Recursive characterization of the NSFP theorems that inherently require strict finitistic negation

Characterize recursively the set SFQp \ (⋃_{n=0}^∞ Ψ^{n}[SFQ_P^{-¬}]), i.e., the set of theorems of the prevalent strict finitistic logic (the NSFP theorems) that cannot be obtained from the negation-free NSFP theorems by finitely many applications of the operator Ψ which replaces an occurrence of B by ¬B inside a formula context, thereby identifying precisely which NSFP theorems inherently require the strict finitistic negation connective.

Background

The paper studies interchangeability between intuitionistic-style weak negation (→ ⊥) and strict finitistic negation (¬) in prevalent models. It shows that while every NSFP theorem can be obtained from a ¬-free theorem by iteratively replacing weak negations with strict negations in suitable contexts (closure under the Φ operation), the converse fails: not all strict-negation theorems are so reducible via inserting ¬ (the Ψ operation).

The author observes that the complement of ⋃_{n=0}^∞ Ψ^{n}[SFQ_P^{-¬}] within SFQp is nonempty but lacks a recursive characterization, leaving open the problem of precisely describing which NSFP theorems fundamentally require strict finitistic negation.