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:
A CONTRARIES E +————————+ |\ / | | \ /S | | C\ /E | | O\ /I | | N\ /R | | T\ /O | S| R\ /T |S U| A\ /C |U B| \ /I |B A| \ /D |A L| \ / |L T| / \ |T E| / D\ |E R| / I\ |R N| /A C\ |N S| /R T\ |S | /T O\ | | /N R\ | | /O I\ | | /C E\ | | / S\ | |/ \ | +————————+ I SUB-CONTRARIES O
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.