Key Metalogical Propositions on a Variant of Hilbert’s Epsilon-Calculus

Hilbert’s 𝜀-operator, a foundational device for forming indefinite descriptions, has long been overshadowed by standard quantifiers in first-order logic. However, its capacity to eliminate quantifiers and reframe logical derivations makes it a compelling tool for alternative proof strategies and automated reasoning. This paper revisits the 𝜀-calculus, offering a streamlined proof of completeness adapted from Hasenjaeger’s 1953 approach. Building on earlier work by Leisenring, Davis, and Fechter, we present a variant of the 𝜀-calculus that omits all predicate symbols aside from equality. The development follows the conventional structure of logical systems—syntax, semantics, and deductive calculus—culminating in a soundness and completeness result. The aim is to reaffirm the 𝜀-operator’s relevance in the foundations of logic through a simplified and accessible formal treatment.