  Before he became aware of Russell’s paradox, Frege attempted to construct a logical foundation for mathematics. Using the logical system containing Basic Law V (1893/1903), he attempted to demonstrate the truth of the philosophical thesis known as logicism, i.e., the idea not only that mathematical concepts can be defined in terms of purely logical concepts but also that mathematical principles can be derived from the laws of logic alone. But given that the crucial definitions of mathematical concepts were stated in terms of extensions, the inconsistency in Basic Law V undermined Frege’s attempt to establish the thesis of logicism. Few philosophers today believe that mathematics can be reduced to logic in the way Frege had in mind. Mathematical theories such as set theory seem to require some non-logical concepts (such as set membership) which cannot be defined in terms of logical concepts, at least when axiomatized by certain powerful non-logical axioms (such as the proper axioms of Zermelo-Fraenkel set theory). Despite the fact that a contradiction invalidated a part of his system, the intricate theoretical web of definitions and proofs developed in the Grundgesetze nevertheless offered philosophical logicians an intriguing conceptual framework. The ideas of  Bertrand Russell  and  Alfred North Whitehead  in  Principia Mathematica  owe a huge debt to the work found in Frege’s Grundgesetze.