Epsilon calculus helps fix partial and flawed mathematical proofs
Towards a Proof-Theoretic Analysis of Incorrect/Incomplete Proofs
Logic in Computer Science
Summary
Sometimes, mathematical proofs have mistakes or gaps but still hold some truth. The authors study how such imperfect proofs can be analyzed and fixed using a logic system called the epsilon calculus. They show that this approach allows identifying when errors can be repaired and how to recover valid conclusions from flawed reasoning. Their work extends traditional logic methods to include handling errors and corrections in proofs.
What this means in practice
- •For automated theorem provers: Improve theorem provers by enabling them to recognize and repair partial errors in proofs to produce valid conclusions.
- •For formal verification engineers: Enhance formal verification systems to tolerate incomplete or flawed proofs while preserving semantic correctness for safety-critical software.
A theory result. No direct application yet.
Authors
Matthias Baaz, Mariami Gamsakhurdia
Abstract
We investigate the proof-theoretic structure of incorrect/incomplete proofs, that is, derivations containing syntactic errors or incomplete inferential steps that nonetheless preserve partial semantic validity. Building on Hilbert's epsilon calculus, we formalize how such derivations can be corrected through semantic projection and weakest preconditions, leading to valid Herbrand disjunctions. We show that the epsilon calculus provides a natural framework for analyzing tolerance of falsity in proofs and for identifying conditions under which an incorrect proof can be semantically repaired. This approach extends Hilbert's program beyond correctness, toward a logic of error and recovery. Moreover we show that the extended first epsilon theorem is false-tolerant.