the face side will always be, say, between 49 and 51. Let's say that you challenge my statement and we decide to test it empirically as much as possible. This means that the theorem states that the longer we continue testing, the more it will seem that my statement is generated by facts and that as the number of throws increases, this probability will approach certainty as a limit. Suppose that by this experiment you are convinced that, from a certain number of throws, the percentage of faces always remains between 49 and 51, but now I assert that, from some more throws, this percentage will always remain between 49.9 and 50.1. We repeat our experiment, and after some time you are convinced of this again, although this time, perhaps, after a longer time than before. After any given number of tosses there will remain a chance that my statement will not be confirmed, but this chance will continually decrease as the number of tosses increases, and may become less than any value assigned to it if the tossing continues long enough.

The above propositions are the basic propositions of pure probability theory, which are of great importance in our study. I do want to say something else, however, about a+1 bags, each containing n white and n black balls, with the r+1th bag containing r white balls and n–r black balls. We start from the following data: I know that the bags contain different numbers of white and black balls, but there is no way to distinguish these bags from each other by external features. I choose one bag at random and take m balls out of it one by one, and when I take out these balls I do not put them back in the bag. It turns out that all the drawn balls are white. Given this fact, I want to know two things: first, what is the chance that I chose a bag containing only white balls? Secondly, what is the chance that the next ball I draw will be white?

We reason as follows. Path h will be the fact that the bags have the form and content described above, and q will be the fact that m white balls were drawn; let also p r be the hypothesis that we have chosen a bag containing r white balls. It's obvious that r should be at least as big as m, that is, if r less than m, then p r /qh=0 and q/p r h=0. After some calculations, it turns out that the chance that we chose a bag in which all the balls are white is equal to (m+1)/(n+1).

Now we want to know the chance that the next ball will be white. After some further calculations, it turns out that this chance is equal to (m+1)/(m+2). Note that it does not depend on n and what if m is large, then it is very close to 1.

CHAPTER 3. INTERPRETATION USING THE CONCEPT OF FINITE FREQUENCY. In this chapter we are interested in one interpretation of “probability,” which I will call “finite frequency theory.” Let B be any finite class, and A any other class. We want to determine the chance that a member of class B, chosen at random, will be a member of class A, for example, that the first person you meet on the street will have the last name Smith. We define this probability as the number of members of class B who are also members of class A divided by the total number of members of class B. We denote this by A/B. It is clear that the probability defined in this way must be either a rational fraction, or 0, or 1.

A few examples will make the meaning of this definition clear. What is the chance that any integer less than 10, chosen at random, will be a prime number? There are 9 integers less than 10, and 5 of them are prime; therefore, this chance is 5/9. What is the chance that it rained in Cambridge on my birthday last year, assuming you don't know when my birthday is? If m is the number of days it rained, then the chance is m/365. What is the chance that a person whose surname appears in the London telephone book has the surname Smith? To solve this problem, you must first count all the entries in this book with the last name "Smith", and then count all the entries in general and divide the first number by the second. What is the chance that a card drawn at random from a deck will be of spades? It is clear that this chance is 13/52, that is, 1/4. If you draw a card of the spade suit, what is the chance that the next card you draw will also be a spade? Answer: 12/51. What is the chance that a roll of two dice will result in a total of 8? There are 36 dice combinations, and 5 of them will total 8, so the chance of rolling a total of 8 is 5/36.

Let us consider Laplace's justification for induction. There are N+1 bags, each containing N balls. Of these bags, the r+1st one contains r white balls and N–r black balls. We pulled out n balls from one bag, and they all turned out to be white.

What's the chance

• that we chose a bag with only white balls?
• that the next ball will also be white?

Laplace says that (a) there is (n+1)/(N+1) and (b) there is (n+1)/(n+2). We illustrate this with several numerical examples. First, let's say that there are 8 balls in total, of which 4 are drawn, all white. What are the chances (a) that we have chosen a bag containing only white balls, and (b) that the next ball drawn will also be white?

