Part II, Chapter 15: Of Two Kinds of Propositions which are of great use in the Sciences: Division and Definition. And firstly, of Division
IT is necessary to say something in detail of two propositions which are of great use in the sciences—division and definition.
Division is the separation of a whole into its parts.
But as there are two kinds of wholes, there are also two kinds of division. There is a whole composed of parts really distinct, called, in Latin, totum, and whose parts are called integral parts. The division of this whole is called properly partition: as when we divide a house into its apartments, a town into its wards, a kingdom or state, into its provinces man into body and soul, the body into its members. The sole rule of this division is, to make the enumeration of particulars very exact, and that there be nothing wanting to them.
The other whole is called, in Latin, omne, and its parts, subjected or inferior parts, inasmuch as the whole is a common term, and its parts are the terms comprising its extension. The word animal is a whole of this nature, of which the inferiors—as man and beast—which are comprehended under its extension, are subjected parts. This division obtains properly the name of division, and there are four kinds of division which may be noticed.
The first is, when we divide the genus by its species: every substance is body or mind; every animal is man or beast. The second is, when we divide the genus by its differences: every animal is rational or irrational; every number is even or uneven; every proposition is true or false; every line is straight or curved.
The third is, when we divide a common subject into the opposite accidents of which it is susceptible, these being according to its different inferiors, or in relation to different times: as, every star is luminous by itself, or by reflection only; every body is in motion or at rest; all the French are nobles or commoners; every man is well or ill; all nations employ, for the purpose of expressing themselves, either speech alone, or writing together with speech.
The fourth is that of an accident into its different subjects, as division of goods into those of mind and body.
The rules of division are—1st, That it be complete, that is to say, that the members of the division comprehend the whole extent of the terms into which it is divided: as, even and uneven comprehend the whole extent of the term number, there being no number which is not either even or uneven. There is scarcely anything which leads us to make so many false reasonings as want of attention to this rule. What deceives us here is, that there are often terms which appear so opposed that they seem to allow no medium, but which, nevertheless, have one. Thus, between ignorant and learned, there is a certain medium of knowledge which removes a man from the rank of the ignorant, but which, still, does not place him in the rank of the learned; between vicious and virtuous there is a certain state of which we may say, what Tacitus said of Galba, magis extra vitia quam cum virtutibus—for there are some people who, having no gross vices, are not called vicious, and who, doing no good, cannot be called virtuous, although, before God, not being virtuous, may be a great vice; between sick and well there is the state of the man indisposed, or convalescent; between day and night there is twilight; between opposite vices there is a mean of virtue, as piety between impiety and superstition; and sometimes this mean is twofold, as between avarice and prodigality there is liberality and a laudable frugality; between the timidity which fears every thing, and the rashness which fears nothing, there is the bravery which is not frightened at dangers, and the reasonable prudence which leads us to avoid those which it is not fitting we should be exposed to.
The second rule, which is a consequence of the first, is that the members of the division be opposed: as, even, uneven, rational, irrational. But what we have already said in the First Part, must be here noticed, viz., that it is not necessary for the differences, which constitute its opposed members, to be positive, but it is sufficient for one to be so, and for the other to be the genus alone with the negation of another difference. It is, indeed, in this very way that we make the members more certainly opposed. Thus, the difference between a beast and a man, is only the absence of reason, which is nothing positive; the unevenness of a number is only the negation of its divisibility into two equal parts. The first number has nothing which the compound number has not, unity being the measure of each, and that number which is called first, differs from the compound one only in this, that it has no other measure save unity.
Nevertheless, it must be confessed that it is better to express the opposed differences by positive terms, when this can be done, inasmuch as this explains better the nature of the members of the division. This is why the division of substance into that which thinks, and that which is extended, is much better than the common one, into that which is material, and that which is immaterial, or equally into that which is corporeal, and that which is not corporeal, inasmuch as the words immaterial, or incorporeal, furnish us with an idea, only very imperfect and confused, of that which is understood much better by the expression, substance that thinks.
The third rule, which is a consequent of the second, is that one of the members be not so contained in the other, that the other may be affirmed of it, although it may sometimes be contained in it after another manner, for line is included in superficies, as a term of superficies, and superficies in solid, as a term of solid. But this does not prevent extension from being divided into line, superficies, and solid, because we cannot say that line is superficies, or that superficies is solid. We cannot, on the other hand, divide number into equal, unequal, and square, since every square number being even or uneven, it is already contained in the first two numbers. Neither ought we to divide opinions into true, false, and probable, since every probable opinion is true or false; but we may first divide them into true and false, and then divide each into certain and improbable.
Ramus and his followers have laboured very hard to show that no divisions ought to have more than two members. When this may be done conveniently, it is better; but clearness and ease being that which ought first to be considered in the sciences, we ought not to reject divisions into three members, and especially when they are more natural, and when it would require forced subdivisions in order to reduce them to two members; for thus, instead of relieving the mind, which is the principal effect of division, we should load it with a great number of subdivisions, which it is much more difficult to retain than if we had made at once more members in that which we divide. For example, is it not more short, simple, and natural, to say, All extension is either line, or superficies, or solid, than to say with Ramus, Magnitudo est linea, vel lineatum, lineatum et superficies, vel solidum?
Finally, we may remark that it is an equal defect not to make enough, and to make too many divisions; the one does not sufficiently enlighten the mind, the other dissipates it too much. Crassotus, who is a philosopher of worth among the interpreters of Aristotle, has injured his book by too great a number of divisions. We fall thus into the confusion which we seek to avoid. Confusum est quidquid in pulverem sectum est.