234 H. m'coll on the gkowth and 



tainty respecting to what complex statement any element or 

 relational symbol belongs. Thus, the compojind statement 

 a{b + c), formed by the two factors a and b + c, is a. very 

 different statement from the disjunctive statement ab + c, 

 formed by the two terms ab and c. So, again, is [a : b + c)', 

 the denial of the whole implicational complex statement 

 a : b + c, a very different statement from a: b + c', in which 

 the symbol of denial affects only the element c. 



Reciprocal implications — that is, compound implications 

 of the form {a :b){b: a) — occur so frequently that a symbol 

 of abbreviation is convenient. Borrowing again from the 

 existent mathematical stock, we may use'^ either : : or =. 

 Thus, either the symbol a::b or the symbol a = b may be 

 taken as an abbreviation for the reciprocal implication 

 {a :b){b: a). 



The symbol f{x) has been already considered, and is 

 employed in logic in the same sense as in mathematics ; 

 that is to say, it denotes any statement whatever that con- 

 tains X as one of its constituents; but the symbol f{x;), 

 for which no logical meaning analogous to its mathematical 

 one is likely to turn up, may be conveniently employed as 

 an abbreviation for {f{x)}'. 



These are the only symbols that need be employed in 

 the system of symbolical logic which I advocate, and they 

 are amply sufficient not only for the complete solution of 

 any logical problem that I have ever seen solved by any 



* To avoid the employment of brackets and repetition as much as pos- 

 sible, it will be convenient to use both, vrith this distinction, that the symbols 

 : and : : should be coordinate (i. c. of equal reach in regard to the state- 

 ments affected by them), but both subordinate {i. e. of inferior reach) to the 

 symbol =. Thus, a:h+c::d-\-e:f::g is an abbi-eviation for the complex 

 statement 



{a:b+c){b + c:: d+c){d+c:f){f:: g), 



while a : b-\-c — d+c :/: :^ is au abbreviation for 



(«,;6 + ,.) = (cf + C;/::y). 



