Representation of Propositions and Reasonings. 169 



of five terms, but with, problems of six, seven, eight, or even 

 more terms ; and it does so because it does not oblige one to 

 take into separate consideration all those perplexing possibili- 

 ties with which Mr. Venn's and similar methods are hampered. 

 That the readers of this Magazine may be able to judge fairly 

 as to the respective capabilities of Mr. Yenn's method and 

 mine, I will first solve one of his four-letter problems, and 

 then a six-letter problem of my own, which though exceed- 

 ingly easy by my method, would, if I am not greatly mistaken, 

 subject his diagrammatic method to a severe strain. 



" Every X is either Y or Z ; every Y is either Z or W ; 

 every Z is either W or X; and every W is either X or Y: 

 what further condition, if any, is needed to ensure that every 

 XY shall be W?" 



This is a special case of the following more general pro- 

 blem : — 



Given a series of implications, A: a, B : b, C : c, &c; what 

 is the weakest implication that need be added to these data to 

 justify the inference m : n ? 



The answer is ran' : Ka r + ~BV + CV + . . . 



When A, a, B, b, &c. are complex expressions involving m 

 or n or both, great simplification may be effected by substitu- 

 ting in these expressions 1 for m and nf, and therefore for w! 

 and n. In Mr. Yenn's problem the data are (when expressed 

 in my notation) 



x : y + z, y : z + w, z : iv + x, w : x+y, 



and the weakest addition to the premises to justify the infer- 

 ence xy : iv is therefore 



xyw' : xy'z' -Vyz'w 1 + zw' x' + wx r y f , 



Substituting 1 for every x, y, and w r (and therefore for every 

 x' , y f , and w) in the consequent of this implication, the impli- 

 cation becomes xyw' : z f , which is equivalent to xyw'z : 0, or 

 xyz : to, the result required. In actual practical working these 

 substitutions of unity and zero would be made mentally while 

 writing down the consequent of the required implication, so 

 that the result may fairly be said to follow directly from mere 

 inspection of the data. 



This and the other problems given by Mr. Venn are much 

 too easy : the following problem, involving six letters, would 

 be a fairer test of the power of his method ; and I should much 

 like to see his solution of it. 



Taking ax-\-by : cdf as the symbolical expression of the 

 statement "whenever the event A happens with X, or B with 

 Y, then C happens without D," and so on for similar state- 



