THE SCIENCES OF THE IDEAL 165 



world, equivalent in formal structure to the foregoing, but composed 

 of a totality of possible statements, or thirdly, it is the world, equiva- 

 lent once more, in formal structure, to the foregoing, but consisting 

 of a totality of possible acts of will, of possible decisions. When we 

 proceed to consider the relational structure of such a world, taken 

 merely in the abstract as such a structure, a relation comes into 

 sight which at once appears to be peculiarly general in its nature. 

 It is the so-called illative relation, the relation which obtains between 

 two classes when one is subsumed under the other, or between two 

 statements, or two decisions, when one implies or entails the other. 

 This relation is transitive, but may be either symmetrical or not 

 symmetrical; so that, according as it is symmetrical or not, it may 

 be used either to establish levels or to generate series. In the order 

 system of the logician's world, the relational structure is thus, in 

 any case, a highly general and fundamental one. 



But this is not all. In this the logician's world of classes, or of 

 statements, or of decisions, there is also another relation observable. 

 This is the relation of exclusion or mutual opposition. This is a 

 purely symmetrical or reciprocal relation. It has two forms 

 obverse or contradictory opposition, that is, negation proper, and 

 contrary opposition. But both these forms are purely symmetrical. 

 And by proper devices each of them can be stated in terms of the 

 other, or reduced to the other. And further, as Kempe incidentally 

 shows, and as Mrs. Ladd Franklin has also substantially shown in 

 her important theory of the syllogism, it is possible to state every 

 proposition, or complex of propositions involving the illative relation, 

 in terms of this purely symmetrical relation of opposition. Hence, 

 so far as mere relational form is concerned, the illative relation itself 

 may be wholly reduced to the symmetrical relation of opposition. 

 This is our first result as to the relational structure of the realm of 

 pure logic, that is, the realm of classes, of statements, or of deci- 

 sions. 



It follows that, in describing the logician's world of possible classes 

 or of possible decisions, all unsymmetrical, and so all serial, relations 

 can be stated solely in terms of symmetrical relations, and can be entirely 

 reduced to such relations. Moreover, as Kempe has also very prettily 

 shown, the relation of opposition, in its two forms, just mentioned, 

 need not be interpreted as obtaining merely between pairs of objects. 

 It may and does obtain between triads, tetrads, n-ads of logical en- 

 tities; and so all that is true of the relations of logical classes may 

 consequently be stated merely by ascribing certain perfectly sym- 

 metrical and homogeneous predicates to pairs, triads, tetrads, n-ads 

 of logical objects. The essential contrast between symmetrical 

 and unsymmetrical relations thus, in this ideal realm of the logi- 

 cian, simply vanishes. The categories of the logician's world of 



