In this paper we present a model for argumentative causal and counterfactual reasoning in a logical setting. Causal knowledge is represented in this system using Pearl's causal model of a set of structural equations and a set of assumptions expressed in propositional logic. Queries concerning observations or actions can be answered by constructing an argumentation framework and determining its extensions. For counterfactual queries we propose an argumentation-based implementation of the twin network method and analyse its expressiveness.
corresponds directly to the solutions of causal models. In addition to that, this representation is also adequate for answering interventional queries for the causal model.
In this paper we contribute to these lines of research by investigating an explainable,
argumentation-based account of reasoning with causal models. The starting point in our
approach is a propositional knowledge base that encodes a causal model in the sense of Pearl [
Our work can be understood as an initial investigation into the possibility of causal reasoning within Dung’s model of argumentation.
To summarise, the contributions of this paper are the following. 1. We introduce the notion of causal knowledge bases (Section 3). 2. We provide a novel approach for argumentation-based explanations for counterfactual reasoning implementing the twin network method (Section 4). 3. We prove the standard three-step procedure and the twin network method for evaluating counterfactuals by Pearl are equivalent under our notion (Section 4). 4. We discuss our approach and related work (Section 5).
In Section 2 we present the necessary background on knowledge bases and formal argumentation and Section 6 concludes the paper.
The causal reasoning framework we develop builds on a simple and well-known form of default
reasoning based on maximal consistent subsets [
Definition 1. Let Δ = (, ) be a knowledge base and , ∈ ℒ. A set Σ ⊆ is a maximal -consistent subset of whenever Σ ∪ is consistent and Σ′ ∪ is inconsistent for all Σ′ ⊆ such that Σ ⊂ Σ′. We say that: • Δ entails (written Δ |∼ ) whenever Σ ∪ ⊢ for every maximal -consistent subset of . • Δ-entails (written |∼ Δ ) whenever ( ∪ {}, ) entails .
In the next section we define a special class of knowledge bases used to represent causal
knowledge. Our goal is to provide an argumentative method for causal and counterfactual
reasoning based on such knowledge bases. The argumentative aspect of this method relies
on a characterisation of reasoning with maximal consistent subsets based on the notion of
argumentation framework or AF, for short [
Definition 2. An argumentation framework is a pair = (A, ⇒) where is a set whose elements are called arguments and where ⇒ ⊆ × is called the attack relation.
We mostly use infix notation for attacks and write ⇒ for (, ) ∈ ⇒ and ̸ ⇒ for
(, ) ̸∈ ⇒. If ⇒ we also say that attacks . Given an AF, a semantics determines sets of
jointly acceptable arguments called extensions. The admissible, preferred and stable semantics
are defined as follows [
Definition 3. Let = (A, ⇒) be an AF. A set ⊆ A is: • conflict-free if for all , ∈ we have ̸⇒ . • self-defending if for all ∈ and ∈ A ∖ such that ⇒ , there is a ∈ such that ⇒ . • admissible if is conflict-free and self-defending. • preferred if is admissible and there is no admissible set ′ ⊆ A such that ⊂ ′. • stable if is conflict-free and for every ∈ A ∖ there is a ∈ such that ⇒ .
Following Cayrol [
• An argument induced by Δ is a pair (Φ, ) such that – Φ ⊆ , – Φ ∪ ⊬ ⊥, – Φ ∪ ⊢ , and if Ψ ⊂ Φ then Ψ ∪ ⊬ . • An argument (Φ, ) undercuts an argument (Φ′, ′) if for some ′ ∈ Φ′ we have ′ ≡ ¬ . • The AF induced by Δ is the AF (A, ⇒) where A consists of all arguments induced by Δ and ⇒ holds whenever undercuts . We denote the AF induced by Δ by (Δ).
Cayrol et al. [
1Note that this construction method allows arguments to have syntactically diferent, but semantically equivalent conclusions. Proposition 1. Let Δ = (, ) be a knowledge base. Then |∼ Δ if and only if every stable extension of ( ∪ {}, ) contains an argument with conclusion .
We conclude this section by noting that the form of argumentation we consider here can also
be captured by structured argumentation formalisms such as ASPIC and ABA [
Our definition of a causal model is essentially that of Pearl [
Definition 5. A causal model is a triple (, , ) where and partition the set of atoms into, respectively, a set of background and explainable atoms. consists of a set of Boolean structural equations, one for each atom ∈ . A Boolean structural equation for is a formula of the form ↔ where is a propositional formula that does not contain .
