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.