2. EXTENSION AND INTENSION DEFINED.

This double function of connotative terms furnishes an important topic for the student of logic—the Extension and Intension of Terms. In short, some authorities claim that to master the extension and intension of terms is virtually to master the entire subject of logic. Though this position may be an exaggerated one, yet it tends to emphasize the importance of the topic.

A term is used in extension when it is employed with reference to the objects for which the term stands.

When the term triangle is used to refer to the objects isosceles triangle, scalene triangle, right triangle, it is employed in extension.

A term is used in intension when it is employed with reference to the attributes for which the term stands.

The term triangle is employed in intension when we use it to refer to the qualities, three sided and three angled.

3. EXTENDED COMPARISON OF EXTENSION AND INTENSION.

A connotative term seems to be two dimensional—it has extent or length and intent or depth.

“Extension consists of the things to which the term applies,” while “intension consists of the properties which the term implies.”

Extension is quantitative, while intension is qualitative. An extensional use means to point out or number objects, while an intensional use means to describe by naming qualities. To name is to use a term in extension—to describe is to use a term in intension.

To divide a term into its kinds we must regard it in an extensional sense; e.g., the term man may be divided into Caucasian, Mongolian, Malay, Ethiopian, American Indian.

To define a term we must regard it in an intensional sense; e.g., man is a rational animal.

Etymologically considered extension means to stretch out, intension, to stretch within. To use a term extensionally one must look out. To use a term intensionally one must look in.

In attempting to use a term in extension we may ask ourselves the question, “What are the kinds?” or “To what objects may the term be applied?” While if we would use a term in intension the question should be, “What does it mean?” or “What are the qualities?” Let us, for example, use the term metal in the two senses, first in extension, second in intension. Question: To what objects may the term metal be applied? Answer: Metal may be applied to the objects silver, gold and iron. Thus has metal been employed in extension.

Question: What are the qualities of metal? Answer: The qualities are element, metallic lustre, good conductor of heat and electricity. Thus has metal been used in intension.

NOTE. Since an attribute is anything which belongs to a subject, then the parts of a subject must be classed as attributes. Hence, a term is used intensionally when reference is made to its parts.

4. A LIST OF CONNOTATIVE TERMS USED IN EXTENSION AND INTENSION.

5. OTHER FORMS OF EXPRESSION FOR EXTENSION AND INTENSION.

Formerly the words extension and intension were applied to concepts while denotation and connotation were applied to terms representing the concepts, but now the words are interchangeable. Denotation, the noun, and denote, the verb, signify, etymologically, a marking off. To denote is to mark off or indicate the objects or classes of objects for which the term stands. Connotation, the noun, and connote, the verb, signify to mark along with. To connote is to mark along with the object, its attributes.

The terms which should be remembered are

6. LAW OF VARIATION IN EXTENSION AND INTENSION.

It has been noted that the intension of a term has reference to its qualities while extension considers its application to various objects. It may be wise to experiment with the extension and the intension of certain terms as types with a view of ascertaining how the two ideas are related to each other. For the sake of definiteness let us make use of the following scheme:

In considering the first illustration we observe that as the number of qualities is decreased, the number of objects increases. While in the second example as the qualities are increased, the number of objects decreases. It would appear from this that the intension and extension of a term are inversely related to each other. As the one increases the other decreases and vice versa. It is customary to state this relation in the form of a law known as the law of variation. “As the intension of a term is increased its extension is decreased and vice versa,” or the extension and intension of a term vary in an inverse ratio to each other. To further illustrate: this book refers to a large number of objects; add to the qualities of book those of text book and the application is much reduced. In other words as we increase the intension, the extension is diminished. Increase the intension further by adding the quality English text book and the extension becomes still less.

6a. TWO IMPORTANT FACTS IN THE LAW OF VARIATION.

In studying the law of variation two facts are especially evident. (1)The law applies only to a series of terms representing notions of the same family. The extension and intension of “text book,” for example, could not be compared with the extension and intension of “house” as they belong to a different class of words, the genus of text book being book, while the genus of house is building.

To illustrate the law of variation, determine upon any class name, then think of its proximate genus (the next higher-up class to which it belongs). Continue this till the series is sufficiently complete to illustrate the law. Or proceed in the opposite direction. That is, after selecting the class name think of the next lower term in the class and thus continue till series is complete. Illustration: The class name man is determined upon; the proximate genus of man is biped, the proximate genus of biped is animal, and so on. Or thinking downward: a proximate species of man is white man, of white man. European, etc.

Thus the series:

animal
biped
man
white man
European

(2) As a second fact: the increase and decrease is not a mathematical one. That is, by doubling the extension the intension is not halved. Or if the intension is decreased by one quality the extension is not necessarily increased by one object. Thus “man” stands for one billion seven hundred million beings or objects. Decrease the intension of “man” by the one quality of rationality and the extension would include all bipeds—many billion objects.

