In the present paper we investigate in some detail two systems closely related to those contained in [4]. The system K1 of [4] is simplified by eliminating conjunction and disjunction, giving rise to a system Σ whose primitives are two individuals, concatenation, identity, existential and limited universal quantification. Any relation expressible in Σ may be expressed in the form (Eα1)(β)α1(Eα2)…(Eαn)(γ = δ). A further reduction of primitives leads to a system Σμ, which differs from Σ by containing the operator μ (i.e. “the first string, such that” with respect to a certain ordering) in place of the quantifiers. Reasons are given for restricting Myhill's definition of constructive ideas to those expressible in Σμ.

Acquaintance with [4], [5], [6] is assumed. We deviate from [4] by dropping concatenation-signs and parentheses in chain matrices from the official notation, and, furthermore, by using Greek and German letters as metalinguistic variables standing for chain matrices and chains, respectively, while retaining small Roman letters (excluding a, b) for use as variables of the systems.