2. THE PREDICABLES.

A predicable is a term which can be affirmed or predicated of any subject. In the proposition, “A man is a rational animal,” the term “rational animal” is a predicable, because it can be affirmed of the subject man.

To gain a clear knowledge of the definition it is quite necessary to understand the five predicables which we shall consider in the following order:

(1) Genus and (2)Species.

Genus and species are relative terms and can best be defined together.

A genus is a term which stands for two or more subordinate classes.

A species is a term which represents one of the subordinate classes.

The genus may be subdivided into species; the species together form the genus.

To illustrate: The term man stands for five subordinate classes or species, as white, black, brown, yellow and red. “Man” is, therefore, a genus, while “white man” and “black man,” etc., are species. The term “polygon” is a genus with reference to “trigon,” “tetragon,” “pentagon,” etc., while “trigon” is a species of “polygon.”

Any given genus may be a species of some higher class. That is, “man,” which is a genus with reference to the kinds of men, is a species of the higher class “biped,” while “biped” is a species of “animal,” “animal” a species of “organized being,” “organized being” of “material being,” “material being” of “being.” But here we stop, as there is no higher grade to which “being” can be referred. This highest genus takes the name of summum genus.

Similarly any given species may be a genus of some lower class. “White man,” for example, which is a species of “man,” is a genus of “American,” “Englishman,” “German,” “Frenchman,” etc. “American” is a genus of “New Yorker,” “Californian,” etc., while “New Yorker” is a genus of “Smith of Jamaica.” This last term is an individual and cannot be subdivided. It represents the lowest possible species and is referred to in logic as infima species.

It is obvious that the highest genus cannot become a species, neither can the lowest species become a genus.

PROXIMATE GENUS.

The proximate genus is the next class above. To illustrate: “Animal” is a genus of “man,” but “biped” is the proximate genus of “man.” “Quadrilateral” is the genus of “square,” but “rectangle” is the proximate genus. The next class above “trigon” is polygon not figure. Hence “polygon” is the proximate genus of “trigon.”

GENUS AND SPECIES OF NATURAL HISTORY.

In natural history the following terms are used to denote the various grades of kinship in any scheme of classification: (1)kingdom, (2)class, (3)order, (4)family, (5)genus, (6)species, (7)variety, (8)the individual thing. Here “genus” and “species” are absolute not relative and occupy a fixed place in the scheme, while from a logical viewpoint any of the grades indicated between the lowest and highest would be the species of the next higher grade or a genus of the next lower; e.g., order is a species of “class,” while it is the genus of “family.”

GENUS, A DOUBLE MEANING.

We recall that any class name or genus has a double use, extensional and intensional. When considered from the standpoint of its extension, a genus represents a group of objects or is mathematical in its application, but when used in an intensional sense it represents a group of qualities or is logical in its application.

Considered extensionally the genus refers to a larger number of objects than the species. But when viewed intensionally the species refers to more qualities than the genus. This was made clear when discussing the law of variation in the extension and intension of terms.

(3) Differentia.

The differentia of a term is that attribute which distinguishes a given species from all the other species of the genus.

It has been observed that the species refers to more qualities than the genus. In fact, it represents all the attributes of the genus plus those which distinguish the particular species from the other species of the genus. These additional qualities are the differentiÆ of the particular species.

TO ILLUSTRATE:

The attribute which distinguishes man from the other bipeds of the world is his rationality. That which distinguishes the rectangle from the other parallelograms is its four right angles. The attributes rationality and right angles are differentiae.

(4) Property.

A property of a term is any attribute which helps to make the term what it is. Thus “consciousness” is a property of man, “binding” a property of book, “angles” a property of triangle. Deprive the terms of these attributes and their true nature is altered.

A differentia is a property according to the foregoing definition. However, Jevons defines “property” as “Any quality which is common to the whole of a class, but is not necessary to mark out the class from other classes.” This viewpoint excludes “differentia” from the notion of property. The difference in opinion is of slight importance.

(5) Accident.

An accident of a term is any attribute which does not help to make the term what it is. It may indifferently belong or not belong to the term. Deprive a term of an accident and the nature of the term remains unchanged. Thus, a teacher’s position, a man’s watch, the fact that the angle is one of 80° are all accidents.

It is obvious that a property is a constant attribute while an accident is variable. This gives to the former a universal validity while the latter is more or less shifting and uncertain. All triangles must have three angles (property) while the value of each angle in degrees (accident) admits of unlimited variation.