6b. THE LAW OF VARIATION DIAGRAMMATICALLY ILLUSTRATED.

In a general way lines may be used to represent the variation in extension and intension. For example: we may let a line an inch long represent the extension of man, one two inches long represent the extension of biped, three inches long represent the extension of animal, etc. While on the other hand, if a line an inch long represents the intension of man, a line one-half inch long may be used to represent the intension of biped, one a quarter of an inch long to represent the intension of animal, etc. The following illustrates this scheme in connection with another series of words:

In the foregoing scheme building refers to a greater number of objects than barn, hence the line under extension representing building should be longer than the line for barn. Likewise structure, referring to a greater number of objects than building, is represented by a longer line. Thus when the series is viewed from top to bottom a gradual increase in extension is noted. Giving attention to the intensional use of the series we note that building has fewer qualities than barn, structure fewer than building and object fewer than structure. Therefore, from top to bottom, the intension of the terms gradually decreases.

The variation may be made still more apparent if triangles are used, one triangle being placed upon the other, vertex to base, like the following:

“Biped” is written near the base or in the broadest part of the extension triangle because it denotes the greatest number of objects, and is, therefore, broadest in extension. “Man” occupies a narrower part of the extension triangle because it refers to fewer objects or is narrower in extension than “biped.” “Arnold” occupies the narrowest part of the extension triangle because it is the narrowest in extension. On the other hand “Arnold” occupies the broadest part of the intension triangle because intensionally it possesses more qualities than the others, while “biped,” having the least depth in intension or possessing the fewest qualities, occupies the narrowest portion of the intension triangle.

7. OUTLINE.

THE EXTENSION AND INTENSION OF TERMS.

8. SUMMARY.

1. Connotative terms are used in a two-fold sense: first, to denote objects; second, to imply qualities.

2. A term is used in extension when it is employed with reference to the objects for which the term stands. Aterm is used in intension when it is employed with reference to the qualities for which the term stands.

3. The answer to either of the following questions will lead one to use any term in extension: First, what are the kinds? or second, to what objects may the term be applied?

The answer to either of the following questions will lead to the use of any term in intension: First, what does it mean? or second, what are the qualities?

4. To illustrate extension and intension it is best to use the class-names in every day speech.

5. The word denotation is commonly used for extension and connotation for intension.

6. “As the intension of a term is increased its extension is decreased and vice versa,” is a statement of the Law of Variation in the extension and intension of terms.

6a. The law of variation applies only to a series of terms representing notions of the same class or family, the words being arranged in a species-genus order. The increase and decrease of the extension and intension of a series is not proportional.

6b. The law of variation is best explained by using two triangles, one super-imposed upon the other vertex to base and base to vertex.

9. ILLUSTRATIVE EXERCISES.

1a. Employ the following terms in extension—European, flower, term, truth.

1b. Employ the following in extension—grain, rock, soil, precious stone.

2a. Use intensionally bird, quadruped, letter, John.

2b. Use the following in intension—word, table, purity, government.

3a. The use of a term in extension follows when attempting to answer two questions: First, what are the kinds? Second, to what objects may the term be applied? Make application of this with reference to the term man.

1. What are the kinds of men? Caucasian, Malay, Mongolian, Ethiopian, Redman.

2. To what objects does the term man refer? George Washington, Chas. Hughes, John Smith.

In both 1 and 2 the word man is used to denote objects, hence it is employed in extension.

3b. Use the term vegetable in extension by answering the two questions in 3a.

4a. Decrease one by one the qualities of some common object with a view of noting how when the intension is decreased the extension is increased.

4b. With a view of noting how when the intension is decreased the extension is increased, decrease one by one the common qualities of peach tree.

5a. In the following series what word could be substituted for “mammal” and why? Being, organized being, animal, vertebrate, mammal. Answer: Fish, reptile, or bird; because there are at least seven classes of animals which belong to the vertebrate family, any one of which could be used to complete the series.

5b. Form a series of which “Baldwin apple” has the narrowest extension. What terms may be substituted for “Baldwin apple?”

6a. In a series of which “pupil” is a member show that the increase and decrease is not proportional. The series: logic pupil, pupil, youth, human being, being. In decreasing the intension of “logic pupil” by dropping the one quality, logic, the extension is made larger by many more than one, as “pupil” represents many more objects than “logic pupil.” Therefore, the increase is not in proportion to the decrease.

6b. In a series in which “ruler” appears, show that the increase and decrease is not proportional.

7. From the following list select the proper words of the series; arrange them; draw and name the triangles: Caesar, brute, man, Roman, American, biped, sensuous being, animal, individual.

10. REVIEW QUESTIONS.

11. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.