THE DOCTRINE OF DIVISION.
409. Logical Division.—The term division, as technically used in logic, may be defined as the setting forth of the smaller groups which are contained under the extension of a given term. It is also defined as the separation of a genus into its constituent species. These two definitions are practically equivalent to one another. Division is to be distinguished from the setting forth of the individual objects belonging to a species, which is technically described as enumeration.
In logical division, the larger class which is divided is called the totum divisum, the smaller classes into which it is divided being the membra dividentia (dividing members). By the ground or principle of division (fundamentum sive principium divisionis) is meant that attribute or characteristic of the totum divisum upon whose modifications the division is based. A given class may of course be divided in different ways according to the particular attribute or attributes whose variations are selected as differentiating its various species. Thus, having regard to the equality or inequality of the sides, triangles may be divided into equilateral, isosceles, and scalene; or, having regard to the size of the largest angle, into obtuse-angled, right-angled, and acute-angled. Again, propositions are divisible according to their truth or falsity, or according to their quantity, or their quality, and so on.
It is sometimes said that the principle of division must be present throughout the dividing members, though constantly varied. On the other hand, it is said that in division we invariably try to think of some attribute which is predicable of certain members of the group, but not of others. The former of these statements does not 442 very well apply when we simply divide a class according to the presence or absence of some attribute (for example, candidates for the Civil Service into successful and unsuccessful) or when the attribute in question may be entirely wanting in some instances whilst present in varying degrees in other instances. In other words, given the attribute whose variations constitute our principle of division, we may have to recognise a limiting case in which it is altogether absent; thus, in dividing undergraduates according to their colleges, we may have to recognise a class of non-collegiate students. The second statement is always true when we simply contrast any given species with all the remaining species, and it may be considered adequate where we have division by contradictories. In other cases, however, it is inadequate; as, for instance, when we divide candidates who are successful in the Indian Civil Service Examination according to the province to which they are assigned.
410. Physical Division, Metaphysical Division, and Verbal Division.—Following the older logicians, we may distinguish division as defined in the preceding paragraph, that is, logical division in the strict sense, from other senses in which the term is used.
The division of an individual thing into its separate parts is called physical division or physical definition (Whately, Logic, p. 143) or partition ; as, for example, if we divide a watch into case, hands, face, and works; or a book into leaves and binding. We have, on the other hand, a logical division if we divide watches into gold, silver, &c., or into English, Swiss, American, &c.; or if we divide books into folios, quartos, &c. Bain (Logic, II. p. 197) gives the analysis of a chemical compound as an instance of logical division. It is rather an instance of physical division. In logical division the totum divisum is always predicable of all the individuals belonging to each of the membra dividentia ; for example, All men are animals, All squares are rectangles. But this is not the case in chemical analysis. We cannot say that oxygen is water, or that sulphur is vitriol, or that sodium is salt.
Distinct both from logical division and from physical division is the mental division of a thing into its separate qualities. This is called metaphysical division. We have an example when we enumerate the separate qualities of a watch, its size, accuracy, the material of which its case is composed, &c.; or when we specify the size of a book, its thickness, colour, the material of its binding, the quality of the paper of which its leaves are composed, and so on. 443 A physical division can be actually made; a watch, for example, can be taken to pieces. A metaphysical division, on the other hand, is only possible mentally. It should be added that the metaphysical division of individual objects may be made the basis of a logical division of the class to which they belong.
One further kind of division may be noticed, namely, the division of an ambiguous or equivocal term into its several significations. This is called verbal division (Clarke, Logic, p. 331) or distinction (Mansel’s Aldrich, p. 37). For example, we have to distinguish between a watch in the sense of a vigil, in the sense of a guard, and in the sense of a time-piece.
411. Rules of Logical Division.—The fundamental rules of logical division are (1) that the members of the division shall be mutually exclusive; and (2) that collectively they shall be exactly coextensive with the class that is divided. Thus if the class X is correctly divided into XA, XB, XC, the following propositions must hold good, namely, No XA is B or C, No XB is C or A, No XC is A or B, Every X is A or B or C.
The two following rules are generally added: (3) Each distinct act of division should proceed throughout upon one and the same basis or principle; (4) If the division involves more than one step, it should proceed gradually and continuously from the highest genus to the lowest species, that is to say, it should not pass suddenly from a high genus to a low species.
It may be objected that (1) and (2) ought not in a strict sense to be described as rules, but rather as constituting between them a precise statement of what is implied when we speak of a logical division. They become rules, however, in the sense that a professed logical division which fails to satisfy either of them implies relations between the members of the division which do not as a matter of fact hold good. Rules (3) and (4) are of a different character. They are rules in the sense that they must be complied with if a division is to have practical utility.
Rule (3) is not intended to condemn the processes of sub-division and co-division. Having made a division upon one principle, we may proceed to subdivide the classes thus arrived at in accordance with another principle, and so on indefinitely. A scientific classification will always consist of a hierarchy of classes thus obtained. There is again no reason why the same class should not for different purposes be divided in accordance with two or more different 444 principles, so long as these are kept distinct from one another, and the members of the different resulting divisions not confused together.
