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.

中文翻译：

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$$\Sigma _2$$-前提的熵极限（猜想）

应用最大熵原理来确定有限域上的概率是很好理解的。它在无限领域的应用仍然缺乏一种经过充分研究的综合方法。将最大熵原理应用于一阶谓词语言有两种不同的策略：(i) 将其应用于有限子语言并取一个极限；(ii) 比较在整个语言上定义的概率函数的有限熵。熵极限猜想粗略地说，这两种策略导致相同的概率。虽然已知该猜想适用于一元语言以及仅包含存在量词或仅包含全称量词的前提句子，但其对于具有更大量词复杂性的前提句子的状态通常是未知的。我在这里表明，第一种方法无法为某些 $$\Sigma _2$$ -premiss 句子提供合理的答案。我讨论了这种失败对第一个策略的影响以及熵极限猜想的后果。