The present note suggests how the computation of complexity-values outlined in The logical simplicity of predicates may be so modified as to satisfy a more stringent requirement than was there imposed.

By the method of computation described in that article, a basis consisting of two two-place predicates has in general the value 6. Any such basis may be replaced by one four-place predicate that is self-complete with respect to two of its places and also with respect to the remaining two. Such a predicate has but one “joint,” and therefore a value of 5. The consolidation of the two predicates into one thus results in a basis having a lower complexity-value. This does not violate the cardinal rule laid down in that article for testing proposed ways of assigning complexity-values; for, conversely, a four-place predicate that is self-complete in the way described can always be replaced by two two-place prediates. The rule requires a four-place predicate of the kind described to have a higher value than two two-place predicates only if replacement of two two-place predicates by such a four-place predicate is always possible, while replacement of such a four-place predicate by two two-place predicates is not.

However, we might reasonably strengthen our cardinal rule for complexity-valuation by adding as a second requirement:

If each basis of kind A is always replaceable by some basis of kind B, and each basis of kind B is always replaceable by some basis of kind A, then a basis of kind A and a basis of kind B must, in the absence of contrary indications, have the same complexity-value.