Interventionist Counterfactuals on Causal Teams

Abstract

We introduce an extension of team semantics ([13], [22]) which provides a framework for the logic of manipulationist theories of causation based on structural equation models, such as Woodward's ([25]) and Pearl's ([18]); our causal teams incorporate (partial or total) information about functional dependencies that are invariant under interventions. We give a unified treatment of observational and causal aspects of causal models by isolating two operators on causal teams which correspond, respectively, to conditioning and to interventionist counterfactual implication. We then introduce formal languages for deterministic and probabilistic causal discourse, and show how various notions of cause (e.g. direct and total causes) may be defined in them.Through the tuning of various constraints on structural equations (recursivity, existence and uniqueness of solutions, full or partial definition of the functions), our framework can capture different causal models. We give an overview of the inferential aspects of the recursive, fully defined case; and we dedicate some attention to the recursive, partially defined case, which involves a shift of attention towards nonclassical truth values. Structural equation modelsThe most basic objects in the structural equation modeling approach are variables, which we will denote with capital letters X,Y.... Each variable V can assume values (tipically denoted as v, v , v ...) within a certain range of objects, Ran(V ). Variables are related to each other by structural equations, for example Y := f Y (X 1 , . . . , X n )

Team Semantics for Interventionist Counterfactuals: Observations vs. Interventions

“…A summary of some of the ideas presented in the paper appeared in conference proceedings [2], but the present paper largely extends those ideas in both scope and depth. The omitted details of some of the proofs can be found in the Arxiv draft [1].…”

“…Some authors claim that this relationship is a form of reduction: for example, D. Lewis in [28] claims that causation can be defined in terms of an appropriate form of counterfactual dependence, which is in turn definable in non-causal terms. In Pearl [32] the direction of the reduction is reversed 1 : interventionist counterfactual statements are assigned a meaning in terms of basic causal assumptions, which are encoded by structural equations. This picture is made more complicated by the observation that more sophisticated notions of cause are then definable in terms of the counterfactuals (see [21,45]); and by Hitchcock's observation that a structural equation can be seen, in turn, as encoding a set of counterfactuals.…”

“…The latter will represent factors whose causes are not further analysed. 2 To 1 The idea that causal assumptions are parts of the semantic definition of counterfactuals also underlies some recent developments of premise semantics: see [8,26,36,42]. 2 The word "exogenous" is sometimes used in the literature with different meanings, for example to denote unobserved variables.…”

“…2 To 1 The idea that causal assumptions are parts of the semantic definition of counterfactuals also underlies some recent developments of premise semantics: see [8,26,36,42]. 2 The word "exogenous" is sometimes used in the literature with different meanings, for example to denote unobserved variables. To appreciate the difference, consider the hour displayed on a clock: it is an observable variable, but we may treat it as exogenous if we omit from the model any description of the internal mechanisms of the clock.…”

“…holds in the state where X 1 takes value m. We take note of the fact that after the intervention do(X 2 = 1), the two other equations of the system, (2) and 3, have remained unaltered, a property referred to in the literature as modularity.…”