Mr Wiener, A Simplification of the Logic of Relations. 387 
A Simplification of the Logic of Relations. 
By N. WIENER, Ph.D. (Communicated by Mr G. H. Hardy.) 
[Read 23 February 1914.] 
Two axioms, known as the axioms of reducibility, are stated on 
page 174 of the first volume of the Principia Mathematica of 
Whitehead and Russell. One of these, *12°1, is essential to the 
treatment of identity, descriptions, classes, and relations: the 
other, *12°11, is involved only in the theory of relations. +*12:11 
is applied directly only in 
*20°701°702'703 and *21:12°13:151:3°'701-702°703. 
It states that, given any propositional function ¢ of two variable 
individuals, there is another propositional function of two variable 
individuals, involving no apparent variables, and having the same 
truth-value as @ for the same arguments, or in symbols: 
Fis) $(@ y)-=-f1@ y). 
In *20 and *21:701°702'703 all that is done with *12°11 is to 
extend it to cases where the arguments of ¢ and fare classes and 
relations: *12°11 is essential to the development of the calculus of 
relations only owing to its application in *21:12:13:151'3. Here 
it is needed to make the transition between the definition of a 
binary relation and its uses. This is due to the fact that a binary 
relation itself is not defined, but only propositions about it, and 
*12:11 is needed to assure us that these propositions about it 
behave as if there were a real object with which they concern 
themselves. The authors of the Principia wish to treat a binary 
relation as the extension of a propositional function of two 
variables: that is, when they speak about the relation between « 
and y when ¢(2, y), they mean to speak of any propositional 
function which holds of those values of # and y, and only those 
values, of which ¢ holds. Now, as it leads one into vicious-circle 
paradoxes to speak directly of “any propositional function which 
holds of those values of « and y, and those only, of which ¢ holds,” 
they first define a proposition concerning the relation between # 
and y when ¢$(a, y) as a proposition concerning a@ propositional 
function involving no apparent variables which holds of « and y 
when and only when ¢(a, y). Then they need to use *12°11 
to assure us that, whatever ¢ may be, there always is some such 
propositional function. Now, if we can discover a propositional 
function of one variable so correlated with @ that its extension 
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