Further Analysis of the Concept

In connection with the predicables and definition, the term “essence has a more restricted sense than it often had in previous chapters. We have often used “essence” in a broad sense, in which it signifies “what a thing is” in any way whatsoever. According to the broad meaning of “essence,” if we grasp what a thing is, no matter how vaguely or indeterminately, even if we grasp it only as a vague and indeterminate “something”—or as “something big,” something colored, “something far away,” and so on—we grasp its essence or quiddity. For instance, if we know a horse only as “something brown, we know what the horse is at least insofar as it is something brown; hence, we grasp its essence in this broad sense of the word.

But in connection with the predicables and definition we use “essence” in the strict or proper sense, in which it includes only the basic intelligible elements of the comprehension of a concept but does not include the derived elements. For instance, the essence, in this sense, of “triangle” includes “figure bounded by three straight lines meeting by twos on a plane” but does not include “enclosing three interior angles.” The three lines of a triangle are looked upon as prior to the three angles and more basic than they. If you think of the three lines meeting by twos at three points on a plane, you think of what a triangle is primarily and necessarily; as a consequence of having this essence—as a consequence, that is, of the three lines meeting by twos at three points on a plane—a triangle has three interior angles. We do not think of the three angles as the reason why a triangle has three straight lines as sides, but of the three straight lines as the reason why it has three angles.

Similarly, the essence (in the strict sense) of “man” is “rational animal” rather than “speaking animal” or “tool-using animal,” although the notes “speaking” and “tool-using” suffice to differentiate man from all other kinds of animals. The reason for this is that “rational animal” is looked upon as prior to, and more basic than, “speaking animal,” “tool-using animal,” and so on. Man is not a rational animal because he can speak and use tools; rather, he can speak and use tools because he is a rational animal. Speaking and using tools may be reasons for our knowing that he is a rational animal—since we can reason from man’s properties and activities to his essence—but they are not the reason for his being a rational animal.

Our explanation of universals, of the predicables, and of definition will throw further light on the nature of essence.

In previous chapters we often used the word “universal” as synonymous with “distributed” and as opposed to “particular” and “singular.” A term and concept are universal in this sense if they stand for each of their inferiors (that is, if they stand for each of the subjects that they can be applied to). We shall now use the word “universal,” not as synonymous with “distributed,” but as synonymous with one meaning of “abstract” (except that “universal,” unlike “abstract,” is used both as a noun and an adjective).

Abstraction, as we saw in Chapter 11, consists in considering one aspect of a thing while omitting other aspects. Now a UNIVERSAL, as we understand the word in the present chapter, is a concept that expresses the essence (quiddity, or nature) of many really distinct individual subjects but leaves their differences entirely unexpressed. It expresses just an essence, and nothing else. The concept “man,” for instance, is a universal; and what is signified by “man” is such an essence. The essence “man” is found simultaneously in Peter, Paul, John, and Mary—Peter is a man, Paul is a man, John is a man, and Mary is a man (at least in the sense that she is a subject having a human nature). Peter, Paul, John, and Mary are four (many) completely distinct individual subjects—Peter is not Paul, Paul is not John, and so on. They not only differ from one another in being each a different individual but by innumerable other differences as well. They differ, for instance, in sex, age, height, and weight; in ability, virtue, and attainments; in race, place of residence, and so¬ cial status. “Man” expresses the essence (nature, or quiddity) of each of them insofar as they are rational animals and nothing more, and leaves innumerable common attributes, together with all their individual differences, entirely unexpressed. The concept “man,” then, is universal, because it expresses only what can exist concretely and at the same time in each of many distinct individual subjects; and, however much these subjects differ from one another in other respects, the essence “man” can be predicated of each of them in exactly the same sense.

Universals, as understood here, must be distinguished from transcendental concepts such as “being,” “good,” and “true.” Transcen¬ dental concepts are similar to universals in that what they signify can be realized concretely and at the same time in each of many subjects—there are, for instance, many beings, many good things, and so on. Transcendental concepts differ from universals, however, in that they express the differences among things, as well as their similarities, and consequently are never predicated of two things in exactly the same sense.

Let us compare the way the transcendental concept “being” is predicated of Peter, Paul, John, and Mary with the way “man” is predicated of them. We have already seen that “man” has exactly the same sense when it is predicated of each of them, although they differ from one another in age, sex, height, weight, and so on. These differences are attributes of men, but none of these differences is man, and consequently none of these-differences is expressed by the concept “man.” The concept of “being,” however, expresses all that is in its inferiors; it expresses not only the notes in which they are similar to one another, but also their differences. These differences are being—since they are something—and consequently they are expressed by the concept “being.” Now, since things actually are different from one another, and since “being” expresses their differences, the meaning of “being” shifts according to the differences among things.

From this it is clear that “being” is analogous in relation to its inferiors, whereas “man” is univocal.

The distinction between DIRECT and REFLEX UNIVERSALS is of importance in logic and in other branches of philosophy as well. It is a special case of the more general division of concepts into first and second intentions.

A DIRECT UNIVERSAL is a concept signifying an absolute essence—that is, an essence (nature, or quiddity) as such, abstracted from all individualizing conditions, and considered without reference to the mode of existence it has either in the mind or in things. You grasp the absolute essence “man,” for instance, if you think of “man” and nothing else—not of Peter or Mary, or this man or that man, or some man or all men, but just “man”—completely prescinding from all orders of existence and expressing only the comprehension of the essence “man” without any consideration of its extension.

Nothing but individual, fully determined natures can exist in the real order. Consequently, direct universals, or absolute essences, cannot exist as such except in the mind. Nevertheless, all that a di¬ rect universal signifies can exist in the real order; all that it signifies has been abstracted from things and can therefore be realized in things and predicated of them.

A direct universal, as we have seen, is considered without reference to its inferiors. It can be referred to its inferiors, but its mean¬ ing is independent of any such reference. Suppose, now, that we go one step further and actually refer it to its inferiors; suppose we consider an essence as capable of being realized in many subjects and of being predicated of them. What we grasp then is a reflex universal. A REFLEX UNIVERSAL, therefore, is an essence, nature, or quiddity, considered with reference to the individuals in which it is verified and to its potential predicability. It includes not only the comprehension of an essence but also its multiplicability and its predicability (that is, its extension).

Before studying the predicables it will be helpful to review pages 17-19 on comprehension, on the distinction between basic and derived notes, on the meaning of absolute extension, and on the inverse ratio of comprehension and extension. To a great extent, the treatment of the predicables is nothing but an elaboration of these notions.

The predicables are a classification of reflex universals based on the five ways in which they express the nature of subjects of which they are predicated. They are listed as species, genus, specific difference, logical property, and logical accident. These names primarily signify the relationships of universals to their inferiors, or the five ways in which they are used as predicates; but these names also signify the universals themselves. Thus, we not only say that “man” is predicated of John as his species, but also that “man” is his species.

