# Contrastive Explanations for Argumentation-Based Conclusions

AnneMarie Borg  Department of Information and Computing Sciences, Utrecht
University  Floris Bex  Department of Information and Computing Sciences,
Utrecht University  Department of Law, Technology, Markets and Society,
Tilburg University

( {A.Borg, F.J.Bex} @uu.nl)

###### Abstract

In this paper we discuss _contrastive explanations_ for formal argumentation –
the question why a certain argument (the fact) can be accepted, whilst another
argument (the foil) cannot be accepted under various extension-based
semantics. The recent work on explanations for argumentation-based conclusions
has mostly focused on providing minimal explanations for the (non-)acceptance
of arguments. What is still lacking, however, is a proper argumentation-based
interpretation of contrastive explanations. We show under which conditions
contrastive explanations in abstract and structured argumentation are
meaningful, and how argumentation allows us to make implicit foils explicit.

##  1 Introduction

_Explainable AI_ (XAI) has become an important research direction in AI [24].
AI systems are being applied in a variety of real-life situations in different
domains and with different users. It is therefore essential that such systems
are able to give explanations that provide insight into the underlying
decision models and techniques, so that users can understand, trust and
validate the system, and experts can verify that the system works as intended.
Most of the research in XAI is directed at explaining decisions of subsymbolic
machine learning algorithms (cf. [25]), but explanations also play an
important role in clarifying the decisions of symbolic algorithms [15],
particularly as such algorithms are all-pervasive in everyday systems.

One area in symbolic AI that has seen a number of real-world applications is
formal argumentation [1]. Two central concepts in formal argumentation are
_abstract argumentation frameworks_ [10], sets of arguments and the attack
relations between them, and _structured_ or _logical argumentation frameworks_
[4], where arguments are constructed from a knowledge base and a set of rules,
and the attack relation is based on the individual elements in the arguments.
Common for argumentation frameworks, abstract and structured, is that we can
determine their extensions, sets of arguments that can collectively be
considered as acceptable, under different semantics [10]. The combination of
an argumentation framework and its extensions is a _global_ explanation: what
can we conclude from the model as a whole? However, often we would prefer
simpler, more compact explanations for the acceptability of an individual
argument, a _local_ explanation for a particular decision or conclusion [11]).
A number of methods for determining local explanations for the
(non-)acceptability of arguments have been proposed [7, 12, 13, 14, 16, 26].
What is still lacking, however, is an argumentation-based interpretation of
_contrastive explanations_.

Contrastiveness is central to local explanations [17, 18, 19]: when people ask
_‘Why P?’_ , they often mean _‘Why P rather than Q?’_ – here $P$ is called the
_fact_ and $Q$ is called the _foil_ [17]. The answer to the question is then
to explain as many of the differences between fact and foil as possible. Like
for XAI in general, much of the research on contrastive explanations is done
in the context of machine learning (e.g. [9, 27, 28]). In the literature on
formal argumentation, there has been no such work, the existing work focusing
on _‘Why is argument A (not) acceptable?’_ instead of the contrastive question
_‘Why is argument A acceptable and argument B not?’_ (or vice versa). While
there are other forms of contrastive questions, we choose this one since it is
intuitive, allows for a variety of foils and it can be interesting for both
expert and lay users of an argumentation-based application.

To study contrastive explanations for argumentation-based conclusions, we
extend the basic framework from [7]. With that framework, explanations for
accepted and non-accepted arguments or formulas can be formulated in a variety
of ways, e.g., by only returning necessary or sufficient arguments [8]. The
main idea of the introduced contrastive explanations is that these return the
common elements of the basic acceptance explanation of the fact and the basic
non-acceptance explanation of the foil (or vice versa). By taking into account
the foil, the explanation is tailored towards a specific explanation direction
and other possible explanations that are not relevant for this direction are
not included. We show that in almost all situations these explanations are
meaningful, i.e., that such common elements exist. Moreover, in the context of
a real-life application at the Dutch National Police, we show that contrastive
explanations can be smaller than the basic explanations, by requesting an
explanation for a specific exception to a rule rather than all possible
exceptions. Additionally we show that, due to the explicit notion of conflict
within argumentation, we can provide contrastive explanations when the foil is
not explicitly known. This is an advantage of formal argumentation, since
determining the foil is a challenge for an AI system.

The paper is structured as follows: we briefly discuss some directly related
work and then recall abstract argumentation as introduced in [10]. Then, in
Section 4, the framework from [7] is recalled and some new results for
acceptance and non-acceptance explanations are shown. In Section 5 contrastive
explanations are introduced and it is shown how, in formal argumentation, the
foil can be determined when it is not explicitly stated. In Section 6 the
introduced contrastive explanations are applied to ASPIC+ [23] and in Section
7 a real-life application is discussed. We conclude in Section 8.

