The Entropy-Limit (Conjecture) for Σ₂-Premisses [The Entropy-Limit (Conjecture) for Sigma(2)-Premisses]

The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: (i) applying it to finite sublanguages and taking a limit; (ii) comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same probabilities. While the conjecture is known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some Sigma(2)-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture.