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:

1.

Original Proposition.

I. All men are mortal. (A)

II. No men are immortal. (E)

III. Some men are wise. (I)

IV. Some men are mortal. (I)

V. Some men are not wise. (O)

VI. Some men are not immortal. (O)

2.

Opposite in Quantity.

I. Some men are mortal. (I)

II. Some men are not immortal. (O)

III. All men are wise. (A)

IV. All men are mortal. (A)

V. No men are wise. (E)

VI. No men are immortal. (E)

3.

Opposite in Quality.

I. No men are mortal. (E)

II. All men are immortal. (A)

III. Some men are not wise. (O)

IV. Some men are not mortal. (O)

V. Some men are wise. (I)

VI. Some men are immortal. (I)

4.

Opposite in Both.

I. Some men are not mortal. (O)

II. Some men are immortal. (I)

III. No men are wise. (E)

IV. No men are mortal. (E)

V. All men are wise. (A)

VI. All men are immortal. (A)

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, “Somemen 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 ofa 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:

Accepted truth: All planets rotate. (A)

Accepted inference: Some planets rotate. (I)

or

Accepted truth: No planet is a sun. (E)

Accepted inference: Some planets are not suns. (O)

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 notimmortal,” 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 propositionthere 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).

CHAPTER 10.
IMMEDIATE INFERENCE (CONTINUED)—​OBVERSION, CONVERSION, CONTRAVERSION AND INVERSION.

(2) IMMEDIATE INFERENCE BY OBVERSION.

Obversion is the process of changing a proposition from the affirmative form to its equivalent negative or from the negative form to its equivalent affirmative.

Some authorities refer to this process as “Inference by Privitive Conception,” but Obversion seems to be a better term.

Obversion is based upon the principle that two negatives are equivalent to one affirmative. With this double negative principle in mind let us experiment with the four logical propositions, A, E, I, O.

The A Proposition.

Example: “All thoughtful men are wise.” Insert the double negative and the proposition reads: “All thoughtful men are not not-wise.” Changed to the logical form this becomes: “No thoughtful men are not-wise.” Simplified and we have, finally: “No thoughtful men are unwise.” Thus by the process of obversion we have passed from the original proposition, “All thoughtful men are wise,” to “No thoughtful men are unwise.” In the first proposition the subject “thoughtful men” is denied of the predicate “unwise.” Assuming that “unwise” is the contradictory of “wise,” then: “What is affirmed of a predicatemay be denied of its contradictory.” Recourse to circles will make this clearer. In the previous chapter it has been suggested that not bisects the world. For example: What can not be included in the wise class may be placed under the not-wise or unwise class. Likewise a circle bisects space—there is the space inside the circle and the space outside the circle. Let the space inside the circle represent all wise beings, then the space outside the circle would represent all not-wise or unwise beings; e. g.,

FIG. 5.

Now representing thoughtful men by a smaller circle and placing it inside the larger we have,

FIG. 6.

Referring to [Fig. 6] we note that all of the smaller circle belongs to the larger or that none of the smaller circle belongs to the space outside of the larger. Hence the two propositions: “All thoughtful men are wise” (A), and “No thoughtful men are unwise” (E) have virtually the same meaning though the same subject is related to different predicates.

The use of the positive or negative form depends upon circumstances. Often the negative puts the thought in a more forceful way.

In passing from, “All thoughtful men are wise,” to “No thoughtful men are unwise,” it was necessary to prefix not to the predicate wise and substitute for not its equivalent un. If the original predicate were unwise or not-wise, then the reverse order of dropping the un or not could be followed. This process of prefixing the not to an affirmative predicate or of dropping the not from a negative predicate is referred to as negating the predicate. Before substituting in, im, un, etc., for not, one must make sure that the substitution really gives the contradictory; there are some logicians who claim that unwise, for instance, is not the contradictory of wise.

In comparing the first proposition with the second it is observed that the first is an A, while the second is an E, also that the predicate of the first was negated to form the predicate of the second. Thus the rule: Negate the predicate and change A to E.

To sum up:

The obversion of an A proposition.

1. Principle:

Two negatives are equivalent to one affirmative.

2. Rule:

Negate the predicate and change the A to an E by using the sign no instead of all.

3. Process illustrated.

The Original Proposition (A) The Obverse (E)
All men are mortal.No men are immortal.
All maples are trees.No maples are not-trees.
All teachers should be sympathetic. No teacher should be un-sympathetic.
All pain is unpleasant.No pain is pleasant.
All men are imperfect.No men are perfect.
All birds are feathered animals. No birds are non-feathered animals.
All men are not-trees.No men are trees.
All scalene triangles are non-equilateral. No scalene triangles are equilateral.

