DERIVED JUDGMENTS

As we have seen, a Judgment is obtained by comparing two objects of thought according to their agreement or difference. The next higher step, that of logical Reasoning, consists of the comparing of two ideas through their relation to a third. This form of reasoning is called mediate, because it is effected through the medium of the third idea. There is, however, a certain process of Understanding which comes in between this mediate reasoning on the one hand, and the formation of a plain judgment on the other. Some authorities treat it as a form of reasoning, calling it Immediate Reasoning or Immediate Inference, while others treat it as a higher form of Judgment, calling it Derived Judgment. We shall follow the latter classification, as best adapted for the particular purposes of this book.

The fundamental principle of Derived Judgment is that ordinary Judgments are often so related to each other that one Judgment may be derived directly and immediately from another. The two particular forms of the general method of Derived Judgment are known as those of (1) Opposition; and (2) Conversion; respectively.

In order to more clearly understand the logical processes involved in Derived Judgment, we should acquaint ourselves with the general relations of Judgments, and with the symbolic letters used by logicians as a means of simplifying the processes of thought. Logicians denote each of the four classes of Judgments or Propositions by a certain letter, the first four vowels—A, E, I and O, being used for the purpose. It has been found very convenient to use these symbols in denoting the various forms of Propositions and Judgments. The following table should be memorized for this purpose:

Universal Affirmative, symbolized by "A."
Universal Negative, symbolized by "E."
Particular Affirmative, symbolized by "I."
Particular Negative, symbolized by "O."

It will be seen that these four forms of Judgments bear certain relations to each other, from which arises what is called opposition. This may be better understood by reference to the following table called the Square of Opposition:

Thus, A and E are contraries; I and O are sub-contraries; A and I, and also E and O are subalterns; A and O, and also E and I are contradictories.

The following will give a symbolic table of each of the four Judgments or Propositions with the logical symbols attached:

(A) "All A is B."

(E) "No A is B."

(I) "Some A is B."

(O) "Some A is not B."

The following are the rules governing and expressing the relations above indicated:

I. Of the Contradictories: One must be true, and the other must be false. As for instance, (A) "All A is B;" and (O) "Some A is not B;" cannot both be true at the same time. Neither can (E) "No A is B;" and (I) "Some A is B;" both be true at the same time. They are contradictory by nature,—and if one is true, the other must be false; if one is false, the other must be true.

II. Of the Contraries: If one is true the other must be false; but, both may be false. As for instance, (A) "All A is B;" and (E) "No A is B;" cannot both be true at the same time. If one is true the other must be false. But, both may be false, as we may see when we find we may state that (I) "Some A is B." So while these two propositions are contrary, they are not contradictory. While, if one of them is true the other must be false, it does not follow that if one is false the other must be true, for both may be false, leaving the truth to be found in a third proposition.

III. Of the Subcontraries: If one is false the other must be true; but both may be true. As for instance, (I) "Some A is B;" and (O) "Some A is not B;" may both be true, for they do not contradict each other. But one or the other must be true—they can not both be false.

IV. Of the Subalterns: If the Universal (A or E) be true the Particular (I or O) must be true. As for instance, if (A) "All A is B" is true, then (I) "Some A is B" must also be true; also, if (E) "No A is B" is true, then "Some A is not B" must also be true. The Universal carries the particular within its truth and meaning. But; If the Universal is false, the particular may be true or it may be false. As for instance (A) "All A is B" may be false, and yet (I) "Some A is B" may be either true or false, without being determined by the (A) proposition. And, likewise, (E) "No A is B" may be false without determining the truth or falsity of (O) "Some A is not B."

But: If the Particular be false, the Universal also must be false. As for instance, if (I) "Some A is B" is false, then it must follow that (A) "All A is B" must also be false; or if (O) "Some A is not B" is false, then (E) "No A is B" must also be false. But: The Particular may be true, without rendering the Universal true. As for instance: (I) "Some A is B" may be true without making true (A) "All A is B;" or (O) "Some A is not B" may be true without making true (E) "No A is B."

The above rules may be worked out not only with the symbols, as "All A is B," but also with any Judgments or Propositions, such as "All horses are animals;" "All men are mortal;" "Some men are artists;" etc. The principle involved is identical in each and every case. The "All A is B" symbology is merely adopted for simplicity, and for the purpose of rendering the logical process akin to that of mathematics. The letters play the same part that the numerals or figures do in arithmetic or the a, b, c; x, y, z, in algebra. Thinking in symbols tends toward clearness of thought and reasoning.

Exercise: Let the student apply the principles of Opposition by using any of the above judgments mentioned in the preceding paragraph, in the direction of erecting a Square of Opposition of them, after having attached the symbolic letters A, E, I and O, to the appropriate forms of the propositions.

Then let him work out the following problems from the Tables and Square given in this chapter.

1. If "A" is true; show what follows for E, I and O. Also what follows if "A" be false.

2. If "E" is true; show what follows for A, I and O. Also what follows if "E" be false.

3. If "I" is true; show what follows for A, E and O. Also what follows if "I" be false.

4. If "O" is true; show what follows for A, E and I. Also what happens if "O" be false.

CONVERSION OF JUDGMENTS

Judgments are capable of the process of Conversion, or the change of place of subject and predicate. Hyslop says: "Conversion is the transposition of subject and predicate, or the process of immediate inference by which we can infer from a given preposition another having the predicate of the original for its subject, and the subject of the original for its predicate." The process of converting a proposition seems simple at first thought but a little consideration will show that there are many difficulties in the way. For instance, while it is a true judgment that "All horses are animals," it is not a correct Derived Judgment or Inference that "All animals are horses." The same is true of the possible conversion of the judgment "All biscuit is bread" into that of "All bread is biscuit." There are certain rules to be observed in Conversion, as we shall see in a moment.