Let p r represent the hypothesis that we have chosen a bag with r white balls. These data exclude p 0, p 1, p 2, p 3. If we have p 4 , then there is only one case where we could draw 4 whites, leaving 4 cases to draw black and none - white. If we have p 5, then there are 5 cases where we could draw 4 whites, and for each of them there was 1 case of drawing the next white and 3 cases of drawing a black one; Thus, from p 5 we get 5 cases where the next ball will be white, and 15 cases where it will be black. If we have p 6 , then there are 15 cases of choosing 4 whites, and when they are drawn, there are 2 cases of choosing one white and 2 cases of choosing black; so from p 6 we have 30 cases of the next one being white and 30 cases of the next one being black. If we have p 7, then there are 35 cases of drawing 4 whites, and after they are drawn, there are still 3 cases of drawing white and one of drawing black; Thus, we get 105 cases of drawing the next white one and 35 cases of drawing the next black one. If we have p 8, then there are 70 cases of drawing 4 whites, and when they are drawn, then there are 4 cases of drawing the next white and none of drawing black; Thus, from p 8 we get 280 cases of taking out the fifth white and none of taking out the black one. Summing up, we have 5+30+105+280, that is, 420 cases in which the fifth ball is white, and 4+15+30+35, that is, 84 cases in which the fifth ball is black. Therefore the difference in favor of white is the ratio of 420 to 84, that is, 5 to 1; this means that the chance of the fifth ball being white is 5/6.

The chance that we have chosen a bag in which all the balls are white is the ratio of the number of times we get 4 white balls from this bag to the total number of times we get 4 white balls. The first, as we have seen, are 70; the second ones are 1+5+15+35+70, that is 126. Therefore, the chance is 70/126, that is 5/9. Both of these results are consistent with Laplace's formula.

Let us now take Bernoulli's law of large numbers. We can illustrate it as follows. Suppose we toss a coin n times and write 1 whenever it lands on the front side, and 2 whenever it lands on the back side, thus forming a number out of the nth number of single-digit numbers. Let's assume that each possible sequence appears only once. Thus, if n = 2, then we get four numbers: 11, 12, 21, 22; if n =3, then we get 8 numbers: 111, 112, 121, 122, 211, 212, 221, 222; if n=4 we get 16 numbers: 1111, 1112, 1121, 1122, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2221, 2222 and so on

Taking the last one from the above list, we find: 1 number with all ones, 4 numbers with three ones and one two, 6 numbers with two ones and two twos, 4 numbers with one one and three twos, t number with all twos.

These numbers - 1, 4, 6, 4, 1 - are coefficients in the expansion of the binomial (a + b) 4. It is easy to prove that for n single-digit numbers the corresponding numbers are coefficients in the binomial expansion (a + b) n. Bernoulli's theorem boils down to the fact that if n is large, then the sum of the coefficients near the middle will be almost equal to the sum of all coefficients (which is equal to 2n). Thus, if we take all possible sequences of the front and back sides in a large number of tosses, then the vast majority of them will have almost the same number on both (that is, on the front and back sides); this majority and approach to complete equality will, moreover, increase indefinitely as the number of throws increases.

Although Bernoulli's theorem is more general and more precise than the above statements with equally probable alternatives, it should still be interpreted, according to our present definition of "probability", in a manner similar to the above. It is a fact that if we make up all the numbers that have 100 digits, each of which is either 1 or 2, then about a quarter of them will have 49, or 50, or 51 digits equal to 1, almost half will have 48 , or 49, or 50, or 51, or -52 digits equal to 1, more than half will have between 47 and 53 digits equal to 1, and about three-quarters will have between 46 and 54 digits. As the number of signs increases, so will the predominance of cases in which ones and twos are almost completely balanced.

I want to clarify my own view regarding the connection of mathematical probability with the natural course of things in nature. Let's take Bernoulli's law of large numbers as an example, choosing the simplest possible case. We have seen that if we collect all possible integers of n digits, each of which is either 1 or 2, then if n is large, say at least 1000, the vast majority of possible integers will have approximately the same number of ones and twos. This is only an application of the fact that when expanding the binomial (x + y) n, when n is large, the sum of the binomial coefficients near the middle will differ little from the sum of all coefficients, which is equal to 2 n. But what does this have to do with the statement that if I toss a coin enough times, I will probably get about the same number of flips on the front side and the back side? The first is a logical fact, the second is obviously an empirical fact; what is the connection between them?

