| * | Proposition symbolized | Name of Process | Inference symbolized | Principle involved |
| A | All S is P† | Opposition | Some S is P (I) | What is said of all may be said of some. |
| Obversion | No S is not-P (E) | Two negatives are equivalent to one affirmative. | ||
| Conversion by Limitation | Some P is S (I) | An undistributed term cannot be distributed. | ||
| Contraversion | No not-P is S (E) | Same principles which obtain in obverting A and converting E. | ||
| E | No S is P | Opposition | Some S is not P (O) | What is said of all may be said of some. |
| Obversion | All S is not-P (A) | Two negatives are equivalent to one affirmative. | ||
| Simple Conversion | No P is S (E) | Distribution not affected. | ||
| Contraversion | Some not-P is S (I) | An undistributed term cannot be distributed. | ||
| I | Some S is P | Opposition | Doubtful | None. |
| Obversion | Some S is not not-P (O) | Two negatives are equivalent to one affirmative. | ||
| Conversion | Some P is S (I) | Distribution not affected. | ||
| Contraversion | Impossible | None. | ||
| O | Some S is not P | Opposition | Doubtful | None. |
| Obversion | Some S is not-P (I) | Two negatives are equivalent to one affirmative. | ||
| Conversion | Impossible | None. | ||
| Contraversion | Some not-P is S (I) | Same as in obversion of O and conversion of I. |
* – 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.
| The Given Proposition. | The Inverse. | |
| I. | All S is P. (A) | Some not-S is not P. (O) |
| All planets rotate. | Some not-planets do not rotate. | |
| II. | No S is P. (E) | Some not-S is P. (I) |
| No men are immortal. | Some not-men are immortal. |
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.”
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.
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 an E.
The crucial fact made obvious by the square of opposition is that A and O are mutually contradictory; likewise E and I.
(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 an I.
(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.
(1a) From the antecedent judgment, “All weeds are plants,” I am 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 I recall 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 I recall 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 that the second is an A, the third an I and the fourth an E. It is now expedient to recall the rules regarding conversion. These are, (1) do not distribute an undistributed term; (2) do not change the quality. We may now attempt to interchange the subject and predicate of each proposition, with the following results:
(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), I find 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 and E’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 to E.
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.”
(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.
(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 an I?
(7) Why cannot an O be derived from an A?
(8) The basic principle of obversion is “Two negatives are equivalent to one affirmative.” Show by means of circles that this is not absolutely true; take as an illustrative proposition, “No men are not mortal.”
(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.