Principal Type-Schemes and Condensed Detachment

and introduction. The condensed detachment rule, or rule D, was first proposed by Carew Meredith in the 1950's for propositional logic based on implication. It is a combination of modus ponens with a "minimal" amount of substitution. We shall give a precise detailed statement of rule D. (Some attempts in the published literature to do this have been inaccurate.) The D-completeness question for a given set of logical axioms is whether every formula deducible from the axioms by modus ponens and substitution can be deduced instead by rule D alone. Under the well-known formulae-as-types correspondence between propositional logic and combinator-based type-theory, rule D turns out to correspond exactly to an algorithm for computing principal typeschemes in combinatory logic. Using this fact, we shall show that D is complete for intuitionistic and classical implicational logic. We shall also show that D is incomplete for two weaker systems, BCKand BCI-logic. In describing the formulae-as-types correspondence it is common to say that combinators correspond to proofs in implicational logic. But if "proofs" means "proofs by the usual rules of modus ponens and substitution", then this is not true. It only becomes true when we say "proofs by rule D"; we shall describe the precise correspondence in Corollary 6.7.1 below. This paper is written for readers in propositional logic and in combinatory logic. Since workers in one field may not feel totally happy in the other, we include short introductions to both fields. We wish to record thanks to Martin Bunder, Adrian Rezus and the referee for helpful comments and advice. ?1. Implicational logic. 1.1. DEFINITION. Propositional formulae (denoted by a, /3, y,...) are built from propositional variables (denoted by a, b, c,...) using the connective A. We assume there is an infinite sequence of propositional variables. There are no propositional constants in this paper, and no other connectives than -A. (Relaxing this restriction would complicate the correspondence with combinatory logic.} Received January 12, 1988; revised January 24, 1989. (D 1990, Association for Symbolic Logic 0022-4812/90/5501 -0007/$02.60