The culmination of Gerhardt's work lay in his 1853 theory of types, the last of the ongoing attempts at organic classification by Laurent and Gerhardt.
Only if one regards the theory of types as a logical truth, and about that there is much more room for doubt than there was about the trivial truisms upon which Russell had originally intended to build mathematics.
In their place he substituted a bewilderingly complex theory known as the ramified theory of types, which, though it successfully avoided contradictions such as Russell's Paradox, was (and remains) extraordinarily difficult to understand.
By the time he and his collaborator, Alfred North Whitehead, had finished the three volumes of Principia Mathematica (1910 13), the theory of types and other innovations to the basic logical system had made it unmanageably complicated.
It will also show what we desire from a theory of types, by exhibiting the facts that a theory of types has to accommodate.
And Linsky regiments the epistemic principles and reasoning with a theory of types.
This view of set theory gave rise to the Theory of Types.
To block this sort of circularity, we can turn to the ramified theory of types.
The full theory can be seen as a development of the simple theory of types.
Russell blocks the contradiction in the Principles by the introduction of a theory of types.
It will also illuminate some of the things we want from a theory of types.