It has been said that a breach of rule (1) necessarily involves a breach of rule (3), since there cannot be any overlapping of classes so long as a division proceeds correctly upon a single principle. This does not, however, always hold good unless we interpret the word “correctly” as implying that precautions are taken to avoid any overlapping, which of course begs the question. Thus, if we divide triangles into those which have (a) a right angle, (b) an obtuse angle, (c) an acute angle, we may be said to proceed upon one principle, and yet the resulting classes are not mutually exclusive. It may, again, be argued that the classes equilateral triangle, isosceles triangle, scalene triangle (which result from a division based upon a single principle) are not mutually exclusive, since all equilateral triangles are isosceles.
This argument can only be met by saying that, in the first case, we are not proceeding upon any clear principle unless we make our division into triangles whose largest angle is an obtuse angle, a right angle, or an acute angle, respectively; nor unless, in the second case, our principle is the maximum number of sides that are equal to one another, so that an isosceles triangle is defined as a triangle that has two and only two sides equal. Any overlapping of classes is then in each case provided against; but only, it may be argued, because special precautions have been taken to attain this end. By the adoption of similar precautions, a division which proceeds “correctly” upon a single principle will also be exhaustive.
Looking at the question from the other side we may note that a division which satisfies both rule (1) and rule (2) may nevertheless be a cross-division; for it may happen that two different principles of division yield coincident results. For example, an isosceles triangle being defined as a triangle that has two and only two sides equal, there is a cross-division, but no overlapping of classes, or omission of any class contained in the totum divisum, if we divide triangles into scalene, isosceles, and equiangular; or if we divide plants into acotyledons, monocotyledons, and exogens.
As regards rule (4), it is to be observed that a division which proceeds per saltum will usually be much less effective than one in which the intermediate steps are filled in. The worst violation of this rule occurs when the division is disparate, that is, when “one of the classes into which we divide is an immediate and proximate 445 class, while others are mediate and remote” (Clarke, Logic, p. 242); as, for example, if we divide animals into invertebrates, fishes, amphibians, reptiles, birds, elephants, horses, dogs, &c.
Another rule of division is sometimes added, namely, that “none of the dividing members must be equal in extent to the divided whole” (Clarke, Logic, p. 236). When this rule is broken, the division is said to become null and void, because one of the sub-divisions contains no members. From the formal point of view, however, the observance of this rule can hardly be insisted upon. We need not regard a division as necessarily implying the actual occurrence of all its members in the universe of discourse; and the rule in question would deprive the logician of the right to employ the powerful method of division by contradictories. It may be a different matter when we are considering scientific classification from the material standpoint.
412. Division by Dichotomy.—Division by dichotomy or, as it is sometimes called more distinctively, dichotomy by contradiction is the division of a class simply with reference to the presence or absence of a given attribute or set of attributes; as, for example, when X is divided into XA and Xa (where a = not-A). An illustration is afforded by the Tree of Porphyry or Ramean Tree, in which Substances are first divided into Corporeal Substances (Bodies) and Incorporeal Substances, Bodies being then divided into Animate Bodies (Living Beings) and Inanimate Bodies, Living Beings being next divided into Sensitive Living Beings (Animals) and Insensitive Living Beings, and Animals being in their turn divided into Rational Animals (Men) and Irrational Animals. At each step in this scheme we proceed by taking contradictories. It was in praise of dichotomal division that Jeremy Bentham, who is here quoted with approval by Jevons (Principles of Science, 30, § 12), spoke of “the matchless beauty of the Ramean Tree.” When this method is employed we ensure formally that the members of our division shall be mutually exclusive and collectively exhaustive. For, by the law of contradiction, No X is both A and a ; and, by the law of excluded middle, Every X is either A or a.
It is pointed out by Spalding (Logic, p. 146) and by Jevons (Principles of Science, 30, § 9) that all logically perfect division is ultimately reducible to dichotomy, usually with the implication that some of the sub-classes which are à priori possible are not as a matter of fact to be found in the universe of discourse. Thus, 446 if we take the class X and divide it into XA and XB we imply that in the class X, A and B are never found either both present or both absent. Hence the division is equivalent to the following dichotomal division:—
Any other division, however complicated in its character, may be reduced to dichotomy in a similar way. This is interesting and important and brings out the value of dichotomy as a method of testing divisions. It must be understood, however, that in speaking of all division as ultimately reducible to dichotomy, it is not intended to imply that dichotomy usually represents our actual procedure in making divisions. Each sub-class is usually arrived at immediately by reference to some positive modification of the fundamentum divisionis ; and the different sub-classes are co-ordinate with one another. Consider, for example, the division of conic sections into parabolas, hyperbolas, ellipses, circles, and pairs of straight lines. It must be added that from the material standpoint, pure division by dichotomy is of little scientific value, because of the indefinite character of the sub-classes which are determined negatively.
413. The place of the Doctrine of Division in Logic.—The doctrine of division, as treated by the older logicians, receives little recognition by some modern writers on two very different grounds: (1) by Mill, taking the material standpoint, it is regarded as too purely formal, and hence is merged in the doctrine of scientific classification; (2) by some writers belonging to the conceptualist school, e.g., Mansel, it is rejected as not being sufficiently formal.
