Binary induction and Carnap’s continuum

We consider the problem of induction over languages with binary predicates and show that a natural generalization of Johnson’s Sufficientness Postulate eliminates all but two solutions. We discuss the historical context and connections to the unary case.

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

Translation Invariance and Miller's Weather Example

In his 1974 paper "Popper's Qualitative Theory of Verisimilitude" published in the British Journal for the Philosophy of Science David Miller gave his so called `Weather Example' to argue that the Hamming distance between constituents is flawed as a measure of proximity to truth since the former is not, unlike the latter, translation invariant. In this present paper we generalise David Miller's Weather Example in both the unary and polyadic cases, characterising precisely which permutations of constituents/atoms can be effected by translations. In turn this suggests a meta-principle of the rational assignment of subjective probabilities, that rational principles should be preserved under translations, which we formalise and give a particular characterisation of in the context of Unary Pure Inductive Logic

The Principle of Signature Exchangeability

We investigate the notion of a signature in Polyadic Inductive Logic and study the probability functions satisfying the Principle of Signature Exchangeability. In the binary case, we prove a representation theorem for such functions and show that they satisfy a binary version of the Principle of Instantial Relevance. We discuss polyadic versions of the Principle of Instantial Relevance and Johnson�s Sufficientness Postulate