Chapter 5: The Extension and Intension of Terms

=1. TWO-FOLD FUNCTION OF CONNOTATIVE TERMS. (See page 52.)=

It has been indicated that a connotative term is one which possesses the double function of signifying a subject as well as an attribute. It may be observed here that an attribute of a notion is any mark, property or characteristic of that notion. Attribute, then, represents quality, relation or quantity. By a subject is meant anything which possesses attributes. Most subjects stand for objects and most attributes are qualities; consequently, for the sake of simplicity, we may use subject and object interchangeably; likewise, attribute and quality.

A connotative term, therefore, denotes an object at the same time it implies a quality. To illustrate: The symbol _man_ stands for the various individual men of the world, such as Lincoln, Washington, Alfred the Great, etc., or for certain qualities like rationality, power of speech and power of locomotion. The connotative term _teacher_ may be used to _denote_ Socrates, Pestalozzi, Thomas Arnold, or _connote_ such qualities as ability to instruct, sympathy, and scholarship. The term _planet_ stands for such objects as Venus, Earth, and Mars, and for such qualities as rotation upon axis, revolution about sun, and opaque or semi-opaque bodies. In each of the three illustrations the term is employed in the two-fold sense of denoting objects and of implying qualities.

=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 _ap_plies,” while “intension consists of the properties which the term _im_plies.”

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.=

_The Term._ _Extensional Use._ _Intensional Use._

{ roots, branches, trunk.
tree. maple, oak, beech. { or
{ woody-fiber, sap, bark.

house. stone, brick, cement. foundation, frame-work,
roof.

dog. shepherd, fox terrier, carnivorous, quadruped,
bull. propensity to bark.

book. textbook, dictionary, cover, leaves, binding.
encyclopaedia.

quadrilateral. trapezium, trapezoid, four sides, four angles,
parallelogram. limited plane.

logic. theoretical logic, science of thinking, art of
applied logic, right thinking, treats of
educational logic. laws of thought.

star. Sirius, Arcturus, Vega. heavenly body, gives light
and heat, twinkles.

force. gravitation, molecular, { produces motion
atomic. { changes motion
{ destroys motion.

term. general, singular, word or group of words,
non-connotative. definite idea.

government. monarchy, aristocracy, body of people,
democracy. established form of law,
banded together for
mutual protection.

bird. crow, robin, pigeon. biped, feathered, winged.

=5. OTHER FORMS OF EXPRESSION FOR EXTENSION AND INTENSION.=

_Extension._ _Intension._
comprehension content
extent intent
breadth depth
denotation connotation
application implication

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

extension } { intension
or } and { or
denotation } { connotation

=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:

I.

_Intensional_ _Extensional_

(1) four sides }
(2) parallel sides } common qualities of { (1) squares
(3) equal sides }
(4) right angles }

(1) four sides }
(2) parallel sides } common qualities of { (1) squares
(3) equal sides } { (2) rhombs

{ (1) squares
(1) four sides } common qualities of { (2) rhombs
(2) parallel sides } { (3) rectangles
{ (4) rhomboids

{ (1) squares
{ (2) rhombs
(1) four sides } common qualities of { (3) rectangles
{ (4) rhomboids
{ (5) trapezoids
{ (6) trapeziums

II.

{ (1) nebulae
{ (2) fixed stars
(1) heavenly body } common qualities of { (3) sun
{ (4) comets
{ (5) meteors
{ (6) moon

{ (1) nebulae
(1) heavenly body } common qualities of { (2) fixed stars
(2) self-luminous } { (3) sun
{ (4) comets

(1) heavenly body } { (1) nebulae
(2) self-luminous } common qualities of { (2) fixed stars
(3) fixed } { (3) sun

(1) heavenly body }
(2) self-luminous } common qualities of { (1) nebulae
(3) fixed } { (2) fixed stars
(4) twinkle }

(1) heavenly body }
(2) self-luminous }
(3) fixed } common qualities of { (1) nebulae
(4) twinkle }
(5) foggy }

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:

_Extension_ _Intension_
―― barn ――――――――
―――― building ――――――
―――――― structure ――――
―――――――― object ――

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:

Illustration: ( ‡ Intension and Extension )

“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.

1. Two-fold Function of Connotative Terms.

2. Extension and Intension Defined.

3. Extended Comparison of Extension and Intension.

4. A List of Connotative Terms used in Extension and Intension.

5. Other Forms of Expression for Extension and Intension.

6. Law of Variation in Extension and Intension.

6a. Two Important Facts in the Law of Variation.

6b. The Law of Variation Diagrammatically Illustrated.

=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. A term 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.

{ Russian
European { Englishman
{ Scotchman

{ lily
flower { rose
{ pansy

{ singular
term { distributive
{ collective

{ Truth has no extension.
truth { Since it refers to a
{ quality only, it is
{ non-connotative.

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

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

{ two feet
bird { ability to fly
{ feathers

{ four feet
quadruped { back bone
{ hairy covering

{ heading
letter { body
{ complimentary close

{ John has no intension.
John { Since it refers to
{ an object only, it is
{ non-connotative.

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.

_Intension_ _Extension_

binding } {
leaves } {
cover } { school arithmetic
printed matter } {
designed for instruction } {
instruction in arithmetic } {

binding } {
leaves } { school arithmetic
cover } { school grammar
printed matter } { school speller, etc.
designed for instruction } {

binding } { school arithmetic
leaves } { school grammar
cover } { school speller, etc.
printed matter } { encyclopaedia
} { novel

} { school arithmetic
binding } { school grammar
leaves } { school speller, etc.
cover } { encyclopaedia
} { novel
} { note book

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.=

1. What is a connotative term? Illustrate.

2. Which is the broader term, quality or attribute? Why?

3. When is a term used in extension?

4. Use the term triangle in intension.

5. As an aid to using a term in extension or intension what questions
may one ask himself?

6. By asking these questions use the term _clock_ in both extension
and intension.

7. By experimenting with the qualities of a rectangle show that as
the intension is decreased the extension is increased.

8. Write a list of five connotative terms. Prove that they are
connotative by illustrating their extension and intension.

9. The term metal { denotes } such qualities as element, metallic
{ connotes }
lustre, conductor of heat and electricity. In the foregoing which
of the two words following the brace should be used? Give reasons.

10. State the law of variation in two ways.

11. As one studies the law of variation what two facts are especially
evident? Explain fully.

12. For the purpose of illustrating the law of variation form a series
of which _desk_ is a member. Draw and name the triangles.

=11. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=

1. Which is the broader term, subject or object? Prove it.

2. If a term like _Caesar_ is given extension does it become a
general term? Why?

3. Using “man” as a member of each, arrange at least three different
series.

4. Why may it be said that a connotative term is _two_ dimensional?

5. Is there a word which has a broader extension than “being”? Why?

6. Prove that _youth_ has less intension than _human being_.

7. Devise a series of words in which the variation is proportional.

8. Advance arguments supporting the hypothesis that the term _John_
has neither extension nor intension.

9. Suggest arguments to prove that “George Washington” has both
extension and intension.