Given a causal model (, , ) we can reconstruct and from as follows; (1) for every
formula ↔ in , add to , and (2) add all the remaining atoms that appear in but
not in to . Therefore we can refer to a causal model as . For every Boolean structural
equation ↔ , we call an atom appearing in a parent (background or explainable) of . Like
in the standard case, a causal model induces a causal graph whose vertices are the explainable
atoms of the model [
Example 1. Let = {corona, influenza , at-risk} and = {covid, flu , short-of-breath, fever, chills}. The set consisting of the following equations represents a causal model for a patient diagnosis scenario.
short-of-breath ↔ covid ∧ at-risk covid ↔ corona lfu
↔ influenza fever ↔ covid ∨ flu chills ↔ fever corona
influenza
In words: corona virus causes covid, influenza virus causes flu , covid and flu cause fever, fever causes chills, and covid causes short-of-breath, but only if the patient is at risk for this condition (represented by the background atom at-risk). Note that corona, influenza and at-risk are background atoms and thus assumed unobservable.
Figure 1 depicts the causal graph for this model. This figure also includes the background atoms of the model, drawn using dotted lines.
We define a causal knowledge base to be a knowledge base where the set of facts is a causal
model and where the set of assumptions is restricted to contain assumptions about background
atoms. Here we depart from the probabilistic approach to causal modelling, where belief about
background atoms is represented using probability distributions over their values [
By restricting background assumptions to be literals we prevent the ability to express dependencies among background atoms. There are three possible attitudes towards a background atom , since we can assume just , just ¬, or both. Assuming just or ¬ amounts to assuming that is true or false, unless we have evidence to the contrary. Assuming both and ¬ represents a state of uncertainty where may, depending on the evidence, be true as well as false. If Δ = (, ) is a causal knowledge base then Δ-entailment represents a relation between observations and predictions. These predictions include causes as well as efects of the observation in accordance with the causal model and background assumptions . Example 2. Consider the causal knowledge base Δ = (, ) where is the causal model defined in Example 1 and where = {at-risk, ¬at-risk, ¬corona, ¬influenza }. In words, these assumptions mean that we are uncertain whether the patient is at risk to develop shortness of breath, and a corona or influenza infection is assumed false unless there is evidence to the contrary. Now consider the question of whether observing fever entails shortness of breath, i.e., whether fever |∼ Δ short-of-breath. Flu and covid cause fever. Thus, fever is evidence for flu or covid. Of these two possible causes, covid may cause shortness of breath, but only if the patient is at risk, which may or may not be true. Hence, fever may or may not be evidence for shortness of breath. This reasoning is depicted by the AF = (( ∪ {fever}, )) shown in Figure 2 (we only depict arguments relevant to the conclusion of short-of-breath). This AF has four stable extensions {2, 4}, {3, 4}, {2, 5}, and {3, 5, 1}. The argument 1 with conclusion short-of-breath is included in some but not all of these extensions. We thus have that (1) is false, but note that
fever |∼ Δ ¬short-of-breath is also false. Thus, given fever, shortness of breath is possible but not necessary.
1 : ({at-risk, ¬influenza}, short-of-breath) 2 : ({¬at-risk}, ¬at-risk)
4 : ({¬corona}, influenza) 3 : ({at-risk}, at-risk) 5 : ({¬influenza}, corona)
In this section we show how causal knowledge bases provide a means to evaluate counterfactuals.
Pearl introduces two methods for evaluating counterfactuals in causal models [
Before dealing with counterfactual statements we start with interventional statements of the form if would be then would be true (2) where is a variable we intervene on and a truth value (we use ⊤ for true and ⊥ for false). The antecedent of this statement represents the action of setting to . In our running example, a patient might be given Ibuprofen to treat fever. This is an action that amounts to setting fever to ⊥. Note that this is diferent from observing ¬fever. While the observation ¬fever would, in our model, imply ¬covid and ¬flu , the action of setting fever to ⊥ amounts to overriding the causal mechanism that determines fever, which does not afect covid or flu . Like Pearl, given a causal model , we denote by [=] the new causal model where the equation ↔ is replaced with ↔ .
