Lecture
Argentine writer X. L. Borges cites an excerpt from a certain "Chinese Encyclopedia." It provides a classification of animals and states that they "are divided into: a) belonging to the emperor; b) embalmed; c) tamed; d) milk pigs; e) sirens; e) fabulous; g) stray dogs; h) prisoners in this classification; i) rampaging as in madness; k) innumerable; l) drawn by a very thin brush of camel hair; m) other; m) just broken pitcher; o) from afar seeming flies. "
What affects this classification? Why does it become obvious from the very beginning that one cannot talk about animals or anything else in this way?
The point, of course, is not in separate headings, no matter how unusual they may seem. Each of them has a very specific content. Among animals, however, fantastic creatures are mentioned - fabulous animals and sirens, but this is done, perhaps, with the aim of distinguishing real-life animals from existing ones only in imagination. Animals are drawn and drawn, but in fact we usually call them animals.
It is not possible to separate the specified species of animals, but just to join them into one group, listing them one after another, so that living and dead animals stand next to each other, outrageous and painted, fantastic and tamed, classified and just broken a jug. Immediately there is a feeling that there is no such unified plane on which all these groups could be placed, there is no common, homogeneous space in which all the listed animals could meet.
The classification always establishes a specific order. It divides the considered area of objects into groups in order to organize this area and make it well-visible. But the classification of animals from the “encyclopedia” not only does not outline a specific system, but, on the contrary, destroys even those ideas about the edges between groups of animals that we have. In essence, this classification violates all those requirements that are imposed on the separation of any set of objects on the components of its group. Instead of a system, it introduces inconsistency and confusion.
What is the classification? This question is important, since classification is one of the usual and frequently used operations, a means of giving our thinking rigor and clarity. But before answering the question, we introduce several auxiliary concepts.
Classification is a special case of division - a logical operation on concepts.
The division is the distribution into groups of those objects that are thought in the original concept.
The resulting groups are called division members. The sign on which the division is made, is called the basis of division.
In each division there is thus a divisible concept, a basis for division, and members of division.
For example, triangles can be divided into acute, rectangular and obtuse. The basis of the division is the nature of the angles of the triangle.
Classification is a multistage, branched division. For example, sensations can be divided into visual, auditory, tactile, olfactory, and gustatory pas. Then, within individual groups, subgroups can be distinguished (for example, spatial and color visual sensations), the subgroups themselves can be divided into more subdivisions, etc.
The division operation has to be resorted to in almost every argument. Defining the concept, we reveal its content, indicate the signs of objects, imaginable in this concept. By dividing the concept, we give an overview of the range of objects that is displayed in it. If we have, say, a definition of a “lens”, we know the most important features of a lens. But at the same time we do not have an exact idea of the types of lenses. Only by dividing the lenses into convex, biconvex, concave, biconcave, etc., we will gain knowledge not only of what a lens is, but also of what lenses are.
It is important to be able, therefore, not only to determine the content of the concept, but also to trace the groups from which the class of objects denoted by the concept is composed.
A simple example from entomology - the science of insects - will once again confirm this thought. On the table of the entomologist there are small boxes with silver flies impaled on thin pins. Under a microscope - otherwise you can’t see - a scientist plucks tails from these flies and sticks them on tiny glasses with needle-thin blades. What for? In some cases, only by the “tails” - by the structural features of individual organs - it is possible to determine exactly which species the insect belongs to. And the painting of insects by species and the determination of their habitat is important not only to satisfy scientific curiosity. After all, some of them are potential carriers of a number of diseases, others are pests of cultivated plants, and still others are the enemies of these pests. For example, trichograms are tiny, half a millimeter long relatives of well-known bees, bumblebees and wasps. Trichograms are widely used in the biological control of crop pests. However, recent studies have shown that, until recently, not one species of this insect was bred in biofactories, but a “mixture” of three species. But each has his affections: one prefers the field, the other - the garden, the third - the garden. And in each case it is better to breed exactly the kind that is suitable for local conditions.
This is just one example of the practical impact of the work of taxonomists engaged in the classification of animals.
