1. Fairouz Kamareddine, Twan Laan & Rob Nederpelt (2002). Types in Logic and Mathematics Before 1940. Bulletin of Symbolic Logic 8 (2):185-245.
    In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead's Principia Mathematica ([71], 1910-1912) and Church's simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege's (...)
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  2. Fairouz Kamareddine & Twan Laan (2001). A Correspondence Between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems. Journal of Logic, Language and Information 10 (3):375-402.
    In Russell''s Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Church''s simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT. Despite the disappearance of orders which have a strong correlation with predicativity, predicative (...)
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  3. Twan Laan & Rob Nederpelt (1996). A Modern Elaboration of the Ramified Theory of Types. Studia Logica 57 (2-3):243 - 278.
    The paper first formalizes the ramified type theory as (informally) described in the Principia Mathematica [32]. This formalization is close to the ideas of the Principia, but also meets contemporary requirements on formality and accuracy, and therefore is a new supply to the known literature on the Principia (like [25], [19], [6] and [7]).As an alternative, notions from the ramified type theory are expressed in a lambda calculus style. This situates the type system of Russell and Whitehead in a modern (...)
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