The Logical Simplicity of Predicates

1. The problem in general. In an earlier article,' I proposed a way of determining the relative simplicity of different sets of extralogical primitives. The calculations assumed a fully platonistic logic, committed to an indefinite hierarchy of classes, with sequences and relations defined as classes. Recently2 it has been shown that a nominalistic logic, countenancing no entities other than individuals, can be made to serve many of the purposes for which a platonistic logic had been thought necessary. The question naturally arises how we are to determine the simplicity of extralogical bases of systems founded upon a nominalistic logic. In such bases, the only extralogical predicates will be predicates (of one or more places) of individuals. The present paper offers a general method of measuring the simplicity of such bases. We have first to consider by what standards we are to judge any proposed assignment of complexity-values. To reduce all our intuitive demands to precise principles would obviously be too difficult; but one basic demand is very roughly set forth as follows: The assignment of complexity-values to predicates must be such that if each basis of kind A is always replaceable by some basis of kind B, while it is not the case that each basis of kind B is always replaceable by some basis of kind A, then any basis of kind B must, in the absence of other indications, have a higher complexity-value3 than any basis of kind A. The "always" indicates that the replacement must not depend on additional information that may sometimes not be available. To illustrate the principle, every basis consisting of one one-place predicate is always replaceable by some basis consisting of one two-place predicate; for example, we may adopt as primitive the predicate "Q"4 so explained that Q(x, y) if and only if P(x) and P(y) and then define: "P(x) =,if Q(x, x)."5 But it is not the case that every two-place predicate can always be replaced by a one-place predicate. Accordingly, two-place predicates must have a higher value than one-place predicates. This admittedly vague principle has to be used with the greatest caution.6