Multiteam semantics for interventionist counterfactuals: probabilities and causation

In (Barbero and Sandu 2020 Journal of Philosophical Logic, 50, 471-521), we showed that languages encompassing interventionist counterfactuals and causal notions based on them (as e.g. in Pearl's and Woodward's manipulationist approaches to causation) as well as information-theoretic notions (such as learning and dependence) can be interpreted in a semantic framework which combines the traditions of structural equation modeling and of team semantics. 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. We show that many causal-probabilistic notions from the literature on causal inference can already be expressed in the finitary language, and we prove a normal form theorem that throws a new light on Pearl's "ladder of causation". In addition, 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.