=Paper=
{{Paper
|id=Vol-3209/7343
|storemode=property
|title=Argumentation-based Causal and Counterfactual Reasoning
|pdfUrl=https://ceur-ws.org/Vol-3209/7343.pdf
|volume=Vol-3209
|authors=Lars Bengel, Lydia Blümel, Tjitze Rienstra, Matthias Thimm
|dblpUrl=https://dblp.org/rec/conf/comma/BengelBRT22
}}
==Argumentation-based Causal and Counterfactual Reasoning==
https://ceur-ws.org/Vol-3209/7343.pdf
Argumentation-based Causal and Counterfactual
Reasoning
Lars Bengel1 , Lydia Blümel1 , Tjitze Rienstra2 and Matthias Thimm1
1
Artificial Intelligence Group, FernUniversität in Hagen, Germany
2
Department of Advanced Computing Sciences, Maastricht University, The Netherlands
Abstract
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.
Keywords
Causality, Counterfactuals, Abstract Argumentation
1. Introduction
The area of explainable AI (XAI) has received increasing attention in recent years and many
different approaches have been proposed (for an overview see e. g., [1, 2]). A central aspect of
XAI is to provide human-understandable explanations for the output of some AI model. To
provide these kinds of explanations, it seems only natural to utilise the techniques that humans
employ to explain events and actions to each other. One such technique is formal argumentation
[3]. In the past, many works have already suggested argumentation-based approaches for the
purpose of providing explanations [4, 5]. More recently, we have seen different argumentative
approaches to explain the output of various machine learning and other AI methods [6, 7].
Another field of research that is of increasing relevance is causality [8]. Especially interesting
is how humans use causal connections for their explanations [9]. In the context of XAI, there
are many advocates in the literature for using causal explanations [10, 11, 12]. Causality is
furthermore closely linked with the notion of counterfactuals, which are statements that express
what would have been true if something that is the case had been different. Counterfactual
explanations have received much attention in recent research on XAI [13]. Relevant is also [14],
in which the authors provide a logical representation of Pearl’s causal models in the framework
of the causal calculus [15]. They show that the non-monotonic semantics of this causal calculus
1st International Workshop on Argumentation for eXplainable AI (ArgXAI, co-located with COMMA ’22), September 12,
2022, Cardiff, UK
$ lars.bengel@fernuni-hagen.de (L. Bengel); lydia.bluemel@fernuni-hagen.de (L. Blümel);
t.rienstra@maastrichtuniversity.nl (T. Rienstra); matthias.thimm@fernuni-hagen.de (M. Thimm)
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
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http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
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Lars Bengel et al. CEUR Workshop Proceedings 1–12
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 [8]
restricted to boolean variables and default assumptions about background variables. The ap-
proach makes it possible to answer interventional and counterfactual queries by constructing
an argumentation framework whose stable extensions provide the answer to the queries. Our
method of answering counterfactual queries is a reformulation of the dual approach due to
Pearl, where a counterfactual query can be answered equivalently using two different methods:
a three-step abduction-action-prediction procedure, and the so called twin network method [8].
We prove that, in our setting, the two methods are also equivalent.
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 argumen-
tation and Section 6 concludes the paper.
2. Preliminaries
The causal reasoning framework we develop builds on a simple and well-known form of default
reasoning based on maximal consistent subsets [16]. Let ℒ be a set of propositional formulas
generated by a finite set of atoms. We assume there is a set 𝐾 ⊆ ℒ of facts and a set 𝐴 ⊆ ℒ of
assumptions. We call a pair Δ = (𝐾, 𝐴) a knowledge base. Facts are true and thus we assume
that 𝐾 is consistent. Assumptions are statements that we are willing to assume true unless
we have evidence to the contrary. The consequences of a knowledge base Δ = (𝐾, 𝐴) are
determined by 𝐾 together with the maximal 𝐾-consistent subsets of 𝐴.
