This paper proposes a causal inference relation and causal programming as general frameworks for causal inference with structural causal models. A tuple, $\langle M, I, Q, F \rangle$, is an instance of the relation if a formula, $F$, computes a causal query, $Q$, as a function of known population probabilities, $I$, in every model entailed by a set of model assumptions, $M$. Many problems in causal inference can be viewed as the problem of enumerating instances of the relation that satisfy given criteria. This unifies a number of previously studied problems, including causal effect identification, causal discovery and recovery from selection bias. In addition, the relation supports formalizing new problems in causal inference with structural causal models, such as the problem of research design. Causal programming is proposed as a further generalization of causal inference as the problem of finding optimal instances of the relation, with respect to a cost function.

We will explore the use of disjunctive causal rules for representing indeterminate causation. We provide first a logical formalization of such rules in the form of a disjunctive inference relation, and describe its logical semantics. Then we consider a nonmonotonic semantics for such rules, described in (Turner 1999). It will be shown, however, that, under this semantics, disjunctive causal rules admit a stronger logic in which these rules are reducible to ordinary, singular causal rules. This semantics also tends to give an exclusive interpretation of disjunctive causal effects, and so excludes some reasonable models in particular cases. To overcome these shortcomings, we will introduce an alternative nonmonotonic semantics for disjunctive causal rules, called a covering semantics, that permits an inclusive interpretation of indeterminate causal information. Still, it will be shown that even in this case there exists a systematic procedure, that we will call a normalization, that allows us to capture precisely the covering semantics using only singular causal rules. This normalization procedure can be viewed as a kind of nonmonotonic completion, and it generalizes established ways of representing indeterminate effects in current theories of action.

We provide a logical representation of Pearl's structural causal models in the causal calculus of McCain and Turner (1997) and its first-order generalization by Lifschitz. It will be shown that, under this representation, the nonmonotonic semantics of the causal calculus describes precisely the solutions of the structural equations (the causal worlds of the causal model), while the causal logic from Bochman (2004) is adequate for describing the behavior of causal models under interventions (forming submodels).