DIVISION (LOGIC)

The name (Gr. διαίρεσις, Lat. divisio ) for various mental operations or their expressions that have in common the consideration apart from each other of the several parts as a whole. As such the idea of division is analogical, having as many meanings as there are senses of whole and part. This analogicity is consequent not only upon the diversity of objective wholes, but also on the three types of mental act: simple apprehension judgment, and reasoning. The auxiliary or instrumental operations of division have a somewhat different sense according as they play their clarificatory role in conjunction with each of these mental acts.

Historically, division first appeared among the Greeks either as itself a mode of reasoning to a defini tion (plato) or as an operation preparatory to definition and to reasoning from the definition so obtained (aris totle). The scholastics elaborated with special care the nature and kinds of division as a clarification of the terms and contents of the concepts formed by simple apprehension.

Notion. What is usually meant by the word "division" is a kind of clarification of terms and concepts. The objective concept, i.e., the content of the concept, may be taken as a whole in several "physical" or natural senses of whole, or in the more properly logical sense of a universal. An illustration of the former class is the integral whole, man, as composed of the parts: head, trunk, and members. A natural division of this kind is, in a broad sense, logical because it is the consideration (and not the physical cutting) apart from each other of the several parts of a real or natural whole. But universals as such are properly logical wholes, and as such are divided into subjective parts, i.e., the potential subjects of which they are potentially the predicates, e.g., the division of triangle into equilateral, isosceles, and scalene, and the division of animal into man and brute. Dialectically or nominally, such a scientific division of a conceptual content is imitated or approximated by the division of a linguistic term into the several senses in which it is used. Such a preliminary investigative division that exploits the riches of a given language may be called a nominal division; it is usually taken as preparatory to the scientific division; but it often resembles and anticipates the sapiential division of an analogical universal whole, which does not have the unity of one objective meaning (ratio ), into its several only proportionally unified meanings. The terminological divisions drawn from ordinary language and previous philosophical usage that constitute Aristotle's dictionary (Metaphysics bk. 5) anticipate such analogical divisions. The division of division is itself an analogical division.

With respect to extension and comprehension, in the light of a general theory of analysis and synthesis, division in the strict sense of that of a logical whole, or universal concept, is the analysis of the extension of a concept. It is thus contrasted to collection (συναγωγή; see Plato, Phaedrus 265D–266B) or universalization, i.e., the synthesis of the extension of a concept, as when one consciously returns in distinct virtual knowledge to the recognition that animal is the universal including man and brute. The two are in turn contrasted with distinction and definition, which are the analysis and synthesis, respectively, of the comprehension of a concept.

In the domain of judgment, division is contrasted to composition (σύνθεσις); they are, respectively, the negative and the affirmative categorical judgments (see Aristotle, Interp. 16a 12, Meta. 1027b 19–1028a 5, 1051b 1–35, Anim. 430a 26–b 31), in which the mind either divides one object from another or composes one object with another. If one considers the act rather than the object of judgment, however, this is always a composition, since it is always a comparison of one simple objective concept with another (St. Thomas Aquinas, In 1 perih. 3.4; De ver. 1.3, 9; Summa theologiae 1a, 16.2).

Kinds. Excluding division per accidens, i.e., the division of an accidental whole, six species of division are generally recognized among the scholastics: (1) Nominal division is the distinction of the meanings of a term with respect to the various objects it signifies. (2) Integral division is the division of a quantified whole into component parts that are continuous but lie outside each other; it is also called partition. (3) Physical division, in the strict sense, is the division of a natural whole into its proper constitutive physical principles, its matter and form, e.g., the division of man into a rational soul and an organized body of such and such specifications. In both integral and physical division the parts are really distinct from each other and therefore really compose the whole. (4) Essential division. e.g., of man into rational and animal, is a division into parts that are not really distinct and compositive, being differentiated only by a distinction of reason. Between rational and animal as understood of man there obtains, according to St. Thomas, only a distinction of reasoned reason. According to Duns Scotus, these are in reality actually distinct metaphysical grades or formalities that together constitute man's essence. The essential whole and its division are sometimes referred to as metaphysical. This usage of Scotists and Suarezians was widely shared by Thomists of the second scholasti cism; it is infelicitous, however, in that it misleadingly suggests that the parts or grades being distinguished are metaphysically (i.e., really) distinct from each other. (5) Logical division, in the proper sense, is the division of the universal, or extensive whole, into the parts of its extension, which are called its subjective parts. Such division usually refers to a univocal whole; analogical division is sometimes termed unequal division (per prius et posterius ), whereas univocal division is referred to as equal division (ex aequo, see De malo 7.1 ad 1). (6) Dynamic, or potestative, division is the division of a functional whole into its functional parts or powers, e.g., the soul into rational, animal, and vegetative. Both logical and dynamic division have in common that their members are in a broad sense subjective, or subordinated to the whole. Accordingly, as both St. albert the great (In 1 sent. 3.34) and Thomas (Summa theologiae 1a, 77.1 ad 1) remark, the dynamic whole may be called a potential whole since it may be predicated distributively of its several parts. Thus the dynamic whole mediates in a certain sense between the integral whole and the universal or logical whole, and consequently its mode of division also mediates between their modes of division.

