Equivalences between Pure Type Systems and systems of Illative Combinatory Logic

Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt’s lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led to four kinds of PTSs. Most importantly PTSs are about statements of the form M : A, where M is a term and A a type. In ICLs there are no explicit types and the statements are terms. In this paper we show that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold.