 Although Russell first introduced his theory of types in his 1903 Principles of Mathematics, he recognized immediately that more work needed to be done since his initial account seemed to resolve some but not all of the paradoxes. Among the alternatives he considered was a so-called substitutional theory (Galaugher 2013). This in turn led to type theory’s more mature expression five years later in Russell’s 1908 article, “Mathematical Logic as Based on the Theory of Types,” and in the monumental work he co-authored with  Alfred North Whitehead,  Principia Mathematica (1910, 1912, 1913). Russell’s type theory thus appears in two versions: the “simple theory” of 1903 and the “ramified theory” of 1908. Both versions have been criticized for being too ad hoc to eliminate the paradox successfully. 