Towards the entropy-limit conjecture

• Published in: Annals of pure and applied logic Vol. 172; no. 2; p. 102870
• Main Authors: Landes, Jürgen, Rafiee Rad, Soroush, Williamson, Jon
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2021
• Subjects: Default models; Inductive logic; Maximum entropy; Normal models; Objective Bayesianism; Probabilistic constraints on predicate languages; 03A10; 03B42; 60A05; 03B60; Objective Bayesianism; Probabilistic constraints on predicate languages; Inductive logic; Normal models; Default models; Maximum entropy
• ISSN: 0168-0072
• DOI: 10.1016/j.apal.2020.102870

The maximum entropy principle is widely used to determine non-committal probabilities on a finite domain, subject to a set of constraints, but its application to continuous domains is notoriously problematic. This paper concerns an intermediate case, where the domain is a first-order predicate language. Two strategies have been put forward for applying the maximum entropy principle on such a domain: (i) applying it to finite sublanguages and taking the pointwise limit of the resulting probabilities as the size n of the sublanguage increases; (ii) selecting a probability function on the language as a whole whose entropy on finite sublanguages of size n is not dominated by that of any other probability function for sufficiently large n. The entropy-limit conjecture says that, where these two approaches yield determinate probabilities, the two methods yield the same probabilities. If this conjecture is found to be true, it would provide a boost to the project of seeking a single canonical inductive logic—a project which faltered when Carnap's attempts in this direction succeeded only in determining a continuum of inductive methods. The truth of the conjecture would also boost the project of providing a canonical characterisation of normal or default models of first-order theories. Hitherto, the entropy-limit conjecture has been verified for languages which contain only unary predicate symbols and also for the case in which the constraints can be captured by a categorical statement of Σ1 quantifier complexity. This paper shows that the entropy-limit conjecture also holds for categorical statements of Π1 complexity, for various non-categorical constraints, and in certain other general situations.