On some interpretations of "probability", a statement containing the word "probable" can never be an empirical statement. It is recognized that what is not likely to happen may happen, and what is considered likely may not happen. It follows that what actually happens does not show that the previous probability judgment was either correct or false; any imagined course of events is logically compatible with any prior estimate of probability imaginable. This can only be denied if we hold that what is highly improbable does not happen, which we have no right to think. In particular, if induction states only probabilities, then everything that can happen is logically compatible with both the truth and falsity of the induction. Consequently, the inductive principle has no empirical content. It's there reductio ad absurdum and shows that we must connect the probable with the actual more closely than is sometimes done.

CHAPTER 5. KEYNES' PROBABILITY THEORY. Keynes's Treatise on Probability puts forward a theory that is in some ways the antithesis of frequency theory. He holds that the relation used in deduction, namely “p implies q,” is an extreme form of the relation which can be called “p more or less implies q.” “If knowledge of h,” he says, justifies rational belief in a of degree α, then we say that there is a probability relation of degree α between a and h.” We write this: a/h=α. “Between two sets of propositions there is a relation in virtue of which, if we know the first, we can ascribe to the second some degree of rational belief.” Probability is essentially a relation: "It is as useless to say 'b is likely' as it is to say 'b is equal to' or 'b is greater than.' From "a" and "a implies b" we can infer "b"; this means that we can omit any reference to the premises and simply state the conclusion. But if A this applies to b that knowledge A turns probable belief into b into a rational one, then we cannot conclude anything at all about b, which has nothing to do with A; there is nothing corresponding to the omission of a true premise in a demonstrative conclusion.

I conclude that the main formal flaw in Keynes' theory of probability is that he views probability as a relation between propositions rather than as a relation between propositional functions. I would say that its application to sentences refers to the application of the theory, not to the theory itself.

CHAPTER 6. DEGREES OF LIKELIHOOD

Although any part of what we would like to consider as "knowledge" may be to some extent doubtful, it is clear that some is almost certain, while some other is the product of risky assumptions. To a reasonable person there is a scale of doubt from simple logical and arithmetic sentences and perceptual judgments at one end to questions such as asking what language the Mycenaeans spoke or "what song did the Sirens sing" at the other end. Any proposition about which we have reasonable grounds for some degree of belief or disbelief can theoretically be placed on a scale between certain truth and certain falsehood.

There is a certain relationship between mathematical probability and degrees of likelihood. This connection is as follows: when, in relation to all the evidence available to us, a proposition has a certain mathematical probability, then this determines the degree of its likelihood. For example, if you are about to throw dice, the sentence “it will be a double six” has only one-thirty-fifth the likelihood assigned to the sentence “it will not be a double six.” Thus, a reasonable person who assigns the correct degree of likelihood to each proposition will be guided by the mathematical theory of probability where it applies. The concept of "degree of likelihood", however, is used much more widely than the concept of mathematical probability.

A proposition which is not a given can receive its plausibility from many different sources; a person who wants to prove his innocence of a crime can argue on the basis of both an alibi and his previous good behavior. The reasons for a scientific hypothesis are almost always complex. If it is recognized that something may not be reliable, the degree of its plausibility can be increased by some argument or, on the contrary, it can be greatly reduced by some counterargument. The degree of credibility conveyed by evidence cannot be easily assessed.

I intend to discuss plausibility first in relation to mathematical probability, then in relation to data, then in relation to subjective certainty, and finally in relation to rational behavior.

