The consistency of a system of combinatory restricted generality

Abstract

The system , proposed without a claim as to its consistency in [CLg, §15C], has turned out to entail a consequence which is extremely suspicious. For from the axiom ⊦LH it follows thatholds for arbitrary X; hence for X ≡ YH, where Y is a fixed-point combinator, we haveWhether or not this leads to an actual contradiction it would be interesting to know; but, no matter whether it does or not, (2) seems highly counterintuitive.The aim of this note is to point out that a related system, here called , is demonstrably consistent and sufficient for all practical purposes served by , and at the same time to correct an error in [CLg, §15D2], which was discovered too late to be corrected on the proofs. This is formed from by dropping H from the list of θ's, and hence deleting ⊦LH and (1). The new system does not allow inferential rules to be converted into formulas with the same ease as does ; but if one is content with stating the results as rules, is adequate for all the main results deduced for in [CLg, §15C–D].