The probabilistic logic FP$(\mathrm{Ł},\mathrm{Ł})$ was axiomatized with the aim of presenting a formal setting for reasoning about the probability of infinite-valued Łukasiewicz events. Besides several attempts, proving that axiomatic system to be complete with respect to a class of standard models, remained an open problem since the first paper on FP$(\mathrm{Ł},\mathrm{Ł})$ was published in 2007. In this article we give a solution to it. In particular we introduce two semantics for that probabilistic system: a first one based on Łukasiewicz states and a second one based on regular Borel measures and we prove that FP$(\mathrm{Ł},\mathrm{Ł})$ is complete with respect to both these classes of models. Further, we will show that the finite model property holds for FP$(\mathrm{Ł},\mathrm{Ł})$.

概率逻辑FP$（\mathrm{Ł}，\mathrm{Ł}）$公理化，目的是为推理无限值Łukasiewicz事件的概率提供正式的环境。自从有关FP的第一篇论文以来，除了几次尝试，证明公理系统相对于一类标准模型而言都是完整的，仍然是一个悬而未决的问题。$（\mathrm{Ł}，\mathrm{Ł}）$于2007年发布。在本文中，我们提供了解决方案。特别是，我们介绍了该概率系统的两种语义：第一个基于Łukasiewicz状态，第二个基于常规Borel测度，我们证明了FP$（\mathrm{Ł}，\mathrm{Ł}）$关于这两种模型都是完整的。此外，我们将证明有限模型属性对FP成立$（\mathrm{Ł}，\mathrm{Ł}）$。