We show that in Polyadic Pure Inductive Logic the Invariance Principle, based on consideration of symmetry with respect to automorphisms, has only a trivial solution, namely the polyadic equivalent of Carnap’s \(c_0\) . (This extends a result proved earlier in the unary case.) We then consider the Exchangeable Invariance Principle, a symmetry principle which is a weakening of the Invariance Principle and has been proven to be strictly stronger than the Permutation Invariance Principle. We show that the Exchangeable Invariance Principle follows from Spectrum Exchangeability, a principle not obviously based on symmetry but based on irrelevance, that is treating certain features as irrelevant for the purpose of belief assignment, and that the converse does not hold. We conclude that Spectrum Exchangeability is the strongest currently known rational principle of belief assignment in Pure Inductive Logic that does not lead to the aforementioned trivial solution.
