An observation on Carnapʼs Continuum and stochastic independencies

We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnsonʼs Sufficientness Principle holds, thus giving Carnapʼs Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnapʼs Continuum for the language with just a single predicate.