« PreviousContinue »
ACD are triangles having two sides of the one equal to two sides of the other, each to each, and the included angle equal.
Conclusion.—They are therefore equal in all their corresponding parts, and hence B E= CD and the angle d= the angle e.
Major Premise. — Prop. IV.
Minor Premise. — The triangles C B E and B C D are triangles having two sides of the one equal to two sides of the other, each to each, and the included angle equal.
Conclusion.—They are therefore equal in all their corresponding parts, and hence the angle f = the angle g.
Major Premise, a=d - f. (Judgment, or intuitive proposition.)
Minor Premise, d-fre- g.
SUMMARY OF THE TRADITIONAL LOGIC
OF THE TRADITIONAL LOGIC GENERALLY
$ 85. As explained in the preface, one of the principal objects of this work is to vindicate, as against modern innovations, the old or traditional Logic; and accordingly, in all that has been said — with exceptions to be noted presently-I have kept close to the traditional view, as expounded by Aristotle and the most approved of the older logicians. I have, in. deed, repudiated the doctrine advocated by Whately, and by modern logicians generally, that would distinguish between the formal and the material relations of terms, and restrict the scope of Logic to the former; but in this also I follow Aristotle and the better authorities.
The only particulars, therefore, in which I have departed from the traditional view of Logic are: (1) that I reject the “ Particular Propositions " of the old Logic and those parts of the old doctrine of the Proposition and of the Syllogism that are founded on this view of the proposition; and (2) that I have adopted, in place of the Dictum, the Principle of Substitution ; which is an obvious corollary from the Dictum, and is more readily understood and applied.
At the same time, it must be admitted, the old doctrines of the Proposition and the Syllogism are remarkable for the accurate analysis upon which they rest, and the wonderful ingenuity and acuteness with which they have been developed. They have thus become part of the accepted philosophy of the world; and there has thus been developed a technical language that has come to be universally received and so generally used that, without an understanding of it, all the literature on the subject must be a closed book to us. I now propose, therefore, to give a brief exposition of these doctrines.
THE TRADITIONAL DOCTRINE OF THE PROPOSITION
$ 86. QUALTITY OF PROPOSITIONS.- Propositions are said to differ in quality accordingly as they are affirmative or negative. Thus the propositions “ All Y is X” and “ Some Y is X” are affirmative; the propositions “No Y is X" and " Some Y is not X," negative.
“ $ 87. QUANTITY OF PROPOSITIONS.-Again, propositions, whether affirmative or negative,
are said to differ in quantity accordingly as the predicate is asserted, or denied universally of all individuals of the class denoted by the subject or only part of such individuals. In the former case the subject is said to be distributed, and the proposition is called universal; in the latter, the subject is undistributed, and the proposition is said to be particular. Thus, l e. g., the propositions, “ All Y is X" and
· ” “No Y is X” are both universal; and the propositions, Some Y is X” and “ Some Y is not X,” both particular.
$ 88. TABLE OF PROPOSITIONS.- Hence, four forms of propositions are recognized by the old logicians, viz. : (1) the Universal Affirma. tive; (2) the Universal Negative; (3) the Particular Affirmative; and (4) the Particular Negative ; which are designated respectively by the letters A, E, I, and 0; and, with their expressions in Euler's Symbols, are as fol. lows, viz. : A: Y is X (i. e., All Y is X)
E: Y is not X (i. e., No Y is X)'
Some Y is X
O: Some Y is not X
1 The above differs somewhat from the ordinary notation; according to which it is thought necessary in A and E to use the signs “ All” and “No,” in order to indicate that the subject is distributed, as, e. g., “All Y is X,” “No Y is X." But, properly speaking, the signs “ All” and “No” are unnecessary and redundant. For when we say, e.g.,
In the negative propositions, E and O, it will be observed, the predicate is distributed or taken universally; in the affirmative propositions it is undistributed.
$89. OPPOSITION OF PROPOSITIONS.-Two propositions are said to be opposed to each other when, having the same subject and predicate, they differ in quantity or quality, or both.
Propositions that differ both in quality and quantity, as A and O, or E and I, are called contradictories, as, e. g.,“ Y is X," and " Some Y is not X”; or “Y is not X” and Some Y is X.” Those that differ in quality only, if
“Man is mortal,” or “ Man is not mortal," we mean, when we speak properly, that in the former case the class man" is wholly included in, and in the latter that it is wholly excluded from, the class “mortal"; or, in other words, as the case may be, that All men are mortal," or that “ No man is mortal" (8 53, n.). The last expression is also objectionable on account of the liability to confound the expression “No man with the term “Not-man" in converting either of the above propositions by contraposition (for which see in fra, & 91); or (more generally) the negative proposition “No Y is X" is liable to be confounded with the affirmative proposition, “Not-Y is X.” Hence it will be preferable to regard the subject as always distributed, except where it is preceded by the adjective “ some"; and, in place of the sign “no” before the subject, to use the particle “not" after the copula.