« PreviousContinue »
that “ No X is Y.” So with I (“Some Y is X''),—as both subject and predicate are undistributed,- the proposition may be simply converted, i. e., if “ Some Y is X,” then it is necessarily true that“ Some X is Y."
By one or the other of these methods, i. e., either simply or per accidens, all propositions of the forms A, E, and I may be converted. But O (“Some Y is not X") cannot be thus converted, Thus, e.g., it cannot be inferred from the proposition Some Greeks are not Athenians” that “Some Athenians are not Greeks.” But such conversion may be effected by simply regarding the negative particle (not) as part of the predicate; by which expedient O is changed into I, and may be simply converted, as, l.g., “Some Greeks are Not-Athenians”; which may be converted into the proposition “ Some Not-Athenians are Greeks." So from the proposition "Some men are not learned," though we may not infer that “ Some learned are not men,” we may infer that “Some unlearned are men.”
This is called by the old logicians “Conversion by Contraposition,” and by Whately, “ Conversion by Negation.”
This method of conversion is applicable to A and E as well as O, and, as it is of very extensive use, we append a table of such conversions, taken, with some alterations, from De Morgan (Formal Logic, p. 67). In this table
(altering De Morgan's notation) the original terms of the proposition are denoted by the capital letters Y and X, and their contraries respectively by prefixing the Greek privative a. We append also for illustration the symbolical expressions for the several propositions:
A: “Y is X”; “Y is not aX”; “aX is not Y”;
“aX is aY":
The righteous are happy
E: “Y is not X"; "Y is aX"; "Some aX is Y";
“Some aX is not a Y." “X is not Y”; “X is aY”; “Some aY is X”;
Some aY is not aX":
Perfect virtue is not human
Human virtue is not perfect
Some imperfect virtue is not unhuman.
aX is Y”; Some aX is not aY”:
Some possible cases are not probable
possible. It will be observed from the above table that a universal affirmative proposition can always be converted into another universal affirmative between the contradictories of its original terms by simply reversing the order of the terms and substituting for them their contradictories.
$ 92. OF MATERIAL CONVERSIONS.-It will be observed that the conversions of propositions treated by logicians have regard to the distinction, heretofore explained, between the formal and the material relations of terms (8 66 (2)), and are confined exclusively to what may be called formal conversions, i. e., to cases where the equivalence of the converted and original propositions results from the formal or general relations of terms. But conversions of propositions based upon the material relations of terms are of essentially the same nature, as, e. g., where the proposition “ John is the son of William " is converted into the proposition “ William is the father of John"; or the proposition “ Cain murdered Abel ” into the proposition “ Abel was murdered by Cain," or into the proposition “ Cain is the man that
murdered Abel." These, having regard to the received distinction between the formal and the material relations of terms, may be called material conversions; and are infinitely the more numerous class, and equally deserving of attention. But though conversions of this kind are in constant use, and though, indeed, we cannot proceed a step in our logical processes without them, yet the subject has received but little attention, and remains as yet a vast, unexplored domain.' It can only be said, therefore, in the present condition of logical doctrine, that as the distinction between the formal and the material relations of terms has been found unessential, so must the distinction between formal and material conversions be regarded. Both classes of conver
"To this domain belong such subjects as the “Categories,” “ Intensive Propositions,' Hypothetical Propositions,” and, in short, all forms of expression that differ from the ordinary logical proposition. With these Logic is concerned only in so far as is involved in their conversion into logical forms. Otherwise, neither the Intensive nor the Hypothetical Logic (if we may give either the name) can be regarded as part of Logic as traditionally received; which is based exclusively upon the logical form of the proposition and its extensive interpretation. With regard to the Hypothetical Logic, it will be observed, it has no place in Aristotle's treatises; and Mansel is of the opinion—in which I agree—that in this he showed a juster notion of the scope of Logic than his successors. The subject is well treated in the current works on Logic, and is worthy of some attention from the student.
sions rest equally for their validity simply upon judgments as to the equivalence of expressions.
THE TRADITIONAL DOCTRINE OF
$ 93. The following epitome of the doctrine of the syllogism as traditionally received, brief as it is, will—with what has already been said - be found amply sufficient to expound it. It will, indeed, require the same close attention and thought as is usually given to mathematical demonstrations; but it may be said that to those who are unwilling to give, or are incapable of giving, to it this kind of thought, the study of Logic cannot be of much benefit.
1. Of the Moods and Figures of the Syllogism
$94. MOODS OF THE SYLLOGISM.—The syllogism is said to be in different moods, according to the occurrence and arrangement in it of the several forms of the proposition-A, E, I, and O; as, l. g., in the syllogism “ Y is X, Z is Y,.. Z is X,” which consists of three universal affirmative propositions, and is, therefore, said to be in the mood A A A.
The four forms of the proposition, A, E, I, O, may be arranged, in sets of three each, in sixty-four different ways, but upon examination it is found that of these there are eleven