Then () ((_ having two external universals, is valid, and gives ).). That is, from " For every z there is an a which is v, and for every x there is a Y which is not a," we deduce'' Sonic as are not vs." Again, 0) Thas an external and an internal universal, and given ( (. "For every a there is an x which is T, and every v is z" gives Every z is x. And )( (*(,haring an external universal and a particular, with middle term Of different quantities in the two, is valid, and gives )). That is, from "For every a there is something neither x nor T; some vs are not as," we deduce " Some as are not xs." Few propositions that ever come before the human mind require closer attention than these transposed syllogisms. They are easily demonstrated from the general forms of numerical syllogism ; not so easily by independent thought.
We shall now proceed to compare the mathematical and metaphysical sides of logic. A term may be formed by junction of terms in two ways. 1. By aggregation, as in what we represent by (a, c), which stands for all that is in a, in xi, in c, or in two or more. 2. By composition, as in what we represent by (x -n-c), or (ape), meaning all that is common to all the three terms A, B, c. The contrary of an aggregate is the compound of the contraries of the aggregants : either (A, n) or (a b). The contrary of a compound is the aggregate of the contraries of the components : either (a a) or (a, b).
The term, consitlered as aggregated of aggregants, is viewed in extent ; as a compound of components, in intent. For total and partial we now substitute the words full and vague, to get rid of the wrong opposition which total and partial suggest ; for the term called partial is not necessarily partial at all, it is only not known to be tot.al. And this ambiguity has led acute logical writers into absolute mistake. In this matter we follow Hamilton, who justly and acutely remarked that " It is only as indefinite that particular, it is only as definite that indi vidual and general, quantities have any (and the smile) logical avail." The word indefinite completely describes the word particular, or partial, as commonly used ; we only use vague as a shorter word. But the universal or total of logicians is more than definite : it is the maximum of definiteness, and we use full as more correctly descriptive. Tire wraiths is as definite as all.
Extent and intent arc the two logical tensions. The elements of
extent are aggregants ; the elements of intent are components. When one tension is full, tho other ie vague; when one tension is vague, the other is full. Thus x) signifies that x is taken fully in extent, vaguely in intent ; and )x signifies that x is taken vaguely in extent, and fully in intent. When a tension is full, an existing element may be dis missed, but a new one cannot be admitted ; when a tension is vague, no element can be dismissed, but any new one may be admitted. Thus A,B)) I' and x )).An allow A )) Y and x ))A.; but not A,n,C))Y nor x ) ) A 13 c. Again, A ) ) and x ) ) A, 13 do not allow A ) ) Ynor x ) ) A ; but they do allow A c ) ) and x ) ) D, c.
Elements may, under certain conditions, be transposed from one term to another, without alteration of the import of the proposition. Universal propositions allow the elements of vague terms to be trans posed : particular propositions allow the elements of full terms to be transposed. In universal propositions the transposition is made directly in negatives, by contraversion in affirmatives. In particular proposi tions the transposition is made directly in affirmatives, by contraversion in:negatives. As instances x ) ) A, B gives xa)) n; .A.B))Y gives ) ) Y, a; AD)(y gives n)(AY; An( )Y givesn( )A Y; A n((Y gives B ( . ( y,A, B ( )Y gives A( ) Y, B, &c. And in all these cases the result of transposition is identical with the original.
Both extent and intent are to be considered in both sides of logic, the mathematical and the metaphysical A class may be an aggregate or a compound ; the class animal is aggregate of man and brute ; the class animal is compound (common part of ) the class body and the class living. An attrihute may be an aggregate or a compound : the attrihute useful is aggregate of the naturally and the artificially useful : the attribute useful is compounded of, or at least has among its com ponents, attainable and applicable. But the conveniences of thought, though not its necessities, and whether by mere habit or by human constitution is not a question of logic, dictate an almost exclusive confinement of the notion of aggregation to that of class, and of the notion of composition to that of attribute. Accordingly, intent is of predominant importance in the metaphysical side of logic, and extent in the mathematical.