We investigate extensions of Heyting's predicate calculus (HPC). We relate geometric properties of the trees of Kripke models (see [2]) with schemas of HPC and thus obtain completeness theorems for several intermediate logics defined by schemas. Our main results are:

(a) ∼(∀x ∼ ∼ϕ(x) Λ ∼∀xϕ(x)) is characterized by all Kripke models with trees T with the property that every point is below an endpoint. (From this we shall deduce Glivenko type theorems for this extension.)

(b) The fragment of HPC without ∨ and ∃ is complete for all Kripke models with constant domains.

We assume familiarity with Kripke [2]. Our notation is different from his since we want to stress properties of the trees. A Kripke model will be denoted by (Aα, ≤ 0), α ∈ T where (T, ≤, 0) is the tree with the least member 0 ∈ T and Aα, α ∈ T, is the model standing at the node α. The truth value at α of a formula ϕ(a1 … an) under the indicated assignment at α is denoted by [ϕ(a1 … an)]α.

A Kripke model is said to be of constant domains if all the models Aα, α ∈ T, have the same domain.