Definition 1. Let Δ = (𝐾, 𝐴) be a knowledge base and 𝜑, 𝜓 ∈ ℒ. A set Σ ⊆ 𝐴 is a maximal
𝐾-consistent subset of 𝐴 whenever Σ ∪ 𝐾 is consistent and Σ′ ∪ 𝐾 is inconsistent for all Σ′ ⊆ 𝐴
such that Σ ⊂ Σ′ . We say that:
• Δ entails 𝜓 (written Δ |∼ 𝜓) whenever Σ ∪ 𝐾 ⊢ 𝜓 for every maximal 𝐾-consistent subset
of 𝐴.
• 𝜑 Δ-entails 𝜓 (written 𝜑 |∼Δ 𝜓) whenever (𝐾 ∪ {𝜑}, 𝐴) entails 𝜓.
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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 [3].
Definition 2. An argumentation framework is a pair 𝐹 = (A, ⇒) where 𝐴 is a set whose
elements are called arguments and where ⇒ ⊆ 𝐴 × 𝐴 is called the attack relation.
We mostly use infix notation for attacks and write 𝑎 ⇒ 𝑏 for (𝑎, 𝑏) ∈ ⇒ and 𝑎 ̸ ⇒𝑏 for
(𝑎, 𝑏) ̸∈ ⇒. If 𝑎 ⇒ 𝑏 we also say that 𝑎 attacks 𝑏. Given an AF, a semantics determines sets of
jointly acceptable arguments called extensions. The admissible, preferred and stable semantics
are defined as follows [3].
Definition 3. Let 𝐹 = (A, ⇒) be an AF. A set 𝐸 ⊆ A is:
• conflict-free if for all 𝑎, 𝑏 ∈ 𝐸 we have 𝑎 ̸⇒ 𝑏.
• self-defending if for all 𝑎 ∈ 𝐸 and 𝑏 ∈ A ∖ 𝐸 such that 𝑏 ⇒ 𝑎, there is a 𝑐 ∈ 𝐸 such that
𝑐 ⇒ 𝑏.
• admissible if 𝐸 is conflict-free and self-defending.
• preferred if 𝐸 is admissible and there is no admissible set 𝐸 ′ ⊆ A such that 𝐸 ⊂ 𝐸 ′ .
• stable if 𝐸 is conflict-free and for every 𝑎 ∈ A ∖ 𝐸 there is a 𝑏 ∈ 𝐸 such that 𝑏 ⇒ 𝑎.
Following Cayrol [17] we define an argument induced by a knowledge base Δ = (𝐾, 𝐴)
to be a pair (Φ, 𝜓) where Φ ⊆ 𝐴 is a minimal set of assumptions (called the premises of the
argument) that, together with 𝐾, consistently entails 𝜓 (called the conclusion of the argument).
One argument undercuts another if the conclusion of the former is the negation of a premise of
the latter. The set of all arguments induced by Δ, together with undercut as the attack relation,
forms the AF induced by Δ 1 . Formally:
Definition 4. Let Δ = (𝐾, 𝐴) be a knowledge base.
• An argument induced by Δ is a pair (Φ, 𝜓) such that
– Φ ⊆ 𝐴,
– Φ ∪ 𝐾 ⊬ ⊥,
– Φ ∪ 𝐾 ⊢ 𝜓, and if Ψ ⊂ Φ then Ψ ∪ 𝐾 ⊬ 𝜓.
• An argument (Φ, 𝜓) undercuts an argument (Φ′ , 𝜓 ′ ) if for some 𝜑′ ∈ Φ′ we have 𝜑′ ≡ ¬𝜓.
• The AF induced by Δ is the AF (A, ⇒) where A consists of all arguments induced by Δ
and 𝑎 ⇒ 𝑏 holds whenever 𝑎 undercuts 𝑏. We denote the AF induced by Δ by 𝐹 (Δ).
