Substitutions of Σ10-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic Original Research Article

This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ10-sentences over Heyting arithmetic (HA). On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ10-substitutions over HA coincides with NNIL-preservativity over intuitionistic propositional logic (IPC). Here NNIL is the class of propositional formulas with no nestings of implications to the left. The identical embedding of IPC-derivability (considered as a preorder and, thus, as a category) into a consequence relation (considered as a preorder) has in many cases a left adjoint. The main tool of the present paper will be an algorithm to compute this left adjoint in the case of NNIL-preservativity. In the last section, we employ the methods developed in the paper to give a characterization the closed fragment of the provability logic of HA.