Eduction

Conversion is the formulation of a new proposition by interchanging the subject and predicate of an original proposition but leaving its quality unchanged. “No cat is a dog,” for instance, is converted to “No dog is a cat.” The original proposition is called the convertend, the new proposition the converse, and the process itself conversion. Conversion is either simple or partial.

a. Simple Conversion

Conversion is simple if the quantity of the converse is the same as the quantity of the convertend. Hence, in simple conversion, if the convertend is universal, the converse is also universal; if the convertend is particular, the converse is particular; and if the convertend is singular, the converse is singular.

By simple conversion we can convert E propositions, I propositions, and singular propositions whose predicates are singular terms. Thus, “No cat is a dog” is converted by simple conversion to “No dog is a cat”; “Some houses are white” to “Some white things are houses”; and “The man near the door is John” to “John is the man near the door.”

A propositions cannot be converted by simple conversion. The original predicate, on being made the subject of a universal proposition, would be changed from a particular term to a universal term. For instance, you cannot argue “All dogs are animals; therefore all animals are dogs.”

O propositions cannot be converted at all. The original subject, on being made the predicate of a negative proposition, would be changed from a particular term to a universal term. Thus, you cannot convert Some dogs are not hounds” to “Some hounds are not dogs.”

b. Partial Conversion

Conversion is partial if the quantity of the proposition is reduced from universal to particular. Partial conversion is also called accidental conversion, conversion by limitation, and reduced conversion. A is converted by partial conversion to I, and E is converted to O. A propositions, if we regard their form alone, can be converted only by partial conversion. For instance, “All men are mortal beings” is converted by partial conversion to “Some mortal beings are men.” Note again how the subject of an affirmative proposition is drawn into the extension of the predicate and embraces an indeterminately designated portion of it.

Men do not include all the mortal beings—plants and the other animals can die too—hence, we may not argue: “All men are mortal beings; therefore all mortal beings are men.” In doing this, we would extend the term “mortal being” from particular to universal. We would assert something about all mortal beings in the converse after having given information about only some mortal beings in the convertend.

An A proposition whose predicate is the definition or a characteristic property of the subject may be converted by simple conversion because of the special character of the matter or thought content. Such a predicate is interchangeable with its subject. Thus, “Every man is a rational animal” may be converted by simple conversion to “Every rational animal is a man.” This conversion is materially valid but formally invalid.

Since E propositions can be converted by simple conversion, they can obviously be converted by partial conversion. If no cat is a dog, it is obvious that some dog is not a cat.

c. Four Notanda

1--Often it is advisable to reduce propositions to logical form before attempting conversion. This will save you from mistakes as illogical as the attempted conversion of "The dog bit the man" to "The man bit the dog."

2--Beware of converting A propositions by simple conversion. This is a common fallacy and is intimately related to the fallacies of the undistributed middle and the illicit process of a term, which we shall study when we take up the categorical syllogism.

3--O propositions, as we explained above, cannot be converted. Their original subject, on becoming the predicate, would be changed from a particular to a universal term. You must be especially careful when the apparent converse of an O proposition is, or at least seems to be, true. Take the example Some cats are not black; therefore some black things are not cats.” It is most assuredly true that some black things are not cats; but this cannot be inferred from the mere fact that some cats are not black. The formal invalidity of this inference is made obvious by a comparison with the following example, which has identical form, “Some dogs are not hounds; therefore some hounds are not dogs.”

4--The actual real existence of a subject may not be asserted in the converse if it has not been asserted in the convertend. There is special danger of doing this in converting A to I or E to O.

Synopsis

• Brief rules for conversion:

  1. Interchange S and P.

  2. Retain quality.

  3. Do not extend any term.

• Kinds of conversion:

  1. Simple (E to E, I to I).

  2. Partial (A to I, E to O).

• Examples of conversion:

A to I: "Every cat is an animal" to "Some animal is a cat."

E to E: "No cat is a dog" to "No dog is a cat."

I to I: "Some house is white" to "Some white thing is a house."

O cannot be converted.

Obversion is the formulation of a new proposition by retaining the subject and quantity of an original proposition, changing its quality, and using as predicate the contradictory of the original predicate. “Every dog is an animal,” for instance, is obverted to “No dog is a non-animal.” Notice that obversion involves either the use or removal of two negatives: the use or omission of the one negative changes the quality, the use or omission of the other negative changes the predicate to its contradictory. The original proposition is called the obvertend, the new proposition the obverse, and the process itself obversion.

