2. IMMEDIATE AND MEDIATE INFERENCE.

It has been noted that a truth may be derived from a consideration of one or two antecedent judgments. To illustrate further: From the judgment, “All men are fallible,” we may derive the conclusion that “No men are infallible”; or, from the two judgments, “All men are fallible,” and “Socrates was a man,” we may readily infer that “Socrates was fallible.” These two modes of inference take the names of immediate inference and mediate inference. Let us express these two kinds in equation form:

Giving attention to the antecedent judgments of the second argument it is noted that the terms “f” and “S” are referred to the common term “m.” In logic this common term is known as the middle term. As there is but one antecedent judgment in the first argument there can be no common or middle term. The first argument is an illustration of immediate inference; the second of mediate inference. This suggests the definitions:

Immediate inference is inference without the use of a middle term.

Mediate inference is inference by means of a middle term.

3. THE FORMS OF IMMEDIATE INFERENCE.

Many logicians recognize four forms of immediate inference. These four forms are (1)opposition, (2)obversion, (3)conversion, (4) contraversion.8

(1) IMMEDIATE INFERENCE BY OPPOSITION.

We have learned that to be logical all categorical assertions must be reduced to some one of the four propositions, A, E, I, O. If these four logical propositions be given the same subject and predicate, certain definite relations will become evident; therefore, Opposition is said to exist between propositions which are given the same subject and predicate, but differ in quality, or in quantity, or in both.

The following illustrative outline will make this clear:

Granting the truth of the propositions in the first column, it follows that those in the second column differ in quantity. That is, in “Some men are mortal” a smaller number of men is referred to than in “All men are mortal.” A similar variation in quantity obtains with the other propositions in the second column. Moreover, the propositions in the third column are the negative of the corresponding ones in the first; while the fourth column propositions differ from the first in both quantity and quality. Thus opposition exists to a greater or less degree between all. We may now ask ourselves the question, “When the propositions are related to each other in opposition which ones are true and which ones are false?” Giving attention to the propositions in row “I,” we note that if the universal affirmative, “All men are mortal,” is true, then the particular affirmative, “Some men are mortal,” is likewise true; because of the principle, “What is true of the whole of the class is true of a part of that class.” But the universal negative, “No men are mortal,” and the particular negative, “Some men are not mortal,” are both false. Briefly stated: If A is true, then I is true, but, both E and O are false.

Regarding row “II” we may conclude that if E is true, then O is likewise true, but both A and I are false.

As to rows “III” and “IV,” granting the truth of the I propositions, “Some men are wise” and “Some men are mortal,” we are able to assert that of the two A propositions, “All men are wise,” and “All men are mortal,” the first is false while the second is true. A is, therefore, indeterminate, or doubtful. Of the O propositions, “Some men are not wise,” is true while, “Some men are not mortal,” is false. Therefore, O is doubtful. Both of the E propositions are false. Hence, the conclusion relative to rows “III” and “IV” is: If I is true, A and O are doubtful, while E is false.

Concerning rows “V” and “VI” it will be seen without further explanation that if O is true, then E and I are doubtful and A is false.

THE SCHEME OF OPPOSITION.

The conditions of opposition are easily comprehended and remembered when recourse is made to the following scheme:

To use the above scheme, read horizontally from left to right. For example: If A be true, then all in the row opposite obtains; that is, A is true, E is false, I is true, and O is false. (We take it for granted that the student will see that the first column belongs to A, the second to E, the third to I, and the fourth to O.) If E be true, then A is false, E is true, I is false, O is true, etc.

The whole of opposition is comprehended in two facts which are based upon one principle. This is the principle: Whatever may be said of the entire class may be said of a part of that class. To put it in another way: Whatever is affirmed of all may be affirmed of some, or, Whatever is denied of all may be denied of some. To illustrate:

These are the two facts: First, a particular affirmative may be derived from a universal affirmative. Second, a particular negative may be derived from a universal negative. Or, more briefly: An I may be derived from an A, and an O from an E.

SQUARE OF OPPOSITION.

Aristotle represented the relations of the four logical propositions by what is termed the square of opposition. Viewed from the standpoint of the square, the relations may be summed up as follows:

1. Contrary Propositions.

Why so named.

As related to each other, A and E are said to be contrary because they seem to express contrariety to the greatest degree.

Relation stated.

If one is true, the other must be false, but both may be false.