The predicables are a classification of reflex universals, since we are here viewing universals with reference to their inferiors (with reference, that is, to the subjects in which the essence they signify can be realized and of which they can be predicated); we are considering not only their comprehension, but their extension and predicability as well.

First we shall give a general survey of the predicables, defining each of them very briefly and explaining why there are five predicables and only five; then we shall give a detailed explanation of each of them.

a. General Survey

The following analysis contains brief definitions of each of the predicables and shows why there are exactly five predicables:

Every universal predicate expresses the nature of the subject of which it is predicated in one of the following ways. Either it expresses its essence (in the strict sense), or else it does not. If it expresses the essence of the subject, it either expresses all the basic constitutive notes (that is, the fully determined essence) and is predicated of the subject as its (1) SPECIES; or else it expresses only some of the basic constitutive notes (that is, the partially determined essence). If it expresses only some of the basic constitutive notes of the essence, it expresses either a determinable constitutive element and is predicated of the subject as its (2) GENUS; or else it expresses the determining constitutive element that distinguishes the essence from other essences belonging to the same genus and is predicated of the subject as its (3) SPECIFIC DIFFERENCE. Suppose, now, that the universal predicate does not express the essence of the subject it is predicated of. Then it either expresses an attribute that belongs to the subject necessarily (and convertibly) and is predicated of the subject as a (4) PROPERTY; or else it expresses an attribute that does not belong to the subject necessarily but only contingently and is predicated of the subject as c (5) LOGICAL ACCIDENT.

The following schema displays the structure of this analysis and sets off certain key words that will help us understand and remember the definitions of each of the predicables. This schema also emphasizes two very important distinctions—(a) the distinction between the notes that constitute an essence and those that do not constitute it but merely accompany it and (b) the distinction between the first four of the predicables, which are necessary to the subject, and the fifth predicable, which is not necessary to it. An understanding of these distinctions is an absolutely necessary prerequisite to an understanding of definition and of the very important division of propositions into necessary propositions and contingent propositions.

b. Detailed Explanation of Each Predicable

We shall prefix a schema of the category of substance to our detailed explanation of each of the predicables. This will help us see the relationship of species, genus, and difference among one another and understand various distinctions made in the definitions given below.

All the predicables are universals. Consequently all of them ex¬ press the nature of many really distinct individuals and can be predicated of each of these individuals in exactly the same sense. This is taken for granted in the following definitions.

1) SPECIES. A species, in the strict and proper sense, is a universal that expresses the completely determined essence of its inferiors and only that. It gives a complete answer to the question, What is a thing essentially? but does not express attributes by which individuals having the same essence differ from one another.

The words “completely determined essence” must be understood correctly. They do not refer to the determinations that an essence, or nature, must have in order to exist in the real order. They refer, rather, to the basic notes involved in the intelligible structure of various kinds of things. The concept “man,” for instance, is a species in relation to all individual men. It expresses only what a thing has to be in order to be thought of as a man, but does not express the innumerable attributes-such as nationality, parentage, place of birth, place of residence, sex, height, weight, age, and so on-that belong to a man because he is this or that man rather than because he is simply man.

A species in the strict and proper sense is predicated directly of individuals but (unlike a genus) only of individuals that do not differ from one another essentially. Between a species in the strict sense and the individuals of which it is predicated no other species in the strict sense can intervene. No species, for instance, intervenes between “man” and Peter, Paul, James, Mary, and so on.

In a broader sense, each genus beneath the supreme genus of a category (for instance, beneath “substance”) is called a species in relation to the genus immediately above it. Thus, “animal,” which is the genus of the species “man” is itself a species in relation to the genus “organism.” Subdivisions of universals (both of genera and species) based on the presence or absence of accidental attributes are likewise called species in a broad sense. In this sense, “green triangle” and “non-green triangle” are species of “triangle.”

Up to the present, we have considered species only from the point of view of comprehension. We can also consider it from the point of view of extension. Viewed under this aspect, “species,” both in the strict and in a broader sense, is defined as a class of things that is a sub-division of a broader class (that is, of a genus).

Since “species” includes both “genus” and “difference” as its constitutive elements, our understanding of it will be more perfect after we have studied them.

2) GENUS. A genus is a universal that expresses the incompletely determined essence of its inferiors, giving an incomplete answer to the question. What is a thing essentially? A genus expresses the essence of its inferiors so indeterminately that what it signifies can be predicated of things that differ from one another specifically (of things, that is, that belong to different species). Thus, “animal,” which is the genus of “man,” expresses the nature of man so indeterminately that what it signifies can be predicated not only of men, but also of dogs, horses, elephants, whales, and so on, which differ from man specifically. “Organism,” again, expresses the nature of “animal” so indeterminately that it can be predicated alike of animals and plants.

Considered from the point of view of extension, “genus” is defined as a broader class made up of narrower classes (that is, of more than one species).

A genus must be either a supreme genus, an intermediate genus, or a lowest genus. A supreme genus is one that is not a subdivision of any other genus (for instance, “substance”). A lowest genus, or genus infimum, has no other genus intervening between it and a species in the strict sense (for instance, “animal,” in relation to “man”). An intermediate genus is one that intervenes between a supreme genus and a lowest genus (“body” and “organism”). (Refer to the schema on p. 245.)

From another point of view, genus is divided into proximate and remote. A proximate genus is directly above a species or another genus. Thus, “animal” is the proximate genus of the species “man,” and “organism” is the proximate genus of the lower genus “animal.” (“Proximate” is derived from proximus, the Latin word for “nearest” or “closest.”) A remote genus is one that has one or more other genera intervening between it and the species or lower genus in relation to which it is being considered. Thus, “organism,” “body,” and “substance” are remote genera of “man.” (See the schema on p. 245.)

Note that what a genus signifies cannot exist in the real order unless it has essential attributes that are not expressed by the genus. “Triangle,” for instance, is the genus of “equilateral triangle,” “isosceles triangle,” and “scalene triangle.” Now a triangle that is only triangle and nothing more, cannot exist; if a triangle is to exist, it must be either equilateral, isosceles, or scalene. The relationship of a genus with a species is not like that of “barn” with “painted barn.” A barn can exist without paint and without any other determination in place of paint. Hence, “barn” is not the genus—at least not in the strict sense—of “painted barn.”

3) DIFFERENCE. A difference is a universal that expresses the constitutive note that distinguishes a species from its genus or a genus from a higher genus.

A specific difference is a difference that distinguishes a species from its proximate genus, as “rational” distinguishes “man” from animal.

A generic difference is a difference that distinguishes a species or a lower genus from things that differ from it generically but belong to the same higher genus. Thus, “sentient,” “animate,” and “corporeal” are generic differences of “man,” since each of them distinguishes a genus that man belongs to from the next higher genus.