Plausibility and frequency. It seems clear to ordinary common sense that in typical cases of mathematical probability it is equal to the degree of likelihood. If I draw a card at random from a deck, then the degree of likelihood of the sentence “the card will be red” will be exactly equal to the degree of likelihood of the sentence “the card will not be red,” and therefore the degree of likelihood of each sentence is 1/3, if 1 represents certainty. In relation to a die, the degree of likelihood of the sentence “you will roll a 1” is exactly the same as that of the sentences “you will roll a 2”, or 3, or 4, or 5, or 6. From here all the inferred frequencies of the mathematical theory can be interpreted as inferred degrees of likelihood.

In this translation of mathematical probabilities into degrees of likelihood, we use a principle that mathematical theory does not need. This principle is required only when mathematical probability is considered as a measure of likelihood.

Credibility of the data. I define a "given" as a proposition which in itself has some degree of reasonable plausibility, independent of any evidence derived from other propositions. The traditional view is accepted by Keynes and expounded by him in his Treatise on Probability. He says: “In order for us to have a rational belief in p, which has no certainty but only some degree of probability, it is necessary that we know a series of propositions h, and also know some secondary proposition q asserting the probability relation between p and h".

Degrees of subjective reliability. Subjective credibility is a psychological concept, while credibility is, at least in part, a logical concept. Let us distinguish three types of reliability.

1. A propositional function is valid with respect to another function when the class of members satisfying the second function is part of the class of members satisfying the first function. For example, “x is an animal” is valid relative to “x is a rational animal.” This confidence value refers to mathematical probability. We will call this type of certainty “logical” certainty.
2. A proposition is credible when it has the highest degree of plausibility, which is either intrinsic to the proposition or the result of evidence. It may be that no sentence is certain in this sense, that is, no matter how certain it may be relative to the person's knowledge, further knowledge may increase its degree of plausibility. We will call this type of reliability “epistemological.”
3. A person is confident in a proposition when he does not feel any doubt about its truth. This is a purely psychological concept, and we will call it “psychological” certainty.

Probability and behavior. Most ethical theories fall into one of two types. According to the first type, good behavior is that behavior which obeys certain rules; according to the second, it is behavior that is aimed at achieving certain goals. The first type of theory is represented by Kant and the Ten Commandments of the Old Testament. When ethics is viewed as a set of rules of conduct, then probability plays no role in it. It acquires significance only in the second type of ethical theory, according to which virtue consists in the pursuit of certain goals.

CHAPTER 7. PROBABILITY AND INDUCTION. The problem of induction is complex and has various aspects and ramifications.

Induction by simple enumeration is the following principle: “If given a number n instances of a that turn out to be p, and if there is not a single a that is not p, then two statements: (a) “the next a will be p” " and (b) "all a's are p" - both have a probability that increases as n increases and approaches certainty as a limit as n approaches infinity."

I will call (a) “particular induction” and (b) “general induction”. Thus (a) asserts, on the basis of our knowledge of the mortality of people in the past, that it is probable that Mr. So-and-So will die, while (6) asserts that it is probable that all men are mortal.

Since the time of Laplace, various attempts have been made to show that the probable truth of an inductive inference follows from the mathematical theory of probability. It is now generally admitted that all these attempts were unsuccessful, and that if inductive proofs are to be effective, it must be by virtue of some extra-logical characteristic of the actual world in its opposition to the various logically possible worlds that can present themselves to the mind's eye of the logician.

The first such proof comes from Laplace. In its true, purely mathematical form, it looks like this:

There are n+1 bags, similar in appearance to each other, each of which contains n balls. In the first, all the balls are black; in the second - one is white and all the rest are black; r +1st bag r balls are white and the rest are black. From these bags, one is selected, the composition of which is unknown, and m balls are taken from it. They all turn out to be white. What is the probability (a) that the next ball drawn will be white, (b) that we have chosen a bag consisting of only white balls?

The answer is: (a) the chance that the next ball will be white is (n+1)/(m +2), (b) the chance that we chose a bag in which all the balls are white is (m+1)/ (n+1). This correct result has a direct interpretation based on finite-frequency theory. But Laplace concludes that if m members of A happen to be members of B, then the chance that the next A will be equal to B is equal to (m+1)/(m+2), and that the chance that all A are B is equal to (m +1)/(n +1). He obtains this result by the assumption that, given a number n of objects about which we know nothing, the probabilities that 0, 1, 2, ..., n of these objects are B are all equal. This, of course, is an absurd assumption. If we replace it with the somewhat less absurd assumption that each of these objects has an equal chance of being or not being B, then the chance that the next A will be a B remains equal to 1/2, no matter how many A's happen to be B's.