##  2 Related Work

XAI has been investigated in many directions, for a variety of approaches to
AI, including formal argumentation. As mentioned in the introduction, we are
interested in _contrastive local explanations for conclusions derived from
formal argumentation_ , where the idea is that the proposed method can be
applied to any Dung-style argumentation framework to generate contrastive
explanations. While contrastive explanations for learning-based decisions have
been investigated extensively (see [27] for a recent overview), there are no
results on contrastive explanations for argumentation-based conclusions.

In the direction of local explanations for argumentation-based conclusions
some research already exists. For example, [14] introduce explanations for
claims as triples of sets of dialectical trees for abstract argumentation and
DeLP and Fan and Toni introduced explanations as dispute trees for accepted
arguments in abstract argumentation and ABA in [12] and for non-accepted
arguments in abstract argumentation in [13]. Even more recently, explanation
semantics, where accepted arguments are labeled with sets of explanation
arguments, were introduced in [16] and explanations for non-accepted arguments
as minimal subframeworks are studied in [20, 26].

For this paper, we take the framework from [7], as it is the only one that
allows for acceptance and non-acceptance explanations in terms of sets of
arguments. While acceptance and non-acceptance is necessary when defining
contrastive explanations (see Section 5), explanations in terms of sets of
arguments make it easier to process the explanations. Additionally, unlike the
other frameworks, the explanations from [7] make it possible to present
explanations derived from a structured setting in terms of elements of
arguments (e.g., premises or rules), rather than full arguments. Therefore, to
the best of our knowledge, this is the first research on _contrastive_ local
explanations for conclusions derived from either abstract or structured formal
argumentation.

##  3 Preliminaries

In this paper we focus on explanations for conclusions that can be derived
from Dung-style argumentation frameworks. Due to space restrictions this
section is very compact, see, e.g., [10] for a more gentle introduction.

An _abstract argumentation framework_ (AF) [10] is a pair
$\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$, where ${\sf
Args}$ is a set of _arguments_ and ${\sf Att}\subseteq{\sf Args}\times{\sf
Args}$ is an _attack relation_ on these arguments. An argumentation framework
can be viewed as a directed graph, in which the nodes represent arguments and
the arrows represent attacks between arguments.

$A$$B$$C$$D$$E$$F$ Figure 1: Graphical representation of the AF
$\mathcal{AF}_{1}$.

######  Example 1.

Figure 1 represents $\mathcal{AF}_{1}=\langle{\sf Args}_{1},\allowbreak{\sf
Att}_{1}\rangle$ where ${\sf Args}_{1}=\\{A,B,C,D,E,F\\}$ and ${\sf
Att}_{1}=\\{(B,A),\allowbreak(B,C),\allowbreak(C,B),\allowbreak(D,A),\allowbreak(E,D),\allowbreak(E,F),\allowbreak(F,E)\\}$.

Dung-style semantics [10] can be applied to an AF, to determine what
combinations of arguments (called _extensions_) can collectively be accepted.

######  Definition 1.

For $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$, ${\sf
S}\subseteq{\sf Args}$ _attacks_ $A\in{\sf Args}$ if there is an
$A^{\prime}\in{\sf S}$ such that $(A^{\prime},A)\in{\sf Att}$; ${\sf S}$
_defends_ $A$ if ${\sf S}$ attacks every attacker of $A$; ${\sf S}$ is
_conflict-free_ if there are no $A_{1},A_{2}\in{\sf S}$ such that
$(A_{1},A_{2})\in{\sf Att}$; and ${\sf S}$ is _admissible_ (${\sf Adm}$) if it
is conflict-free and it defends all of its elements. We denote by ${\sf
S}^{+}$ the set of all arguments attacked by ${\sf S}$.

An admissible set that contains all the arguments that it defends is a
_complete extension_ (${\sf Cmp}$). The _grounded extension_ (${\sf Grd}$) of
$\mathcal{AF}$ is the minimal (with respect to $\subseteq$) complete
extension; a _preferred extension_ (${\sf Prf}$) of $\mathcal{AF}$ is a
maximal (with respect to $\subseteq$) complete extension; and a _semi-stable
extension_ (${\sf Sstb}$) of $\mathcal{AF}$ is a complete extension ${\sf S}$
where ${\sf S}\cup{\sf S}^{+}$ is maximal. An extension will be denoted by
$\mathcal{E}$ and ${\sf Sem}(\mathcal{AF})$ denotes the set of all the
extensions of $\mathcal{AF}$ under the semantics ${\sf Sem}\in\\{{\sf
Adm},\allowbreak{\sf Cmp},\allowbreak{\sf Grd},\allowbreak{\sf
Prf},\allowbreak{\sf Sstb}\\}$.

In what follows, given an argumentation framework $\mathcal{AF}$, we will
denote by ${\sf Sem}{\sf With}(A)=\\{\mathcal{E}\in{\sf Sem}(\mathcal{AF})\mid
A\in\mathcal{E}\\}$ the set of all ${\sf Sem}$-extensions of $\mathcal{AF}$ of
which $A$ is a member and by ${\sf Sem}{\sf Without}(A)=\\{\mathcal{E}\in{\sf
Sem}(\mathcal{AF})\mid A\notin\mathcal{E}\\}$ the set of all ${\sf
Sem}$-extensions of $\mathcal{AF}$ of which $A$ is not a member.

