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The Rules of Logical Division.

I. There should be one fundamentum divisionis, and one only, for each complete act of division.

II. The species or alternatives into which a genus is divided must be mutually exclusive.

III. If the division involves more than one step, it should proceed gradually from the summum genus towards the infimæ species. Divisio ne fiat per saltum.

IV. The division, within the limits of relevancy, must be disjunctively exhaustive.

Rule I. There should be one fundamentum divisionis, and one only, for each complete act of division.*

The division of a genus is complete when the genus has been differentiated, and the process of successive differentiation continued until the degree of distinction required by the purpose of the division has been precisely attained. In this process each subdifferentiation, or subdivision, should help to develop, more and yet more distinctly, that one indeterminate aspect of the genus of which the differentiation was the original aim in dividing.

The principle which is here involved is that the F.D. must be a mark of the meaning that we aim at developing through division. We may find it convenient to change the F.D. after a first division, and to carry out the 'subdivisions' upon fresh bases. But in this case our division is no longer a single process, but a chain of divisions, and the term 'subdivision' becomes a misnomer. For, in assuming a fresh basis, we have started a fresh division. A division of the species is therefore not necessarily a subdivision of the genus.

And yet we must not misinterpret the function of the F.D. in Division by insisting that it is itself incapable of any development. Discontinuity between one basis and another implies, indeed, a corresponding change of the interest which gives unity and direction to the dividing process, and so implies also a corresponding break in the division. But there is an important via media between discontinuity and a static continuity. The F.D. may legitimately be changed, provided the change is a change within its own original meaning. Thus, after dividing 'human being' into 'male or female,' the F.D. being 'sex,' we do not necessarily abandon this F.D. when we proceed to subdivide 'male' into 'man or boy,' and 'female' into 'woman or girl,' for the age-basis may be here brought forward in its bearing on sex differences. What is essential is that the sex interest should dominate the division into its most detailed differentiations, and that all variations in divisional basis

* It is, of course, possible (as in the last illustration) to divide a genus according to more than one principle of division, provided that we keep the divisions distinct. We then have what is called co-division. Thus, again, adopting the fundamenta of age and sex, we may co-divide 'human being' into 'young, middle-aged, or old,' and into 'man or woman.'

should be variations on the sex-theme. It is in this sense that the F.D. must be one and constant throughout the development of any given division. There may be many sub-fundamenta, but these must themselves be developed in the service of the original fundamentum.

In so far as the 'sub-fundamenta' are developed on their own account, each initiating a new interest, the division is broken up into component parts, which are only loosely and, as it were, externally connected with each other. The organic unity of the division is lost. Moreover, overlapping is almost certain to ensue, for the supreme preventive against the overlapping of the various parts of a division lies in making sure that the parts stand for the various modes in which a single general meaning-e.g., the sex of a human being-can be differentiated or developed.

When two or more bases of division are simultaneously adopted and developed, the resulting overlapping is known as cross-division. The different divisions cross each other, and the confusion which ensues bears witness to the importance of the first rule of logical Division.

Rule II. The second rule of logical Division follows naturally upon the first rule. It is directed against the errors which result in overlapping, whether of the cross-division kind or not. The species or alternatives into which a genus is divided must be mutually exclusive-i.e., no part of the division must overlap or be included under any other part. The only security for observing this rule lies in holding to a single fundamentum. If we divide 'human being' into 'male or female or young or old,' employing simultaneously the two fundamenta of sex and age, we obviously break this rule. It is possible, however, to break Rule II. without breaking Rule I.-namely, through carelessness in the statement of alternatives. Thus I may divide 'man' (F.D. ' means ') into 'rich, easy, or poor,' but may define 'easy' in such a way as to cause it to overlap with 'rich' or 'poor,' or both.

Rule III. If the division involves more than one step, it should proceed gradually from the highest genus towards the lowest species. Divisio ne fiat per saltum.

In each step of the division the species must stand in the same order or rank of generality. Let G be divided into S1, S2, S3; and S2 again into S1, S2, S'3. Were we to divide G into S1, S2, S3, we should have two ranks of generality under one and the same genus. The division would clearly be inadequate, since no account would have been taken of S'1 or S'3.

Consider the old-fashioned division of 'Digitigrade' into 'weasel, civet, hyæna, the cat-kind, fox, wolf, dog.' Here the species are not in the same order of generality. Thus 'fox' and 'wolf' are species of the genus Canis (the dog kind), just as 'lion,' 'tiger,' etc. are species of the genus Felis (the cat kind). Had we given the genus Canis, and thereby kept in the same order of generality the members of one step in the division, we should have been secure against omitting the jackal, which would have been included as being under that genus.

Rule IV. The division, within the limits of relevancy, must be disjunctively exhaustive.

We have already had occasion to point out the essentially disjunctive character of Division. When we divide G into S1, S2, S3, we mean that G may be developed either into S1 or into S2 or into S3; we do not mean that G may be developed into S1 and S2 and S3. Hence, when we say that the division must be disjunctively exhaustive, we mean that S1, S2, S3 must-within the limits of relevancy-exhaust the alternatives.