Even if his proof were accepted, general induction remains improbable if n is much larger than m, although particular induction may be highly probable. In reality, however, his proof is only a historical rarity.

Induction has played such a large role in debates about scientific method since Hume's time that it is very important to be completely clear about what - if I'm not mistaken - the above arguments lead to.

Firstly: there is nothing in the mathematical theory of probability that would justify our understanding of both general and particular induction as probable, no matter how large the established number of favorable cases may be.

Secondly: if no constraint is placed on the nature of the intentional determination of the classes A and B involved in the induction, then it can be shown that the principle of induction is not only doubtful, but also false. This means that if given that n members of some class A belong to some other class B, then the values ​​of "B" for which the next member of class A does not belong to class B are more numerous than the values ​​for which the next member belongs to B, if n is not very different from the total number of things in the universe.

Thirdly: what is called “hypothetical induction,” in which some general theory is considered probable because all its hitherto observed consequences have been confirmed, does not differ in any significant way from induction through simple enumeration. For if p is the theory in question, A is the class of relevant phenomena, and B is the class of consequences of p, then p is equivalent to the statement 'all A's are B', and the evidence for p is obtained by mere enumeration.

Fourth: in order for an inductive argument to be effective, the inductive principle must be formulated with some hitherto unknown limitation. Scientific common sense in practice avoids various types of induction, in which, in my opinion, it is right. But what guides scientific common sense has not yet been formulated.

PART SIX. POSTULATES OF SCIENTIFIC CONCLUSION

CHAPTER 1. TYPES OF KNOWLEDGE. What is recognized as knowledge has two varieties; firstly, knowledge of facts, secondly, knowledge of general connections between facts. Very closely connected with this distinction is another, namely, that there is knowledge which may be described as “reflection,” and knowledge which consists in the capacity for intelligent action. Leibniz's monads "reflect" the universe and in this sense "know" it; but since monads never interact, they cannot "act" on anything external to them. This is the logical extreme of one concept of “cognition”. The logical extreme of another concept is pragmatism, which was first proclaimed by K. Marx in his “Theses on Feuerbach” (1845): “The question of whether human thinking has objective truth is not a theoretical question at all, but a practical question. In practice, a person must prove the truth, that is, the reality and power, the this-worldliness of his thinking... Philosophers have only explained the world in different ways, but the point is to change it.”

In what sense can we say that we know the necessary postulates of scientific inference? I believe that knowledge is a matter of degree. We may not know that “of course A is always followed by B,” but we may know that “probably A is usually followed by B,” where the word “probably” is to be taken in the sense of “degree of likelihood.” In some sense and to some extent, our expectations can be considered "knowledge".

What do animal habits have to do with people? According to the traditional concept, there is no “knowledge”. According to the concept that I want to defend, it is very large. According to the traditional concept, knowledge at its best is an intimate and almost mystical contact between subject and object, of which some may in a future life have a full experience in a beatific vision. Some of this direct contact - we are assured - exists in perception. As for the connections between facts, the old rationalists equated natural laws with logical principles, either directly or indirectly, with the help of divine goodness and wisdom. All this is outdated, except as regards perception, which many still regard as giving direct knowledge, and not as the complex and bizarre mixture of sensation, habit and physical causation, which, as I have argued, perception is. Belief in generalities, as we have seen, has only a rather indirect relation to what is said to be believed; when I believe without words that there will soon be an explosion, it is completely impossible to say with accuracy what is happening in me. Belief actually has a complex and somewhat indeterminate relation to what is believed, just as perception has to do with what is perceived.

