This paper is concerned with the logical structure of intuitionistic equality theories. We prove that De Jongh theorem holds for the theory of decidable equality, but uniform De Jongh theorem fails even for the theory of weakly decidable equality. We also show that the theory of weakly decidable equality is the weakest equality theory which enjoys Glivenko theorem.

This paper is a sequel to the papers Baaz and Iemhoff [4] and [6] in which an alternative skolemization method called eskolemization was introduced that, when restricted to strong existential quantifiers, is sound and complete for constructive theories. In this paper we extend the method to universal quantifiers and show that for theories satisfying the witness property it is sound and complete for all formulas. We obtain a Herbrand theorem from this, and apply the method to the intuitionistic theory of (...) equality and the intuitionistic theory of monadic predicates. (shrink)

Based on the paper [4] we show that true second‐order arithmetic is interpretable over the real‐algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras.

From classical, Fraïissé-homogeneous, ($\leq \omega$)-categorical theories over finite relational languages, we construct intuitionistic theories that are complete, prove negations of classical tautologies, and admit quantifier elimination. We also determine the intuitionistic universal fragments of these theories.

We prove some preservation theorems concerning inductive and model-complete theories in the framework of semi-classical logic introduced in [1].

In this paper an alternative Skolemization method is introduced that, for a large class of formulas, is sound and complete with respect to intuitionistic logic. This class extends the class of formulas for which standard Skolemization is sound and complete and includes all formulas in which all strong quantifiers are existential. The method makes use of an existence predicate first introduced by Dana Scott.