Program Semantics and Classical Logic

Definition 6 (Types) The set of types is the smallest set such that:

1. al l basic types are types;

2. if α and β are types then α → β and α × β are types.

Each type may correspond to a set of objects of that type. In this way
hierarchies of type domains, called (standard) frames are obtained.

Definition 7 (Standard Frames) A (standard) frame is a set of non-empty
sets {D_α | α is a type} such that, for all α, β

Dt = {0, 1};

Dα→β ⁼ Dα ^→ Dβ ^;

Dα×β = Dα × Dβ.

Frames will be the backbones of our models. We shall use them in order to
interpret the language which we are about to define. For each type α, let us
assume the existence of a denumerably infinite set of variables V ARα and
some set of constants CON_α . From these basic terms complex ones can be
built.

Definition 8 (Terms) Define, for each α, the set Tα of terms of type α by
the following induction.

1. CONα ⊆ Tα, VARα ⊆ Tα;

2. A ∈ Tα→β, B∈Tα ⇒ A(B) ∈Tβ;

3. A∈Tβ, ξ∈VARα⇒λξ.A∈Tα→β;

4. A∈Tα, B∈Tβ⇒ hA, Bi ∈ Tα×β;

5. A∈ Tα×β ⇒ (A)0 ∈Tα and (A)1 ∈ Tβ;

6. A∈Tα, B∈Tα⇒ (A=B) ∈Tt.

If A ∈ Tα we may (but need not) indicate this by writing Aα . Terms of type
t are called formulae. In (A)0 and (A)1 the parentheses are often omitted if
this is not likely to lead to confusion. Note that the first five clauses in the
above definition give the syntax of a simply typed λ-calculus with pairing and

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