Arithmetic is well known for dividing numbers. The division of concepts, or logical division, is another mental operation that has not only a name, but also a structure with the first: both operations have a “dividend”, a “divisor” and a “result of division”. Logical division is applied to concepts, the result of such a division is several new, specific concepts. The content of the latter includes all those signs that were conceived in the original, generic concept, and, moreover, signs that distinguish one species from another.
Logical division happens to be mixed with another operation, which is also sometimes referred to as “division,” with the dismemberment of an object into its component parts.
We say that all trees are divided into coniferous and deciduous. This is a logical division. But we can also say that the tree is divided into crown, trunk and roots. This is no longer a division of the concept “tree”, but the dismemberment of the tree itself into its parts.
The distinction here is important and at the same time simple. About each of the parts of the logical division, you can express all that is said in the content of the divisible concept. And conifers and deciduous trees are trees. And with respect to the first and with respect to the second, everything that is true of trees in general is fair. But the parts resulting from the dismemberment of a tree are not trees at all. It is impossible to say about the crown, trunk or roots: “This is a tree”, the general characteristics of trees cannot be extended to parts of a single tree.
Kings are divided into hereditary and elective. And about the hereditary and electoral king can say: "This is the king." But when, as it happened, the king was beheaded, none of the formed parts could be called the king.
This difference between logical division and dismemberment is played up by Polish humorist S. Lec in his “Non-Combed Thoughts”: “People can be divided in different ways! This is known to all. It is possible for people and nonhumans. And the astonished executioner said: "And I divide them into heads and bodies!"
In one of Aesop's fables, it tells how animals divided prey. Leo demanded a quarter for himself as the head of the beasts, another quarter for his incomparable courage and another quarter for his wife and children. As for the last quarter, Lev concluded, "any of the animals can argue with me because of her."
Hence the expression "lion's share." The division of production is, of course, not a logical division of the concept “production”, but the division of production into parts, in this case into four parts.
The word "division" is used in other senses. They are associated with the main only through shaky momentary associations.
In the tale of L. Carroll, the White Queen asks Alice if she knows the arithmetic division operation:
- Share the loaf of bread with a knife - what will happen?
“I think ...” began Alice, according to which the Scooping Queen intervened.
“Sandwiches, of course,” she said. - And here is another example of subtraction. Take the bone from the dog - what's left?
Alice thought.
- Bone, of course, will not remain - because I took it away. And the dog also will not remain - she will run after me to bite me ... Well, I, of course, will not stay either!
- So you think there will be nothing left? The Black Queen asked.
- It must be nothing ...
This kind of comic "division" and "subtraction" even if desired, can not be confused with the usual operations on numbers and concepts.
In the future we will talk only about logical division. There will be no danger of confusing this division with any other operation and there is no need therefore to single it out with the word “logical”.
The rules that must be followed when dividing concepts are elementary. Usually they formulate four such rules.
First, the division should be carried out only on one basis.
This requirement means that an individual or a set of features selected at the beginning as a basis should not be replaced by other features during the division.
That's right, for example, to divide the climate into cold, temperate and hot. Dividing it into cold, moderate, hot, sea and continental will already be wrong: first, the division was made according to the average annual temperature, and then - to a new basis. The division of people into men, women and children is wrong; shoes - for men, women and rubber; substances - liquid, solid, gaseous and metals, etc.
Secondly, the division must be proportionate, or exhaustive, i.e. the sum of the volumes of the division members must be equal to the volume of the dividend concept. This requirement cautions against missing individual members of the division.
Erroneous, non-exhaustive will be, in particular, the division of triangles into acute-angled and rectangular (missing obtuse triangles); division of people from the point of view of the level of education in those with primary, secondary and higher education (those who have no education are missed); division of sentences into narrative and motivating (interrogative sentences are missing).
Wrong and humorous division of people depending on who can do what and what is not allowed: one can do everything, even what is impossible; everything is possible except that which is impossible; the third can not do anything except what is possible; and, finally, the fourth cannot be anything, even that which is possible. Here are those who can not miss anything, except that it is impossible.