Definition 7. Let be a causal model, let be an explainable atom, and let ∈ {⊤, ⊥}. We denote by [=] the causal model defined by [=] = {(′ ↔ ) ∈ | ′ ̸= } ∪ {( ↔ )}. Example 3. Let Δ = (, ) be the causal knowledge base from Example 2. Since short-of-breath is caused by covid, covid causes fever, and fever causes chills, we have that shortness of breath leads to the prediction of chills
short-of-breath |∼ Δ chills.
But what if we observe short-of-breath in a patient that has been given Ibuprofen? The answer is given by the causal knowledge base Δ′ = ([fever=⊥], ). Here, the action of setting fever to ⊥ blocks the efect of covid on chills. We now have that shortness of breath leads to the prediction of no chills
short-of-breath |∼ Δ′ ¬chills.
We now turn to counterfactual statements. These are statements of the form
given , if had been then would be true
where is an observed piece of evidence and is the counterfactual conclusion. An example of
a counterfactual in our running example would be: given that the patient has fever, would the
patient have had fever if we had administered a covid vaccine? This amounts to the statement
given fever, if covid had been ⊥ then fever would be true. The standard causal modelling approach
of evaluating a counterfactual statement of the form (5) is based on a three-step procedure [
Definition 8. Given a causal knowledge base Δ = (, ), the truth of a counterfactual statement (5) is determined by: • Step 1 (abduction) Determine the maximal ∪ {}-consistent subsets Σ1, . . . , Σ of .
These sets represent the possible configurations of the background atoms consistent with evidence . • Step 2 (action) Set to by updating the causal model to [=]. • Step 3 (prediction) For each Σ obtained in step 1, determine whether Σ ∪ [=] ⊢ . If the answer is yes for all ∈ 0, . . . , then the counterfactual statement (5) is true.
The dificulty with this procedure is that it requires the computation of the maximal ∪
{}consistent subsets of . The same dificulty arises in probabilistic causal models, where the
abduction step requires the computation of a probability distribution over configurations of
the background atoms. The twin network method overcomes this dificulty [
Definition 9. The twin model for a causal model is the causal model * defined by * = ∪ {(* ↔ * ) | ( ↔ ) ∈ } .
While the abduction step takes place in the actual network, the action and prediction steps take place in the counterfactual network. Let us now prove this intuition of separated areas in the twin model holds up formally during the evaluation, i. e., the three step procedure is equivalent to the evaluation of the twin model.
Proposition 2. Let Δ = (, ) be a causal knowledge base. The counterfactual statement given , if had been then would be true is true in Δ if and only if |∼ Δ*[* =] * .
Proof. We prove that the twin model method is equivalent to the three-step procedure for evaluating counterfactuals from Pearl as given in Definition 8.
Let Δ = (, ) be a causal knowledge base. First, we recall that |∼ Δ if and only if ( ∪ {}, ) entails . ( ∪ {}, ) entails whenever for every maximal K-consistent subset Σ of , we have that Σ ∪ ⊢ .
(⇐) Now, for the given counterfactual statement, consider Δ* = (* , ) with * as given in Definition 9. Furthermore, we modify Δ* by setting * = in * , i.e., we have Δ*[* =]. That means, * fully contains as well as an exact copy of (each copy of a formula of is denoted with * ). In this copy, we also have set * = . Therefore, we can essentially also consider * as the union of and [=].
With the twin model method, we have that |∼ Δ*[* =] * . That means, according to Definition 1 we determine the * ∪{}-consistent subsets Σ*1, ..., Σ* of . Now, since ⊂ * , it follows that Σ*1 ∩ , ..., Σ* ∩ are maximal consistent with ∪ {} (as required by step 1 abduction).
The action step of Definition 8 is also executed in the twin model by setting * = in * .
In the prediction step, we have to determine whether Σ ∪ [=] ⊢ for all -consistent Σ. In the twin model method, we consider the * ∪ {}-consistent subsets. As already explained above, [=] is fully contained in * and therefore, similarly to above, if we have that Σ* ∪ * ⊢ * it follows that Σ* ∪ [=] ⊢ . Thus, the twin model method is equivalent to the three-step procedure for evaluating counterfactual statements.
