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 ${\mathrm{\Sigma }}_1$ quantifier complexity. This paper shows that the entropy-limit conjecture also holds for categorical statements of ${\mathrm{\Pi }}_1$ complexity, for various non-categorical constraints, and in certain other general situations.

最大熵原理被广泛用于确定有限域上的非承诺概率，但要受一组约束的约束，但是其在连续域中的应用却出了问题。本文涉及一种中间情况，其中域是一阶谓语。提出了两种策略来在这种域上应用最大熵原理：（i）将其应用于有限的子语言，并随着子语言的大小n的增加，对结果概率进行逐点限制；（ii）从整体上选择一种概率函数，该函数对于大小为n的有限子语言的熵不被足够大的n的任何其他概率函数的熵所控制。熵极限猜想说，在这两种方法产生确定概率的情况下，这两种方法产生相同的概率。如果发现这个猜想是正确的，那将有助于寻求单一规范归纳逻辑的项目—当Carnap在此方向上的尝试仅成功确定了一系列归纳方法时，该项目便步履蹒跚。猜想的真相还将促进提供一阶理论的正常或默认模型的规范表征的项目。

迄今为止，已经对仅包含一元谓词的语言以及在其中可以通过以下类别的语句捕获约束的情况下，验证了熵极限猜想。 ${\mathrm{\Sigma }}_{\mathrm{1个}}$量词的复杂性。本文表明，熵极限猜想也适用于${\mathrm{\Pi }}_{\mathrm{1个}}$ 复杂性，各种非分类约束以及某些其他一般情况。