Invariance Principles in Polyadic Inductive Logic

We show that the Permutation Invariance Principle can be equivalently stated to involve invariance under finitely many permutations, specified by their action on a particular finite set of formulae. We argue that these formulae define the polyadic equivalents of unary atoms. Using this we investigate the properties of probability functions satisfying this principle, in particular, we examine the idea that the Permutation Invariance Principle provides a natural generalisation of (unary) Atom Exchangeability. We also clarify the status of the Principle of Super Regularity in relation to invariance principles