Let Δ be a set of axioms of a theory Tc(Δ) of classical predicate calculus (CPC); Δ may also be considered as a set of axioms of a theory TH(Δ) of Heyting's predicate calculus (HPC). Our aim is to investigate the decision problem of TH(Δ) in HPC for various known theories Δ of CPC.
Theorem I(a) of §1 states that if Δ is a finitely axiomatizable and undecidable theory of CPC then TH(Δ) is undecidable in HPC. Furthermore, the relations between theorems of HPC are more complicated and so two CPC-equivalent axiomatizations of the same theory may give rise to two different HPC theories, in fact, one decidable and the other not.
Semantically, the Kripke models (for which HPC is complete) are partially ordered families of classical models. Thus a formula expresses a property of a family of classical models (i.e. of a Kripke model). A theory Θ expresses a set of such properties. It may happen that a class of Kripke models defined by a set of formulas Θ is also definable in CPC (in a possibly richer language) by a CPC-theory Θ′! This establishes a connection between the decision problem of Θ in HPC and that of Θ′ in CPC. In particular if Θ′ is undecidable, so is Θ. Theorems II and III of §1 give sufficient conditions on Θ to be such that the corresponding Θ′ is undecidable. Θ′ is shown undecidable by interpreting the CPC theory of a reflexive and symmetric relation in Θ′.