1880.] representation of logical propositions. 51 



figures to illustrate special cases. Thus a collection of small circles 

 included in a large one represents a number of species (mutually 

 exclusive) comprehended under one genus, though, since the small 

 circles cannot fill up all the contents of the large one we cannot 

 thus conveniently represent the exhaustion of the genus by the 

 aggregate of the species; a row of such circles, each of them 

 interlinked with the next, represents the case of a succession of 

 species each of which has something in common with the next, 

 and so forth. 



The most ingenious of his figures for four terms is the follow- 

 ing. I give it here in order to show the necessary shortcomings 

 of this method : — 



It is offered in representation of the proposition " A which is B 

 is the same as C which is D". It must have taken some trouble 

 to arrange it, so that as regards economy of time any such resort 

 would be decidedly the reverse of an aid. Moreover, as the reader 

 will readily perceive, it is not quite correct ; one possible sub- 

 division, viz. ABCD, having been omitted. There is nothing in 

 the statement to forbid the occurrence of BD which is neither A 

 nor C, so that the correct state of things would be better exhibited 

 thus, on the plan described in my Article in the Philosophical 

 Magazine (July, 1880). 



With the exception of that of Bolzano, I have seen no attempt 

 to extend diagrammatic notation to the results of four terms, and 

 it is only quite recently that really adequate figures have been 

 proposed for those of three terms :— for instance both Drobisch 

 and Schroder have used what I have called, in the article in ques- 

 tion, the three-circle diagram*. In saying this, I do not of course 

 mean to imply that the problem was one of any particular diffi- 

 * These writers merely represent in this way the class combinations or 

 subdivisions as such : they do not adopt the subsequent step of using them an 

 a basis for representing propositions. 



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