Epsilon Calculi

This paper covers the history of the development of various epsilon calculi, and their applications, starting with the introduction of epsilon terms by Hilbert and Bernays. In particular it describes the Epsilon Substitution Method and the First and Second Epsilon Theorems, the original Epsilon Calculus of Bourbaki, several Intuitionistic Epsilon Calculi, and systems that have been constructed to incorporate epsilon terms in modal, and general intensional structures. Standard semantics for epsilon terms are discussed, with application to Arithmetic, and it is shown how epsilon terms give distinctive theories of descriptions and identity, through providing complete individual terms for individuals, which are rigid across possible worlds. The Epsilon Calculus' problematic thereby extends that of the predicate calculus primarily through its applicability to anaphoric reference, both in extensional and also intensional constructions. There are higher-order applications, as well, some of which resolve paradoxes in contemporary logic through allowing for indeterminacy of reference.