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upon consideration, it will be found that the conversion of the (universal) affirmative proposition-i. e., conversion per accidens-is not an exception to the rule, but an application of it; for the process consists simply in substituting for the predicate another term precisely equivalent to the subject in signification, as, 1.g., in the proposition “ Y is X,” the expression
some X” for “ X,” meaning, by the expression some X,” that part of X which coincides with Y; which is but an application of Rule 3. And when this substitution is made, the proposition becomes equational, and means the same thing whether we convert it or not.
$ 80. OF IMMEDIATE INFERENCES GENERALLY.-Propositions derived from other propositions by conversion, and also those derived by opposition (explained infra, $ 89), are regarded by recent logicians as inferences, and to distinguish them from syllogistic inferences are called immediate. This innovation we regard as unfortunate, though of too general use to be neglected, for, according to our view, only one kind of inference is allowed, namely, syllogistic. This, as we have shown, includes the case of conversion per accidens; and it also includes other, and perhaps all, cases of socalled immediate inference; as may be readily shown.
(1) SUBSTITUTION OF CONTRADICTORY.– One of these is what is called by Bishop Thompson, “ Immediate Inference by Means of Privative Conceptions,” and by other logicians, improperly, “Infinitation.” It is, in fact, identical with the process treated hereafter under the head of “Conversion by Contraposition" (S 91). It consists in substituting for the predicate its negative, or contradictory, and in changing the quality of the proposition,i. e., making the copula of the negative proposition affirmative, or that of the affirmative proposition negative. Thus, denoting the terms by the capital letters Y and X, and their negatives or contradictories by aY and aX, the negative proposition “ Y is not X" may be converted into the affirmative proposition, “Y is aX”; and similarly the affirmative proposition, “ Y is X,” into the negative proposition, “Y is not aX" (i. e., is not Not-X). The validity of the process, as may be illustrated by the following diagrams, rests upon ciple that any negative proposition, as, e.g., “Y is not X," may always be regarded either as denying that the class Y is included in the class X, or as affirming that it is included in the class aX, or “ Not-X”; and conversely the affirmative proposition, “ Y is X,” may be regarded either as affirming that the class Y is included in the class X, or as
the prindenying that it is included in the class ax, or "Not-X."
But when from the affirmative proposition “ Y is X” we conclude that “ Y is not NotX," there is a syllogistic inference; which, denoting the negative or contradictory of X by aX, may be thus expressed:
X is not aX (i.e., not Not-X)
Y is X .:. Y is not aX.
The inference, therefore, rests upon the judgment that the term “X” is equivalent to the term “ Not-aX,” and consists in substituting the latter for the former. Hence the principle of inference involved may be stated generally by saying that a term is always equivalent in signification to the contradictory of its contradictory, or, as otherwise expressed, the negative of its negative; which is but a different expression of the maxim that “ two negatives make an affirmative."
It is, indeed, said that the major terms in the two propositions are the same--the propositions differing only in quantity, and hence that no third term is introduced. But this is incorrect; for the major term in the former proposition is X, and in the latter “not Not-X”; and it is a fundamental logical doctrine that no two terms are identical that differ, either in denotation or connotation, or vocal sign; and also that the very essence of ratiocination consists in the recognition of identity of signification in terms having different connotations or vocal signs, and in the substitution of the one for the other (SS 77 et seq.).
(2) IMMEDIATE INFERENCE BY ADDED DETERMINANTS, AND (3) THE SAME BY COMPLEX CONCEPTIONS.—These kinds of supposed immediate inference were introduced into Logic by Leibnitz (Davis, Theory of Thought, p. 104). The former is stated in the proposition that the same mark may be added to both terms of a judgment; the latter, in the proposition that the two terms of a judgment may be added to the same mark. Of the former, the example given by Thompson is: “A negro is a fellowcreature," therefore, " A negro in suffering is a fellow-creature in suffering”; of the latter:
Oxygen is an element," and therefore, “ The decomposition of oxygen would be the decomposition of an element.'
The two processes seem to be in substance the same, and both may be expressed symbolically by saying that “If Y is X," then" ZY will be ZX," or (what ; as may be
is the same) “ YZ will be YX thus symbolically illustrated :
This process is erroneously regarded by logi. cians as an immediate inference; but it is, in fact, mediate, and may be stated in syllogistic form as follows:
Y is X ZY is Y .:. ZY is X
The conclusion “ZY is X,” fully expressed, is that ZY is that part of X with which it coincides; or, in other words, that“ ZY is ZYX." But ZYX is ZX; and hence ZY is ZX.
In this case the observations made with reference to infinitation (supra) will apply a fortiori; for here a new term, “ZY," is introduced, differing from Y in denotation, in connotation, and in verbal sign.
But the converse is not true, -i. e., from the proposition, ZY is ZX, we cannot infer that Y is X; as will appear from the following diagram :