Team Semantics for Interventionist Counterfactuals: Observations vs. Interventions

Abstract

Team semantics is a highly general framework for logics which describe dependencies and independencies among variables. Typically, the (in)dependencies considered in this context are properties of sets of configurations or data records. We show how team semantics can be further generalized to support languages for the discussion of interventionist counterfactuals and causal dependencies, such as those that arise in manipulationist theories of causation (Pearl, Hitchcock, Woodward, among others). We show that the “causal teams” we introduce in the present paper can be used for modelling some classical counterfactual scenarios which are not captured by the usual causal models. We then analyse the basic properties of our counterfactual languages and discuss extensively the differences with respect to the Lewisian tradition.

Thinking About Causation: A Causal Language with Epistemic Operators

“…In this section we will compare our proposal to the Causal Team Semantics developed in [5,6,7]. Causal Team Semantics was proposed with the intention of supporting languages that discuss both accidental and causal dependencies.…”

“…Adding contradictory negation (defined by T | = ∼ψ iff T | = ψ) would lead to a more expressive language and to an unintended reading of negation. As observed in [6], the language can be extended -without changes in expressivity -with a dual negation, defined by the clause: (T , F ) | = ¬ψ iff, for all s ∈ T , (s, F ) | = ψ. The dual negation has the intended reading on formulas without dependence atoms.…”

“…is the unique valuation where (i) the value of each exogenous variable not in #-X is exactly as in A, (ii) the value of each each exogenous variable X i in #-X is the provided x i , and (iii) the value of each endogenous variable complies with its new structural function (that in F #-X = #x ). 6 We can now extend a propositional language with a new type of sentence for describing the effect of an intervention. The expression [ #-X= #x ]γ should be read as the counterfactual conditional if the variables in #-X were set to the values #x , respectively, then γ would be the case.…”

“…Case ψ ∨ χ. As a preliminary observation, note how the causal team language is downward closed, in the sense that if T , F | = θ and T ′ ⊆ T , then T ′ , F | = θ (see [6] for a proof). By downward closure, it is easy to see that the statement that there are T 1 , T 2 such that T For the next step, notice that the first of these public announcement formulas holds trivially on valuations from T 2 (where the announcement is false); analogously, the second formula holds trivially on valuations from T 1 Thus, the statement above is equivalent to the assertion that both formulas hold on each valuation of T .…”

“…Section 4 extends the standard causal modeling with means to express knowledge and external communication, and Section 5 provides a sound and complete axiomatization for the new system. Section 6 concludes the paper discussing the relationship with Causal Team Semantics [5,6].…”