236 H, M^COLL ON THE GROWTH AND 



question in dispute is this, does the conditional statement 

 " If a is true h is true/^ as I define and symbolize it, con- 

 vey a meaning in any way different from the disjunctive 

 statement '' Either a is false or b is true/^ as I define and 

 symbolize it ? 



My argument in ' Mind ^ was^ that since the denial of 

 the firsts namely " a may be true without b being so,^' 

 conveys less information than '' a is true and b is false/' 

 which is the denial of the second^ the conditional disjunc- 

 tive statements of which these are the respective denials 

 cannot be equivalent. As the non-equivalence of the 

 denials^ however^ is much more evident than that of the 

 affirmative statements, it will be weU worth while to give, 

 if possible, a more direct proof of the non-equivalence of 

 the latter. 



As it can easily be shown that a i-b is equivalent to 

 I : a' -\-b, the question may be narrowed to this, is the 

 implication i : a, in which a denotes a' + b, equivalent to 

 the simple affirmation « ? It seems to me that i : a and a 

 differ in pretty much the same way as the statement " It 

 is well-known that tin is heavier than zinc," and the 

 simpler affirmation ^'^Tin is heavier than zinc;''' that is to 

 say, the former implies the latter, but is not implied in it. 

 The statement i : a, in addition to claiming the symbol i 

 for itself, asserts that its protege a. has fairly made good 

 its right to it ; whereas a only claims this symbol on its 

 own account. The symbol (i : a)', which is the denial of 

 I : a, maybe read, ^'^ais not necessarily true /'' whereas a', 

 the denial of a , is much stronger, and asserts positively that 

 a is false. It follows from the law of logic called contra- 

 position that the denial of the weaker (or implied) state- 

 ment is stronger than and implies the denial of the stronger 

 (or implying) statement, ■ 



The disjunctive statements of ordinary language may be 



