A Class Room Logic / Deductive and Inductive, with Special Application to the Science and Art of Teaching

by George Hastings McNair

8. THE RELATION BETWEEN SUBJECT AND PREDICATE.

In Chapter 5 the extension and intension of terms was explained. The student recalls, for instance, that the term “man” may be used to denote objects, as “white man,” “black man,” “red man,” etc. In this sense the term “man” is used extensionally. But when made to stand for the attributes “rationality,” “power of speech,” etc., the term “man” is used intensionally.

In considering the relation between subject and predicate it is customary to employ the terms in an extensional sense only, since such a restriction serves the purpose of syllogistic reasoning and conversion.

Let us, then, give attention to the extension of the subject and predicate of the categorical propositions A, E, I, O.

(1) The Universal Affirmative or A Proposition.

All S is P symbolizes the A proposition. This may be interpreted as meaning that all of the subject belongs to a part of the predicate, or that all of the subject belongs to all of the predicate. The first interpretation is the usual one and may be illustrated by the following propositions:

1. “All men are mortal.”

2. “All trees grow.”

3. “All metals are elements.”

It is obvious that the subjects of these propositions include every specimen of the particular class mentioned. For example: The subject all men includes every specimen of the human family; all trees includes every object of that class; all metals covers everything which the scientist classes as such. In the three propositions, then, reference is made to the whole subject but to only a part of the predicate, as other beings beside men, such as the horse, are mortal; and other plants aside from trees, such as the sun flower, grow; other substances, namely oxygen, are elements.

For the sake of making the logical meaning of the four propositions clearer, recourse may be made to Euler’s diagrams, so named because the Swiss mathematician and logician, Leonhard Euler, first used them.

The first illustration of the A proposition, “All men are mortal,” may be represented by two circles, a larger circle standing for the predicate, mortal, and a smaller circle entirely inside the larger representing the subject, men. Thus:

FIG. 1.

It is evident that all of the smaller circle belongs to the larger. This diagram will then fit any proposition where it may be said that all of the subject belongs to a part of the predicate, or which may be symbolized as “All S is some P.” (All the subject is some of the predicate.)

The student knows that circles are plane surfaces and when such a statement as “All men are mortal” is given, reference is made to only that part of the “mortal” circle which is directly underneath the “men” circle. Nothing has been said relative to the remaining part of the “mortal” circle.

“A” propositions which may be interpreted as meaning “All S is all P” are called co-extensive A’s because the subject and predicate are exactly equal in extension. Such propositions are best illustrated by definitions; e. g.:

1. “A man is a rational biped.”

2. “A trigon is a polygon of three sides.”

3. “Teaching is the art of occasioning those activities which result in knowledge, power and skill.”

To represent the meaning of the co-extensive A by the Euler diagram, two circles of the same size may be drawn, one coinciding at every point with the other. If the first circle is drawn heavily in black and the second dotted in red, it will make clear to the eye that there are two circles.

(2) The Universal Negative or E Proposition.

“No S is P” best symbolizes the E proposition, though sometimes the universal negative is written “All S is not P.” This latter form, as has been explained, is ambiguous and therefore illogical.

“No S is P” surely means that no part of the subjectbelongs to any part of the predicate and no part of the predicate belongs to any part of the subject. The subject and predicate are mutually exclusive.

The following illustrate the E proposition:

1. “No man is immortal.”

2. “No true teacher works for money.”

3. “No thorough student can remain unwise.”

The E proposition may be represented by two circles, the one entirely without the other as in [Fig. 2]:

FIG. 2.

(3) The Particular Affirmative or I Proposition.

This may be symbolized as “Some S is P,” and considered as meaning that a part of the subject belongs to a part of the predicate. It has already been noted that “some” is ambiguous and that its logical signification is “some at least.” (It may be all or it may not be all.) For example, the only logical interpretation which can be placed on “Some men are wise” is, that the investigation has resulted in finding only a part of the man family wise. Whether or not all are wise is unknown as the entire field has not received attention. In no case can it be assumed that all the others are not wise.

The I proposition illustrated:

1. “Some men are wise.”

2. “Some animals are vertebrates.”

3. “Some teachers are inspiring.”

The meaning of the I proposition may be represented by two circles intersecting each other:

FIG. 3.

The significant feature of the diagram is the shaded part which represents a part of the “men” circle as belonging to a part of the “wise” circle. The unshaded part of each circle is the unknown field.

(4) The Particular Negative or O Proposition.

The common symbolization of the O is “Some S is not P.” Put in statement form: Some of the subject is excluded from the whole of the predicate.Here, as in the I, the same logical import must be given to some; e. g., in the proposition, “Some men are not wise,” our knowledge is confined to the group who are not wise. Whether or not the others are wise or not-wise is unknown.

Illustrations of the O proposition:

1. “Some men are not wise.”

2. “Some laws are not just.”

3. “Some novels are not helpful.”

The significance of the O proposition may be shown by two intersecting circles as in [Fig. 4]:

FIG. 4.

A similar diagram represents the I proposition, the only difference being in the part shaded. In the O proposition the investigated field is all of the “men” circle outside of the “wise” circle, while in the I proposition the known field is that part of the “men” circle inside the “wise” circle.