Rules. The conditions or rules for good division are framed in view of the logical division of a univocal whole, just as the rules of definition are framed in view of the real physical definition, but each set of rules is applicable analogically to the other forms of division or definition. The rules of division are: (1) that the members taken singly be inferiors, i.e., less than the thing divided, since every whole is greater than its part; (2) that all the members that divide the whole taken together fill up or equal the whole divided, the reason being that the whole is equivalent to all its parts taken together; and (3) that the members that divide the whole be opposed, at least formally, to each other. If they had no opposition, they would not have distinction, but rather identity; and therefore they would not be diverse.

A corollary of the first rule is that the genus is not divided into differences but, through the differences, into the several species. A fourth rule is sometimes added, namely, that the opposition between the members should be one of contradiction; actually, however, since this opposition fails within the genus to be divided, it amounts to one of contrariety, and thus this rule is reducible to the third. The fourth rule is sometimes referred to as the rule of dichotomous division, which insists that each division have only two members. A better, and more general, rule is that "a division have only two dividing members, or at least as few as possible" (John of St. Thomas, Summulae 2.4). In his Parts of Animals (624b 5–664a 11, esp. 643a 24–27) Aristotle criticizes dyadic division as allowing only a single differentia to each species, whereas what is often needed to distinguish the several biological species of a genus is a set of differentiae or properties that can uniquely define a species.

A set of topics or sources (Gr. τóποι Lat. loci ) for arguing, or rules of consequences, can be framed in accordance with these three (or four) rules of division. They are (1) the rule of equivalence or substitution of the dividendum for the division, and vice versa—of use for materially converting the minor premise in induction; (2) the rule of negating one alternate and then affirming the other, i.e., modus tollendo ponens, for the so-called weak disjunctive or alternative; (3) the rule of affirming one disjunct and then negating the other, i.e., modus ponendo tollens, for the strong or strict disjunctive—see John of St. Thomas, loc. cit. ; (4) the rules of subalternation and categorical seriality or transitivity, e.g., if a is a subjective part of b, and b of c, then a is a subjective part of c. This is of importance as the dictum de omni of the syllogism, formulated either according to extension ("to be in the whole of") or comprehension ("to belong to"); it has also a negative formulation, the dictum de nullo.

See Also: distinction, kinds of; logic.

Bibliography: john of st. thomas, Cursus philosophicus thomisticus, ed. b. reiser, 3 v. (new ed. Turin 1930–37), v. 1 Ars Logica, pt. 1 Summulae 2.4, 20b 42–22a 41; Eng. Outlines of Formal Logic, tr. f. c. wade (Milwaukee 1955) 49–51; The Material Logic, tr. y. r. simon et al. (Chicago 1955). j. a. oesterle, Logic: The Art of Defining and Reasoning (2nd ed. Englewood Cliffs, N.J.1963). v. e. smith, The Elements of Logic (Milwaukee 1957). j. a. mourant, Formal Logic (New York 1963). f. h. parker and h. b. veatch, Logic as a Human Instrument (New York 1959). e. d. simmons, The Scientific Art of Logic (Milwaukee 1961). h. f. cherniss, Aristotle's Criticism of Plato and the Academy (Baltimore 1944).

[j. j. glanville]