The E Proposition.

It is obvious that the process of obverting an E is simply the reverse of obverting an A. Consequently, the same principle obtains; whereas the process may be illustrated by reading the foregoing illustrations reversely.

The rule for obverting E is: Negate the predicate and change the E to an A by changing the sign no to all.

The I Proposition.

Let us note the result when the double negative principle is applied to the I proposition.

Original: “Some men are wise.”

Adding two negatives: “Some men are not not-wise.”

The foregoing simplified: “Some men are not unwise.”

In comparing the first proposition with the last it is observed that the first is an I while the last is an O; it is also observed that the predicate of the first was negated in order to form the predicate of the last. Thus the rule: “Negate the predicate and change the I to an O.”

The use of circles may make this clearer:

FIG. 7.

The significant part of [Fig. 7] is that which is inked. Here we have represented the part of the “men” circle which is common to the “wise” circle. Thus the inked part represents “Some men are wise.” If the inked part is entirely inside of the “wise” circle, no part of it can belong to the “unwise” space without. Thus the obverse, “Some men are not unwise.”

Summary.

The obversion of an I proposition.

1. Principle:

Same as with A.

2. Rule:

Negate the predicate and change the I to an O.

3. Process illustrated.

The Original Proposition (I) The Obverse (O)
Some water is pure.Some water is not impure.
Some curves are perfect.Some curves are not imperfect.
Some friends are loyal.Some friends are not disloyal.
Some men are true.Some men are not not-true.
Some precious stones are imperfect.Some precious stones are not perfect.
Some plants are not-trees.Some plants are not trees.
Some boys are not-honest.Some boys are not honest.

It must be borne in mind that when “not” is used without the hyphen it makes the proposition negative, because when “unhyphened,” “not” must be thought of in connection with the copula and not in connection with the predicate; while “not” attached to the predicate with a hyphen simply makes the predicate negative without affecting the quality of the proposition; e. g., “Some plants are not trees” is a negative proposition, while “Some plants are not-trees” is an affirmative proposition with a negative predicate.

It may not be clearly seen how it is possible, by following the rule given, to pass from such a proposition as “Some plants are not-trees,” to “Some plants are not trees.” Let us illustrate the steps:

1. The original: “Some plants are not-trees.”

2. Negating predicate: “Some plants are trees.”

3. Changing to an O: “Some plants are not trees.”

Dropping the not from “1” and then adding it again to “2” is simply putting into operation the double negative idea, so that there is no violation of the principle.

The O Proposition.

O bears the same relation to I that E bears to A. The principle involved is the same. The process is illustrated by reading reversely the scheme of illustrations under I. The rule is as follows: To obvert an O negate the predicate and change the O to an I by eliminating the not.

Summary of Obverting the Four Logical Propositions.

1. Principle:

Two negatives are equivalent to one affirmative.

2. Rules:

Negate the predicate and change(1) A to E
(2) E to A
(3) I to O
(4) O to I

(3) IMMEDIATE INFERENCE BY CONVERSION.

Conversion is the process of inferring from a given proposition another which has, as its subject, the predicate of the given proposition, and, as its predicate, the subject of the given proposition. It is simply a matter of transposing subject and predicate. The original proposition is called the convertend while the derived proposition is named the converse.

The process of conversion is limited by two rules. First rule. No term must be distributed in the converse which is not distributed in the convertend. Second rule. The quality of the converse must be the same as that ofthe convertend. More briefly: (1) Do not distribute an undistributed term. (2) Do not change the quality.

We recall that a term is distributed when it is referred to as a definite whole. An undistributed term is referred to only in part. The principle underlying rule “1,” therefore, is the one which forms the basis of inference by opposition; namely, “Whatever may be said of the entire class may be said of a part of that class.” The converse of this is not true, that is, “What is said of part of a class cannot be said of the whole of that class.” When we distribute an undistributed term we are saying of the whole class what was said only of a part of that class. This is fallacious. On the other hand, we may say of a part what was said of the whole, or “undistribute” a distributed term.

We recall that the conclusion of the whole matter of inference by opposition was, that only an I could be inferred from an A and only an O from an E, or to put it in another way: Only an affirmative from an affirmative and only a negative from a negative. This establishes the truth of the second rule in conversion: “Do not change the quality.”

Let us apply the two rules to the four logical propositions.

Converting an A proposition.

