MATHEMATICAL CONCEPTS OF THE MATEKIAL WOELD. 473 



substituted for x, makes (j>lx to be a true proposition. It is not necessary for the 

 above symbolism that the proposition involving x should be in the symbolic form 

 <j>\x. Again, L'X denotes the class possessing x as its sole member, and i'x denotes 

 the sole member of the class x, and uuv denotes the logical sum of u and v, that is, the 

 class possessing all members of u and all members of v and no other members. Thus, 

 i'aui'& denotes the class whose sole members are a and b. Again, unv denotes the 

 logical product of u and v, that is, the complete common subclass of u and v; and u'w 

 denotes the class which is the logical sum of all members of u, that is, the class which 

 has as members all members of members of u; and n'w denotes the class which is 

 the logical product of all members of u. The exact symbolic definition of n'n is 



n'w = x {v f. u . o v . x e v} Df 



It follows from this definition that, if u possess no members, r\'u is the class of all 

 entities. 



On A, els', -, Nc' 



Again, A denotes the null class, that is, the class with no members ; els'", denotes 

 the class ivhose members are the subclasses contained in u, including u itself and the 

 null class. It follows that the propositions, vecls'u and vcu, have practically 

 identical meanings. Again, u-v denotes the class u with the exception of those 

 members which it possesses in common ivith v. 



The cardinal numbers* are themselves classes. Thus, 1 is the class whose members 

 are the unit classes, 2 is the class whose members are couples. Accordingly, xc2 

 means x is a class with two members ; Nc'it denotes the cardinal number of the 

 class u. 



On fi, fi", i,", i", u", and so on. 



The general form for a non-propositional function whose value depends on x is fix, 

 where fi receives different forms for different functions, as has been illustrated by the 

 particular cases considered above. The apostrophe may be read as "of"; it is the 

 general symbol for the connection of the preceding functional sign with the succeeding 

 argument. According to this rule we should write sin'x for sin x and log'x for log ,r. 

 Again, fi"u denotes the class of values of fix, ivhen the various members of u are 

 substituted for x ; it may be read " the class of fi's of u's." Thus, if fix is " the head 

 of x," and u is "the class of horses," then fi'u is " the class of heads of horses." The 

 exact symbolic definition of fi'u is as follows : 



fi'u = z {(333) . xeu . z = fix} Df 



It follows from the definition, by substituting for fi, that i", i"u, u", n", cls'X 

 Nc"w are now defined. 



* Cf. RUSSELL, 'Principles of Mathematics,' chap. XL, and FKEGE, 'Grundlagen der Arithmetik,' 

 Breslau, 1884, pp. 79, 85. 



VOL. CCV, A, 3 P 



