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 predicate may 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 of the 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 an undistributed 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 predicate without 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 obverse of an I is an O, and it will be remembered that “O” cannot be converted. All the other propositions admit of contraversion.