(1) It is true that the rules of logical division lead us a very little way in practical science. They give certain conditions which must be complied with; but they neither help us towards making good divisions, nor provide us with a test which is capable of being formally applied. Leaving dichotomy on one side, we cannot, without the aid of material knowledge, even determine whether the members of a given division are mutually exclusive and collectively 447 exhaustive. When, however, we avowedly pass beyond purely formal considerations and take up a material standpoint, then the doctrine of division should rightly give place to a doctrine of classification, which is not content with such rules as those laid down above, but seeks to indicate the principles that should serve as a guide in the classification of objects scientifically.
In regard to the use of the terms division and classification, Miss Jones draws a distinction which is of value and to which it might be well systematically to adhere. “Division and classification are the same thing looked at from different points of view; any table presenting a division presents also a classification. A division starts with unity and differentiates it; a classification starts with multiplicity, and reduces it to unity, or at least to system” (Elements of Logic, p. 123).
(2) It remains to be considered how far any treatment of division whatever can properly fall under the consideration of formal logic. From this point of view division is usually contrasted with definition. The latter of these—using the phraseology of the conceptualist logicians—expounds the intension of a concept; the former expounds its extension. But the intension of a concept is said to be far more intrinsic to it than its extension. Given a concept its intension is necessarily given; but knowledge of its extension, such as may serve to determine its division, will require a fresh appeal to the subject-matter. “Division,” says Mansel, “is not, like definition, a mental analysis of given materials: the specific difference must be added to the given attributes of the genus; and to gain this additional material, it is necessary to go out of the act of thought, to seek for new empirical data” (Prolegomena Logica, p. 192). For example, the division of members of the University of Cambridge into those in statu pupillari and members of the Senate could not be obtained without something more being given than the mere conception of a member of the University. Moreover, unless we proceed by contradictories, we cannot, when we have got our division, formally determine whether it complies with our rules or not.
The above position may be accepted, if an exception is made for division by dichotomy. Mansel, however, and some other logicians will not even allow that division by dichotomy is a formal process; and here they lay themselves open to criticism. The grounds on which their view is based are twofold:—(i) It is not sufficient that 448 the genus to be divided be given; the principle of division must be given also. “Even in the case of dichotomy by contradiction the principle of division must be given, as an addition to the attributes comprehended in the concept, before the logician can take a single step” (Prolegomena Logica, p. 207). “The division of A into B and not-B is not strictly formal; for the dividing attribute, not being part of the comprehension of A, has to be sought for out of the mere act of thought, after A has been given” (Mansel’s Aldrich, p. 38). (ii) We cannot tell à priori that both the sub-classes obtained by dichotomy really exist. How, for example, can we divide A into B and not-B when for anything we know to the contrary all A may be B? “Logically, the division of animal into mortal and immortal is as good as that into rational and irrational” (Mansel’s Aldrich, p. 38). Both these arguments are summed up in the following quotation from Mr Monck: “It is alleged indeed that Logic enables us to divide all the B’s into the B’s which are C’s and the B’s which are not C’s…… But Logic does not supply us with the term C and after we have obtained this term there are two cases in which the proposed division fails, namely, where all the B’s are C’s and where none of them are so. In either of these events the class B remains as whole and undivided as before; and whether they have occurred or not cannot be ascertained by Logic. This Division by Dichotomy, as it is called, is as much outside the province of Logic as any other kind of division” (Logic, p. 174).
As regards the first of the above arguments, there is no reason why the principle of division (A) should not be assumed given as well as the totum divisum (X). The question is whether we can then formally divide X into XA and Xa. The fact that A must be given as well as X does not prevent the possibility of formal division by dichotomy, any more than the fact that the conclusion of a syllogism is not contained in one premiss alone prevents the syllogism from being a formal process.
The force of the second argument depends upon the implication that all the sub-classes obtained as the result of a division necessarily exist in the universe of discourse. If this implication is granted, then dichotomy is certainly not a formal process; but why need we assume the existence of all the sub-classes obtained by dichotomy? Without such an assumption, our division may not have much practical utility, but its formal validity will remain unaffected. 449 We have only to make it clear that we are dividing the extension of a term, not its denotation, in the sense in which extension and denotation have been already distinguished.[453] This is in keeping with the general standpoint of formal logic, which can deal with classes without regarding their existence as necessarily guaranteed in any assigned universe of discourse. If we are not allowed to apply the principle of excluded middle in formal logic and say Every X is A or a, until we know that there actually exist both XA’s and Xa’s, we shall be exceedingly hampered, and can make but little progress, especially in the treatment of complex inferences. Some schemes of symbolic logic (e.g., Jevons’s) depend essentially and explicitly upon an antecedent scheme of dichotomal division.
We may then regard division by dichotomy as a formal process, but only on the understanding (1) that the principle of division is given as well as the genus to be divided; (2) that the division is not assumed to be more than hypothetical so far as concerns the existence of the resulting sub-classes in any assigned universe of discourse.