Intuitionistic Completeness of First-Order Logic

TL;DR

This paper proves the completeness of intuitionistic first-order logic by linking provability to uniform validity in minimal logic, providing an effective proof procedure, and resolving a long-standing open question.

Contribution

It establishes a new completeness theorem for iFOL using an intuitionistic proof method and implements an effective procedure Prf for converting evidence into formal proofs.

Findings

Proves that a formula is provable iff its embedding is uniformly valid.

Provides an effective procedure Prf for converting evidence into proofs.

Resolves Beth's open question from 1947.

Abstract

We establish completeness for intuitionistic first-order logic, iFOL, showing that a formula is provable if and only if its embedding into minimal logic, mFOL, is uniformly valid under the Brouwer Heyting Kolmogorov (BHK) semantics, the intended semantics of iFOL and mFOL. Our proof is intuitionistic and provides an effective procedure Prf that converts uniform minimal evidence into a formal first-order proof. We have implemented Prf. Uniform validity is defined using the intersection operator as a universal quantifier over the domain of discourse and atomic predicates. Formulas of iFOL that are uniformly valid are also intuitionistically valid, but not conversely. Our strongest result requires the Fan Theorem; it can also be proved classically by showing that Prf terminates using Konig's Theorem. The fundamental idea behind our completeness theorem is that a single evidence term evd witnesses the uniform validity of a minimal logic formula F. Finding even one uniform realizer guarantees intuitionistic validity because Prf(F, evd) builds a first-order proof of F, establishing its intuitionistic validity and providing a purely logical normalized realizer. We establish completeness for iFOL as follows. Friedman showed that iFOL can be embedded in minimal logic (mFOL) by his A-transformation, mapping formula F to FA. If F is uniformly valid, then so is FA, and by our completeness theorem, we can find a proof of FA in minimal logic. Then we intuitionistically prove F from FFalse, i.e. by taking False for A and for \bot of mFOL. Our result resolves an open question posed by Beth in 1947.