The RULES OF OBVERSION may be briefly expressed as follows:

Rule 1. Retain the subject and the quantity of the obvertend. Rule 2. Change the quality. If the obvertend is affirmative, the obverse must be negative; and if the obvertend is negative, the obverse must be affirmative.

From Rules 1 and 2 we see that A propositions are obverted to E, E to A, I to O, and O to I.

Rule 3. As predicate, use the contradictory of the predicate of the original proposition. Under certain conditions you can make a materially valid obversion by using as predicate the immediately opposed contrary of the predicate of the original proposition.

First we shall explain and exemplify obversion in which the predicate of the obverse is the contradictory of the predicate of the obvertend.

A to E. “Every dog is an animal” is obverted to “Every dog is NOT a NON-animal,” which is normally expressed, “No dog is a non-animal.” Note that both the obvertend and the obverse are universal propositions and that the one negative particle negates the copula and the other negates the predicate.

E to A. “No dog is a cat” is obverted to “Every dog is a NON-cat.” The subject “dog” is universal in the obvertend; so it stays universal in the obverse. The quality is changed from negative to affirmative. The negative particle is transferred from the copula to the predicate.

I to O. “Some man is a voter” is obverted to “Some man is NOT a NON-voter.”

O to I. “Some man is not a voter” is obverted to “Some man is a NON-voter. The negative, again, is transferred from the copula to the predicate.

Sometimes it is helpful to reduce propositions to logical form before attempting obversion. The logical form of “He will not go” is He is not one who will go or “Ss is not P.” By changing the quality and negating the predicate, you get the obverse, “He is one who will not go” or “Ss is non-P.”

We shall now explain and exemplify materially valid obversion in which the immediately opposed contrary of the predicate of the obvertend becomes the predicate of the obverse. Such obversion may be made when the subject of the propositions belongs to the same genus that the two immediately opposed contraries belong to. For instance, “living being” is the genus (or quasi genus) of both mortal and “immortal,” and within this genus whatever is not mortal is immortal and vice versa. Hence, when the subject of the proposition is a living being, if what is signified by the subject is not mortal, it is immortal; if it is mortal, it is not immortal; and so on. Thus, you can legitimately obvert “Angels are not mortal” to “Angels are immortal.” But you cannot legitimately obvert “A stone is not mortal” to “A stone is immortal,” because a stone—a non-living being —does not belong to the genus of beings that must be either mortal or immortal. It is legitimate, though, to obvert “A stone is not mortal” to “A stone is non-mortal.”

Mediately opposed contraries are of no use in obversion, because there is a middle ground between them. You can obvert “The house is not white” to “The house is non-white,” but not to “The house is black.” The reason for this is that there are many other colors besides white and black; hence, a house that is not white is not necessarily black.

You can easily be deceived in the use of terms prefixed by “in-,” “im-,” “un-,” and so on. Often a term and the corresponding term having one of these prefixes are contraries, but often they are not. For instance, “flammable” and “inflammable,” “habitable” and “inhabitable,” “vest” and “invest,” and so on, are not contraries.

Synopsis

• Brief rules for obversion:

  1. Retain subject and quantity.

  2. Change quality.

  3. As predicate, use contradictory of original predicate.

• Examples of obversion:

A to E: "Every cat is an animal" to "No cat is a non-animal."

E to A: "No cat is a dog" to "Every cat is a non-dog."

I to O: "Some house is white" to "Some house is not non-white."

O to I: "Some house is not white" to "Some house is non-white."

Contraposition is the formulation of a new proposition whose subject is the contradictory of the original predicate. It is a combination of obversion and conversion. Like conversion, it involves the interchange of the subject and predicate; and like obversion, it involves either the use or the removal of negatives affecting the copula and terms. The original proposition is called the contraponend, the new proposition the contraposit or contrapositive, and the process itself contraposition. There are two types.

a. Type 1

The first type of contraposition (which is sometimes called partial, or simple, contraposition) consists in the formulation of a new proposition (1) whose subject is the contradictory (or, in certain circumstances, the immediately opposed contrary) of the original predicate, (2) whose quality is changed, and (3) whose predicate is the original subject. Thus, “Every dog is an animal” becomes Every NON-animal is NOT a dog,” which is normally expressed as “No non-animal is a dog.”

To get Type 1:

• 1. Obvert

  2. Then convert the obverse

Thus, beginning with "Every dog is an animal," first obvert this to "No dog is a non-animal," and then convert this to "No non-animal is a dog."