We must not think that a difference is added to a genus as paint is added to a barn or cream to coffee. We do not have an “animal’ in us that is somehow or other overlaid with “rational,” so that the two of them together make the “man” in us, as “barn” plus “paint” makes a painted barn. There is a real distinction between a barn and the paint on it, as well as between coffee and cream. A barn and paint—as well as coffee and cream—are different things. We do not just think of them, the one without the other, but they are two distinct things independently of our thought. There is, however, no real distinction between a genus and logical difference but only a mental distinction. A genus does not express the nature of one real thing, and a difference the nature of another real thing; but both express the nature of the same real thing—the genus and difference together expressing it more completely than does the genus alone. For instance, when we say “John is a rational animal,” we give a more complete answer to the question, What is John? than when we say “John is an animal,” since we not only state that he is an animal but also what kind of animal he is. But we do not imply that one part of John—say his head—is rational and the rest animal but not rational, or that what is signified by “rational” is mixed with what is signified by “animal,” as cream is mixed with coffee. The very same concrete reality that is signified by “animal” is likewise signified by “rational animal,” and vice versa. This reality (“John”) is signified by each of them whole and entire; but its nature, or intelligible structure, is expressed more determinately and completely by “rational animal” than by “animal” alone.

Let us consider another example. The genus of “equilateral triangle” is “triangle” and its difference is “equal-sided.” We can think of the number of the sides without thinking of their equality. Hence, there is a mental distinction between the two. Nevertheless the equal sides are really the same as the three sides, and vice versa.

4) LOGICAL PROPERTY. In an earlier chapter, when we treated of the comprehension of terms and concepts, we distinguished between the basic, constitutive notes of an essence and the notes that are implied in and deducible from these. The basic, constitutive notes, as we have seen, are the genus and specific difference of an essence. The notes that are implied in and deducible from the genus or specific difference, or from both of them together, are called properties. Thus, “animal” and “rational” are the genus and specific difference of man; “capable of speech,” “social,” “risible,” “tool¬ using,” and so on, are properties.

A logical property is a universal that expresses, not the essence of the subject of which it is predicated, but an attribute that accompanies this essence necessarily. A property in the strictest sense is connected with its subject conceptually—in such a way, that is, that its subject cannot even be thought of, without contradiction, as not having the property. In this sense, “three-angled” is a property of “triangle.” The essence of triangle, as we explained above, is “figure bounded by three straight lines meeting by twos in a plane. Be¬ cause its three straight lines meet by twos in a plane, a triangle must likewise have three interior angles. A triangle without three interior angles involves a contradiction and is therefore absolutely impossible. In other words, the very notion of a triangle requires that a triangle have three interior angles, and a triangle without three interior angles not only cannot exist but cannot even be thought of.

We shall refer to the necessity arising from such a conceptual connection as logical necessity.

A logical specific property (in the strictest sense) expresses an attribute that results with logical necessity from the completely determined essence of the subject of which it is predicated. It arises from the combined genus and specific difference but not from the genus alone. It is therefore realized concretely in every individual having the specific nature and only in such individuals. Capable of speech,” “social,” “risible,” and so on, are specific properties of man.

A generic property (in the strictest sense) expresses an attribute that results with logical necessity from the incompletely determined essence of the subject of which it is predicated. It arises from the basic, constitutive elements that one species has in common with other species (that is, from its generic nature). “Mortal,” for instance, is a generic property of man. “Mortality” flows from man’s nature insofar as man is a living organism; mortality is not distinctive of man but common to all living organisms. The grounds, in other words, for the possibility of dying in both animals and plants is in their generic nature as organisms. Since organisms consist of parts, they can be resolved into them; and such resolution is death. Similarly, to be “three-angled” is a generic property of the species “equilateral triangle.” An equilateral triangle is not three-angled because it is equilateral but because it is a triangle—that is, because of its generic nature of “triangle.” It shares the attribute of “three¬ angled” with “isosceles triangle” and “scalene triangle,” the other species of the genus “triangle.”

We often hear the term “physical property” used with reference to the various chemical elements and compounds and to the various kinds of animals and plants. A physical property is an attribute that belongs to its subject with physical necessity but is not conceptually connected with it—or at least, if it is conceptually connected with it, we are unable to see the connection.

Physical properties are both specific and generic. Atomic weight, atomic number, boiling point, and so on, are specific properties of the various chemical elements and compounds. To be subject to the law of gravitation is a generic property of all species of bodies. To be white is a generic property of common salt—common salt has to be white, but there are many other things that also are white. Inasmuch as physical properties are not conceptually connected with the subjects of which they are predicated, they are not logical properties in the strict sense but logical accidents.

The term “property” is also applied to attributes that are realized concretely in individuals of only one species (or genus), but not in every individual or at all times. To write poetry, to play the piano, to give speeches, and so on, are properties of man in this sense. Only men do these things but not all men; and even the men who do them are not doing them all the time. Note that the basic ability of a man to do these things is a specific property in the strict sense; but the actual doing of them is a property only in a broader sense, since a man can be thought of, without contradiction, as not doing them.

5) LOGICAL ACCIDENT. A logical accident is an attribute that is not conceptually connected with the essence of the subject that it is predicated of. It is compatible with the subject—for otherwise it could not be predicated of it at all—but the subject can be thought of, without contradiction, as not having the attribute. To be white, standing, an American, six feet tall, and so on, are logical accidents in relation to the essence “man.” A man can be white, standing, an American, six feet tall, and so on; but a man can be thought of, without contradiction, as not being any of these.

The schema on Page 251 throws further light on the nature of the predicables. Pay special attention to the questions. They suggest the key words in the definitions of each of the predicables. Notice that rational” is not the specific difference of John, but of the kind of being he is. Similarly, “risible,” “able to speak,” and so on, are not properties of John as an individual but rather of the kind of being he is (that is, of “man”).

When you do the exercises, remember that the same universal can be classified variously as it is considered in relation to various subjects. “Animal,” for instance, is a genus in relation to “man” but a logical accident in relation to “organism” (since an organism can be thought of, without contradiction, as not being an animal—a plant, for instance). Consequently, a question like “To which of the predicables is animal to be referred?” does not make sense unless you are told the subject in relation to which you are to consider the concept “animal.”

A brief treatment of the Aristotelian categories, or predicaments, will complete our background for the study of definition and logical division. It will do this by clarifying the notion of “genus” and by deepening our understanding of the law of the inverse ratio of comprehension and extension, showing how we abstract from differences as we rise from individual beings through species and genera to a supreme genus and how we unify our knowledge of things by referring them to a minimum number of univocal concepts. A study of the categories will likewise throw light on the function of the copula bv revealing how its meaning varies according to variations in the meaning of the predicate. A final fruit of our study of the categories will be an acquaintance with certain very important philosophical terms which we simply must understand in order to read Aristotelian or Thomistic philosophy intelligently.

