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.