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. A division 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 isthe 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 theproximate 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.