 
(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 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 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.
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:
(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 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 Conversion by Limitation Exemplified Further.
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 anI. 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, “Aman 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 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.
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.
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.
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 *–Name of proposition †–“S” represents any subject and “P” any predicate. INFERENCE BY INVERSION. Some logicians treat of a form of immediate inference known as inversion though it is of small importance and of little practical value. The process can be applied only to propositions A and E. In the one case the contradictory subject is limited by “some” and then denied of the predicate, whereas, in the other case, the contradictory subject is merely affirmed of the predicate. Illustrations.
From the foregoing we are able to conclude that the inverse of “A” is found by negating the subject and changing to an “O”; while the inverse of “E” is found by negating the subject and changing to an “I.” 5. OUTLINE.IMMEDIATE INFERENCE—?OPPOSITION—?OBVERSION, CONVERSION, CONTRAVERSION AND INVERSION. 1. The Nature of Inference. 2. Immediate and Mediate Inference. 3. The Forms of Immediate Inference. (1) Opposition. (a) Scheme of Opposition. (b) Square of Opposition. (2) Obversion. (3) Conversion. (a) Simply. (b) By Limitation. (4) Contraversion. Inversion. 6. SUMMARY.1. Inference is the thought process of deriving a judgment from one or two antecedent judgments. 2. Immediate inference is inference without the use of a middle term. Mediate inference is inference by means of a middle term. 3. The four common forms of immediate inference are (1)opposition, (2)obversion, (3)conversion, (4)contraversion. (1) The name opposition stands for certain definite relations which exist between the logical propositions when they are given the same subject and predicate. The one principle underlying opposition is: Whatever is said of the entire class may be said of a part of that class. The two statements which sum up opposition are first, an I may be derived from an A; and second, an O may be derived from anE. The crucial fact made obvious by the square of opposition is that A and O are mutually contradictory; likewise E andI. (2) Obversion is the process of passing from an affirmative to its equivalent negative or from a negative to its equivalent affirmative. “Two negatives are equivalent to one affirmative,” is the basic principle of obversion. The proposition A may be obverted by negating the predicate and changing to an E. “E” is obverted by negating the predicate and changing to an A. “I” is obverted by negating the predicate and changing to an O. “O” is obverted by negating the predicate and changing to anI. (3) 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. Conversion is limited by the two rules, (1)do not distribute an undistributed term; (2)do not change the quality. To convert an A interchange subject and predicate, limiting the latter by some, or a word of like significance. This is called conversion by limitation. The co-extensive A may be converted without limiting the predicate. This is called simple conversion. An E proposition may be converted either simply or by limitation. When converted by limitation the inference is a weakened one. An I proposition is converted simply only. The O proposition does not admit of conversion. (4) Immediate inference by contraversion is a process involving first obversion and then conversion. “A,” “E” and “O” may be contraverted; “I” cannot be contraverted. 7. ILLUSTRATIVE EXERCISES.(1a) From the antecedent judgment, “All weeds are plants,” Iam able to derive by immediate inference these judgments: (1)“All weeds are not not-plants,” or “No weeds are not plants.” (2)“No not-plants are weeds.” (3)“Some plants are weeds.” (4)“Some weeds are plants.” (1b) “All vertebrates have a backbone.” From the foregoing judgment derive immediately five different conclusions. (2a) “All good citizens try to vote,” “Albert White is a good citizen,” Hence, “Albert White will try to vote.” I know that the above is an example of mediate inference because the two antecedent judgments make use of the middle term, “good citizen.” (2b) Why is the following illustrative of mediate inference? “All wise men are close observers,” “All wise men are thoughtful,” Hence, “Some thoughtful men are close observers.” (3a) Derive immediate inferences by opposition from the following: (1) “Good men are wise.” (2) “No teacher can afford to be unjust.” (3) “All birds fly.” (4) “None of the inner planets are as large as the earth.” I first determine that “1” and “3” are A propositions, while “2” and “4” are E’s. Then Irecall that by opposition an I may be derived from an A and an O from an E. Hence, the inferences are: (1) “Some good men are wise.” (2) “Some teachers cannot afford to be unjust.” (3) “Some birds fly.” (4) “Some of the inner planets are not so large as the earth.” (3b) Derive by opposition inferences from the following: (1) “No true woman will neglect her home for society.” (2) “All patriotic men love the flag.” (3) “Fools rush in where angels fear to tread.” (4a) Obvert the following: (1) “All earnest teachers are diligent students.” (2) “No self-respecting man can afford to be careless in his personal appearance.” (3) “Some of the great teachers of the past did not practice what they preached.” (4) “Some weeds are beautiful.” I determine first the logical character of each proposition, finding the first to be an A, the second an E, the third an O and the fourth an I. Then Irecall that in obversion the predicate must always be negated and an A must be changed to an E or an E to an A; also an I must be changed to an O or an O to an I. Hence, the obverse of each proposition is: (1) “No earnest teacher is a not-diligent student.” (2) “All self-respecting men can afford to be not-careless (careful) in their personal appearance.” (3) “Some of the great teachers of the past did not-practice (failed to practice) what they preached.” (4) “Some weeds are not not-beautiful.” (4b) Infer by obversion from