Must probabilities be countably additive? On the one hand, arguably, requiring countable additivity is too restrictive. As de Finetti pointed out, there are situations in which it is reasonable to use merely finitely additive probabilities. On the other hand, countable additivity is fruitful. It can be used to prove deep mathematical theorems that do not follow from finite additivity alone. One of the most philosophically important examples of such a result is the Bayesian convergence to the truth theorem, which says that conditional probabilities converge to 1 for true hypotheses and to 0 for false hypotheses. In view of the long-standing debate about countable additivity, it is natural to ask in what circumstances finitely additive theories deliver the same results as the countably additive theory. This paper addresses that question and initiates a systematic study of convergence to the truth in a finitely additive setting. There is also some discussion of how the formal results can be applied to ongoing debates in epistemology and the philosophy of science.

中文翻译：

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收敛于没有可数可加性的真理

概率必须是可数可加的吗？一方面，可以说，要求可数可加性过于严格。正如 de Finetti 所指出的，在某些情况下，仅使用有限可加概率是合理的。另一方面，可数可加性是富有成效的。它可用于证明不能仅从有限可加性得出的深层数学定理。这种结果在哲学上最重要的例子之一是贝叶斯收敛到真值定理，它说条件概率收敛到 1 为真假设和 0 为假假设。鉴于关于可数可加性的长期争论，很自然地会问在什么情况下有限加性理论会产生与可数加性理论相同的结果。本文解决了这个问题，并启动了在有限可加设置中收敛到真理的系统研究。还有一些关于如何将正式结果应用于认识论和科学哲学中正在进行的辩论的讨论。