From Calculus of Proof Functions to the Logic of Partial Terms

Abstract

In this chapter some derived logical constants, ∧, ●, ∀, >, and ⊂, are defined and their CPF properties are established. Next, formulae are introduced: these are expressions in the style of predicate calculus, built up from atomic formulae (which are simply terms) using the logical constants ∧, ∨, ⊂, ∃, and ∀. Every formula is interpreted as a proof function and thus inherits a notion of intuitionistic proof. The Calculus of Proof Functions (CPF), when restricted to interpreted formulae, turns into Logic of Partial Terms (LPT). LPT is an axiomatisation of first-order intuitionistic predicate calculus with equality, reduction and induction, adapted to take account of the fact that not all terms have values. Every theorem of LPT (without free variables) has an intuitionistic proof.