MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 475 



Thus, R : (; >- ) denotes the class of terms, such as x (say), such that there exist terms, 

 y and z say, such that ~R-(xyz) holds. 



Again, R : (;;z) denotes the class which is the logical sum o/R ; (;-z) and R : (-;z); 

 and R ; (;; - ) denotes the class which is the logical sum q/"R ; (;") and R ; ( - ; - ); and 

 R : (;;;) denotes the class which is the logical sum of R ; (;") and R ; (v) and R ; (--;), 

 that is, the "field" of the relation R. The symbolic definitions of the above, and of 

 similar entities, are : 



R'(;;z) = R'(;-z)uR'(-;z) Df 



(;y;) = R'(;jr)uR ; (-y;) Df 



R'(z;;) = R'(* ;>&(*;) Df 



R'(;;-) = R i ("')uR i (-;-) Df 



arid so on, and 



R'(;;;) = R'(;--)uR'(-;-)uR'(--;) Df 



This notation, which has been explained for triadic relations, can obviously be 

 extended to any polyadic relations. Thus, ~R-(abcd) and ~R'(abcdt) are defined in a 

 similar manner, and so are the symbols for the allied propositions and classes. 



On 1 1. 



A dyadic relation, S say, is called one-one, when each referent has only one relatum, 

 and each relatum has only one referent. The class of one-one relations is denoted 

 by 1 *!. The symbolic definition is 



Df 



On I- 



The Assertion Sign. A proposition, which is stated in symbols as being true, i.e., 

 which is asserted as distinct from being considered, has the symbol " prefixed to it, 

 with as many dots following as will serve to bracket off the proposition. This 

 symbol I" is called the assertion sign* 



PART II.- THE PUNCTUAL CONCEPTS. 



Those concepts of the material world in which the class of objective reals is 

 composed of points, or particles, or of both, will be called the punctual concepts. 

 The classical concept is a punctual concept, and will be considered first. The other 

 punctual concepts can be explained briefly by reference to the classical concept. 



Concept I. (The Classical Concept). This is dualistic, the class of objective reals 



* This symbol is due to FREGE, who first drew attention to the necessity of the idea which it 

 symbolizes; cf. his ' Begriffschrift,' HALLE, 1879, and RUSSELL, ' Principles of Mathematics,' p. 35. 



3 P 2 