If an animal has such a habit that, in the presence of a particular A, it behaves in the same way as, before acquiring the habit, it behaved in the presence of a particular B, then I will say that the animal believes the general proposition: “Every (or almost every) particular instance of A is accompanied by (or is followed by) case B'. This means that the animal believes what this form of words means. If this is so, then it becomes clear that animal habit is essential to understanding the psychology and biological origin of common beliefs.

Returning to the definition of "knowledge", I will say that the animal "knows" the general proposition: "A is usually followed by B if the following conditions are met:

1. The animal repeatedly experienced how A was followed by B.
2. This experience caused the animal to behave in the presence of A more or less in the same way as it had previously behaved in the presence of B.
3. A is indeed usually followed by B.
4. A and B are of such a character or relation to each other that in most cases where this character or relation is present, the frequency of the observed consequences is an evidence of the probability of a general, if not an invariable law of consequence.

CHAPTER 3. POSTULATE OF NATURAL SPECIES OR LIMITED DIVERSITY. Keynes's postulate arises directly from his analysis of induction. Keynes's formulation of his postulate reads as follows: “Consequently, as a logical basis for the analogy, we seem to need some assumption which would say that the amount of variety in the universe is so limited that there is no single object so complex that its qualities would fall into an infinite number of independent groups (that is, groups that could exist either independently or in conjunction); or rather, that none of the objects about which we generalize is so complex as this one; or at least that, although some objects may be infinitely complex, we sometimes still have a finite probability that the object about which we are trying to generalize is not infinitely complex.”

During the 18th and 19th centuries, it was discovered that a colossal variety of substances known to science could be explained by the assumption that they were all composed of ninety-two elements (some of which were not yet known). Each element was believed down to our century to have a number of properties which were found to coexist, although for an unknown reason. Atomic weight, melting point, appearance, etc. made each element a natural appearance as definitely as in biology before the theory of evolution. Finally, however, it turned out that the differences between the elements are differences in structure and consequences of laws that are the same for all elements. It is true that there are still natural species - currently electrons, positrons, neutrons and protons - but they are thought to be not finite and can be reduced to differences in structure. Already in quantum theory their existence is somewhat vague and not so significant. This suggests that in physics, as in biology after Darwin, it can be proven that the doctrine of natural species was only a temporary phase.

CHAPTER 5. CAUSAL LINES."Cause", as it appears, for example, in John Stuart Mill, may be defined as follows: all events may be divided into classes in such a way that each event of some class A is followed by an event of some class B, which may or may not be different. different from A. If two such events are given, then the event of class A is called "cause" and the event of class B is called "effect".

Mill believes that this law of universal causation, more or less the same as we have formulated it, is proved, or at least made extremely probable, by induction. His famous four methods, which are designed in a given class of cases to discover what is cause and what is effect, presuppose causation, and depend on induction only in so far as induction is supposed to confirm the supposition. But we have seen that induction cannot prove causation unless causation is previously probable. However, for inductive generalization, causation is perhaps a much weaker basis than is usually thought.

We feel that we can imagine, or sometimes perhaps even perceive, a cause-effect relation which, when it occurs, ensures an invariable effect. The only weakening of the law of causation that is easy to recognize is not that the causal relation is not immutable, but that in some cases there may be no causal relation.

The belief in causing - right or wrong - is deeply ingrained in language. Let us remember how Hume, despite his desire to remain a skeptic, allows the use of the word “impression” from the very beginning. The "impression" must be the result of some effect on someone, which is a purely causal understanding. The difference between "impression" and "ideen" must be that the former (but not the latter) has a proximate external cause. True, Hume states that he also found an internal difference: impressions differ from ideas by their greater “liveness.” But this is not so: some impressions are weak, and some ideas are very vivid. As for me, I would define an “impression” or “sensation” as a mental event, the proximate cause of which is physical, while an “idea” has a psychic proximate cause.

A "line of causation," as I am going to define the term, is a temporal sequence of events so related to each other that if some of them are given, something can be inferred about the others, no matter what happens elsewhere.

The great importance of statistical laws in physics began to be felt with the kinetic theory of gases, which made, for example, temperature a statistical concept. Quantum theory has greatly strengthened the role of statistical law in physics. It now seems likely that the basic laws of physics are statistical and cannot tell us, even in theory, what an individual atom will do. Moreover, the replacement of individual patterns with statistical ones turned out to be necessary only in relation to atomic phenomena.