In comparing the four diagrams the student will note that the affirmative propositions are inclusive, while the negative propositions are exclusive.

(5) The Distribution of Subject and Predicate.

A term is said to be distributed when it is referred to as a definite whole.

In the proposition, “All men are mortal,” the subject all men is considered as a whole. “All men” stands for every specimen of the human race; not a single one has been left out. Again, the whole is definite; any one, if he were given the time and opportunity, could ascertain by actual count just how many “all men” represented.

It should be observed that if the word definite is not incorporated in the definition of a distributed term, thereis afforded an opportunity for error. The attending illustrations will make this clear:

1. “All the students except John and James are dismissed.”

2. “All the students except John, James, etc., are dismissed.”

The subject of the first proposition is distributed, while the subject of the second is undistributed. Reasons: The first subject, “All the students except John and James,” is referred to as a whole and that whole is definite, therefore, it is distributed; the second subject, “All the students except John, James, etc.,” is referred to as a whole, but as the whole is not definite, the term is not distributed. Because all is the quantity sign of the second subject the casual observer might easily be misled in designating it as a distributed term.

Here it may be well to explain that when reference is made to subject or predicate the logical subject or predicate is meant. Unless this is constantly kept in mind error results; e. g., in the proposition, “All white men are Caucasians,” the logical subject is “white men,” not “men.” If the subject were “men,” it would be undistributed, as the whole of the man family is not considered, but the actual subject, being “white men,” is distributed because the predicate refers to all white men.

Recurring to the illustration, “All men are mortal,” we have concluded that the subject “all men” is distributed. The predicate, “mortal,” however, is undistributed, as reference is made to it only in part; i. e., there are other beings aside from men that are mortal, such as “trees,”“horses,” “dogs,” etc. In all A propositions of the type of “all men are mortal,” the subject is distributed while the predicate is undistributed. This relation is clearly shown by the diagrammatical illustration, [Fig. 1]. Here all of the “men” circle is identical with only a part of the “mortal” circle. In other words, the whole of the “men” circle is considered, while reference is made to only a part of the “mortal” circle.

In the case of the co-extensive A both subject and predicate are distributed. Relative to the co-extensive “All men are rational animals,” it could likewise be said that “all rational animals are men,” or that “all men are all of the rational animals.” Reference is thus made to all of the definite predicate as well as to all of the definite subject.

In the E propositions, such as “No men are immortal,” the whole of the subject is excluded from the whole of the predicate. This makes evident the fact that both terms are distributed. See [Fig. 2].

The I proposition, such as “Some men are wise,” concerns itself with only a part of the subject and only a part of the predicate, consequently neither subject nor predicate is distributed. This relation is verified by the representation, [Fig. 3].

In the O proposition the subject is undistributed, while the predicate is distributed. For example, in the proposition, “Some men are not wise,” “some men” would indicate that only a part of the logical subject is under consideration. But the predicate is distributed because “some men” is denied of the whole of the predicate,“wise.” This may become clear by studying [Fig. 4]. Here all of the shaded part which stands for the subject, “some men,” is excluded from the whole of the “wise” circle. But all of the shaded part is only a part of the entire “men” circle, consequently the subject which the shaded part represents (some men) is undistributed. The predicate, “wise,” however, is distributed, as the subject is excluded from every part of it. It is well to remember that not, when used with the copula, distributes the predicate which follows it.

If the student is to succeed in testing the value of arguments, he must ever have “at the tip of his tongue” his knowledge of the distribution of the terms of the four logical propositions. With this in view the following schemes are offered:

I.
	Subject	Predicate
A	distributed	undistributed
E	distributed	distributed
I	undistributed	undistributed
O	undistributed	distributed
II.
A	distributed	undistributed
O	undistributed	distributed
E	distributed	distributed
I	undistributed	undistributed
III.
A	All S is P
E	No S is P
I	Some S is P
O	Some S is not P

Referring to scheme II it may be observed that A and O contradict each other; i. e., where A is distributed O is undistributed and vice versa. A similar relation exists between E and I.

In scheme III the underline under the symbol indicates the term which is distributed.

IV. As a fourth scheme a “key word” might be adopted. Any of these three might be used: (1) saepeo, or (2) asebinop, or (3) uaesneop. The significance of “saepeo” is this: “s” stands for subject distributed, “p” for predicate distributed, “a” “e” “o” for the logical propositions where any distribution occurs. Putting the letters together gives this: subject distributed of propositions A and E, predicate distributed of propositions E and O.

Similarly, “asebinop” stands for this: “as,” a distributes its subject; “eb” e distributes both; “in,” i distributes neither; “op,” o distributes the predicate.

In the coined word “uaesneop” appear six letters which compose “saepeo,” and the letters have the same significance. The two additional letters, u and n, stand for universal and negative. The interpretation of the entire word, therefore, is this: “uaes,” the universals a and e distribute their subjects; neop, the negatives e and o distribute their predicates.

It seems to me that the last word is the most helpful as it emphasizes the two facts which are the most used; namely, (1) Only the universals distribute their subjects; (2) Only the negatives distribute their predicates.

If the student will visualize “uaesneop” so thoroughly as never to forget it, he will not experience difficulty indetermining the distribution of the terms of the four logical propositions.