I'd like to conclude by emphasizing what a wonderful eld this is to work in. Logical reasoning plays such a fundamental role in the spectrum of intellectual activities that advances in automating logic will inevitably have a profound impact in many intellectual disciplines. Of course, these things take time. We tend to be impatient, but we need some historical perspective. The study of logic has a very long history, going back at least as far as Aristotle. During some of this time not very much progress was made. It's gratifying to realize how much has been accomplished in the less than fty years since serious e orts to mechanize logic began.

Abstract. The Curry-Howard correspondence connects Natural Deduction derivation with the lambda-calculus. Predicates are types, derivations are terms. This supports reasoning from assumptions to conclusions, but we may want to reason ‘backwards ’ from the desired conclusion towards the assumptions. At intermediate stages we may have an ‘incomplete derivation’, with ‘holes’. This is natural in informal practice; the challenge is to formalise it. To this end we use a one-and-a-halfth order technique based on nominal terms, with two levels of variable. Predicates are types, derivations are terms — and the two levels of variable are respectively the assumptions and the ‘holes ’ of an incomplete derivation. 1