(⇒) Suppose the counterfactual statement is true according to the three step procedure. In the twin model it holds that and {(* ↔ * ) | ( ↔ ) ∈ } have no variables in common. Thus, for any Σ maximal consistent wrt. * ∪{} it holds that Σ′ = Σ∩{(* ↔ * ) | ( ↔ ) ∈ } is maximal consistent wrt. [=] ∪ {}. By the assumption we have Σ′ ∪ [=] ∪ {} ⊢ for any such Σ′. By monotony of classic entailment it follows that Σ ∪ [** =] ∪ {} ⊢ .
We have now proven that the standard three-step procedure can be reduced to a single evaluation step on the twin model. To conclude, let us give some examples of counterfactual reasoning with the AFs build from a twin model.
Example 4. Let Δ = (, ) be the causal knowledge base defined in Example 2. Suppose we have evidence that the patient has fever. Would the patient have had fever if we had administered a covid vaccine (i.e., if covid had been false)? This depends on whether the fever is caused by covid or flu. If fever was caused by covid then the vaccine would indeed have prevented fever. If, on the corona
influenza other hand, fever was caused by flu, then the vaccine would not have had an efect, and the patient would still have had fever. To evaluate this counterfactual statement we check whether fever |∼ Δ*[covid* =⊥] fever* .
fever |∼ Δ*[covid* =⊥] ¬fever* Figure 3 shows the causal graph of the twin model c*ovid* =⊥ and Figure 4 shows the AF = ([*covid* =⊥] ∪ {fever}, ). This AF has two stable extensions {1, 3} and {2}. While the first entails fever, the second does not. Thus, (6) is false, but note that is also false. Thus, given that the patient has fever, the patient may or may not have had fever, had we administered a covid vaccine. 3 : ({¬corona}, influenza)
Example 5. Suppose that, in addition to having evidence that the patient has fever, we also have evidence of shortness of breath. Again, we want to know if the patient would have had fever if we had administered a covid vaccine. This time, the evidence of shortness of breath implies that the fever must have been caused by covid. Hence, the vaccine would indeed have prevented fever. We check this formally by checking whether
fever ∧ short-of-breath |∼ Δ*[covid* =⊥] fever* . (7) (8)
Our work can be understood as an initial investigation into the possibility of reasoning with
causal models using tools of abstract information science and logic. By doing this we contribute
to explaining technically advanced causal reasoning procedures with concepts closer to human
reasoning. Our account of Pearl’s theory was motivated by the non-monotonic interpretation
of his causal model in [
There are several major design diferences between the two approaches. First Rago et al.
use bipolar AFs in contrast to our Dung-style approach. Second, our approach is based on
structured argumentation, while theirs uses causal atoms as arguments. On the evaluation
level, they apply gradual semantics while we make use of classic stable semantics. For both
frameworks an interesting future work direction would be to apply the resp. other type of
semantics. An important diference lies in the resp. attack notions. Rago et al. let causes
contribute negatively or positively towards the status of an argument via the attack and support
relation, respectively. Our focus is on the explanation aspect. Attacks are used to exclude
certain premises as causes. This is done by undercut-attacks on the corresponding argument. A
current shortcoming of our approach is the restriction to propositional logic in the attack notion,
which would have dificulties handling uncertain causal relations. A promising future work
direction would therefore be to investigate ways for representing uncertain causal relations in
some type of probabilistic argumentation framework, e. g. by adding a probability distribution
over possible causal structures [
We have proposed a natural way to model causal reasoning with Dung-style AFs by means of causal knowledge bases. The framework introduced integrates given causal knowledge in form of structural equations into an attack relation and a set of structured arguments constructed from sets of assumptions resp. observations from the real world. By doing this we have laid the groundwork for an implementation of the twin network method for counterfactual reasoning. It ofers an alternative for computing the truth value of counterfactual statements which, as we proved, is equivalent to the standard approach in our framework. The results we obtain demonstrate that causal reasoning can be done as an instantiation of Dung’s model of argumentation. This can be used to provide argumentative explanations for causal and counterfactual queries. That is, the argumentation framework contains not only the arguments used to arrive at a prediction, but also possible counterarguments that lead to alternative predictions.
Acknowledgements The research reported here was partially supported by the Deutsche Forschungsgemeinschaft (grant 375588274).