Some logicians divide accidents into separable and inseparable. Aman’s hat would be a separable accident while his birthplace would be an inseparable accident.

FIVE PREDICABLES ILLUSTRATED.

In the following brief descriptions the five predicables are designated:

3. THE NATURE OF A DEFINITION.

It will be remembered that an individual notion is a notion of a single thing or attribute, while a general notion is a notion of a class of things or a group of attributes. Aterm which represents an individual notion is known as a singular term, while a term which stands for a general notion is referred to as a general term.

One may explain the meaning of a singular term which stands for one thing by enumerating its various attributes. For example, such attributes as a piercing bark, a yellow color, intelligent, companionable, a strong liking for sweetmeats, explain the meaning of the singular term “Fido.” Likewise we may explain the meaning of a general term by enumerating its attributes. To illustrate: power of speech, rationality, ability to laugh, etc., explain the meaning of the general term man. The explanation of the singular term fits only Fido. There is probably no other dog in the world just like Fido. But the explanation of the general term man may be applied to all men.

A brief enumeration of attributes which may be applied to a class of things often takes the form of a definition. The word definition comes from the word definire, meaning to limit or fix the bounds of.

A definition, then, consists of the enumeration of such attributes as distinguish a term from all other terms. In this sense it would seem that the singular term Fido, as well as the general term man, admits of definition, but it is usual for logicians to confine definition to the general term. Singular terms may be described; general terms, defined.

A DEFINITION OF DEFINITION.

A definition of a term is a statement of its meaning by enumerating its characteristic attributes.

That the enumeration must be in terms of its distinguishing or characteristic attributes is implied in the derivation of the term definition. The attributes must establish limits or bounds, just as a line fence limits a land owner’s possessions. To indicate that man is a creature possessing the power of locomotion, sense of sight and ability to eat, is surely not a definition, as the marks are not characteristic of men only. These attributes set no boundary between man and horse, consequently the statement is a faulty description of man, not a definition. But when the enumeration includes such attributes as power of speech, rationality and ability to laugh, then does the description become a definition. To put it differently: Adefinition is a description of a term by means of its distinguishing attributes. This statement may be considered a definition of man, though somewhat faulty: “Aman is a creature who is rational and who possesses the power of speech and ability to laugh.”

4. DEFINITION AND DIVISION COMPARED.

We have learned that general terms when connotative may be used extensionally or intensionally.

A definition indicates the intensional nature of a term, while a statement which points out the extensional nature of a term is known as logical division. More briefly: Adefinition is an intensional statement of the nature of the term, while logical division is an extensional statement of the nature of the term.

To illustrate: The following statements are definitions:

The following represent Logical Division:

5. THE KINDS OF DEFINITIONS.

Generally speaking there are three kinds of definitions, namely, (1)Etymological, (2)Descriptive, (3)Logical.5

(1) An etymological definition is one based upon the derivation of the term.

This kind of a definition, which gives merely the meaning of the symbol, is sometimes called a nominal or verbal definition; while a real definition is regarded as one which gives the meaning of the notion for which the symbol stands. The modern logician is inclined to ignore this classification on the argument that to make a distinction between a symbol and the notion it symbolizes is simply to misunderstand the relation which exists between them. If the definition does not agree with the thing then it cannot correctly explain the term which represents the thing. Define correctly the term and one has defined correctly the notion signified by the term.

The attributes of a term may be separated into three classes: differentia, property and accident. It would appear possible, therefore, to define a term by enumerating the accidents only or by enumerating the properties, or, finally, by stating the differentiae. But if the enumeration is confined to accidents the chances are that the statement will be a description, not a definition, as accidents are seldom sufficiently characteristic to determine the boundaries of a term. This leaves open two distinct ways of defining a term: First, by naming the properties or properties and accidents only; second, by stating the differentiÆ only. The former kind is the so-called descriptive definition, while the latter is the logical.

(2) A descriptive definition of a term is a description of its nature by means of its properties and accidents.

(3) A logical definition of a term is a description of its nature by means of its differentiÆ.

THE THREE KINDS OF DEFINITIONS ILLUSTRATED AND COMPARED.

Etymological Definition of Trigon.

A trigon is a figure of three corners.

Descriptive:

A trigon is a figure which has three sides and three angles, the sum of the latter being equal to two right angles.

Logical:

A trigon is a polygon of three angles.

It is seen that an etymological definition is simply a root-word analysis. In the case of trigon, the prefix comes from the Greek, meaning three, while the root-word comes from the Greek meaning corner.

The descriptive definition of trigon names the properties, “three sides and three angles” (differentiÆ) and the accident, “the sum of the angles of which equals two right angles.”

The logical definition of trigon simply states the proximate genus, “polygon,” and the differentia, “three angles.”

6. WHEN THE THREE KINDS OF DEFINITIONS ARE SERVICEABLE.

The etymological definition is helpful in furnishing a cue for remembering the descriptive and logical definitions. It also leads to precision of expression—the right word in the right place. Here is where the knowledge of a foreign language, particularly Latin, is helpful.

The descriptive definition is best adapted to the child-mind. Children think in the large; are not given to hair-splitting discriminations, and, therefore, many characteristic marks must be mentioned in order to insure a mastery of the content. With children the logical definition is often too brief to be clear. For example, it is easy to see which of the following definitions would be better adapted to the child-mind. Logical: Asquare is an equilateral rectangle. Descriptive: Asquare is a figure of four equal sides and four right angles.

The logical definition may be introduced to the student of the secondary school.

Few exercises are better adapted to the development of powers of discrimination and precision than practice in defining logically the common terms of every-day life. For example: “Abook is a pack of paper-sheets bound together.” “Achair is a piece of furniture with back and seat, designed for the seating of one person.” “Alead pencil is a cylindrical writing implement with lead through the center.” “Adoor is an obstacle designed to swing in and out to open and close an entrance.” “An eraser is an implement made to rub out written or printed characters.”

These definitions, coming from training school students, are not above criticism, yet they illustrate the point in hand.

7. THE RULES OF LOGICAL DEFINITION.

Five rules summarize the requirements to which a logical definition must conform.

FIRST RULE.

A logical definition should state the essential attributes of the species defined.

This means that a logical definition should contain the species, the proximate genus and the differentia. As the terms species, genus and differentia have been explained, it will be sufficient to briefly illustrate this rule.

Logical According to the First Rule.

Illogical According to the First Rule.

The Foregoing Illogical Definitions Made Logical.

SECOND RULE.

A logical definition should be exactly equivalent to the species defined.

This means that the species must equal the genus plus the differentia or the subject and predicate of the definition must be co-extensive—of the same bigness. The subject must refer to the same number of objects as the predicate.

A man upon the witness stand makes the declaration that he will testify to the truth, the whole truth and nothing but the truth. Alogical definition must contain the species, the whole species and nothing but the species. If the definition does not include all the species, it is too narrow; while on the other hand, if it includes other species of the genus it is too broad.

An excellent test of this second requirement is to interchange subject and predicate. If the interchanged proposition means the same as the original then the conditions have been met. To illustrate: Original—A trigon is a polygon of three angles. Interchanged—A polygon of three angles is a trigon.

The very best way of making the definition conform to this rule is to put to oneself these three questions: 1.Does it include all of the species? 2.Does it exclude all other species of the genus? 3.Has it any unnecessary marks?

To exemplify: Let us ask the three questions relative to the following logical definitions:

(1) A parallelogram is a quadrilateral whose opposite sides are parallel.

(2) A bird is a biped with feathers.

Questions:

(1) Does the definition include all the parallelograms? Yes. Does it exclude all other quadrilaterals? Yes. Are there any unnecessary marks? No.

(2) Does it include all birds? Yes. Does it exclude all other bipeds? Yes. Any unnecessary marks? No.

Illogical According to the Second Rule.

(1) A man is a vertebrate animal.

(Too broad. Does not exclude other species of the genus, such as horses, dogs, etc.)

(2) A barn is a building where horses are kept.

(Too narrow. Does not include all of the species, such as cow barn.)

(3) An equilateral triangle is a triangle all of whose sides and angles are equal.

(Equal angles is an unnecessary mark.)

The Foregoing Definitions Made Logical.

(1) A man is a rational biped. (Proximate genus.)

(2) A barn is a building where horses and cattle are kept and hay and grain are stored.

(3) An equilateral triangle is a triangle all of whose sides are equal.

THIRD RULE.

A definition must not repeat the name to be defined nor contain any synonym of it.

A violation of this rule is known as “a circle in defining” (circulus in definiendo).

There are some exceptions to this rule, as in the case of compound words and a species which takes its name from its proximate genus. To say that a hobby-horse is a horse, or that an equilateral triangle is a triangle, is not only allowable but necessary, that the proximate genus may be used.

The following definitions are illogical according to the third rule:

(1) A teacher is one who teaches.