By contraposition of the first type—that is, by partial, or simple, contraposition—A is changed to E, E to I, and O to I. Just as O has no converse, so I has no contraposit. Note that the use of immediately opposed contraries is the same in contraposition as it is in obversion.

A to E. We have already illustrated and diagrammed the contraposition of A to E. As we have seen, we may argue, “Every dog is an animal; therefore no non-animal is a dog.”

E to I. “No dog is a cat” becomes “Some non-cat is a dog.” You may not argue, “No dog is a cat; therefore every non-cat is a dog.” This involves a violation of the rule for conversion that you should not extend a term.

A consideration of this diagram makes it obvious that only some non-cats are dogs and that there at least might be other non-cats that are not dogs. Moreover, we know that, as a matter of fact, there are innumerable non-cats that are not dogs; for instance, horses, stones, angels, and triangles. Let us take each step separately. First, we obvert “No dog is a cat” to “Every dog is a non-cat.” As the predicate of an affirmative proposition, “non-cat” is particular. Then we convert this either to “Some non-cat is a dog” or to “Some non-cats are dogs.”

O to I. “Some man is not a voter” may be changed by simple contraposition to “Some non-voter is a man.” First we obvert “Some man is not a voter” to “Some man is a NON-voter.” Then we convert this by simple conversion of “Some non-voter is a man.”

I propositions have no contraposit. The first step in contraposition is obversion. If we obvert I, we get O. The second step is conversion, and we have seen that O cannot be converted. Hence, since O cannot be converted, I cannot be contraposed.

b. Type 2

The second type of contraposition (which is sometimes called complete contraposition) is the formulation of a new proposition (1) whose subject, just as with Type 1, is the contradictory (or, in certain circumstances, the immediately opposed contrary) of the original predicate, (2) whose quality is unchanged, and (3) whose predicate is the contradictory of the original subject. This is the sense in which the older Scholastic logicians used the term “contraposition.” Frequently they called it “conversion by contraposition.”

To get Type 2:

1. Obvert

2. Then convert the obverse

3. Then obvert the converse of the obverse

In other words, the contraposit of the second type is the obverse of the contraposit of the first type. (Hence, the contraposit of the second type is sometimes called “the obverted contraposit,” as opposed to the first type, which is called the “simple contraposit.”)

By contraposition of the second type, A is changed to A, E to O, and O to O:

A to A: “Every man is mortal” to: “Every non-mortal is a non-man.”

E to O: “No dog is a cat” to: “Some non-cat is not a non-dog.”

O to O: “Some man is not a voter” to: “Some non-voter is not a non¬ man.”

With Type 2, just as with Type 1, I has no contraposit, because one of the steps would involve the conversion of an O proposition.

Synopsis

• Brief rules for contraposition:

Type 1. Simple

• 1. The subject is the contradictory of the original predicate.

  2. The quality is changed.

  3. The predicate is the original subject.

To get Type 1:

• 1. Obvert

  2. Then convert the obverse

Type 2. Complete

• 1. The subject is the contradictory of the original predicate.

  2. The quality is not changed.

  3. The predicate is the contradictory of the original subject.

To get Type 2:

• 1. Obvert

  2. Then convert the obverse

  3. Then obvert the converse of the obverse

• Examples of contraposition (note that Type 2 is the obverse of Type 1):

Type 1. Simple

A to E. "Every S is a P" to "No non-P is an S."

E to I. "No S is a P" to "Some non-P is an S."

O to I. "Some S is not a P" to "Some non-P is an S."

I cannot be contraposed.

Type 2. Complete

A to A. "Every S is a P" to "Every non-P is a non-S."

E to O. "No S is a P" to "Some non-P is not a non-S."

O to O. "Some S is not a P" to "Some non-P is not a non-S."

I cannot be contraposed.

Just as there are two types of contraposition, so too there are two types of inversion. Both types consist in the formulation of a new proposition whose subject is the contradictory of the original subject. In the first type (called partial or simple inversion), the quality is changed, but the predicate is the same as in the original proposition. In the second type (called complete inversion), the quality is unchanged, but the predicate is the contradictory of the original predicate. Immediately opposed contrary terms may be used just as in obversion and contraposition. The original proposition is called the invertend, the new proposition the inverse, and the process itself inversion.

Inversion is effected by a series of obversions and conversions. Experiment will show that only A and E can be inverted. By Type 1, A is inverted to O, and E to I; by Type 2, A is inverted to I, and E to O.