Cayrol et al. [17] showed that there is a one-to-one correspondence between maximal 𝐾-
consistent subsets of a knowledge base and the stable extensions of the induced AF. Thus, given
a knowledge base Δ = (𝐾, 𝐴), the question of whether 𝜑 Δ-entails 𝜓 can be determined by
constructing the AF induced by (𝐾 ∪ {𝜑}, 𝐴) and checking whether every stable extension
contains an argument with conclusion 𝜓.
1
Note that this construction method allows arguments to have syntactically different, but semantically equivalent
conclusions.
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Proposition 1. Let Δ = (𝐾, 𝐴) be a knowledge base. Then 𝜑 |∼Δ 𝜓 if and only if every stable
extension 𝐸 of 𝐹 (𝐾 ∪ {𝜑}, 𝐴) contains an argument with conclusion 𝜓.
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 [18].
3. Causal Knowledge Bases
Our definition of a causal model is essentially that of Pearl [8] except that we restrict our
attention to Boolean-valued variables. A causal model consists of a set 𝑈 of background atoms,
a set 𝑉 of explainable atoms, and a set 𝐾 of formulas which we call Boolean structural equations
(this terminology was adopted from [14]). Background atoms represent variables that are
determined outside of the model. They are typically unobservable and uncontrollable. Every
explainable atom 𝑣 is functionally dependent on other atoms of the model. This dependency
is specified by a Boolean structural equation of the form 𝑣 ↔ 𝜑. Intuitively, this equation
represents the causal mechanism by which 𝑣 is determined by the other atoms in the model.
We use bi-implication because the represented causal mechanism determines not only when 𝑣
is true, but also when 𝑣 is false.
Definition 5. A causal model is a triple (𝑈, 𝑉, 𝐾) where 𝑈 and 𝑉 partition the set of atoms into,
respectively, a set of background and explainable atoms. 𝐾 consists of a set of Boolean structural
equations, one for each atom 𝑣 ∈ 𝑉 . A Boolean structural equation for 𝑣 is a formula of the form
𝑣↔𝜑
where 𝜑 is a propositional formula that does not contain 𝑣.
Given a causal model (𝑈, 𝑉, 𝐾) we can reconstruct 𝑈 and 𝑉 from 𝐾 as follows; (1) for every
formula 𝑣 ↔ 𝜑 in 𝐾, add 𝑣 to 𝑉 , and (2) add all the remaining atoms that appear in 𝐾 but
not in 𝑉 to 𝑈 . Therefore we can refer to a causal model as 𝐾. For every Boolean structural
equation 𝑣 ↔ 𝜑, we call an atom appearing in 𝜑 a parent (background or explainable) of 𝑣. Like
in the standard case, a causal model induces a causal graph 𝐺 whose vertices are the explainable
atoms of the model [8]. This graph contains an edge from atom 𝑣 to atom 𝑣 ′ whenever 𝑣 is an
explainable parent of 𝑣 ′ .
Example 1. Let 𝑈 = {corona, influenza, at-risk} and 𝑉 =
{covid, flu, short-of-breath, fever, chills}. The set 𝐾 consisting of the following equations
represents a causal model for a patient diagnosis scenario.
covid ↔ corona
flu ↔ influenza
fever ↔ covid ∨ flu
chills ↔ fever
short-of-breath ↔ covid ∧ at-risk
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corona influenza
at-risk covid flu
short-of-breath fever
chills
Figure 1: Causal graph for Example 1.
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 condi-
tion (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 [8].
Definition 6. A causal knowledge base is a knowledge base Δ = (𝐾, 𝐴) where 𝐾 is a causal
model and where 𝐴 is a set of background assumptions, at least one for each background atom. A
background assumption for an atom 𝑢 is a literal 𝑙 ∈ {𝑢, ¬𝑢}.
