2. LOGICAL DIVISION DISTINGUISHED FROM ENUMERATION.

When the genus is separated at once into individual objects the process is not logical division, but simple enumeration. Logical division implies a separating into smaller class terms, each term being a genus of still smaller subdivisions. This process may be continued till the last division gives individuals as species. Enumeration takes place when the first subdivision results in a list of individuals. To illustrate:

3. LOGICAL DIVISION AS PARTITION.

Partition is the process of separating an individual thing into its parts.

The partition is quantitative or mathematical when the separation is in terms of space or time, but when otherwise the partition becomes qualitative or logical. Or to put it in another way, the partition is mathematical when the separation gives parts and logical when the separation gives ingredients.

To illustrate:

An easy way to determine that the separation involves logical division proper and not partition is to affirm the connection between a class and a sub-class. To wit: Aman is a biped; a square is a rectangle; a Caucasian is a man, etc. If such an affirmation cannot be made then the separation involved is not properly logical division but probably partition. For example it cannot be said that a roof is a house, or that sap is a tree. It is seen, then, that a logical division of any genus may be summarized in the form of a series of judgments of which a species is the subject and the genus is the predicate. For example, by a logical division quadrilaterals may be divided into trapeziums, trapezoids and parallelograms; this process may then be summarized in a series of three judgments: (1)A trapezium is a quadrilateral; (2)A trapezoid is a quadrilateral; (3)A parallelogram is a quadrilateral.

4. RULES OF LOGICAL DIVISION.

When the logical division of a genus is under consideration there are four rules which should be observed.

FIRST RULE. There must be but one principle of division (fundamentum divisionis). To divide mankind into white man, Australian, yellow man, African and red man is a violation of this rule as the two principles of color and geographical location are involved. Adivision in which more than one principle is used is sometimes referred to as cross division because the various species cross each other. For example in the foregoing there are many white men who are Australians.

This rule applies only to one division. Where there is a series of divisions a new principle may be employed in each division. For example, in dividing triangles into scalene, isosceles and equilateral, the equality of sides is the principle involved, but, in subdividing isosceles triangles into right angled and oblique angled, the principle employed concerns the nature of the angle.

SECOND RULE. The co-ordinate species must be mutually exclusive. There must be no overlapping. The illustration given in the first rule is likewise a violation of this rule. Another example in which this second rule is not obeyed may be found in most geometries where triangles are divided into scalene, isosceles and equilateral. Here the second and third classes are not mutually exclusive since all equilateral triangles are isosceles according to the usual definition, “An isosceles triangle is a triangle having two equal sides.” All equilateral triangles have two equal sides.

THIRD RULE. The division must be exhaustive. That is, the species taken together must equal the whole genus. The sum of the species must be co-extensive with the genus.

Dividing man into Caucasian, Ethiopian and Mongolian would be a violation of this rule, as there are at least two other species of man, Malay and American Indian.

A distinction should be made between an exhaustive division and a complete division as the latter is not a logical requirement. To divide government into monarchy, aristocracy and democracy is exhaustive but incomplete. Exhaustive because there is no other kind of government, all the species are included; but incomplete in that monarchy may be divided into absolute and limited; democracy into pure and representative.

FOURTH RULE. The division must proceed from the proximate genus to the immediate species. There should be no sudden jumps from a high genus to a low species. The division must be gradual and continuous; step by step. To divide government into limited monarchy, absolute monarchy, pure democracy and representative democracy would be a violation of this rule, as government is the proximate genus of monarchy, not of limited monarchy, therefore one step has been omitted. Such an omission involves a step from grandfather to grandchild, so to speak, the generation of father having been left out.

A violation of this rule is most insidious when some of the species of a subdivision are immediate while others are not. To wit: dividing government into monarchy, aristocracy, pure democracy and republic, or dividing quadrilaterals into trapeziums, trapezoids, rectangles, squares, rhomboids and rhombs.

5. DICHOTOMY.

Dichotomy comes from the Greek, meaning to cut in two. Dichotomy is a continual division of a genus into two species which are contradictory in nature.

Contradictory terms are such as admit of no middle ground. They divide the whole universe of thought into two classes. For example, honest and not-honest, pure and impure, perfect and imperfect, are contradictory terms. Dichotomy thus affords an easy opportunity for an exhaustive division as in the use of contradictories nothing in the universe need be omitted.

An historical illustration of dichotomy is the “Tree of Porphyry” named after Porphyrius, a Neo-Platonic philosopher of the third century.

