(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 quant.i.ty 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 quant.i.ty. 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 quant.i.ty 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 ill.u.s.trations 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.

CHAPTER XIII.

REASONING

In the preceding chapters we have seen that in the group of mental processes involved in the general process of Understanding, there are several stages or steps, three of which we have considered in turn, namely: (1) Abstraction; (2) Generalization or Conception; (3) Judgment.