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On Skolemization in constructive theories
 Journal of Symbolic Logic
"... In this paper a method for the replacement, in formulas, of strong quantifiers by functions is introduced that can be considered as an alternative to Skolemization in the setting of constructive theories. A constructive extension of intuitionistic predicate logic that captures the notions of preorde ..."
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In this paper a method for the replacement, in formulas, of strong quantifiers by functions is introduced that can be considered as an alternative to Skolemization in the setting of constructive theories. A constructive extension of intuitionistic predicate logic that captures the notions of preorder and existence is introduced and the method, orderization, is shown to be sound and complete with respect to this logic. This implies an analogue of Herbrand’s theorem for intuitionistic logic. The orderization method is applied to the constructive theories of equality and groups. 1
The eskolemization of universal quantifiers
, 2009
"... 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 ..."
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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 quantifierfree fragment, also the strong existential quantifier fragment is decidable. Keywords: Skolemization, eskolemization, Herbrand’s theorem, constructive theories, intuitionistic logic, decidability.