First we shall treat of the categories insofar as they express modes of being; then we shall treat of them insofar as they are orderly classifications of individuals, species, and subgenera under a supreme genus—which is the special point of view from which logic considers them.

a. The Categories as Expressing Modes of Being

The categories are a classification of predicates, each of which expresses some mode of being of its subject while omitting other modes of being. There are ten categories: substance and the nine accidents. Our first step in explaining the nature of each of the categories will be to list various predicates, or modes of being, of John; then we shall indicate the category illustrated by each of these predicates, adding in parentheses either the question to which each category gives the ultimate and logically irreducible answer or else the key words of a descriptive definition of the category. Finally we shall give brief definitions of substance and accident and of the first three of the accidents—quantity, quality, and relation. The treatment of the categories as expressing modes of being belongs to metaphysics, not to logic; still, since these modes of being are the intelligibilities expressed by the supreme genus of every category, logic cannot ignore them entirely.

Let us now examine the following illustrations of substance and the nine accidents: “John is: (1)> (2), (3), and so on,” on Page 255.

Each of these predicates or groups of predicates declares what (or how) John is but each in a different way. The first predicates (“man,” “animal,” and “organism”) declare what he is substantially; the others declare what he is accidentally. John is necessarily, permanently, and constitutively a man; he cannot exist at all without being a man or without having “man” as predicable of him. But many of the accidents express a perfection that he may have at one time and not have at another. Although John cannot exist without having at least some accidents predicable of him, the way in which he has many of them will vary from day to day and moment to moment; once sick, he is now healthy; once shorter than James, he is now taller; although he must be in some place, he is not necessarily in the street; and so on.

SUBSTANCE is that which exists in itself and for itself, without requiring another being as a subject of inherence. A man, a tree, and an angel are substances, since they exist in and for themselves and not as mere modifications or further perfections of a subject in which they inhere. However, a smile, a wink, and a thought are not substances (but accidents) since their nature is such that they cannot exist except as perfections or modifications of a subject that has existence directly in itself and for itself (that is, of a substance).

A concrete individual thing is called FIRST SUBSTANCE. It cannot be a predicate in the strict and proper sense but is the ultimate subject of all perfections and of all predication. Universal concepts belonging to the category of substance are called SECOND SUBSTANCES; all of these can be predicates, and all but the highest can likewise be subjects.

A predicamental ACCIDENT (that is, any of the accidents of the categories or predicaments) expresses being—or, more accurately, a perfection or mode of being—that cannot exist in and for itself but whose nature is such that it must exist in another as in a subject of inherence. Ultimately every accident must exist in a substance. Accident presupposes substance and cannot even be defined without bringing the notion of substance into its definition. A smile, height, weight, skill, power, place, superiority, inferiority, and similarity are examples of accidents.

We must carefully distinguish the kind of accident we are treating here from the kind we treated in the previous section. The accidents of the categories are first intentions, expressing the nature of things according to their own proper being—what they signify can be predicated of things themselves and not just of concepts— whereas the accidents of the predicables are second intentions. Now, when we reflect on some of the accidents of the categories, considering them in relation to the subjects whose predicates they are, we find that they are also logical accidents because they can be absent from their subject without destroying its nature. However, many accidents of the categories are referred to the predicable of logical property. For instance, man’s intellect is an accident inasmuch as it is a perfection inhering in man’s soul; however, to have an intellect is a logical property of the concept “man,” since it is a necessary consequent of his essence “rational animal.”

Notice that the notion “accident” (like “being”) is predicated analogously, not univocally, of the various kinds of accidents, and is therefore not their supreme genus. The reason for this is that the differences among the various accidents, as well as their similarities, are formally “being that inheres in another as in a subject.”

We shall now give very brief definitions of the three most important kinds of real accidents: quantity, quality, and relation. The other accidents will be understood sufficiently for our present purpose from the examples and the words in the parentheses in the schema given above.

QUANTITY is that by which a material substance has parts outside of parts and on account of which it is big or small, long or short, thick or thin, and so on. A line, a surface, and a solid are continuous quantity; a number is discrete quantity. QUALITY is the accident that characterizes, from within, a na¬ ture that is already substantially complete. RELATION is the order that one thing has towards another. Notice that an existing thing (for instance, a man) is a substance but has quantity, quality, relations, and so on.

b. The Notion of a Logical Category

Now that we have considered the categories as expressing modes of being—that is, from the point of view of metaphysics—we are ready to consider them as orderly classifications of concepts and from the point of view of logic. To prepare ourselves for the definition of “category” we shall build up the category of substance; and then, with an eye on the arrangement of genera, subgenera, species, and individuals as displayed in our schema of this category, we shall give a descriptive definition of category itself.

To build up the category of substance, let us ask the question, What is John essentially? Let us first give the narrowest answer to this question (“man”), then the next broader answer (“animal”), and so on, until we work our way up to the broadest answer short of “being” (which is a transcendental concept and therefore pervades all the categories, without being limited to any one of them). We shall place the word “John” at bottom of our schema and the narrowest answer (“man”) directly above the word “John” but beneath the other answers. We shall now place the broadest answer (“substance”) on top and the other answers in between, arranging all of them in the order of increasing extension and decreasing comprehension. This gives us the series of terms directly above “John” in the schema given on Page 258.

Our next step will be to add the difference that narrows down each genus to the genus below it, beginning at the top and working down, until we come to the species “man” and, under it, to the enumeration of individuals (Peter, Paul, John, and so on). To the right of these differences we shall add terms (“incorporeal,” “inanimate,” and so on) that include all the differences that might narrow down each of the genera to lower genera or species. Now we have the fully worked out category of substance. Every substance, except God, can be fitted into this schema.

With an eye on this schema of the category of substance, we shall now give a descriptive definition of “category.” A category, as the word is usually understood in logic, is an orderly classification of genera, species, and individuals under a supreme genus, all of them so arranged that each of the universal terms can be predicated univocally and essentially of everything under it. For instance, “substance” is predicated univocally and essentially of “body,” “organism,” “animal,” and “man,” of Peter, Paul, and so on, as well as of every animal, every plant, and every body. Similarly, “body” is predicated univocally and essentially of everything under it; and so on. Sometimes, especially in metaphysics, the name “category” signifies only the supreme genus, without explicitly including its inferiors.

The supreme genus, then (not only of substance but of each category), gives the ultimate and logically irreducible answer to the question, What is a thing essentially and in the last analysis? Asked what John is essentially, you reply “A man.” And that? “An animal.” And that? “An organism.” And so on, until you come to “substance,” which is at once the broadest and the simplest concept that can be predicated univocally of John and other beings. Or asked what white is, you reply “A color.” And that? “A kind of quality.” Shortness and bulkiness are essentially and in the last analysis quantities; standing is a situs; and so on.

