 Although today we have more clearly delimited systems of abstraction and rewriting, in its early days \(\lambda\)-calculus and combinatory logic (à la Schönfinkel) were bound up with investigations of foundations of mathematics. In the hands of Curry, Church, Kleene, and Rosser (some of the pioneers in the subject) the focus was on defining mathematical objects and carrying out logical reasoning inside the these new systems. It turned out that these early attempts at so-called illative \(\lambda\)-calculus and combinatory logic were inconsistent. Curry isolated and polished the inconsistency; the result is now known as Curry’s paradox. See the entry on  Curry’s paradox  and appendix B of (Barendregt, 1985).