CHAPTER 6. STRUCTURE AND CAUSAL LAWS. Induction by mere enumeration is not a principle by which non-demonstrative inferences can be justified. I myself believe that the concentration on induction has greatly hindered the progress of the entire study of the postulates of the scientific method.

We have two different cases of identity of the structure of groups of objects: in one case, the structural units are material objects, and in the other, events. Examples of the first case: atoms of one element, molecules of one compound, crystals of one substance, animals or plants of one species. Examples of another case: what different people see or hear at the same time in the same place, and what cameras and gramophone discs display at the same time, simultaneous movements of an object and its shadow, the connection between different performances of the same music and so on

We will distinguish between two kinds of structure, namely "event structure" and "material structure". The house has a material structure, and the performance of music has the structure of events. As a principle of inference, unconsciously applied by ordinary common sense, but consciously in both science and law, I propose the following postulate: “When a group of complex events, more or less adjacent to each other, have a common structure and are grouped according to -apparently around some central event, then it is likely that they have a common predecessor as a cause.”

CHAPTER 7. INTERACTION. Let us take one historically important example, namely the law of falling bodies. Galileo, using a small number of rather crude measurements, found that the distance traveled by a vertically falling body is approximately proportional to the square of the time of fall, in other words, that the acceleration is approximately constant. He assumed that if it were not for air resistance, it would be quite constant, and when a short time later the air pump was invented, this assumption seemed to be confirmed. But further observations suggested that acceleration varies slightly with latitude, and subsequent theory established that it also changes with height. Thus, the elementary law turned out to be only approximate. Newton's law of universal gravitation, which replaced this one, turned out to be a more complex law, and Einstein's law of gravitation, in turn, turned out to be even more complex than Newton's law. Such a gradual loss of elementarity characterizes the history of most of the early discoveries of science.

CHAPTER 8. ANALOGY. Belief in the consciousness of others requires some postulate, which is not required in physics, since physics can be satisfied with the knowledge of structure. We must resort to something that may be called, rather vaguely, "analogy." Other people's behavior is in many ways similar to our own, and we assume that it must have similar causes.

From observing ourselves, we know the causal law of the form “A is the cause of B,” where A is a “thought” and B is a physical event. We sometimes observe B when no A can be observed, then we infer unobserved A. For example, I know that when I say, “I am thirsty,” I usually say it because I am really thirsty, and therefore when I hear the phrase: “I’m thirsty,” - at that moment when I myself am not thirsty, I make the assumption that someone else is thirsty.

This postulate, once accepted, justifies the conclusion about other consciousnesses, just as it justifies many other conclusions that ordinary common sense unconsciously makes.

CHAPTER 9. SUMMATION OF POSTULATES. I believe that the postulates necessary for the recognition of the scientific method can be reduced to five:

1. Postulate of quasi-constancy.
2. Postulate of independent causal lines.
3. Postulate of spatiotemporal continuity in causal lines.
4. The postulate of a common causal origin of similar structures located around their center, or, more simply, a structural postulate.
5. Postulate of analogy.

All these postulates, taken together, are intended to create the prior probability necessary to justify inductive generalizations.

Postulate of quasi-constancy. The main purpose of this postulate is to replace the concepts of ordinary common sense “thing” and “person”, which does not imply the concept of “substance”. This postulate can be formulated as follows: If any event A is given, then it very often happens that at any nearby time in some neighboring place there is an event very similar to A. “Thing” is a sequence of such events. It is precisely because such sequences of events are common that “thing” is a practically convenient concept. There is not much resemblance between a three-month-old fetus and an adult human being, but they are connected by gradual transitions from one state to the next and are therefore considered as stages in the development of one “thing.”