The category of substance has been worked out in much greater detail than any of the other categories. Above (on Page 259) we have indicated a possible arrangement of the category of quantity. All quantities can be fitted into this schema. Many of the headings can be further divided—for instance “line” can be divided into “straight line” and “curved line”; “triangle” can be divided into “isosceles triangle,” “equilateral triangle,” and “scalene triangle,” and so on.

A definition is a statement that gives the meaning of a term. The word “definition” is derived from the Latin word definire, which means “to enclose within limits.” Originally, definire meant “to mark boundaries or limits” as of a field. Later it came to be applied to the act of stating the meaning of a term. The boundary of a field is defined by indicating the limits within which a field is confined and by which it is marked off from other fields; similarly, a term is defined by indicating the limits within which it is used and by which it is marked off from other terms.

“Definition” signifies the act of defining, as well as the finished statement in which the meaning of a term is given.

Notice that a definition is not a proposition but a term, generally a complex term. Thus, the definition of “man” is not “Man is a rational animal,” but simply “rational animal.” A definition is generally expressed by the predicate of a proposition whose subject is the term or thing to be defined. Often, too, a definition is expressed by a formula such as “ ‘Man’ means ‘rational animal’,” or “The definition of ‘man’ is ‘rational animal’.” First we shall treat of the kinds of definition—of nominal and real definitions and of various subdivisions of each—and then of the rules governing definition.

a. Kinds of Definitions

Does a definition merely indicate what thing is signified by a term, without declaring the nature of that thing, or does it also declare its nature? The answer to this question is the basis for the very important division of definitions into nominal and real.

1) NOMINAL DEFINITION. A nominal definition (definitio nominis, “definition of a name”) merely indicates what thing is signified by a term, without declaring the nature of that thing. Its purpose is merely to give the meaning of a term—either its current meaning or a special meaning within some context—so that hearers or readers will know the sense in which the term is used. If you are asked “What does anthropos mean?” and you answer “Anthropos means ‘man’,” you are giving a nominal definition. By substituting a word that your questioner knows for the one that he does not know, you indicate for him what thing is signified by the word anthropos. However, if you are then asked “But what’s a man?,” you will be expected to give a real definition (supposing, of course, that the person asking the question knows English).

Nominal definitions are commonly given at the beginning of debates to ensure agreement among the disputants as to the exact point at issue. Nominal definitions, again, are used to call attention to the equivocal use of terms. To show that a term is used equivocally, you need only show that it signifies a different kind of thing in each of (at least) two occurrences; there is no need of declaring what the essences of these things are. Nominal definitions are also used to introduce new terms or to indicate a special sense in which a speaker or writer intends to use a term that is often used in other senses.

On the basis of the various ways in which they indicate the thing signified by a term, nominal definitions are either etymological definitions, definitions by synonym, definitions by description, or definitions by example.

a) Etymological Definition. An etymological definition defines a word by giving the meaning of the word or words from which it is derived. “Philosophy” is derived from philos, a Greek word meaning “loving,” and sophia, which means “wisdom.” Consequently, the etymological definition of “philosophy” is “love of wisdom. The etymological definition of “martyr” is “witness.” The meaning of many words is so different from the meaning of their parent words that giving their etymology throws little or no light on their present meanings and is therefore really not a definition at all. For instance, the etymological meaning of “scruple” is “small sharp rock,” which is far removed indeed from the uneasiness of conscience that the word “scruple” signifies today.

b) Definition by Synonym. A nominal definition can be made by giving a synonym (either of the same language as the word to be defined or of a different language) that is better known than the word to be defined; for instance, anthropos means “man,” and “to confect” means “to put together.”

c) Definition by Description. A nominal definition can be made by describing the thing signified by the term, not for the purpose of revealing its nature, or essence, but merely to indicate what thing it is that is being spoken of. “Chalk,” for instance, can be defined by this sort of definition as “the material of which is made the little stick that a teacher uses to write on the blackboard.”

d) Definition by Example. A nominal definition can also be made by indicating an example of the thing signified by the term to be defined. For instance, if a child asks “What is chalk?” and you show him a piece of chalk and say “This is chalk,” you are giving him a nominal definition by example.

Notice that many nominal definitions are, to some extent, also real definitions. The purpose of the person giving the definition sometimes determines whether a definition is to be regarded as nominal or real. If his purpose is merely to indicate what thing is signified by a term, a definition is to be regarded as nominal even if it throws some light on the nature of the thing. On the other hand, if the person giving a definition assumes that his hearers or readers already know what things are signified by a term and intends to declare the nature, or essence, of those things, his definition is to be regarded as a real definition, or a definitio rei (“a definition of the thing”).

Except for words that are better known than any words we could use to define them, all words can be defined by nominal definitions.

2) REAL DEFINITION. A real definition (definitio rei, “definition of a thing”) not only indicates what thing is signified by a term but also declares the nature of that thing. It manifests the intelligible structure of the thing by explicitly setting before the mind various notes that are expressed only obscurely by the term that is to be defined.

A real definition is always a complex term, consisting of at least two parts—one part giving the note that the thing has in common with similar kinds of things; the other, the note that differentiates it from them. For instance, in the definition of man as a rational animal, the term “animal” (man’s proximate genus) expresses what man has in common with the kinds of things that resemble him most closely, and the term “rational” (man’s specific difference) expresses the note that differentiates him from them.

Limits of Real Definition. Only common terms—and not all of them—can be defined by real definitions.

Individuals are identified rather than defined. Thus, if you say “Mary is the girl in the front row nearest the window,” you are not defining Mary but merely pointing her out. Again, if you say “John is a rational animal,” you are defining the nature he possesses, but not John himself.

Transcendental concepts (such as “being,” “thing,” “something,” and so on) cannot be defined by a strict definition but can only be described. Such concepts transcend the limits of all genera and ex¬ press the differences among things as well as their similarities. The transcendental are studied at length in metaphysics. To explain them at length now would take us too far afield.

The supreme genera of the categories (“substance,” “quantity,” “quality,” “relation,” and so on) are also incapable of definition in the strict sense. Since they are supreme, there is no higher genus to which they can be referred.

Simple qualities that are the immediate data of sense experience do not admit of definition in the strict sense, but can be known (properly) only by direct experience. If you do not already know what red is, you will still not know what it is if you are told that it is a color that manifests itself by vibrations varying from 350 to 500 billions per second. No description can convey the meaning of taste to one who never had taste buds, or of light to one who was always blind.

a) Definition by Genus and Specific Difference. The notion of real definition is verified most perfectly in a definition that defines by stating a thing’s genus and specific difference. These, as we saw when we treated of the predicables, are the notes or intelligibilities that constitute the intelligible structure or essence of a thing, the genus giving an incomplete answer to the question. What is a thing essentially? and the specific difference indicating the note or intelligibility that distinguishes the thing to be defined from other things of the same genus. The definition of “man” as “rational animal is such a definition, “animal” expressing man’s genus and "rational’ expressing man’s specific difference.

