Multiteam semantics for interventionist counterfactuals: probabilities and causation

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.