Wrong and division with redundant member. For example, the division of chemical elements into metals, non-metals and alloys; the division of the spider into natural, social and mathematical, etc. However, the introduction of extra members does not violate this, the second, the rule, but the first, prescribing to divide on one basis and not to replace it in the process of division.
Thirdly, the members of the division should mutually exclude each other.
According to this rule, each separate item should be in the scope of only one species concept and not included in the scope of other species concepts.
It is impossible, for example, to divide all integers into such classes: numbers that are multiples of two; multiples of three; multiples of five, etc. These classes intersect, and, say, the number 10 falls into both the first and third classes, and the number 6 falls into both the first and second classes. It is a mistake to divide people into those who go to the cinema and those who go to the theater; There are people who go to the cinema and to the theater.
And finally, fourth, the division should be continuous.
This rule requires not to make jumps in the division, to move from the original concept to single-order species, but not to the subspecies of one of these species.
For example, it is right to divide people into men and women, women into those living in the northern hemisphere and living in the southern hemisphere. But it is wrong to divide people into men, women of the Northern Hemisphere and women of the Southern Hemisphere. Among the vertebrates there are such classes: fish, amphibians, reptiles (reptiles), birds and mammals. Each of these classes is divided into further types. If, however, we begin to divide vertebrates into fish, amphibians, and instead of specifying reptiles to list all their species, this will be a leap in division.
It can be noted that the first follows from the third rule. Thus, the division of shoes into men's, women's and children's violates not only the first rule, but also the third: members of the division do not exclude each other. The division of kings into hereditary, elective, and club kings is again inconsistent, both with the first and with the third rule.
Now, using the rules of division, you can specifically answer the question of what defects the classification of animals offered by the “Chinese Encyclopedia”. It is clear that this classification does not adhere to any firm rule at all; there is not even a hint of unity and immutability of the basis during the division. Each new group of animals is distinguished on the basis of its own characteristics, irrespective of how other groups stand apart. The connection between the groups is almost completely destroyed, no coordination or subordination between them can be established. It can be assumed that the sirens are fabulous animals, and dairy pigs and stray dogs do not belong to either. But do sirens, fabulous animals, suckling pigs and stray animals belong to those animals that are rampant, as in madness, or incalculable, or those that are drawn with a thin brush? How do animals that have just broken a jug and animals that seem flies from afar relate to each other? It is impossible to answer such questions, and it is pointless to ask them, since it is obvious that no single principle underlies this classification. Further, the division members here do not exclude each other. All of these animals can be drawn, many of them may seem flies from afar, they are all included in the classification, etc. Regarding the fact that the listed species of animals exhaust the multitude of all animals, we can speak with a stretch: those animals that are not mentioned directly are heaped under the heading "Other". Finally, the jumps allowed in this division are obvious. Fabulous and real-life animals seem to differ, but instead of a special mention of the latter, their individual species are listed - pigs and dogs, and not all piglets, but dairy, and not all dogs, but only stray.
Classifications like this are so messy that even a doubt arises whether they should be considered as divisions of any concepts at all. On the improvement of such classifications, giving them at least the visibility of the system and order can not be said.
But it is interesting that even this kind of division, characterized by confusion and slurredness, can sometimes be practically useless. It is wrong to divide, for example, shoes for men, women and rubber (or children), but in many shoe stores it is divided in this way, and this does not put us in a dead end. There is nothing impossible in a sentence, as the classification of animals, similar to that taken from the Encyclopedia, can quite successfully serve some practical, heterogeneous in nature goals. Theoretically, in terms of logic, it is no good. However, not everything that is used on a daily basis is at the level of the requirements of high theory and meets the standards of flawless logic.
One should strive for logical perfection, but one should not be too rigorous and discard from the threshold everything that seems logically incomplete. Sometimes instead of strict, meeting all the requirements of division, a simple grouping of objects of interest may be used. Not being a division, it can, nevertheless, satisfactorily serve practical purposes. Some of the divisions mentioned can be considered as such kind of groupings.
Comments
To leave a comment
Logics
Terms: Logics