1872.] Mr. A. J. Ellis's Contributions to Formal Logic. 307 



April 25, 1872. 

 GEORGE BIDDELL AIRY, C.B., President, in the Chair. 

 The following communications were read : — 



I. " Contributions to Formal Logic" By Alexander J. Ellis, 

 F,R.S., F.C.P.S., F.S.A., &c. Received March 21, 1872. 



(Abstract.) 



This paper contains the following contributions to Formal Logic : — 

 Statement of the Problem of Deductive Logic, with a classification of 

 its cases. 



Method of solution for three principal cases, of which the second has 

 not been hitherto formally considered. 



New notation and calculus of logical relations. 



Xew system of diagrams, adapted for typography, and coextensive with 

 the system of notation for cases of composition. 



Purely logical solution of all the problems within the scope of Boole's 

 system, with complete results, and without his mathematical analogies or 

 hazardous theories, with a proof that his primary and secondary proposi- 

 tions only partially, and not, as he assumes, wholly obey the same laws. 



Fusion of Hamilton's judgments and De Morgan's propositions, as part 

 of one system of assertions. 



Exhaustive analysis of the syllogism. 



Re-cast of the theory and notation of De Morgan's numerically definite 

 assertions, as the general case of the logic of composition, and a legitimate 

 application of algeb/a to logic. 



Direct passage from the purely logical formulae of consistency to the 

 mathematical formulae of Boole's system of probabilities. 



The above contributions are believed to be entirely original, and are 

 given with the least possible restatement of former theories ; but, for con- 

 venience, frequent reference has been made to Boole, De Morgan, Thom- 

 son as representing Hamilton, Ueberweg, and Jevons. Boole and De 

 Morgan have been constantly before my mind, and whatever is common to 

 this paper and their works must be credited to them. Jevons first led my 

 thoughts in this direction, but all resemblance between us is entirely 

 superficial. 



The problem of deductive logic as here conceived is : — Given any asser- 

 tions, to determine precisely what they affirm, precisely what they deny, 

 and precisely what they leave in doubt, separately and jointly. 



Assertions have respect to Coexistence or Succession, or both. 



Coexistence generates : 



Composition, arising from the coexistence of different attributes in the 

 same thing, and the coexistence of the same attribute in different things ; 



