The following calculus of propositions is based in its essentials upon that given in Principia Mathematica *10. But it differs from the latter in several important respects. The primitive propositions of Principia are not completely formalised, for they contain such words as "proposition," "function," etc., which refer to the meaning of the symbolism, and without an understanding of which the calculus cannot be developed. The reduction of the primitive propositions of *1 to an abstract set of postulates has been understood for some time.' Our paper extends this abstract postulation to the functional calculus. In reformulating our postulates we have followed Russell and Whitehead as closely as possible; we have retained symbols which are real variables representing individuals and predicates, which form the elements of the system; conjunction, negation, implication etc. still appear as operations on these elements. Thus we have not followed the more revolutionary methods of Schbnfinkel2 and Curry' in which all variables are eliminated, and the logical operations themselves become the elements of the system. Our system differs from that of the authors of Principia in that the postulates can be understood without interpreting the elements and operations as the undefined terms of intuitional logic; and from that of the formalists in that our postulates and theorems concern, not the symbols themselves, but the objects which they symbolise; though their interpretation is not prescribed within the postulates. The chief obstacle to this formal reduction is the peculiar nature of the apparent variable. For in the functional calculus, in addition to propositions of the form ox (x has the property 4) we need expressions of the form (x) * ox (every x has the property 4). In Principia this is treated as an operation on Ox, and called generalization of ox. Admirable though this symbolism is for practical use, it raises a theoretical difficulty. For the "x" in ox directly represents some object, while in (x) -(kx the x does not directly represent an object, but rather a collection of objects. So long as the symbolism contains apparent variables, it is impossible to reduce the primitive propositions to an abstract set of postu-