Translation Invariance and Miller’s Weather Example

Abstract

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.

Six Problems in Pure Inductive Logic

“…However, for polyadic languages PIP is genuinely a new principle. Apart from its justification inherited from INV, it has two other quite different claims to rationality: as shown in [19] and [16] (or [18]) respectively, it can be expressed equivalently as the Translation Invariance Principle TIP and as Nathanial's Invariance Principle NIP.…”

“…TIP asserts that any sentence should get the same probability as its 'translations', see [19]. NIP asserts that state descriptions with the same 'structures', see [16,18], should have the same probabilities.…”