USE OF A SYMBOLICAL LANGUAGE, 233 



The denial of any assertion may be conveniently de- 

 noted by an c^ccent, thus : — Let x denote the statement, 

 " He is in England ; " then x^ will denote '' He is not 

 in England." 



The statements hitherto spoken of are simple or ele- 

 mentary statements — that is, statements represented each 

 by a single letter, or a single letter and an accent. Any 

 statement that requires more than one letter to express it 

 may be called a complex statement. The principal rela- 

 tions by virtue of which simple statements combine into 

 complex ones are three — namely, conjunction, disjunction, 

 and implication, corresponding respectively to the three 

 conjunctions and, or, if. The first relation is generally 

 symbolized (like multiplication in ordinary algebra) by 

 simple juxtaposition, and occasionally, though never 

 necessarily, by the symbol x; the second (like addition in 

 ordinary algebra) by the symbol + ; and the third by the 

 symbol : , as in the following examples : — 



Let X denote the statement " He will go to Paris ; " and 

 let y denote the statement " I shall go to York.^^ Then 

 xy denotes the compound statement " He will go to Paris 

 and I shall go to York ; " the symbol x-\-y denotes the 

 disjunctive statement " He will go to Paris or I shall go 

 to York ; " and x : y denotes the implication " If he goes 

 to JParis I shall go to York." 



A compound statement, as ab c, claims the symbol i for 

 every one of its factors; a disjunctive statement, as 

 a-\-b + c, claims the symbol i for one at least of its terms; 

 and the implicational statement a : b claims the symbol i 

 for the consequent b, provided the antecedent a is entitled 

 to it, but neither claims nor disclaims it for Z> if « is not 

 entitled to it. 



Brackets are used when necessary to collect the different 

 elements of a complex statement, and so prevent any unccr- 



