Causal multiteam semantics is a framework where probabilistic dependencies arising from data and causation between variables can be together formalized and studied logically. We discover complete characterizations of expressivity for several logics that can express probabilistic statements, conditioning and interventionist counterfactuals. The results characterize the languages in terms of families of linear equations and closure conditions that define the corresponding classes of causal multiteams. The characterizations yield a strict hierarchy of expressive power. Finally, we present some undefinability results based on the characterizations.

Abstract This paper makes a first step towards a logic of learning from experiments. For this, we investigate formal frameworks for modeling the interaction of causal and (qualitative) epistemic reasoning. Crucial for our approach is the idea that the notion of an intervention can be used as a formal expression of a (real or hypothetical) experiment (Pearl, 2009, Causality. Models, Reasoning, and Inference, 2nd edn. Cambridge University Press, Cambridge; Woodward, 2003, Making Things Happen, vol. 114 of Oxford Studies in the Philosophy of Science. Oxford University Press). In a first step we extend a causal model (Briggs, 2012, Philosophical Studies, 160, 139–166; Galles and Pearl, 1998, An axiomatic characterisation of causal counterfactuals. Foundations of Science, 3, 151–182; Halpern, 2000, Axiomatizing causal reasoning. Journal of Artificial Intelligence Research, 12, 317–337; Pearl, 2009, Causality. Models, Reasoning, and Inference, 2nd edn. Cambridge University Press, Cambridge) with a simple Hintikka-style representation of the epistemic state of an agent. In the resulting setting, one can talk about the knowledge of an agent and information update. The resulting logic can model reasoning about thought experiments. However, it is unable to account for learning from experiments, which is clearly brought out by the fact that it validates the principle of no learning for interventions. Therefore, in a second step, we implement a more complex notion of knowledge (Nozick, 1981, Philosophical Explanations. Harvard University Press, Cambridge, Massachusetts) that allows an agent to observe (measure) certain variables when an experiment is carried out. This extended system does allow for learning from experiments. For all the proposed logics, we provide a sound and complete axiomatization.

Causal team semantics ([2]) supports causal-observational languages, which enrich the languages for deterministic causation ([11], [18]) with dependencies and other team-specific operators. Handling the causal aspects of these languages requires a richer semantics than propositional team semantics; nonetheless, in this paper we show that the causal-observational languages considered in [2] can be embedded into first-order dependence logic by means of a translation and a careful choice of models. We show that, in some significant cases, the translation can be refined to an embedding into the Bernays-Schönfinkel-Ramsey fragment of dependence logic or, in the restricted case of recursive causal models, into the existential fragment. As an application, we use the embeddings to show the decidability of a satisfiability problem for the causal-observational languages. Along the way, we question the correctness of the semantics for interventionist counterfactuals proposed by Halpern ([18]) and propose an alternative one which behaves as usual in the uncontroversial recursive case.