This kind of division is not altogether satisfactory as the negative side is too indefinite. On the other hand, if both subdivisions are made positive then there is danger of making the opposing terms contrary rather than contradictory. This, of course, would be a serious logical fallacy, as contrary terms admit of middle ground while contradictory terms give no choice, it is either the one or the other.

The use of dichotomy becomes evident in situations where new and unexpected discoveries may be made. Without disturbing the classification the new species may be appended to the negative side of the division. The following illustrates:

6. CLASSIFICATION—?COMPARED WITH DIVISION.

Classification is the process of grouping notions according to their resemblances or connections.

So far as results are concerned there is no difference between logical division and classification. Both processes may give us the same orderly scheme of heads and subheads. The difference lies in the process itself. Division is deductive in nature as it proceeds from the more general genus to the less general species. While classification is inductive as it groups the less general species under the more general genus. Division differentiates unity into multiplicity, while classification reduces multiplicity to unity. It follows that the one is the inverse of the other. The difference in the mode of procedure may be illustrated by using the common classification or division of triangles. For example:

Without any knowledge of the kinds of triangles the student discovers by examining the various shapes of many triangles that there is a group in which none of the sides are equal. For the lack of a better name he terms these non-equilateral (scalene). Further observation discloses another group in which two of the sides are equal. These he names bi-equilateral (isosceles). Finally a third group is designated as tri-equilateral (equilateral). This process is classification. Division would consist in separating the genus triangle into the three kinds—scalene, isosceles, equilateral.

7. KINDS OF CLASSIFICATION—?ARTIFICIAL AND NATURAL.

An artificial classification is one in which the grouping is made on the basis of some arbitrary connection. Cataloguing alphabetically the books in a library illustrates this kind of classification. Likewise the arrangement of the names in a directory or a telephone book. The connecting mark being the initial letter of the title or name. The reason why Mills and Meyers are put in the same group is that both names happen to commence with the letter M.

Artificial classifications are resorted to for some special purpose, designed by man, not by nature. Consequently artificial classifications are sometimes called special or working classifications.

A natural classification is one in which the grouping is made on the basis of some inherent mark of resemblance.

Classifications in animal and plant life are the best illustrations of this kind. Such classifications are suggested by nature and not by man, and may, therefore, be called general or scientific. The main aim of natural classification is to derive general truths and arrange knowledge so that it may be easily remembered.

8. TWO RULES OF CLASSIFICATION.

The rules of logical division are applicable in the making of a logical classification. In addition to these an artificial classification should be made to conform to the one rule: The classification must be appropriate to the purpose in hand. Likewise a natural classification should be made to conform to the rule: Every classification should afford opportunity for the greatest possible number of general assertions.

9. USE OF DIVISION AND CLASSIFICATION IN THE SCHOOL ROOM.

It has been stated that classification and division aim at the same result. Classification reduces multiplicity to unity while division differentiates unity into multiplicity. In short, division is deductive while classification is inductive in mode of procedure. Therefore, classification should be used in those situations which call for induction and division in cases where deduction is the better method.

Speaking generally, classification should be used with small children when the essential thing is to present the concrete facts with a view of leading the children to discover for themselves the truths contained therein.

With older pupils division may be used, if the purpose is to set in order facts which are already known.

10. TOPICAL OUTLINE.

LOGICAL DIVISION AND CLASSIFICATION.

11. SUMMARY.

(1) Logical division is the process of separating a proximate genus into its co-ordinate species.

(2) The first subdivision in a logical division gives class terms, while the first subdivision in an enumeration gives individual objects.

(3) Partition is the process of separating an individual thing into its parts. These parts may be either quantitative or qualitative.

A logical division of any genus may be summarized in a series of judgments of which a species is the subject and the genus is the predicate.

(4) The four rules of logical division are: the division must (1)be based on one principle, (2)have species mutually exclusive, (3)be exhaustive and (4)proceed from proximate genus to immediate species.

A violation of the first rule gives a cross division.

Exhaustive division is easily confused with a complete or finished division.

(5) Dichotomy is a continual division of a genus into two species which are contradictory in nature.

An historical illustration of dichotomy is the Tree of Porphyry.

Dichotomy is of service in the field of new and unexpected discoveries.

(6) Classification is the process of grouping notions according to their resemblances or connections.

Classification is inductive in nature, division deductive. Classification unifies, division differentiates.

(7) An artificial classification is made on the basis of some arbitrary connection; a natural classification, on some inherent mark of resemblance.

(8) The rules of logical division are applicable in any classification. In addition to these a classification should (1)be appropriate and (2)afford opportunity for the greatest possible number of assertions.

(9) Classification should be the mode of procedure in the lower grades, division in the higher grades.

12. REVIEW QUESTIONS.

13. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.