The meaning of the word 'exhaustive' can, in fact, be defined only in relation to the requirement of relevancy. When we say that a division of a genus into its species must be exhaustive, we mean that it must give all the differentiations of the genus which are at once possible and relevant. The limit of relevance will be given by the purpose of the division. In the case of the divisions which figure within the classification of the natural sciences, the exhaustiveness cannot be other than provisional, for further investigations may reveal new species, or call for the revision of divisions as previously carried out. Moreover, only those species would be relevant that are also actual, for scientific classifications are not concerned with the laying out of possibilities as such, but only with the ordering of such possibilities as Nature has realized. Thus a division of Man, according to skin-colour, which included blue man and green man, would include irrelevant items, since anthropological science studies not mere possibilities, but facts. It would be more than exhaustive, and break this fourth rule of Division just as much as a division into 'white man or black man' which would be under-exhaustive.

In Division by Dichotomy (vide p. 47) the division will be seen to be implicitly, though not determinately, exhaustive.

In connexion with this rule of exhaustiveness in Division Mr. Joseph (ibid., p. 103) gives an instructive illustration which I take the liberty of quoting in full: 'Suppose that an income-tax is introduced; it is necessary that the Act imposing it should state what forms of wealth are to be regarded as income, and taxed accordingly. The rent of land and houses is clearly a form of income, and would be included in the division of that genus; but if the owner of a house lives in it instead of letting it, he receives no rent. Nevertheless, he enjoys an income, in the shape of the annual value of the house he lives in, just as truly as if he had let that house, and received for it a sum of money sufficient to hire himself another; and he ought to be taxed if he lives in his own house as much as if he lets it. But if the income-tax Act omitted to include among the species of income the annual value of houses occupied by their owners, he would escape payment on that head altogether. Such is the practical importance of making a division exhaustive.

Division by Dichotomy.

In the process known as Dichotomy (δίχα, in two; τέμνω, I cut) we divide the genus into two alternative species-' x or not-x': z is commonly called the positive, and not-x the negative species; but, as the negative species proves on analysis to be negative only in the name, we propose to substitute for the words 'positive' and 'negative' the words 'definite' and 'indefinite.' Thus we may divide 'Animal' into 'vertebrate or non-vertebrate,' when by 'nonvertebrate' we mean 'some animal other than vertebrate.' We then systematically subdivide on the same principle, and continue dichotomizing in this way until it ceases to be purposive to go further. What is known as Porphyry's Tree* illustrates the process in that incomplete form in which only the definite terms are dichotomized.

Being.

Corporeal. Incorporeal.

Animate. Inanimate.

Sensible. Insensible.

Rational. Irrational.

Socrates. Plato. Etc.

* As Mr. Stock points out ('Logic,' ed. 1903, p. 94), the 'Tree of Porphyry' is a device added by later writers.' In Porphyry's treatise there is no division by dichotomy, but simply the logical development of the single category of Substance taken as summum genus :

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Mr. Stock adds the following interesting footnote: 'We might suppose that "thing" or "being" could be predicated of "substance," but Porphyry, following Aristotle, regards each of the ten categories as a distinct summum genus. He will not allow that "being" is predicable of them all in the same sense.'

This rejection of the indefinite term at each step of the division is technically known as an 'abscissio infiniti,' the 'infinitum' or ' indeterminate' being here the indefinite term.

The definite and indefinite terms in their relation to each other are sometimes referred to as Contradictory Opposites, Contradictory Relatives, or Contradictories. Thus 'cold' and 'not-cold' are said to be contradictory opposites. But the name is unfortunate and apt to mislead. A definite term and its counter-indeterminate are not contradictory in the sense of contradicting each other. It is only statements that can contradict or be contradicted. It is true that when such terms are predicated of the same subject in the same relation the assertions within which they thus function as the respective predicates contradict each other; but it is the opposition of the two statements, and not that of the two predicates as such, which constitutes the contradiction.

We shall, in fact, see, when we come to consider what we mean by an indefinite term, that these so-called contradictory opposites are complementary rather than antithetic. They should therefore be carefully distinguished from contrary opposites or contrary relatives, which may be defined as terms markedly opposed under the same head. We say 'markedly' and not 'most,' since under any given head-e.g., that of temperature-we may have more than one pair of contraries. Thus 'cold' and 'hot' are contraries; but so also are 'freezing' and 'broiling.'* It will be seen that each of a given pair of contrary terms is itself a positive term with well-defined positive reference. 'Black' is just as positive in meaning as 'white,' ' miserable' as positive as 'happy,' 'hard' as positive as 'soft.'

A term is, of course, a 'contradictory' or a 'contrary,' not per se, but only in relation to its opposite. In particular the indefinite term 'not-x' is not in itself 'a contradictory term.' It is contradictory only in relation to the complementary definite term 'x, and that only in the derivative sense already indicated.

The Meaning of the Indefinite Term.

The logical significance of Dichotomy depends primarily on the meaning we assign to the indefinite term. We must, therefore, carefully consider what this meaning may be.

An indefinite term is a term of the form 'not-x' or 'non-x.' It indicates what is other than x in a sense that we must now proceed to determine. Some logicians insist that it must be, in character, perfectly and illimitably indefinite. Not-x, they say, must surely take up all that is excluded from x. Out of the sum-total of think

*

This indefiniteness does not extend to contrary propositions. There the opposition exists unambiguously between 'all' and 'none,' between 'All S is P

and 'No S is P.'

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