标题： 通过Dialectica解释的近似零一法则 显示英文标题

标题： An approximate zero-one law via the Dialectica interpretation

摘要： 零一定律指出，某种类型的概率事件必须以$0$或$1$的概率发生，而不会介于两者之间。 我们提出一个句法上的零一定律，它在逻辑性质上表现良好，同时在概率论中具有广泛的应用性。 然后，受哥德尔的Dialectica解释启发，我们对其进行有限化：结果是一个近似的零一定律，该定律指出具有特定有限结构的事件发生的概率接近$0$或$1$，并且可以达到任意精度。 这个近似的零一定律在经典逻辑下等价于原始的零一定律，但与后者不同的是，它完全用事件的有限并集和交集来表述。 此外，根据最近关于概率的逻辑元定理，它允许计算解释，这反过来有助于对那些使用零一定律证明的定理进行定量分析。 在此精神下，讨论了在各种不同设置中的具体应用。 显示英文摘要

摘要： Zero-one laws state that probabilistic events of a certain type must occur with probability either $0$ or $1$, and nothing in between. We formulate a syntactic zero-one law, which enjoys good logical properties while being broadly applicable in probability theory. Then, inspired by G\"odel's Dialectica interpretation, we finitise it: The result is an approximate zero-one law which states that events with a particular finite structure occur with probability close to $0$ or $1$ up to an arbitrary degree of precision. This approximate zero-one law is equivalent - over classical logic - to the original zero-one law, but in contrast to the latter, is formulated entirely in terms of finite unions and intersections of events. Furthermore, in line with recent logical metatheorems for probability, it admits a computational interpretation, which in turn facilitates a quantitative analysis of theorems whose proof makes use of zero-one laws. Concrete applications in this spirit, over a variety of different settings, are discussed.