A definition by genus and specific difference in the strict sense is the ideally perfect type of definition. It is the briefest, fullest, and most precise answer to the question, “What is that (kind of) thing? In the first place, it expresses the notes that are the grounds for the other attributes of a thing. We saw this when we studied genus, specific difference, and species and the relationship of properties to them. The grounds, for instance, of man’s ability to speak, to use tools, to laugh, and so on, lie in his being a rational animal. A man is not a rational animal because he can speak, use tools, and so on; rather, he can speak, use tools, and so on, because he is a rational animal. In the second place, a definition by genus and specific difference is perfectly convertible with the thing defined. For instance, a thing that is a man but not a rational animal, or a rational animal but not a man, would involve a contradiction. Some of the other kinds of definition, however, are not perfectly convertible with the thing defined. Take the definition of “man” as a “two-handed, featherless biped that holds its head erect,” which is not a definition by genus and specific difference but a descriptive definition. There is no conceptual repugnance—no contradiction—in the thought of a man that is not a two-handed, featherless biped holding its head erect, or of a two-handed, featherless biped holding its head erect that is not a man. This will be clearer after we have taken the other kinds of real definitions.

Only a few things can be defined by stating their genus and specific difference in the strict sense. Still, this type of definition is very important because it is a model that we should imitate as closely as possible in other types of definitions.

Notice that “genus” and “specific difference” are frequently used in a broad sense in which “genus” signifies whatever a thing has in common with other kinds of things that resemble it most closely and “specific difference” signifies whatever differentiates a thing from other kinds of things. Since “genus” and “specific difference” constitute the essence of a thing, a definition made by stating a thing’s genus or specific difference is often called an essential definition.

b) Definition by Substitutes for Genus and Specific Difference. Most definitions define by stating the genus or quasi genus of a thing together with some substitute for a specific difference in the strict sense. This substitute may be a description or a cause, as explained below. As opposed to definition by genus and specific difference in the strict sense, these are called nonessential definitions.

1—Descriptive Definitions. A descriptive definition states the genus or quasi genus of the thing defined and uses a description in place of the specific difference. The following classification is based on the various relationships that nonessential attributes have towards the subjects of which they are predicated.

a-Descriptive Definition by Properties. Some descriptive definitions give one or more logical properties in place of the specific difference. The definition of man as animal able to speak is this kind of definition.

Like the definition by genus and specific difference in the strict sense, this type of definition is perfectly convertible with the thing defined. Just as every possible man is an animal able to speak (that is, has the basic powers requisite for speaking-although they may be undeveloped or impeded), so too every possible animal that is able to speak is a man.

b—Descriptive Definition by Logical Accidents. Some descriptive definitions express the genus or quasi genus of the thing defined and state one or more logical accidents in place of the specific difference.

(1) A definition that states a single physically necessary and characteristic property (which, as we saw in the chapter on the predicables, is a logical accident) in place of the specific difference enables us to distinguish a thing from all other kinds of things. “Iron,” for instance, is defined by this type of definition as a metal whose atomic number is 26 and whose atomic weight is 55.85. So far as we know, all iron and only iron has this atomic number and atomic weight; still, we see no intrinsic necessity why iron must have this atomic number and atomic weight or why no other metal can have them.

(2) As a substitute for the specific difference, a definition may state a combination of noncharacteristic properties that are, as a matter of fact, found only in the thing defined. The definition of “man” as a “two-handed, featherless biped that holds its head erect is this sort of definition. As a matter of fact, all men—at least almost all men—and only men are two-handed, featherless bipeds that hold their heads erect; still, there could be men in whom this definition is not verified, and there could be other beings in which it is verified.

(3) As a substitute for the specific difference, a definition can give logical accidents that are found either singly or collectively only in the thing defined. For instance, an “elephant” is “the largest extant land animal.” As a matter of fact, an elephant is the largest extant land animal, but an elephant could cease being the largest extant land animal and still be an elephant.

Notice that all the definitions that define by stating logical accidents merely indicate subjects that have an essence; they do not unfold the essence itself. Hence, such definitions are often nominal definitions rather than real definitions.

2—Causal Definitions. A causal definition gives the genus or quasi genus of the thing defined and states a cause in place of the specific difference.

a—Definition by Final Cause. A definition by final cause, or a final definition, substitutes the final cause, or purpose, of a thing for its specific difference. Articles made by men are commonly defined by final definitions. A barometer, for instance, is an “instrument for determining atmospheric pressure and hence for judging probable changes of weather, for ascertaining the height of an ascent, and so on.” The definition of logic as the “science and art of correct thinking” is a final definition, since it defines logic by stating its proximate end or purpose.

b—Definition by Efficient Cause. A definition by efficient cause substitutes the efficient cause of a thing for its specific difference. An efficient cause produces its effect by its activity. Diseases are often defined by stating the kind of bacterium, germ, or parasite that brings them on. Malaria, for instance, can be defined as a febrile disease that is caused by animal parasites in the red blood corpuscles and is transferred to man by the bite of the anopheles mosquito.”

c—Definition by Material and Formal Cause. A material cause is the stuff out of which a thing is made, and a formal cause is that in a thing that makes it the kind of thing it is. Marble, for instance, is the material cause of a marble statue, and the shape given the statue by the sculptor is its formal cause. The marble, before this particular shape had been given it, could indifferently have become a pedestal, a column, a tombstone, and so on; but once this particular shape has been given it, it is a statue. Now in definition the material cause of a thing may be used as a quasi genus and its formal cause as a substitute for its specific difference.

The definition of man as a being composed of an organized body and a rational soul is this type of definition. In this definition organized body” is the quasi genus, as it expresses the nature of both man and the other animals; “rational soul is the substitute for the specific difference, as it indicates the principle within man that differentiates him from the other animals.

Material and formal causes are treated at length in metaphysics and cosmology. We mention them only in passing, as a thorough explanation would take us out of the field of logic.

d-Genetic Definition. A genetic definition defines a thing by stating the process by which it is produced (or is imagined to be produced) and the elements that concur in its production. Geometric figures are often defined by genetic definitions. Thus, a circle is a “figure formed by revolving a line in a plane around one of its ends.” An eclipse is an “obscuration of a heavenly body by the interposition of another.” Bread is an “article of food made from flour or meal by moistening, kneading, and baking.”

b. Rules Governing Definition

The following rules, which are traditional in treatises on logic, set forth the conditions that a wood definition should fulfill. They will help us distinguish good definitions from defective definitions. Notice that all that logic can do to help us in the construction and evaluation of definitions is to supply us with general norms. In order to make and evaluate definitions we must also have a sufficient knowledge of the terms, or things, to be defined.

