460 



SCIENCE. 



[N. S. Vol. XX. No. 510. 



sible logical classes, or again, it is the ideal 

 world, equivalent in formal structure to 

 the foregoing, but composed of a totality 

 of possible statements, or thirdly, it is 

 the world, equivalent once more, in formal 

 structure, to the foregoing, but consist- 

 ing of a totality of possible acts of ivill, 

 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 en- 

 tails the other. This relation is transitive, 

 but may be either symmetrical or not sym- 

 metrical; so that, according as it is sym- 

 metrical 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 ob- 

 servable. This is the relation of exclusion 

 or mutual opposition. This is a purely 

 symmetrical or reciprocal relation. It has 

 two forms— obverse or contradictory op- 

 position, i. e., negation proper, and con- 

 trary opposition. But both these forms are 

 purely symmetrical. And by proper de- 

 vices 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 substan- 

 tially shown in her important theory of the 

 syllogism, it is possible to state every propo- 

 sition, 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 tlie symmetrical relation of op- 

 position. This is our first result as to the 

 relational structure of the realm of pure 

 logic, i. e., the realm of classes, of state- 

 ments, or of decisions. 



It follows that, in describing the logi- 

 cian's world of possible classes or of pos- 

 sible 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. More- 

 over, 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, «-ads of logical 

 entities; and so all that is true of the rela- 

 tions of logical classes may consequently 

 be stated merely by ascribing certain per- 

 fectly symmetrical and homogeneous predi- 

 cates to pairs, triads, tetrads, m-ads of log- 

 ical objects. The essential contrast be- 

 tween symmetrical and unsymmetrical rela- 

 tions thus, in this ideal realm of the logician, 

 simply vanishes. The categories of the 

 logician's world of classes, of statements, 

 or of decisions, are marvelously simple. 

 All the relations present may be viewed as 

 variations of the mere conception of op' 

 position as distinct from non-opposition. 



All this holds, of course, so far, merely 

 for the logician's world of classes or of de- 

 cisions. There, at least, all serial order 

 can actually be derived from wholly sym- 

 metrical relations. But Kempe now very 

 beautifully shoM's (and here lies his great 

 and original contribution to our topic) — 

 he shows, I say, that the ordinal relations 

 of geometry, as well as of the number-sys- 

 tem, can all be regarded as indistinguish- 

 able from mere variations of those relations 

 which, in pure logic, one finds to be the 

 symmetrical relations obtaining within 

 pairs or triads of classes or of statements. 