The Subject of a judgment is, of course, the term of which something is affirmed; and the Predicate is the term expressing that which is affirmed of the Subject. The Predicate is really an expression of an attribute of the Subject. Thus when we say "All horses are animals" we express the idea that all horses possess the attribute of "animality;" or when we say that "Some men are artists," we express the idea that some men possess the attributes or qualities included in the concept "artist." In Conversion, the original judgment is called the Convertend; and the new form of judgment, resulting from the conversion, is called the Converse. Remember these terms, please.

The two Rules of Conversion, stated in simple form, are as follows:

I. Do not change the quality of a judgment. The quality of the converse must remain the same as that of the convertend.

II. Do not distribute an undistributed term. No term must be distributed in the converse which is not distributed in the convertend.

The reason of these rules is that it would be contrary to truth and logic to give to a converted judgment a higher degree of quality and quantity than is found in the original judgment. To do so would be to attempt to make "twice 2" more than "2 plus 2."

There are three methods or kinds of Conversion, as follows: (1) Simple Conversion; (2) Limited Conversion; and (3) Conversion by Contraposition.

In Simple Conversion, there is no change in either quality or quantity. For instance, by Simple Conversion we may convert a proposition by changing the places of its subject and predicate, respectively. But as Jevons says: "It does not follow that the new one will always be true if the old one was true. Sometimes this is the case, and sometimes it is not. If I say, 'some churches are wooden-buildings,' I may turn it around and get 'some wooden-buildings are churches;' the meaning is exactly the same as before. This kind of change is called Simple Conversion, because we need do nothing but simply change the subjects and predicates in order to get a new proposition. We see that the Particular Affirmative proposition can be simply converted. Such is the case also with the Universal Negative proposition. 'No large flowers are green things' may be converted simply into 'no green things are large flowers.'"

In Limited Conversion, the quantity is changed from Universal to Particular. Of this, Jevons continues: "But it is a more troublesome matter, however, to convert a Universal Affirmative proposition. The statement that 'all jelly fish are animals,' is true; but, if we convert it, getting 'all animals are jelly fish,' the result is absurd. This is because the predicate of a universal proposition is really particular. We do not mean that jelly fish are 'all' the animals which exist, but only 'some' of the animals. The proposition ought really to be 'all jelly fish are some animals,' and if we converted this simply, we should get, 'some animals are all jelly fish.' But we almost always leave out the little adjectives some and all when they would occur in the predicate, so that the proposition, when converted, becomes 'some animals are jelly fish.' This kind of change is called Limited Conversion, and we see that a Universal Affirmative proposition, when so converted, gives a Particular Affirmative one."

In Conversion by Contraposition, there is a change in the position of the negative copula, which shifts the expression of the quality. As for instance, in the Particular Negative "Some animals are not horses," we cannot say "Some horses are not animals," for that would be a violation of the rule that "no term must be distributed in the converse which is not distributed in the convertend," for as we have seen in the preceding chapter: "In Particular propositions the subject is not distributed." And in the original proposition, or convertend, "animals" is the subject of a Particular proposition. Avoiding this, and proceeding by Conversion by Contraposition, we convert the Convertend (O) into a Particular Affirmative (I), saying: "Some animals are not-horses;" or "Some animals are things not horses;" and then proceeding by Simple Conversion we get the converse, "Some things not horses are animals," or "Some not-horses are animals."

The following gives the application of the appropriate form of Conversion to each of the several four kind of Judgments or Propositions:

(A) Universal Affirmative: This form of proposition is converted by Limited Conversion. The predicate not being distributed in the convertend, it cannot be distributed in the converse, by saying "all." ("In affirmative propositions the predicate is not distributed.") Thus by this form of Conversion, we convert "All horses are animals" into "Some animals are horses." The Universal Affirmative (A) is converted by limitation into a Particular Affirmative (I).

(E) Universal Negative: This form of proposition is converted by Simple Conversion. In a Universal Negative both terms are distributed. ("In universal propositions, the subject is distributed;" "In negative propositions, the predicate is distributed.") So we may say "No cows are horses," and then convert the proposition into "No horses are cows." We simply convert one Universal Negative (E) into another Universal Negative (E).

(I) Particular Affirmative: This form of proposition is converted by Simple Conversion. For neither term is distributed in a Particular Affirmative. ("In particular propositions, the subject is not distributed. In affirmative propositions, the predicate is not distributed.") And neither term being distributed in the convertend, it must not be distributed in the converse. So from "Some horses are males" we may by Simple Conversion derive "Some males are horses." We simply convert one Particular Affirmative (I), into another Particular Affirmative (I).

(O) Particular Negative: This form of proposition is converted by Contraposition or Negation. We have given examples and illustrations in the paragraph describing Conversion by Contraposition. The Particular Negative (I) is converted by contraposition into a Particular Affirmative (I) which is then simply converted into another Particular Affirmative (I).

There are several minor processes or methods of deriving judgments from each other, or of making immediate inferences, but the above will give the student a very fair idea of the minor or more complete methods.

Exercise: The following will give the student good practice and exercise in the methods of Conversion. It affords a valuable mental drill, and tends to develop the logical faculties, particularly that of Judgment. The student should convert the following propositions, according to the rules and examples given in this chapter:

1. All men are reasoning beings.
2. Some men are blacksmiths.
3. No men are quadrupeds.
4. Some birds are sparrows.
5. Some horses are vicious.
6. No brute is rational.
7. Some men are not sane.
8. All biscuit is bread.
9. Some bread is biscuit.
10. Not all bread is biscuit.