Thinking About Causation: A Causal Language with Epistemic Operators

In this paper we propose a formal framework for modeling the interaction of causal and (qualitative) epistemic reasoning. To this purpose, we extend the notion of a causal model [11, 16, 17, 26] with a representation of the epistemic state of an agent. On the side of the object language, we add operators to express knowledge and the act of observing new information. We provide a sound and complete axiomatization of the logic, and discuss the relation of this framework to causal team semantics.

1. 1.

   E.g., by adding a probability distribution over a causal model’s exogenous variables.

2. 2.

   See [5] for an exception, though the epistemic element is not made fully explicit in the language. We will come back to this approach in Sect. 6.

3. 3.

   arXiv:2010.16217 [cs.AI].

4. 4.

   Given \((X_1,\ldots ,X_k) \in (\mathcal {U}\cup \mathcal {V})^k\), abbreviate \(\mathcal {R}(X_1)\times \cdots \times \mathcal {R}(X_k)\) as \(\mathcal {R}(X_1,\ldots ,X_k)\).

5. 5.

   This notion of a direct cause is adopted from [16]; it is related to the notion of a variable having a direct effect on another, as discussed in [23] in the context of Causal Bayes Nets. The notions defined here differ from Halpern’s notion of affect [17], and this affects the axiomatization: axiom HP6 (Table 1) has the same function as C6 in [17] (ensuring that the canonical model is recursive), but does so in a slightly different way.

6. 6.

   The reason behind this restriction is that only acyclic relations are thought to have a causal interpretation (see [29] for an argument). The counterfactuals satisfy different logical laws if cyclic dependencies are allowed (see [17]).

7. 7.

   Note that, since \(\mathcal {F}\) is recursive, the valuation is uniquely determined. First, the value of every exogenous variable U is uniquely determined, either from (if U occurs in ) or else from \(\mathcal {A}\) (if U does not occur in ). Second, the value of every endogenous variable V is also uniquely determined, either from (if V occurs in , as V’s new structural function is a constant) or else from the (recall: recursive) structural functions in (if V does not occur in ).

8. 8.

   The other two levels that Pearl distinguishes are the level of association, which is based on observation, and the level of intervention, which is based on doing. Modern AI technology is for him still at the first level: association. Counterfactual reasoning is not possible without a true understanding of why things happen – in our terminology, it is not possible without knowing the causal relationships as determined by \(\mathcal {F}\).

9. 9.

   However, notice that the semantics already allows for nested occurrences of all dynamic operators. We will extend the proofs of sound- and completeness to the unrestricted language in the future.

10. 10.

    Note how the announcement of \(\psi \) is a deterministic action with precondition \(\psi \). Hence the similarities and differences between RP4 and CM.

11. 11.

    Readers familiar with DEL might have noticed that \(\mathsf {L}_{ PAKC }\) does not have a reduction axiom for nested announcements \([\phi _1!][\phi _2!]\phi \). There are (at least) two strategies for dealing with such formulas. The first follows an ‘outside-in’ approach, reducing two announcements in a row into a single one. This requires an axiom for nested announcements. The second follows an ‘inside-out’ strategy, applying the reduction over the innermost announcement operator in the formula until the operator disappears, and then proceeding to the next. For this, the rule of substitution of equivalents (our rule RE) is enough [31, Theorem 11].

12. 12.

    We are presenting here the definition from [7], which, save for implementation details, corresponds to what are called fully defined causal teams in [5] (where a more general notion is considered).

13. 13.

    With some additional machinery, which is not worth exploring here.

14. 14.

    Notice that the syntax allows negation only at the atomic level. Adding contradictory negation (defined by \(T\models \mathord {\sim } \psi \) iff \(T\not \models \psi \)) 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: \((\mathcal {T},\mathcal {F})\models \lnot \psi \) iff, for all \(s\in \mathcal {T}\), \(({s},\mathcal {F})\not \models \psi \). The dual negation has the intended reading on formulas without dependence atoms. Neither negation allows the usual interdefinability of \(\wedge \) and \(\vee \) via the De Morgan laws; for this reason, both \(\wedge \) and \(\vee \) are included in the syntax.

15. 15.

    As far as we know, this has been first observed, independently, in [14] and [1], in the context of epistemic languages with modalities for the knowledge of values.

16. 16.

    A formula is non-nested if, in every subformula of the form , no occurs inside \(\phi \). Providing a translation for these formulas is sufficient, since every formula of the causal team language is provably equivalent to a non-nested one.

17. 17.

    The additional K operator in the definition of e-dependence is needed to deal with the fact that information update always checks first whether the information that the information state is updated with is true. This problem disappears in the case of interventions, because the formula you intervene with is made true in the hypothetical scenario you consider.

Authors and Affiliations

1. University of Helsinki, Helsinki, Finland

   Fausto Barbero

2. ILLC, Universiteit van Amsterdam, Amsterdam, The Netherlands

   Katrin Schulz, Sonja Smets, Fernando R. Velázquez-Quesada & Kaibo Xie

3. Department of Information Science and Media Studies, University of Bergen, Bergen, Norway

   Sonja Smets

Corresponding author

Correspondence to Fernando R. Velázquez-Quesada .

Editors and Affiliations

1. Department of Mathematics, University of Aveiro, Aveiro, Portugal

   Manuel A. Martins

2. Institute of Computer Science, Czech Academy of Sciences, Praha 8, Czech Republic

   Igor Sedlár

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