A good definition must be (1) clear; (2) coextensive with the term or thing defined; (3) positive, when possible; and (4) brief. We shall take each of these conditions in turn.

Rule 1. A definition should be clear.

The purpose of nominal definition is to indicate the thing signified by a term; the purpose of real definition is to declare the nature of the thing signified by a term. This purpose cannot be attained unless a definition is clearer than the term or thing defined.

In order to attain clarity in definition, we must avoid metaphors, circular definition, and excessively difficult terminology.

First, a definition should not be expressed in metaphorical or figurative language. The following definitions (perhaps they should not be called definitions at all) are defective on this count: Sleep is the brother of death,” “Bread is the staff of life,” “Loyalty is the flame of the lamp of friendship,” and “A myth is the voice of a dreamer and an idealist crying Why cannot these things be?

Secondly, a definition should not be circular; that is, it should not be so phrased that you cannot understand it unless you already understand the term that is being defined.

This rule is violated most openly in a tautological definition, which contains a part, or a cognate form, of the term to be defined —as in the definition of a “governor” as “one who performs gubernatorial functions” and of a star as a stellar body.

This rule is violated more subtly by defining a term and then using the same term—generally a few pages later—to define words occurring in its definition. For instance, a writer might define “peace” as “the absence of war ” and then define “war” as “the absence of peace.”

Thirdly, in order to be clear, a definition should not contain excessively difficult terminology—as in Dr. Johnson’s humorous definition of “net” as “a reticulated fabric, decussated at regular intervals, with interstices and intersections.” This definition is also tautological, inasmuch as “reticulated” is simply another word for “netted.”

This rule does not forbid difficult terminology if the matter defined is such as to require it. The definition, for instance, of the soul as “the first entelechy of an organized body having the potency of life” is an accurate technical definition, whose meaning is obvious to anyone familiar with Aristotelian philosophy.

In deciding whether or not a definition (especially a nominal definition) is clear, you must, to some extent, keep in mind the condition of the person for whom the definition is intended. A definition of table salt that will be perfectly satisfactory to a four-year-old will not suit a chemist, and vice versa.

Rule 2. A definition should he coextensive with the term or thing defined.

A definition should set off, or distinguish, the thing defined from all other things. In order to do this, it must be applicable to every example of the thing defined and only to examples of the thing defined. In other words, the term that is defined and the definition should have identical extension and should be perfectly convertible. Hence, supposing that “rational animal” is a correct definition of “man,” then every possible man is a rational animal and every possible rational animal is a man.

Definitions by genus and specific difference and descriptive definitions that define by stating logical properties in the strict sense fulfill this rule most perfectly. Definitions by logical accidents fulfill it less perfectly, because there can be examples of the thing defined to which they do not apply and there can be other things to which they do apply. The definition, for instance, of “cod” as “an important food fish found especially in the Newfoundland Banks and along the New England and the Norwegian coasts” might fit other fish besides cod; and cod would still be cod even if it ceased being used for food, as well as if it ceased to be found in these areas.

This rule is violated by the definition of a “wolf” as “a sheep-killing animal.” This definition is at once too narrow and too broad. It is too narrow because it does not include all wolves, since there are wolves that do not kill sheep; it is too broad because it is applicable to animals other than wolves.

Rule 3. A definition should be positive, when possible.

A definition should state what a thing is rather than what it is not. Of course, negative terms can be defined only negatively. For instance, you cannot define “blindness” except as “the absence of sight in a subject that ought to have it,” or “death” except as “the cessation of life.”

Some things must be defined negatively on account of the limitations of human knowledge. We know so little about spirits, for instance, that we must define “spirit” negatively as “immaterial substance.”

This rule is violated by the definitions of “virtue” as “the absence of vice” and of “wisdom” as “the avoidance of folly.”

Rule 4. A definition should be brief.

A definition should contain no superfluous words. In the definition of “man” as “a rational, social, speaking, mortal animal,” the words “social,” “speaking,” and “mortal” are superfluous, since they are implied in, and deducible from, “rational animal."

The ideally perfect definition is not supposed to give us the fullest possible knowledge of the thing defined but merely to state its essence.

We shall add one caution that might be called a fifth rule. In definition, a thing must be referred to its own proper genus. The definition of “to thresh” as “when you beat out grain” is defective on this count inasmuch as it refers threshing to the category of time although it belongs to the category of action.

Definition, as we saw in the last chapter, manifests the comprehension of a concept. A definition (at least, a definition that defines by stating a thing’s genus and specific difference or its genus and a logical property) explicitly states various notes or intelligible elements that are contained implicitly in the concept it defines. Logical division, on the other hand, has to do with the extension of terms and concepts, and expresses the various kinds of inferiors in which a concept can be realized or of which a term can be predicated.

What we shall say about division is largely a development of what we said in Chapter 2 on the inverse ratio of comprehension and extension.

First, we shall treat of the notion of logical division; then, we shall treat of the rules governing logical division.

a. The Notion of Logical Division

Logical division is the resolution of a logical whole into its logical parts; that is, of a genus into its subgenera or species. Logical division is an answer to the question, In what kinds of subjects is a concept, essence, or quiddity, realized? The concept “triangle,” for instance, is realized in equilateral, isosceles, and scalene triangles which are the logical parts (or species) of the logical whole (or genus) triangle. The concept “animal” is realized in man and in brute, which are the logical parts of the logical whole “animal.”

Notice that the members into which a genus is divided are themselves universal. A species is not, properly speaking, divided into individuals.

The notions of “logical whole” and “logical part” will be clarified if we contrast the relationship of a logical whole to its logical parts with the relationship of a physical whole to its physical parts. We are more familiar with the latter kind of whole and parts; so we shall begin with them.

A physical whole cannot be predicated of its physical parts. Body and soul, for instance, are the physical essential parts of a man; the two of them unite into one substantial whole and constitute one man. But neither of them, taken by itself, is a man: a man’s body is not a man, and a man’s soul is not a man. Or take integral heterogeneous parts of a man—parts, that is, like a man’s head, trunk, arms, and legs. No single one of them is a man but only a part of a man. Or, again, take integral homogeneous parts of a thing—like the pieces into which a pie has been cut. No single piece of a pie is a whole pie.

A logical whole, on the contrary, can be predicated of each of its logical parts. “Man,” as we have seen, is a logical part of the logical whole “animal.” Still, every man is a whole animal and has the complete comprehension of “animal” realized in him in its entirety. Similarly, “equilateral triangle” is a logical part of “triangle.” Nevertheless, an equilateral triangle is a whole triangle, and all that “triangle” signifies is verified in it. In other words, the entire comprehension of a genus is expressed by each of its species; a species adds something to the comprehension of a genus, but does not take anything away from it.