By restricting background assumptions to be literals we prevent the ability to express depen-
dencies among background atoms. There are three possible attitudes towards a background
atom 𝑢, since we can assume just 𝑢, just ¬𝑢, or both. Assuming just 𝑢 or ¬𝑢 amounts to
assuming that 𝑢 is true or false, unless we have evidence to the contrary. Assuming both 𝑢 and
¬𝑢 represents a state of uncertainty where 𝑢 may, depending on the evidence, be true as well
as false. If Δ = (𝐾, 𝐴) is a causal knowledge base then Δ-entailment represents a relation
between observations and predictions. These predictions include causes as well as effects of the
observation in accordance with the causal model 𝐾 and background assumptions 𝐴.
Example 2. Consider the causal knowledge base Δ = (𝐾, 𝐴) where 𝐾 is the causal model
defined in Example 1 and where 𝐴 = {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. (1)
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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 𝐹 = 𝐹 ((𝐾 ∪ {fever}, 𝐴)) shown in Figure 2 (we only depict arguments relevant to the
conclusion of short-of-breath). This AF has four stable extensions {𝑎2 , 𝑎4 }, {𝑎3 , 𝑎4 }, {𝑎2 , 𝑎5 }, and
{𝑎3 , 𝑎5 , 𝑎1 }. The argument 𝑎1 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.
𝑎1 : ({at-risk, ¬influenza}, short-of-breath)
𝑎2 : ({¬at-risk}, ¬at-risk) 𝑎4 : ({¬corona}, influenza)
𝑎3 : ({at-risk}, at-risk) 𝑎5 : ({¬influenza}, corona)
Figure 2: The AF 𝐹 (𝐾 ∪ {fever}, 𝐴).
4. Counterfactuals
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 [8], a three-
step procedure and the twin network method. We now demonstrate these two processes are
equivalent under our notion of causal knowledge bases.
Before dealing with counterfactual statements we start with interventional statements of the
form
if 𝑣 would be 𝑥 then 𝜓 would be true (2)
where 𝑣 is a variable we intervene on and 𝑥 a truth value (we use ⊤ for true and ⊥ for false).
The antecedent of this statement represents the action of setting 𝑣 to 𝑥. 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 different 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 affect covid or flu. Like Pearl, given
a causal model 𝐾, we denote by 𝐾[𝑣=𝑥] the new causal model where the equation 𝑣 ↔ 𝜑 is
replaced with 𝑣 ↔ 𝑥.
Definition 7. Let 𝐾 be a causal model, let 𝑣 be an explainable atom, and let 𝑥 ∈ {⊤, ⊥}. We
denote by 𝐾[𝑣=𝑥] the causal model defined by
𝐾[𝑣=𝑥] = {(𝑣 ′ ↔ 𝜑) ∈ 𝐾 | 𝑣 ′ ̸= 𝑣} ∪ {(𝑣 ↔ 𝑥)}.
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Example 3. Let Δ = (𝐾, 𝐴) 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. (3)
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′ = (𝐾[fever=⊥] , 𝐴). Here, the action of setting fever to ⊥
blocks the effect of covid on chills. We now have that shortness of breath leads to the prediction of
no chills
short-of-breath |∼Δ′ ¬chills. (4)
We now turn to counterfactual statements. These are statements of the form
given 𝜑, if 𝑣 had been 𝑥 then 𝜓 would be true (5)
where 𝜑 is an observed piece of evidence and 𝜓 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 [8]
that we adapt for our setting as follows.
Definition 8. Given a causal knowledge base Δ = (𝐾, 𝐴), the truth of a counterfactual state-
ment (5) is determined by:
• Step 1 (abduction) Determine the maximal 𝐾 ∪ {𝜑}-consistent subsets Σ1 , . . . , Σ𝑛 of 𝐴.
These sets represent the possible configurations of the background atoms consistent with
evidence 𝜑.
• Step 2 (action) Set 𝑣 to 𝑥 by updating the causal model 𝐾 to 𝐾[𝑣=𝑥] .
• Step 3 (prediction) For each Σ𝑖 obtained in step 1, determine whether Σ𝑖 ∪ 𝐾[𝑣=𝑥] ⊢ 𝜓. If
the answer is yes for all 𝑖 ∈ 0, . . . 𝑛, then the counterfactual statement (5) is true.