Take as a type, “All horses are quadrupeds.” Here the subject “horses” is distributed, but the predicate “quadrupeds” is undistributed. In transposing subject and predicate we cannot distribute the term “quadrupeds,” according to the rule which says, “Do not distribute anundistributed term.” Hence in interchanging subject and predicate we cannot say, “All quadrupeds are horses,” but must limit the assertion to, “Some quadrupeds are horses.” Logicians call this process Conversion by Limitation.

Conversion by Limitation Exemplified Further.

Convertend.Converse.
All metals are elements.Some elements are metals.
All bees buzz.Some buzzing insects are bees.
All men are fallible.Some fallible beings are men.
All good teachers are sympathetic.Some sympathetic persons are good teachers.

The conclusions from the foregoing are these: First, the usual mode of converting an A is to interchange subject and predicate, limiting the latter by the word “some” or a word of similar significance. Second, this mode is called conversion by limitation. Third, the converse of an A is an I.

The Co-extensive A.

In the conversion of A propositions there is the one exception of “co-extensive A’s,” such as truisms and definitions. It will be remembered that with these both subject and predicate are distributed; hence, they may be interchanged without limiting the predicate by “some.” To illustrate: The converse of the truism, “A man is a man.” is “A man is a man,” while the converse of the definition, “A man is a rational animal,” is “A rational animal is a man.” This mode of interchanging subject and predicatewithout limiting the latter is called Simple Conversion. The ordinary A proposition is thus converted by limitation, while the co-extensive A is converted simply.

Converting an E proposition.

As both terms of the E proposition are distributed it is not possible to violate the rule of distribution. It is to be remembered that no fallacy is committed by “undistributing” a term which is already distributed.

Illustrations.

Convertend.Converse.
No men are immortal.No immortals are men. Simply.
No birds are quadrupeds.No quadrupeds are birds. Simply.
No metals are compounds.No compounds are metals. Simply.
No men are immortal.Some immortals (at least) are not men. Limitation.
No birds are quadrupeds.Some quadrupeds are not birds. Limitation.
No metals are compounds.Some compounds are not metals. Limitation.

Three facts are evident relative to the converting of an E. First: An E proposition may be converted either simply or by limitation. Second: E may be converted into either E or O. Third: If the converse is an O then is the inference a weakened one, being particular when it could just as well be universal.

Converting an I proposition.

With an I proposition neither term is distributed.Thus care must be used lest an undistributed term in the convertend be distributed in the converse. Illustrations:

Convertend.Converse.
Some men are wise.Some wise beings are men.
Some teachers scold.Some who scold are teachers.
Some high school graduates enter college. Some who enter college are high school graduates.
Some Americans live simply.Some who live simply are Americans.

From the foregoing we conclude first, that I is converted simply; second, that I is converted into I.

The O Proposition.

With an O proposition the subject is undistributed while the predicate is distributed. This condition presents a peculiar difficulty. Consider, for example, the O proposition, “Some men are not wise.” Convert this into, “Some wise beings are not men,” and the undistributed subject of the convertend, which is “men,” becomes the distributed predicate of the converse. Thus the O proposition cannot be converted without violating the rule for distribution.

A Summary of How the Four Logical Propositions May be Converted.

1. A. The ordinary A proposition may be converted by limitation only. The co-extensive A may be converted simply.

2. E. The E proposition is converted simply. The E may also be converted by limitation, but the inference thus obtained is weakened.

3. I. The I proposition may be converted simply only.

4. O. The O proposition cannot be converted.

(4) INFERENCE BY CONTRAVERSION. (Contraposition).

This mode of inference is usually referred to as inference by contraposition, but contraversion, indicating more definitely the nature of the process, is a better term. Contraversion involves two steps: First, obversion; second, conversion. The same principles and rules evident in these two processes obtain in inference by contraversion. The following scheme, therefore, ought to be sufficient to make the matter clear:

Inference by Contraversion.

1.The Given Proposition.2.Obverted.
A.All men are mortal. No men are immortal.
All trees are plants. No trees are not-plants.
E.No men are infallible. All men are fallible.
No men are trees. All men are not-trees.
I.Some men are wise. Some men are not not-wise.
O.Some water is not pure. Some water is impure.
Some houses are not white. Some houses are not-white.

3. Converted; giving the contraverse of the original proposition.

No immortals are men.

No not-plants are trees.

Some fallible beings are men.

Some not-trees are men.

An O cannot be converted, consequently the contraversion of an I is impossible.

Some impure liquids are water.

Some not-white buildings are houses.

It is indicated in the foregoing scheme that “I” cannot be contraverted. This is due to the fact that the obverseof an I is an O, and it will be remembered that “O” cannot be converted. All the other propositions admit of contraversion.