Deductive Logic Part 14

_Rules for Definition._

(1) A definition must be co-extensive with the term defined.

(2) A definition must not state attributes which imply one another.

(3) A definition must not contain the name defined, either directly or by implication.

(4) A definition must be clearer than the term defined.

(5) A definition must not be negative, if it can be affirmative.

Briefly, a definition must be adequate (1), terse (2), clear (4); and must not be tautologous (3), or, if it can be avoided, negative (5).

-- 379. It is worth while to notice a slight ambiguity in the term 'definition' itself. Sometimes it is applied to the whole proposition which expounds the meaning of the term; at other times it is confined to the predicate of this proposition. Thus in stating the first four rules we have used the term in the latter sense, and in stating the fifth in the former.

-- 380. We will now ill.u.s.trate the force of the above rules by giving examples of their violation.

Rule 1. Violations. A triangle is a figure with three equal sides.

A square is a four-sided figure having all its sides equal.

In the first instance the definition is less extensive than the term defined, since it applies only to equilateral triangles. This fault may be amended by decreasing the intension, which we do by eliminating the reference to the equality of the sides.

In the second instance the definition is more extensive than the term defined. We must accordingly increase the intension by adding a new attribute 'and all its angles right angles.'

Rule 2. Violation. A triangle is a figure with three sides and three angles.

One of the chief merits of a definition is to be terse, and this definition is redundant, since what has three sides cannot but have three angles.

Rule 3. Violations. A citizen is a person both of whose parents were citizens.

Man is a human being.

Rule 4. Violations. A net is a reticulated fabric, decussated at regular intervals.

Life is the definite combination of heterogeneous changes, both simultaneous and successive, in correspondence with external co-existences and sequences.

Rule 5. Violations. A mineral is that which is neither animal nor vegetable.

Virtue is the absence of vice.

-- 381. The object of definition being to explain what a thing is, this object is evidently defeated, if we confine ourselves to saying what it is not. But sometimes this is impossible to be avoided. For there are many terms which, though positive in form, are privative in force.

These terms serve as a kind of residual heads under which to throw everything within a given sphere, which does not exhibit certain positive attributes. Of this unavoidably negative nature was the definition which we give of 'accident,' which amounted merely to saying that it was any attribute which was neither a difference nor a property.

-- 382. The violation of Rule 3, which guards against defining a thing by itself, is technically known as 'circulus in definiendo,' or defining in a circle. This rule is often apparently violated, without being really so. Thus Euclid defines an acute-angled triangle as one which has three acute angles. This seems a glaring violation of the rule, but is perfectly correct in its context; for it has already been explained what is meant by the terms 'triangle' and 'acute angle,' and all that is now required is to distinguish the acute-angled triangle from its cognate species. He might have said that an acute-angled triangle is one which has neither a right angle nor an obtuse angle: but rightly preferred to throw the same statement into a positive form.

-- 383. The violation of Rule 4 is known as 'ignotum per ignotius' or 'per aeque ignotum.' This rule also may seemingly be violated when it is not really so. For a definition may be correct enough from a special point of view, which, apart from that particular context, would appear ridiculous. From the point of view of conic sections, it is correct enough to define a triangle as that section of a cone which is formed by a plane pa.s.sing through the vertex perpendicularly to the base, but this could not be expected to make things clearer to a person who was inquiring for the first time into the meaning of the word triangle. But a real violation of the fourth rule may arise, not only from obscurity, but from the employment of ambiguous language or metaphor. To say that 'temperance is a harmony of the soul' or that 'bread is the staff of life,' throws no real light upon the nature of the definiend.

-- 384. The material correctness of a definition is, as we have already seen, a matter extraneous to formal logic. An acquaintance with the attributes which terms imply involves material knowledge quite as much as an acquaintance with the things they apply to; knowledge of the intension and of the extension of terms is alike acquired by experience. No names are such that their meaning is rendered evident by the very const.i.tution of our mental faculties; yet nothing short of this would suffice to bring the material content of definition within the province of formal logic.

CHAPTER VIII.

_Of Division._

-- 385. To divide a term is to unfold its extension, that is, to set forth the things of which it is a name.

-- 386. But as in definition we need not completely unfold the intension of a term, so in division we must not completely unfold its extension.

-- 387. Completely to unfold the extension of a term would involve stating all the individual objects to which the name applies, a thing which would be impossible in the case of most common terms. When it is done, it is called Enumeration. To reckon up all the months of the year from January to December would be an enumeration, and not a division, of the term 'month.'

-- 388. Logical division always stops short at cla.s.ses. It may be defined as the statement of the various cla.s.ses of things that can be called by a common name. Technically we may say that it consists in breaking up a genus into its component species.

-- 389. Since division thus starts with a cla.s.s and ends with cla.s.ses, it is clear that it is only common terms which admit of division, and also that the members of the division must themselves be common terms.

-- 390. An 'individual' is so called as not admitting of logical division. We may divide the term 'cow' into cla.s.ses, as Jersey, Devons.h.i.+re, &c., to which the name 'cow' will still be applicable, but the parts of an individual cow are no longer called by the name of the whole, but are known as beefsteaks, briskets, &c.

-- 391. In dividing a term the first requisite is to fix upon some point wherein certain members of the cla.s.s differ from others. The point thus selected is called the Fundamentum Divisionis or Basis of the Division.

-- 392. The basis of the division will of course differ according to the purpose in hand, and the same term will admit of being divided on a number of different principles. Thus we may divide the term 'man,'

on the basis of colour, into white, black, brown, red, and yellow; or, on the basis of locality, into Europeans, Asiatics, Africans, Americans, Australians, New Zealanders, and Polynesians; or again, on a very different principle, into men of nervous, sanguine, bilious, lymphatic and mixed temperaments.

-- 393. The term required to be divided is known as the Totum Divisum or Divided Whole. It might also be called the Dividend.

-- 394. The cla.s.ses into which the dividend is split up are called the Membra Dividentia, or Dividing Members.

-- 395. Only two rules need be given for division--

(1) The division must be conducted on a single basis.

(2) The dividing members must be coextensive with the divided whole.

-- 396. More briefly, we may put the same thing thus--There must be no cross-division (1) and the division must be exhaustive (2).

-- 397. The rule, which is commonly given, that each dividing member must be a common term, is already provided for under our definition of the process.

-- 398. The rule that the dividend must be predicable of each of the dividing members is contained in our second rule; since, if there were any term of which the dividend were not predicable, it would be impossible for the dividing members to be exactly coextensive with it.

It would not do, for instance, to introduce mules and donkeys into a division of the term horse.

-- 399. Another rule, which is sometimes given, namely, that the const.i.tuent species must exclude one another, is a consequence of our first; for, if the division be conducted on a single principle, the const.i.tuent species must exclude one another. The converse, however, does not hold true. We may have a division consisting of mutually exclusive members, which yet involves a mixture of different bases, e.g. if we were to divide triangle into scalene, isosceles and equiangular. This happens because two distinct attributes may be found in invariable conjunction.

-- 400. There is no better test, however, of the soundness of a division than to try whether the species overlap, that is to say, whether there are any individuals that would fall into two or more of the cla.s.ses. When this is found to be the case, we may be sure that we have mixed two or more different fundamenta divisionis. If man, for instance, were to be divided into European, American, Aryan, and Semitic, the species would overlap; for both Europe and America contain inhabitants of Aryan and Semitic origin. We have here members of a division based on locality mixed up with members of another division, which is based on race as indicated by language.