the following: (1) “All roses are beautiful.” (2) “None of the members of the stock exchange are dishonest.” (3) “Some pupils are not industrious.” (4) “Some teachers are tactful.” (5a) Convert the following: (1) “All that glitters is not gold.” (2) “All good men are wise.” (3) “Some books are to be chewed and digested.” (4) “No man is perfectly happy.” It is first necessary to determine the logical character of each proposition. Carelessness might lead one to call the first proposition an A because it is introduced by the quantity sign “all.” But on second thought we note that the meaning is to the effect that some glittering things are not gold; this is an O. It is clear (1) Conversion impossible. (2) “Some wise men are good men.” (3) “Some things to be chewed and digested are books.” (4) “No perfectly happy being is a man.” When attempting to convert proposition (1), Ifind that the subject which is undistributed becomes distributed, hence the rule pertaining to distribution is violated. This conclusion is verified by recalling the fact that an O proposition cannot be converted. The second proposition, being an A, is converted by limitation; while the third and fourth are converted simply, as is the natural procedure with all I’s andE’s. (5b) Convert these propositions: (1) “Blessed are the meek.” (All the meek are blessed.) (2) “None but material bodies gravitate.” (All gravitating bodies are material.) (3) “Gold is not a compound substance.” (4) “Usually cruel men are cowards.” NOTE.—The first proposition is poetical while the second is an exclusive. (6a) Contravert the following propositions: (1) “All virtue is praiseworthy.” (2) “Some teachers are not tactful.” (3) “A man who lies is not to be trusted.” Contraversion consists in obverting first, and then converting; consequently, the contraverse of the three propositions is as follows: (1) “No unpraiseworthy deed is virtue.” (2) “Some not-tactful persons are teachers.” (3) “Some untrustworthy men are those who lie.” (6b) Write the contraverse of the following: (1) “All honest men pay their debts.” (2) “All men are rational.” (3) “Nearly all the troops have left the town.” (4) “Some teachers are not patient.” (7a) The attending scheme indicates the logical process and rule involved in passing from one proposition to another: A. “All men are imperfect.” Process: Obversion. Rule: Negate predicate and change toE. E. “No men are perfect.” Process: Simple Conversion. Rule: Interchange subject and predicate. E. “No perfect beings are men.” Process: Contraversion. Rule: Obvert and then convert. I. “Some not-men are perfect beings.” (7b) Treat in a manner similar to the above the proposition, “All horses are quadrupeds.” 8. REVIEW QUESTIONS.(1) What is inference? (2) What is the meaning of antecedent? (3) Define (1)judging, (2)a judgment. (4) All roses are beautiful, This flower is a rose, This flower is beautiful. Write this example of mediate inference in equation form. Name the middle term. (5) Define immediate inference. Illustrate. (6) Define mediate inference. Illustrate. (7) Name the five forms of immediate inference. (8) What principle is involved in inference by opposition? (9) Draw the scheme of opposition. (10) Make use of this scheme in deriving inferences from the following propositions: (a) “Good men are wise.” (b) “No king is infallible.” (c) “Cattle are ruminants.” (d) “All who cheat the railroads are not honest.” (11) What are contradictory propositions? Illustrate. (12) What would be the simplest way of disproving the statement that “No great religious teacher has been consistent?” (13) Why are A and E said to be contrary propositions? (14) Define obversion. (15) By what other name is obversion known? (16) State the basic principle of obversion. (17) Illustrate the process known as negating the predicate. (18) State the rule for obverting an A proposition. (19) Obvert the following: (1) “All the boys in my room are industrious.” (2) “Honesty is the best policy.” (3) “Only the industrious are truly successful.” (20) First state the rule and then obvert the following: (1) “Some plants are biennial.” (2) “Planets are not suns.” (3) “Blessed are the merciful.” (4) “These samples are not perfect.” (21) Define conversion. (22) State and illustrate the rules which condition the process of conversion. (23) Convert, if possible, the following: (1) “Some men practice sophistry.” (2) “Few men know how to live.” (3) “Some of the inhabitants are not civilized.” (4) “All the world is a stage.” (5) “None of my pupils failed.” (6) “Experience is a hard taskmaster.” (24) Why may co-extensive propositions be converted simply? (25) Describe the process of inference by contraversion. 9. PROBLEMS FOR ORIGINAL THOUGHT AND INVESTIGATION.(1) What ground is there for the belief that immediate inference, so called, is merely a matter of the interpretation of propositions? (2) Is there any difference between reasoning and inference? (3) When the conclusion is reached that two rooms are of the same width, because each is five yards wide, what is the middle term? (4) Put in equation form: All teachers instruct, John Jones is a teacher, John Jones instructs. Show that the equations are not absolutely true. (5) Indicate the true relation between the subjects and predicates of the foregoing by using the algebraic signs > and<. (6) Why cannot an A be derived from anI? (7) Why cannot an O be derived from anA? (8) The basic principle of obversion is “Two negatives are equivalent to one affirmative.” Show by means of circles that (9) Show that agreeable and disagreeable are not contradictory terms. (10) Why should the logician class individual propositions as universal? (11) Show by circles that there is a difference in signification between, “Some men are not wise” and “Some men are not-wise.” (12) Show by circles that the O proposition cannot be converted. (13) “The I proposition cannot be contraverted.” Make this clear. (14) Is there any difference in meaning between, “All illogical work is unscholarly” and “No illogical work is scholarly?” Explain by circles. (15) State the logical process involved in passing from each proposition to its succeeding one: (1) “All men are imperfect.” (2) “No men are perfect.” (3) “No perfect beings are men.” (4) “Some not-men are perfect beings.” (5) “Some perfect beings are not-men.” (6) “Some perfect beings are not men.” (16) It is sometimes said that in sub-contraries there is really no opposition. Do you agree? Give arguments. |
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