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.