@TECHREPORT{Laan94aformalization, author = {Twan Laan}, title = {A formalization of the Ramified Type Theory}, institution = {}, year = {1994} }

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Abstract

In "Principia Mathematica " [17], B. Russell and A.N. Whitehead propose a type system for higher order logic. This system has become known under the name "ramified type theory". It was invented to avoid the paradoxes, which could be conducted from Frege's "Begriffschrift" [7]. We give a formalization of the ramified type theory as described in the Principia Mathematica, trying to keep it as close as possible to the ideas of the Principia. As an alternative, distancing ourselves from the Principia, we express notions from the ramified type theory in a lambda calculus style, thus clarifying the type system of Russell and Whitehead in a contemporary setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see e.g. [3]. In these formalizations, and also when defining "truth", we will need the notion of substitution. As substitution is not formally defined in the Principia, we have to define it ourselves. Finally, the reaction by Hilbert and Ackermann in [10] on the