In [4], we introduced an extension of team semantics (causal teams) which assigns an interpretation to interventionist counterfactuals and causal notions based on them (as e.g. in Pearl's and Woodward's manipulationist approaches to causation). We now present a further extension of this framework (causal multiteams) which allows us to talk about probabilistic causal statements. We analyze the expressivity resources of two causal-probabilistic languages, one finitary and one infinitary. %establishing a strict hierarchy; we provide a strongly complete (infinitary) axiom system for one of these languages; and We show that many causal-probabilistic notions from the field of causal inference can be expressed already in the finitary language, and we prove a normal form theorem that throws new light on Pearl's ``ladder of causation''. On the other hand, we provide an exact semantic characterization of the infinitary language, which shows that this language captures precisely those causal-probabilistic statements that do not commit us to any specific interpretation of probability; and we prove that no usual, countable language is apt for this task.

翻译：在文献[4]中，我们引入了团队语义学的一种扩展（因果团队），该扩展为干预主义反事实及其基于这些反事实的因果概念（如Pearl和Woodward操纵主义方法中的因果概念）赋予了解释。现在我们提出该框架的进一步扩展（因果多团队），这使得我们能够讨论概率因果陈述。我们分析了两种因果概率语言的表现力资源：一种有限语言和一种无限语言。我们建立了一个严格的层次结构；为其中一种语言提供了一个强完备的（无限）公理系统；并证明因果推断领域中的许多因果概率概念已经可以用有限语言表达。我们还证明了一个范式定理，该定理为Pearl的“因果阶梯”提供了新的视角。另一方面，我们给出了无限语言的精确语义刻画，表明这种语言恰好捕捉了那些不承诺任何特定概率解释的因果概率陈述；并且我们证明了没有通常的可数语言适用于此任务。