Equivalence of WFT_dec with WFT_pred plus DDNS_C

Establish whether the Weak Fan Theorem with decidable paths (WFT_dec) is equivalent to the conjunction of the Weak Fan Theorem with predicate-defined paths (WFT_pred) and the generalised disjunctive Double Negation Shift principle DDNS_C, in the intuitionistic settings considered for constructive completeness of classical first-order logic with possibly-exploding Tarski semantics.

Background

The paper reviews differing formulations of the Weak Fan Theorem (WFT) that arise from how infinite paths in binary trees are represented: a fully intuitionistic version using arbitrary predicates (WFT_pred) and a weakly classical version requiring decidability (WFT_dec). Prior work connects WFT (or Weak König’s Lemma classically) with completeness results in both classical and intuitionistic settings.

Building on these connections and their constructive completeness analysis (including disjunction via double-negation principles) together with results by Krivtsov and Espínola, the authors pose a concrete equivalence conjecture linking WFT_dec to WFT_pred augmented by the disjunctive double-negation shift DDNS_C.