Postulate of independent causal lines. This postulate has many applications, but perhaps the most important of all is its application in connection with perception - for example, in attributing the multiplicity of our visual sensations (when looking at the night sky) to the many stars as their cause. This postulate can be formulated as follows: It is often possible to form such a sequence of events that from one or two members of this sequence something can be deduced that relates to all the other members. The most obvious example here is motion, especially unimpeded motion like the motion of a photon in interstellar space.

Between any two events belonging to the same line of causation there is, as I would say, a relation which may be called the relation of cause and effect. But if we call it so, we must add that the cause does not completely determine the effect even in the most favorable cases.

Postulate of space-time continuity. The object of this postulate is to deny "action at a distance" and to assert that when there is a causal connection between two events that are not contiguous, there must be such intermediate links in the causal chain, each of which must be adjacent to the next, or (alternatively ) such that the result is a process that is continuous in the mathematical sense. This postulate does not concern evidence in favor of causation, but inference in cases where causation is considered to have already been established. It allows us to believe that physical objects exist even when they are not perceived.

Structural postulate. When a number of structurally similar complexes of events are located near a center in a relatively small area, it usually happens that all these complexes belong to causal lines that have their source in an event of the same structure located in the center.

Postulate of analogy. The postulate of analogy may be formulated as follows: If two classes of events A and B are given, and if it is given that, wherever both these classes A and B are observed, there is reason to believe that A is the cause of B, and then, if in any then in this case A is observed, but there is no way to establish whether B is present or not, then it is likely that B is still present; and similarly, if B is observed, but the presence or absence of A cannot be established.

CHAPTER 10. LIMITS OF EMPIRISM. Empiricism can be defined as the statement: “All synthetic knowledge is based on experience.” “Knowledge” is a term that cannot be precisely defined. All knowledge is doubtful to some extent, and we also cannot say at what degree of doubtfulness it ceases to be knowledge, just as we cannot say how much hair a person must lose in order to be considered bald. When faith is expressed in words, we must bear in mind that all words beyond logic and mathematics are indefinite: there are objects to which they definitely apply, and there are objects to which they definitely do not apply, but are (or at least can be) ) intermediate objects for which we are not sure whether these words apply to them or not. Knowledge of individual facts should depend on perception is one of the most basic principles of empiricism.

In my opinion, there is a mistake in the book. This formula is given not as a quotient, but as a product.

It seems that it was not published in Russian. It should be noted that I had read more than once about the theory of probability put forward by Keynes, and hoped that with the help of Russell I could understand it. Alas... this is still beyond my understanding.

This is where I “broke” :)

Philosophy. Cheat sheets Malyshkina Maria Viktorovna

101. Human knowledge

101. Human knowledge

Cognition is the interaction of a subject and an object with the active role of the subject itself, resulting in some kind of knowledge.

The subject of cognition can be an individual, a collective, a class, or society as a whole.

The object of knowledge can be the entire objective reality, and the subject of cognition can be only its part or area directly included in the process of cognition itself.

Cognition is a specific type of human spiritual activity, the process of comprehending the surrounding world. It develops and improves in close connection with social practice.

Cognition is a movement, a transition from ignorance to knowledge, from less knowledge to more knowledge.

In cognitive activity, the concept of truth is central. Truth is the correspondence of our thoughts to objective reality. A lie is a discrepancy between our thoughts and reality. Establishing the truth is the act of transition from ignorance to knowledge, in a particular case - from misconception to knowledge. Knowledge is a thought that corresponds to objective reality and adequately reflects it. A misconception is an idea that does not correspond to reality, a false idea. This is ignorance, presented, accepted as knowledge; a false idea presented or accepted as true.

A socially significant process of cognition is formed from millions of cognitive efforts of individuals. The process of transforming individual knowledge into universally significant knowledge, recognized by society as the cultural heritage of humanity, is subject to complex sociocultural patterns. The integration of individual knowledge into the commonwealth is carried out through communication between people, critical assimilation and recognition of this knowledge by society. The transfer and transmission of knowledge from generation to generation and the exchange of knowledge between contemporaries are possible thanks to the materialization of subjective images and their expression in language. Thus, cognition is a socio-historical, cumulative process of obtaining and improving knowledge about the world in which a person lives.

Cognition

Cognition