Principia mathematica, vol. 1 (of 3)

In this number, the propositions proved for one variable in *10 are to be extended to two variables, with the addition of a few propositions having no analogues for one variable, such as *11·2·21·23·24 and *11·53·55·6·7. "" stands for a proposition containing and containing ; when and are unassigned, is a propositional function of and . The definition *11·01 shows that "the truth of all values of " does not need to be taken as a new primitive idea, but is definable in terms of "the truth of all values of ." The reason is that, when is assigned, becomes a function of one variable, namely , whence it follows that, for every possible value of , "(" embodies merely the primitive idea introduced in *9. But "(" is again only a function of one variable, namely , since has here become an apparent variable. Hence the definition *11·01 below is legitimate. We put:

In addition to the definition *11·01, we need the primitive proposition that "whatever possible argument may be, is true whatever possible argument may be" implies the corresponding statement with and interchanged. Either may be taken as the meaning of " is true whatever possible arguments and may be."

This proposition is useful because it analyses a proposition containing two apparent variables into two propositions which each contain only one. "" is a function of two variables, but is compounded of two functions of one variable each. Such a function is like a conic which is two straight lines: it may be called an "analysable" function.