We establish a completeness theorem for first-order basic predicate logic BQC, a proper subsystem of intuitionistic predicate logic IQC, using Kripke models with transitive underlying frames. We develop the notion of functional well-formed theory as the right notion of theory over BQC for which strong completeness theorems are possible. We also derive the undecidability of basic arithmetic, the basic logic equivalent of intuitionistic Heyting Arithmetic and classical Peano Arithmetic.
"Basic Predicate Calculus." Notre Dame J. Formal Logic 39 (1) 18 - 46, Winter 1998. https://doi.org/10.1305/ndjfl/1039293019