OF ARTS AND SCIENCES. 305 



means the negation of some other, we may express our premises in 

 negative form. Thus, in general, A — < B (every A is B) may be 

 expressed A b -< (A's which are b do not exist). This we express 

 upon our machine by pressing down the letter lieys A and b, and then 

 the destruction key. The falling of the A b pointers indicates exactly 

 the change effected in the logical universe by the introduction of the 

 premise A -<; B. We may then continue to impress as many prem- 

 ises as we please, until all the pointers have fallen. The following 

 formulae* will suffice to illustrate the manner in which premises may 

 be reduced from the positive to the negative form. 



Positive Form. Negative Form. 



(1.) A-< B Ab-<0. 



(2.) A+B-<0 \i:^a 



(3.) A-< B-{- O . . . . . Abc-<0. 



(4.) A-<,+ CD . . . . {ll^:^^,. 



(5.) AB-< O A Bc-<0. 



(«■) ^-<^o {jf:^^; 



(-.) A(B+c-)-<D . . . {:J^j:<2: 



(8.) A-<B(C+D) . . . {il^_^-o. 



Having expressed our premises upon the machine, and their effect 

 being recorded by the pointers, it only remains for us to read off the 

 conclusion. The entire conclusion is represented by the fallen point- 

 ers, and might be expressed as their joint sum ; or it may be viewed 

 as the logical sum of the combinations represented by the horizontal 

 pointers. Thus, the premises of Barbara, A -< B and B -< C, give 

 as the entire conclusion read negatively, 



AbC 



f 



A b c{ ^ 



ABc^ ^ ^• 

 a B c 



* The sign -< is tiiat used by Mr. C. S. Poirce for tlie general sign of in- 

 ference. A -<. B means, if A, then D. Viewed in the light of class extension, 

 it means the class A is included in the class D. The sign of addition is here 

 tised in the non-exclusive sense ; thus, A -\- B means either A or B, or both. 

 The expression A B means, when designating a class, tlie individuals whicli 

 belong to both classes A and B ; when designating a quality, the combination 

 of the qualities A and B. 



VOL. XXI. (n. s. xiii.) 20 