The second type of inverse is the obverse of Type 1, and is therefore sometimes called the “obverted inverse,” as opposed to Type 1 which is called the “simple inverse.”

If you subject an A proposition to the following processes, you finally get its inverse:

Invertend: A. Every S is a P. ("Every cat is an animal.")

Obvert to: E. No S is a non-P. ("No cat is a non-animal.")

Convert to: E. No non-P is an S. (No non-animal is a non-cat.")

Obvert to: A. Every non-P is a non-S. ("Every non-animal is a non-cat.")

Convert to: I. Some non-S is a non-P. ("Some non-cat is a non-animal.")

Obvert to: O. Some non-S is not a P. (Some non-cat is not an animal.") (This is inverse, Type 1.)

If you subject an E proposition to the following processes, you finally get its inverse. Note that you must convert first and then obvert.

Invertend: E. No S is a P. ("No cat is a dog.")

Convert to: E. No P is an S. (No dog is a cat.")

Obvert to: A. Every P is a non-S. ("Every dog is a non-cat.")

Convert to: I. Some non-S is a P. ("Some non-cat is a dog.") (This is inverse, Type 1.)

Obvert to: O. Some non-S is not a non-P. (Some non-cat is not a non-dog.") (This is inverse, Type 2.)

Up to the present we have been treating, for the most part, of formal inferences; that is, of inferences that depend for their validity on the quality of propositions and the quantity of their terms without any regard for the special character of their matter or thought content. There is another kind of inference, of much less importance, known as material eduction, which is based on the meanings of terms.

a. Eduction by Added Determinant

Eduction by an added determinant is the formulation of a new proposition in which both the subject and the predicate of the origi¬ nal proposition are limited by the addition of some modifier which has exactly the same meaning in relation to both of them. For instance, “Citizens are men; therefore honest citizens are honest men” is valid; “honest” has exactly the same meaning with both “citizens” and “men.” But we may not argue: “A mouse is an animal; therefore a big mouse is a big animal,” because “big” does not have exactly the same meaning with “mouse” as it has with “animal” in general.4 A thief is a man, but a good thief is not therefore a good man. A soprano is a woman, and a shrieking soprano is a shrieking woman; but a flat soprano is not necessarily a flat woman, nor is a bad soprano necessarily a bad woman.

b. Eduction by Complex Conception

Eduction by complex conception is the formulation of a new proposition whose subject consists of a term modified by the subject of the original proposition and whose predicate consists of the very same term modified by the predicate of the original proposition. In eduction by added determinants, a new term modifies the original subject and predicate; in eduction by complex conception, a new term is modified by them.

“If a horse is an animal, the head of a horse is the head of an animal, and the tail of a horse is the tail of an animal.” You must be especially careful in the use of words expressing quantitative pro¬ portions; for instance, “Dogs are animals, and ten dogs are ten animals.” Yet it is false that half of the dogs in the world are half of the animals in the world. In “Dogs are animals,” “animals” is particular; that is, “dogs” embraces an indeterminately designated portion of the extension of “animals.” Since the portion is indeterminate to begin with, half of it is also indeterminate.

Ordinarily nothing can be educed by complex conception from a negative proposition. For instance, you may not argue: “A dog is not a cat; therefore the owner of a dog is not the owner of a cat.”

c. Eduction by Omitted Determinant

Eduction by omitted determinant is the formulation of a new proposition in which a modifier of the original predicate is omitted. Care must be had that the meaning of what is left of the original predicate is not altered, as in: “This locket is false gold, therefore it is gold.” Consider also: “It is a pretended fact; therefore it is a fact;” “It is stage money; therefore it is money;” and “It is nothing; therefore it is.”

Nothing can be educed in this way from negative propositions. You may not argue: “A dog is not a rational animal; therefore a dog is not an animal.” The reason for this is clear from what we know of the relationship in comprehension of the subject and predicate of negative propositions.

d. Eduction by Converse Relation

Eduction by converse relation is the formulation of a new proposition in which a relationship is expressed that is the reverse of the relationship expressed in the original proposition. For instance, “A mouse is smaller than an elephant; therefore an elephant is larger than a mouse”; “Johnny is Mary’s nephew; therefore Mary is Johnny’s aunt”; “Since he is my father, I must be his son (or daughter)”; and “Chicago is northeast of St. Louis; therefore St. Louis is southwest of Chicago.”