Axiomatizations of causal reasoning under indeterministic causal laws

We investigate the generalization of causal models to the case of indeterministic causal laws that was suggested in Halpern (2000). We give an overview of what differences in modeling are enforced by this more general perspective, and propose an implementation of generalized models in the style of the causal team semantics of Barbero & Sandu (2020). In these models, the laws are not represented by functions (as in the deterministic case), but more generally by relations. We point out significant differences in the indeterministic vs. the deterministic case, both for what concerns the axiomatization of counterfactuals and the definability of causal notions in terms of them. We notice, for instance, that the notions of parenthood and direct cause, differently from the deterministic case, come apart. We provide two main strongly complete axiomatizations, one for the class of indeterministic causal models and one for their team generalizations, which can also represent uncertainty about the state of the system. Then, we also identify axioms that allow specializing the axiomatizations to a number of significant subclasses, characterized by properties such as acyclicity, totality and determinism of the causal laws. We also see that a minor change in the notion of signature of the models has a dramatic impact; in particular, it makes so that the deterministic subclass of models become undefinable in the usual Halpern-style counterfactual language. We see that the definability of determinism can be recovered either by a restriction of the class of models, by allowing right-nested counterfactuals, or by introducing an operator for information update.