The difficulty with this procedure is that it requires the computation of the maximal 𝐾 ∪ {𝜑}-
consistent subsets of 𝐴. The same difficulty 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 difficulty [8]. It is based on
constructing a so called twin model whose causal network consists of two parts, one to represent
the actual world, and one to represent the counterfactual world. While the two networks share
the same background variables, the counterfactual part consists of a “counterfactual copy” of all
the explainable variables. In what follows we use 𝑣 * to denote a unique new atom representing
the counterfactual copy of 𝑣, and use 𝜑* to denote the formula 𝜑 with each occurrence of an
explainable atom 𝑣 replaced with 𝑣 * .
Definition 9. The twin model for a causal model 𝐾 is the causal model 𝐾 * defined by
𝐾 * = 𝐾 ∪ {(𝑣 * ↔ 𝜑* ) | (𝑣 ↔ 𝜑) ∈ 𝐾}
.
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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 = (𝐾, 𝐴) be a causal knowledge base. The counterfactual statement
given 𝜑, if 𝑣 had been 𝑥 then 𝜓 would be true
is true in Δ if and only if 𝜑 |∼Δ* * 𝜓*.
[𝑣 =𝑥]
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 = (𝐾, 𝐴) be a causal knowledge base. First, we recall that 𝜑 |∼Δ 𝜓 if and only if
(𝐾 ∪ {𝜑}, 𝐴) entails 𝜓. (𝐾 ∪ {𝜑}, 𝐴) entails 𝜓 whenever for every maximal K-consistent subset
Σ of 𝐴, we have that Σ ∪ 𝐾 ⊢ 𝜓.
(⇐) Now, for the given counterfactual statement, consider Δ* = (𝐾 * , 𝐴) with 𝐾 * as given
in Definition 9. Furthermore, we modify Δ* by setting 𝑣 * = 𝑥 in 𝐾 * , i.e., we have Δ*[𝑣* =𝑥] .
That means, 𝐾 * fully contains 𝐾 as well as an exact copy of 𝐾 (each copy of a formula 𝜑 of 𝐾
is denoted with 𝜑* ). In this copy, we also have set 𝑣 * = 𝑥. Therefore, we can essentially also
consider 𝐾 * as the union of 𝐾 and 𝐾[𝑣=𝑥] .
With the twin model method, we have that 𝜑 |∼Δ* * 𝜓 * . That means, according to
[𝑣 =𝑥]
Definition 1 we determine the 𝐾 * ∪{𝜑}-consistent subsets Σ*1 , ..., Σ*𝑛 of 𝐴. Now, since 𝐾 ⊂ 𝐾 * ,
it follows that Σ*1 ∩ 𝐾, ..., Σ*𝑛 ∩ 𝐾 are maximal consistent with 𝐾 ∪ {𝜑} (as required by step 1
abduction).
The action step of Definition 8 is also executed in the twin model by setting 𝑣 * = 𝑥 in 𝐾 * .
In the prediction step, we have to determine whether Σ𝑖 ∪ 𝐾[𝑣=𝑥] ⊢ 𝜓 for all 𝐾-consistent
Σ𝑖 . In the twin model method, we consider the 𝐾 * ∪ {𝜑}-consistent subsets. As already
explained above, 𝐾[𝑣=𝑥] is fully contained in 𝐾 * and therefore, similarly to above, if we have
that Σ*𝑖 ∪ 𝐾 * ⊢ 𝜓 * it follows that Σ*𝑖 ∪ 𝐾[𝑣=𝑥] ⊢ 𝜓. 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 𝐾 and {(𝑣 * ↔ 𝜑* ) | (𝑣 ↔ 𝜑) ∈ 𝐾} have no variables in common. Thus,
for any Σ maximal consistent wrt. 𝐾 * ∪{𝜑} it holds that Σ′ = Σ∩{(𝑣 * ↔ 𝜑* ) | (𝑣 ↔ 𝜑) ∈ 𝐾}
is maximal consistent wrt. 𝐾[𝑣=𝑥] ∪ {𝜑}. By the assumption we have Σ′ ∪ 𝐾[𝑣=𝑥] ∪ {𝜑} ⊢ 𝜓
for any such Σ′ . By monotony of classic entailment it follows that Σ ∪ 𝐾[𝑣 *
* =𝑥] ∪ {𝜑} ⊢ 𝜓.
