1880.] expressing the common propositions of Logic. 43 



(For Grassmann's scheme, see his Begriffslehre, 1872 ; for that 

 of Mr Maccoll, the Proc. Lond. Math. Soc, 1877; and for that of 

 Prof. Peirce, the American Journal of Math., Vol. III.) 



15. Frege's scheme (Begriffsschrift, 1879) deserves almost as 

 much to be called diagrammatic as symbolic. It is one of those 

 instances of an ingenious man working out a scheme, — in this 

 case a very cumbrous one, — in entire ignorance that anything of 

 the kind had ever been achieved before. A word or two only of 

 explanation can be devoted to it here. A horizontal dash with a 

 short vertical stroke at the end signifies a proposition. The line 

 8 running into P means that P is dependent upon S ; — this is in 

 fact his sign of dependence or implication. The little stroke 

 under the P-line marks negation. So that the whole arrangement 

 stands for ' If S then not P.' We can proceed in this way to build 

 up more complicated dependencies. For instance, by joining this 

 whole arrangement on to another such line, we can represent the 

 compound dependency ' The fact that S implies the absence of 

 P, implies P,' and so on. One obvious defect in this scheme is 

 the inordinate amount of space demanded by it ; nearly half a 

 page is sometimes demanded for an implication which any reason- 

 able scheme would compress into half a line. 



The members of the group now before us, expressing a relation 

 between S and not-P, are essentially non-convertible, and therefore 

 appropriately employ non-symmetrical symbols for the copula*. 

 In this respect they depart somewhat from tradition and each is 

 the scheme of a logical innovator. The next group keeps closer to 

 old custom, in respect that its members express directly a relation 

 between S and P, and therefore call for a symmetrical copula-symbol. 



16 — 18. These three of course employ purely arbitrary sym- 

 bols, and are meant to do so, the symbols being mere substitutes 

 for the copula of ordinary Logic. Wundt's symbol is one of a 

 group (Logik, p. 244) some of which mark reciprocal relations 

 between the terms, and some non-reciprocal, and its symmetrical 

 form is meant to shew that it belongs to the former class. Thus 

 S=P marks identity; S>P superordination of 8 to P\ S<P 

 subordination of S to P, and S][P the intersection of S and P. 

 These, with S)(P, make up the five possible distinct forms of 

 class relations. To these however Wundt adds some others 

 which are not so much class relations as dependencies or impli- 

 cations. De Morgan, I suspect, had not this distinction between 



• No. (13) is of course non-symmetrical in its recognized signification in 

 mathematics, though symmetrical in actual form. 



