Hilbert's epsilon as an Operator of Indefinite Committed Choice

TL;DR

This paper proposes a new, flexible semantics for Hilbert's epsilon operator that supports indefinite and committed choice, aligns with natural language interpretation, and enhances proof search in classical logic.

Contribution

It introduces a semantics for the epsilon operator that avoids overspecification, supports indefinite and committed choice, and integrates well with proof search and natural language semantics.

Findings

Supports indefinite and committed choice

Enhances proof search efficiency

Aligns with natural language referential interpretation

Abstract

Paul Bernays and David Hilbert carefully avoided overspecification of Hilbert's epsilon-operator and axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the epsilon-operator underspecified. In the meanwhile, there have been several suggestions for semantics of the epsilon as a choice operator. After reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the epsilon supports proof search optimally and is natural in the sense that it does not only mirror some cases of referential interpretation of indefinite articles in natural language, but may also contribute to philosophy of language. Finally, we ask the question whether our epsilon within our free-variable framework can serve as a paradigm useful in the specification and computation of semantics of discourses in natural language.