Illustrations.

(1) If one is true, the other must be false; e.g., if A is true, as “All metals are elements,” then E is false, as “No metals are elements.” Or, if E is true, as “No birds are quadrupeds,” then A is false, as “All birds are quadrupeds.”

(2) Both may be false. If A is false, as “All men are wise,” then E may be false, as “No men are wise.”

2. Subcontrary Propositions.

Why so named.

Propositions I and O are said to be related to each other in a subcontrary manner because they are contrary as to each other and “under” their universals A and E.

Relation stated.

If one is false, the other must be true, or, both may be true.

Illustrations.

(1) If one is false, the other must be true.

If I is false, as “Some metals are compounds,” then, O is true, as “Some metals (at least) are not compounds.” Or, if O is false, as “Some metals are not elements,” then I is true, as “Some metals are elements.”

(2) Both may be true.

If I is true, as “Some men are wise,” then O also may be true, as “Some men are not wise.”

3. Subalterns.

Why so named.

Etymologically considered subaltern means under the one, thus proposition I is under A, and O is under E.

Relation stated.

First Relation.

Subalterns are related to each other as are the universals and particulars; hence,

(1) If the universal is true, the particular under it is also true; while if the particular is true, the corresponding universal may, or, may not, be true.

Illustrations.

(a) If the universal is true, the particular under it is true.

If A is true, as “All metals are elements,” then I is true, as “Some metals are elements.” Or, if E is true, as “No metals are compounds,” then, O is also true, as “Some metals (at least) are not compounds.”

(b) If the particular is true, the corresponding universal may, or, may not, be true.

If I is true, as “Some men are wise,” or, “Some men are mortal,” then A may be false, as “All men are wise,” or, A may be true, as “All men are mortal.” Or, if O is true, as “Some men are not wise,” or, “Some men are not immortal,” then E may be false, as “No men are wise”; or, true, as “No men are immortal.”

Second Relation.

(2) If the universal is false, the particular under it may or may not be true, but, if the particular is false, the universal above it must be false.

Illustrations.

(a) If the universal is false, the particular under it may or may not be true.

If A is false, as “All metals are compounds,” or “All men are wise,” then I may be false, as “Some metals are compounds,” or, I may be true, as “Some men are wise.” Or, if E is false, as “No men are mortal,” or, “No men are wise,” then O may be false, as “Some men are not mortal,” or, O may be true, as “Some men are not wise.”

(b) If the particular is false, the universal above it must be false.

If I is false, as “Some men are trees,” then A is false, as “All men are trees.” Or, if O is false, as “Some men are not bipeds,” then E is also false, as “No men are bipeds.”

4. Contradictory Propositions.

Why so named.

The propositions A and O, likewise E and I, are called contradictory propositions because they oppose each other in both quantity and quality. They are mutually opposed to each other or absolutely contradictory.

Relation stated.

If one is true the other must be false.

Illustrations.

(1) A and O compared.

If A is true, as “All metals are elements,” then, O is false, as “Some metals are not elements.” Or, if O is true, as “Some metals are not compounds,” then A is false, as “All metals are compounds.”

(2) E and I compared.

If E is true, as “No birds are quadrupeds,” then I is false, as “Some birds are quadrupeds.” Or, if I is true, as “Some birds are bipeds,” then E is false, as “No birds are bipeds.”

The chief value of the square of opposition springs from the contradictory propositions. The square shows conclusively that any universal affirmative assertion (an A) may best be contradicted by proving a particular negative (an O). For example: To satisfactorily refute the statement that, in this section, all birds migrate to the south in winter, it would be sufficient to prove that the English sparrow and starling do not migrate to the south. The square likewise makes evident that any universal negative (an E) may be conclusively denied by establishing the truth of a particular affirmative (an I). To illustrate: The easiest way to prove the falsity of “No trusts are honest” is to present facts showing that at least trusts A and B are honest.

The Individual Proposition.

An individual proposition is one with an individual subject such as “Aristotle was wise.” In logic, the individual proposition is classed as a universal. This seems to be a bit irregular, as with the individual proposition there is no particular, while, the strictly logical universal always implies a particular. Because of this variation from the true logical form the relations, as indicated by the square of opposition, do not apply to the individual proposition. For example: According to the square A and E are contrary, but, when individual, A and E contradict each other, as “Aristotle was wise” (A)—“Aristotle was not wise” (E).