On Logics and Semantics of Indeterminate Causation

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.