The Epsilon Calculus and Herbrand Complexity

Abstract

Abstract. Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.

Al-Sijistani's and Maimonides's double negation theology explained by constructive logic

“…According to this book, in Binah (Understanding), one of ten emanations (sefirot) through which God reveals himself, God exists as the great "Who" who stands at the beginning of each question. This is perfectly in line with the main idea behind Hilbert's epsilon calculus, when we introduce a symbol εx P whose meaning is as follows: if ∃x P (x), then εx P is an object for which the property P (x) is true, i.e., for which P (εx P ); see, e.g., [17]. Of course, this formal idea does not help us actually construct this object in situations when not such construction is possible, but it helps us apply techniques of constructive logic to such situations -i.e., in effect, similar to double negation, it helps us apply techniques of constructive logic in situations when usual finite algorithms do not lead to constructions.…”

Non‐elementary speed‐ups in logic calculi

“…[19,20,7,6,9]: 1. If P is an FP-proof of a sequent in which no predicate for fixed points occurs, we can find a proof FP(P ) of the same sequent in first-order logic.…”

“…As in [6,9] the rank of second-order epsilon terms is defined to be the nesting of cross-binding of epsilons, and the degree to be the nesting of applications of epsilons as functionals.…”

The Epsilon-Reconstruction of Theories and Scientific Structuralism

“…39 Let us briefly look at two different accounts to deal with this kind of semantic indeterminedness based on two versions of eliminative structuralism.…”