Induction of permutations and the invariance condition

ABSTRACT: Set-theoretic objects to be judged for invariance are of various kinds, some quite simple, others of high structural complexity. Formation and/or transformation processes may be conceived, which go through classes of objects of varying complexity; these processes are paralleled by induction of permutations which act on the classes and become critical in judging the invariance of objects. This paper studies the connection between formation and induction processes, by presuming four basic ways of passing from class to class (restriction, transformation, the domain and codomain of a set-theoretic power) and four corresponding ways of permutation induction. The theoretical system thus defined is then applied in examining specifications of the invariance condition for various kinds of set-theoretic constructs (subsets, families of subsets, relations, operations, patterns, statistical measures, decision rules, etc.) which are encountered when discussing, for example, meaningfulness in measurement theory, symmetry of perceptual patterns, and properties of rules for statistical inference, and are thus of concern to some parts of psychological science.