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 = (𝐾, 𝐴) 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
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corona influenza
at-risk covid flu covid* flu*
short-of-breath fever short-of-breath* fever*
chills chills*
Figure 3: Twin causal diagram for Example 4.
other hand, fever was caused by flu, then the vaccine would not have had an effect, and the patient
would still have had fever. To evaluate this counterfactual statement we check whether
fever |∼Δ* fever* . (6)
[covid* =⊥]
Figure 3 shows the causal graph of the twin model 𝐾covid *
*
=⊥ and Figure 4 shows the AF 𝐹 =
*
(𝐾[covid* =⊥] ∪ {fever}, 𝐴). This AF has two stable extensions {𝑎1 , 𝑎3 } and {𝑎2 }. While the first
entails fever, the second does not. Thus, (6) is false, but note that
fever |∼Δ* ¬fever* (7)
[covid* =⊥]
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.
𝑎1 : ({¬corona}, fever* ) 𝑎2 : ({¬influenza}, corona)
𝑎3 : ({¬corona}, influenza)
*
Figure 4: The AF 𝐹 ((𝐾[covid* =⊥] ∪ {fever}, 𝐴)).
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* fever* . (8)
[covid* =⊥]
*
Figure 5 shows the AF 𝐹 = (𝐾covid *
=⊥ ∪ {fever ∧ short-of-breath}, 𝐴). This AF is the same as
the one shown in Figure 4 except that there is an extra argument (∅, corona) which is based on the
evidence short-of-breath. This AF has one stable extension {𝑎4 , 𝑎2 } which does not entail fever* .
Hence, (8) is true, i.e., given that the patient has fever and shortness of breath, the patient would
not have had fever, had we administered a covid vaccine.
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Lars Bengel et al. CEUR Workshop Proceedings 1–12
𝑎1 : ({¬corona}, fever* ) 𝑎2 : ({¬influenza}, corona)
𝑎4 : (∅, corona) 𝑎3 : ({¬corona}, influenza)
*
Figure 5: The AF 𝐹 (𝐾covid* =⊥ ∪ {fever ∧ short-of-breath}, 𝐴).
5. Discussion
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 [14]. The need to further explain causal reasoning, e. g., using Bayesian
networks, has also been pointed out in [19]. Their contribution towards this is making d-
separation explicit in so called support graphs, which eliminate circular causal structures
and help to explain interdependent causes. Implementing a similar mechanism within our
framework has proven nontrivial. In [20] they illustrate the power of their framework by
applying it to legal reasoning. As the authors point out under future work, the support graph
is more of a visualization tool while the actual evaluation of arguments is still done in the
original Bayesian network formalism. A sophisticated approach towards a framework capable
of intrinsic evaluations of causal statements is the method for generating bipolar argumentation
frameworks from causal models by Rago et al. [21], which is also based on Pearl’s notion of
a causal model. For that they create so called explanation moulds, that reinterpret desirable
properties of semantics of argumentation frameworks. Rago et. al. point out that, compared to
[22], they make better use of the AF representation in their causal reasoning by basing their
semantic evaluation on the support and attack relations themselves instead of relying on an
additional ontological structure. This criticism does not apply to our framework, we only rely
on the causal model. Once the AF is constructed, we do not need the inference apparatus of the
underlying knowledge base formalism for causal reasoning in our framework.
There are several major design differences 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 difference 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 difficulties 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
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Lars Bengel et al. CEUR Workshop Proceedings 1–12
over possible causal structures [23]. Another option could be to utilize a form of probabilistic
instantiation like PABA [24].
6. Conclusion
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 offers 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).
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