Simple proof of the completeness theorem for second order classical and intuitionictic logic by reduction to first-order mono-sorted logic

TL;DR

This paper introduces a simplified method to prove the completeness theorem for second-order classical and intuitionistic logic by reducing it to first-order mono-sorted logic, streamlining existing proofs.

Contribution

It provides a new, simpler reduction technique to derive completeness for second-order logics from first-order logic, applicable to both classical and intuitionistic cases.

Findings

Simplified proof of the second-order classical logic completeness theorem

Extension of the method to second-order intuitionistic logic

Reduction to first-order mono-sorted logic as a unifying approach

Abstract

We present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic.