We offer an alternative approach to the existing methods for intuitionistic formalization of reasoning about probability. In terms of Kripke models, each possible world is equipped with a structure of the form \(\langle H, \mu \rangle \) that needs not be a probability space. More precisely, though H needs not be a Boolean algebra, the corresponding monotone function (we call it measure) \(\mu : H \longrightarrow [0,1]_{\mathbb {Q}}\) satisfies the following condition: if \(\alpha \), \(\beta \), \(\alpha \wedge \beta \), \(\alpha \vee \beta \in H\), then \(\mu (\alpha \vee \beta ) = \mu (\alpha ) + \mu (\beta ) - \mu (\alpha \wedge \beta )\). Since the range of \(\mu \) is the set \([0,1]_{\mathbb {Q}}\) of rational numbers from the real unit interval, our logic is not compact. In order to obtain a strong complete axiomatization, we introduce an infinitary inference rule with a countable set of premises. The main technical results are the proofs of strong completeness and decidability.

我们为概率推理的直观形式化的现有方法提供了一种替代方法。就 Kripke 模型而言，每个可能的世界都配备有\(\langle H, \mu \rangle \)形式的结构，该结构不一定是概率空间。更准确地说，虽然H不一定是布尔代数，但相应的单调函数（我们称之为测度）\(\mu : H \longrightarrow [0,1]_{\mathbb {Q}}\)满足以下条件：如果\(\alpha \)、\(\beta \)、\(\alpha \wedge \beta \)、\(\alpha \vee \beta \in H\)，则\(\mu (\alpha \vee \beta ) = \mu (\alpha ) + \mu (\beta ) - \mu (\alpha \wedge \beta )\)。由于\(\mu \)的范围是来自实单位区间的有理数的集合\([0,1]_{\mathbb {Q}}\) ，因此我们的逻辑并不紧凑。为了获得强大的完整公理化，我们引入了具有可数前提集的无限推理规则。主要技术成果证明了较强的完整性和可判定性。