Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms

Abstract

The Curry-Howard correspondence connects derivations in natural deduction 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 a partial derivation, with holes.

This is natural in informal practice but it can be difficult to formalise. The informal act of filling holes in a partial derivation suggests a capturing substitution, since holes may occur in the scope of quantifier introduction rules. As other authors have observed, this is not immediately supported by the lambda-calculus. Also, universal quantification requires a ‘fresh name’ and it is not immediately obvious what formal meaning to assign to this notion if derivations are incomplete. Further issues arise with proof-normalisation; this corresponds with lambda-calculus reduction, which can require alpha-conversion to avoid capture when beta-reducing, and it is not immediately clear how to alpha-convert a name in an incomplete derivation. We apply a one-and-a-half level technique based on nominal terms to construct a Curry-Howard correspondence for first-order logic. This features two levels of variable, but with no lambda-abstraction at the second level. Predicates are types, derivations are terms, proof-normalisation is reduction — and the two levels of variable are, respectively, the assumptions and the holes of an incomplete derivation.

We give notions of proof-term, typing, alpha-conversion and beta-reduction for our syntax. We prove confluence, we exhibit several admissible rules including a proof that instantiation of level two variables is type-safe — this corresponds with the act of filling holes in an incomplete derivation, and can be viewed as a form of Cut-rule — and we explore the connection with traditional Curry-Howard in the case that the derivation is in fact complete.

Our techniques are not specifically tailored to first-order logic and the same ideas should be applicable without any essential new difficulties to similar logical systems.