(2) Life is the sum of the vital functions.

(3) A sensation is that which comes to the mind through the senses.

FOURTH RULE.

A definition must not be expressed in obscure, figurative or ambiguous language.

A violation of this rule is referred to in logic as “defining the unknown by the still more unknown” (ignotum per ignotius).

It is known that the purpose of definition is to make clear some obscure term, consequently unless every word used is understood the chief aim of the definition has been defeated.

From this it must not be inferred that all definitions should be free from technical terms. Such a restriction would make the defining of many terms unsatisfactory and in a few cases practically impossible. To the student of evolution the following definition by Spencer is intelligible while to the uninitiated it would appear obscure: “Evolution is a continuous change from an indefinite, incoherent homogeneity to a definite coherent heterogeneity through successive differentiations and integrations.”

This rule insists upon simple language when it is possible to use such in giving an accurate and comprehensive meaning to the term defined.

Illogical Definitions According to the Fourth Rule.

(1) “A net is something which is reticulated and decussated, with interstices between the intersections.” Dr.Johnson.

(2) “Thought is only a cognition of the necessary relations of our concepts.”

(3) “The soul is the entelechy, or first form of an organized body which has potential life.” Aristotle.

FIFTH RULE.

When possible the definition must be affirmative rather than negative.

The fact that there are a considerable number of terms which admit of a negative definition only, takes from the force of this rule. Such terms as deafness, inexpressible, infidel and the like can best be defined negatively.

It likewise happens that when words are used in pairs it is expedient to define one affirmatively and the other negatively. Recall, for example, the definitions of relative and absolute terms: “Arelative term is one which needs another term to make its meaning clear.” “An absolute term is one which does not need another term to make its meaning clear.”

Illogical Definitions According to the Fifth Rule.

(1) A gentleman is a man who is not rude.

(2) An element is a substance which is not a compound.

(3) An univocal term is a term which does not have more than one meaning.

8. TERMS WHICH CANNOT BE DEFINED LOGICALLY.

A logical definition insists upon a proximate genus and differentia. But as there is no genus higher than the highest genus (summum genus) then surely such cannot be defined logically. The words being and thing illustrate terms of this class. Moreover, it is impossible to give a satisfactory definition of an individual (infima species) as no attributes can be mentioned which will distinguish definitely and permanently the individual from others of the class. We may perceive the attributes but not those that are possessed solely by the individual. To say that Abraham Lincoln was a man who was simple and honest is not a definition, as other men have had the same characteristics.

Again there are a few terms such as life, death, time and space which cannot be defined satisfactorily. These terms seem to be in a class by themselves or of their own genus (sui generis).

Since a definition of a term is a brief explanation of it by means of its attributes, it follows that collective terms and terms standing for a single attribute are incapable of definition. Such terms as group, pain, attribute, belong to this class.

We may say, then, that there are some terms too high, some too low and some too peculiar to come within the province of logical definition. In short, “summum genus,” “infima species” and “sui generis” are incapable of definition.

9. DEFINITIONS OF COMMON EDUCATIONAL TERMS.

10. OUTLINE.

DEFINITION.

THREE KINDS ILLUSTRATED AND COMPARED.

11. SUMMARY.

(1) To be logical one must acquire the habit of accurate definition.

This topic ought to appeal strongly to the school teacher, who should above all others make his work stand for clearness, pointedness and continuity.

(2) A predicable is a term which can be affirmed or predicated of any subject.

The five predicables are Genus, Species, Differentia, Property and Accident.

(3) A definition of a term is a statement of its meaning by enumerating its characteristic attributes.

(4) Definitions explain a term intensionally, while logical division explains a term extensionally.

(5) There are three kinds of definitions: (1)etymological, (2)descriptive, (3)logical.

An etymological definition is based upon the derivation of the term; a descriptive definition states the characteristic properties and accidents of a term, while a logical definition is simply a statement of the differentia of a term.

(6) The etymological definition leads to precision of expression, the descriptive definition is best adapted to the child-mind, while the logical definition belongs to the realm of secondary education.

(7) Five rules summarize the requirements to which a logical definition must conform. In a word or two these five rules are: Every logical definition must (1)state the genus and differentia, (2)be equivalent to the species defined, (3)not repeat the name to be defined, (4)not be expressed in obscure language, (5)commonly be affirmative.

(8) Some terms are too high (summum genus), some too low (infima species), some too peculiar (sui generis) to come within the province of logical definition.