Skolemization in intuitionistic logic

In [2] an alternative skolemization method called eskolemization was introduced that is sound and complete for existence logic with respect to existential quantifiers. Existence logic is a conservative extension of intuitionistic logic by an existence predicate. Therefore eskolemization provides a skolemization method for intuitionistic logic as well. All proofs in [2] were semantical. In this paper a proof-theoretic proof of the completeness of eskolemization with respect to existential quantifiers is presented. Keywords: Skolemization, eskolemization, orderization, Herbrand’s theorem, intuitionistic logic, existence logic, Gentzen calculi

Eskolemization in intuitionistic logic

In [2] an alternative skolemization method called eskolemization was introduced that is sound and complete for existence logic with respect to existential quantifiers. Existence logic is a conservative extension of intuitionistic logic by an existence predicate. Therefore eskolemization provides a skolemization method for intuitionistic logic as well. All proofs in [2] were semantical. In this paper a proof-theoretic proof of the completeness of eskolemization with respect to existential quantifiers is presented

The eskolemization of universal quantifiers

This paper is a sequel to the papers [4, 6] in which an alternative skolemization method called ekolemization was introduced that, when applied to the strong existential quantifiers in a formula, is sound and complete for constructive theories. Based on that method an analogue of Herbrand’s theorem was proved to hold as well. In this paper we extend the method to universal quantifiers and show that for theories satisfying the witness property the method is sound and complete for all formulas. We prove a Herbrand theorem and, as an example, apply the method to several constructive theories. We show that for the theories with a decidable quantifier-free fragment, also the strong existential quantifier fragment is decidable. Keywords: Skolemization, eskolemization, Herbrand’s theorem, constructive theories, intuitionistic logic, decidability