Logical division is effected by removing the indeterminacy of a logical whole, or genus, in either of two ways.

First, the indeterminacy of a logical whole, or genus, can be removed by adding to its comprehension an attribute that is found in some, but not in all, of the inferiors of the genus. This method of division is illustrated by the division of “animal” into “man” and “brute” through the addition of “rational” and “irrational.” “Rational” removes the indeterminacy of the genus “animal” by the addition of a note, or thought element, that is obviously something positive. Note, however, that “irrational” also adds something positive to “animal,” even though it is expressed negatively. “Irrational” does not merely signify the absence of rationality; it also signifies the presence of some other attribute, whose exact nature we do not know or do not care to express, in place of rationality. Hence, “rational” and “irrational” are related to one another, not as contradictories, but as immediately opposed contraries within the genus animal.

This method of division, which is known as dichotomy (“a cutting into two”), is valid and useful if the added attribute actually serves as a basis for division. For instance, having webbed feet or not having them is a legitimate basis for the division of birds, since all birds are actually divided into those that have webbed feet and those that do not have them; but having or not having webbed feet is not a legitimate basis for dividing men, since all men would be included in one of the intended divisions, namely, among those who do not have webbed feet. These examples show that divisions cannot be made without a consideration of the special character of the matter, or thought content, of the concepts to be divided. Division, in other words, is not a purely formal process.

Secondly, division can be effected by removing the indeterminacy of a logical whole, or genus, by adding to the comprehension of the genus the various ways in which some attribute found in every member of a genus is realized in each of them. This method of division is exemplified by the division of “triangle” into equilateral, isosceles, and scalene triangles on the basis of the comparative length of their sides. If all three sides are of equal length, the triangle is equilateral; if two and only two are equal, the triangle is isosceles; and if no sides are equal, the triangle is scalene. In every triangle the sides must stand in some relationship with one another on the basis of the comparative length of their sides; but it does not spring from the nature of a triangle that its sides have any one of these three possible relationships rather than the others.

A division is expressed by the predicate of a disjunctive proposition in the strict sense, whose subject is the genus, or logical whole, that is divided. Thus, the division of triangles on the basis of the comparative length of their sides is expressed, “Triangles are either equilateral, isosceles, or scalene.” Sometimes, of course, the following formula is used, “Triangles are divided into equilateral, isosceles, and scalene.”

The logical parts of a logical whole are sometimes called its subjective parts. The reason for this is that they are subjects. In the first place, they are subjects in which a genus is looked upon as inhering, somewhat as an accident inheres in a substance. In the second place —and this is the principal reason—they can be the subject of a proposition whose predicate is the logical whole.

Subdivision consists in submitting the parts of a logical whole to another process of division. “Term,” for instance, can be divided into “univocal term,” “equivocal term,” and “analogous term. The last can be subdivided into terms that are analogous by analogy of proportionality and those that are analogous by analogy of attribution.

Co-division consists of more than one division of the same logical whole, each being made according to a different basis. Co-division is exemplified in the divisions of triangle on the basis of the relative length of the sides into equilateral, isosceles, and scalene triangles and on the basis of the size of the largest angle into obtuse-angled triangle (which has an angle of over ninety degrees), right-angled triangle (which has an angle of ninety degrees), and acute-angled triangle (all of whose angles are smaller than ninety degrees).

Notice that in connection with logical division, the terms “genus” and “species” are often used in a very broad sense. For instance, “barns” can be considered a quasi genus that is divided into “painted barns” and “unpainted barns” as quasi species.

Notice, too, that classification is the reverse of division. If you start with what is less general and work up to what is more general, you classify. If you begin with what is more general and work down to what is less general, you divide.

b. Rules Governing Logical Division

Four rules of division are traditionally given in treatises on logic. These rules aim at an ideal that, for practical reasons, cannot always be attained in actual divisions. We should observe them as perfectly as the matter allows.

Rule 1. A division should be made on a single basis.

The basis of a division is an attribute whose presence or absence, or whose modifications, in the various inferiors of a genus differentiate the inferiors from one another and thus divide the genus into species. Thus, on the basis of the presence of rationality or the absence of rationality (and the presence of something else in its place), the genus “animal” is divided into man and brute. On the basis of modifications in the comparative length of the sides, the genus “triangle” is divided into equilateral, isosceles, and scalene triangles. On the basis of modifications in the size of the largest angle, the genus “triangle” is divided into obtuse-angled, right-angled and acute-angled triangles.

Violation of this rule is known as cross division. You make a cross division if you divide triangles into equilateral, isosceles, and right-angled triangles, since you shift the basis of division from the comparative length of the sides to the size of the angles. You also make a cross division of Americans if you divide them into Republicans, Democrats, and Christians, since adherence to a political party is the basis for calling a man a Republican or Democrat, but religious belief is the basis for calling a man a Christian.

Rule 2. A division should be exhaustive.

A division is exhaustive if there is place for everything belonging to the genus in one or other of the members into which the genus has been divided. In other words, the members, when they are taken collectively, must be equal to the logical whole and coextensive with it. The division of Americans into Republicans and Democrats violates this rule. There are Americans who belong to neither of these parties.

Rule 3. The members of a division should be mutually exclusive.

This rule is violated when the members of a division overlap, so that something belonging to the genus can be referred to more than one of the members into which the genus has been divided.

Such overlapping takes place when a cross division is made, as in the division of Americans into Republicans, Democrats, and Christians. Some Americans are both Republicans and Christians; others are both Democrats and Christians; and so on. Such overlapping also takes place when a subordinate species is included in a series of coordinate species, as in the division of human beings into males, females and girls.

Rule 3 is a corollary of Rule 1 and of the rule that follows.

Rule 4. A genus should be divided into its proximate species.

Not only Rule 3 but also Rule 4 is violated in the division of human beings into males, females, and girls. “Human beings” should first be divided on the basis of sex into male and female, which are its proximate species or subclasses. Then “females” should be subdivided on the basis of age and maturity into women and girls.

Rule 4 is sometimes violated together with Rule 2, as in the division of rectilinear plane figures into equilateral triangles, squares, pentagons, hexagons, and so on. A complete list of the remote species of a genus is often difficult to make and there is great danger of omitting a species. This danger is eliminated if the genus is first divided into its proximate species and these are then subdivided. “Rectilinear plane figure,” for example, should first be divided on the basis of the number of sides into three-sided, four-sided, five¬ sided plane figures, and so on. Then these can be subdivided— triangle,” for instance, being subdivided into equilateral, isosceles, and scalene triangles.