Causal multiteam semantics is a framework where probabilistic dependencies arising from data and causation between variables can be together formalized and studied logically. We consider several logics in the setting of causal multiteam semantics that can express probability comparisons concerning formulae and constants, and encompass interventionist counterfactuals and selective implications that describe consequences of actions and consequences of learning from observations, respectively. We discover complete characterizations of expressivity of the logics in terms of families of linear equations that define the corresponding classes of causal multiteams (together with some closure conditions). The characterizations yield a strict hierarchy of expressive power. Finally, we present some undefinability results based on the characterizations.

翻译：因果多元组语义是一个框架，在该框架中，由数据产生的概率依赖性与变量间的因果关系可以被共同形式化并进行逻辑研究。我们考虑了因果多元组语义设置下的几种逻辑，这些逻辑能够表达涉及公式和常数的概率比较，并涵盖干预性反事实和选择性蕴含，分别描述行动后果和基于观察的学习后果。我们发现了这些逻辑表达性的完全刻画，刻画形式为定义相应因果多元组类别（连同一些闭包条件）的线性方程组族。这些刻画导出了表达力的严格层级。最后，我们基于这些刻画给出了若干不可定义性结果。