######  Definition 2.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an
argumentation framework, $A\in{\sf Args}$ and ${\sf Sem}\in\\{{\sf
Adm},\allowbreak{\sf Grd},\allowbreak{\sf Cmp},\allowbreak{\sf
Prf},\allowbreak{\sf Sstb}\\}$. It is said that, for ${\sf
Sem}(\mathcal{AF})\neq\emptyset$, $A$ is, w.r.t. ${\sf Sem}$:

  * •

_skeptically accepted_ iff ${\sf Sem}{\sf With}(A)={\sf Sem}(\mathcal{AF})$;

  * •

_credulously accepted_ iff ${\sf Sem}{\sf With}(A)\neq\emptyset$;

  * •

_skeptically non-accepted_ iff ${\sf Sem}{\sf Without}(A)\neq\emptyset$;

  * •

_credulously non-accepted_ iff ${\sf Sem}{\sf Without}(A)={\sf
Sem}(\mathcal{AF})$.

We will denote skeptical [resp. credulous] (non-)acceptance by $\cap$ [resp.
$\cup$] and when $\cap$ or $\cup$ is clear from the context or not relevant
simply write _accepted_ and _non-accepted_.

The notions of attack and defense can also be defined between arguments and
can be generalized to indirect versions: for $\mathcal{AF}=\left\langle{\sf
Args},{\sf Att}\right\rangle$: $A\in{\sf Args}$ _defends_ $B\in{\sf Args}$ if:
there is some $C\in{\sf Args}$ such that $(C,B)\in{\sf Att}$ and $(A,C)\in{\sf
Att}$, in this case $A$ _directly defends $B$_; or $A$ defends $C\in{\sf
Args}$ and $C$ defends $B$, in this case $A$ _indirectly defends $B$_. It is
said that _$A$ defends $B$ in $\mathcal{E}$_ if $A$ defends $B$ and
$A\in\mathcal{E}$. Similarly, $A\in{\sf Args}$ _attacks_ $B\in{\sf Args}$ if:
$(A,B)\in{\sf Att}$, in this case $A$ _directly attacks_ $B$; or $A$ attacks
some $C\in{\sf Args}$ and $C$ defends $B$, in this case $A$ _indirectly
attacks_ $B$.

We will require that an explanation for an argument $A$ is _relevant_ , in
order to prevent that explanations contain arguments that do not influence the
acceptance of $A$.

######  Definition 3.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ and
$A,B\in{\sf Args}$. It is said that $A$ is _relevant_ for $B$ if $A$
(in)directly attacks or defends $B$ and $A$ does not attack itself. A set
${\sf S}\subseteq{\sf Args}$ is relevant for $B$ if all of its arguments are
relevant for $B$. A relevant argument $A$ for $B$ is _conflict-relevant_ for
$B$ if $A$ (in)directly attacks $B$ and it is _defending-relevant_ for $B$ if
$A$ (in)directly defends $B$.

######  Example 2.

In $\mathcal{AF}_{1}$ $B$ and $C$ attack each other and both defend
themselves. Example conflict-free sets are $\\{A_{,}C\\}$ and $\\{A,F\\}$.
There are four preferred and semi-stable extensions:
$\mathcal{E}_{1}=\\{A,C,E\\}$, $\mathcal{E}_{2}=\\{B,D,F\\}$,
$\mathcal{E}_{3}=\\{B,E\\}$ and $\mathcal{E}_{4}=\\{C,D,F\\}$ and $\emptyset$
is the grounded extension.

No argument is skeptically accepted and all arguments are credulously accepted
and skeptically non-accepted for ${\sf Sem}\in\\{{\sf Cmp},\allowbreak{\sf
Prf},\allowbreak{\sf Sstb}\\}$. Argument $F$ defends itself and $D$ directly,
it attacks $E$ directly and $A$ indirectly, it is conflict-relevant for $E$
and $A$ and defending-relevant for $F$ and $D$.

##  4 The Basic Framework

In this section we recall the basic framework of explanations from [7] and
present some new results. The explanations in that paper are defined in terms
of two functions: $0pt$, which determines the arguments that are in the
explanation and $\mathbb{F}$, which determines what elements of these
arguments the explanation presents. To avoid clutter, we instantiate $0pt$
immediately with the following functions, while instantiations of $\mathbb{F}$
are discussed in Section 6:111We refer the interested reader to [7] for
suggestions of other variations of these functions.

######  Definition 4.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF,
$A\in{\sf Args}$ and $\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ for some
semantics ${\sf Sem}$. Then:

  * •

${\sf Defending}(A)=\\{B\in{\sf Args}\mid B\text{ defends }A\\}$ denotes the
set of arguments in ${\sf Args}$ that (in)directly defend $A$ and ${\sf
Defending}(A,\mathcal{E})={\sf Defending}(A)\cap\mathcal{E}$ denotes the set
of arguments that (in)directly defend $A$ in $\mathcal{E}$;

  * •

${\sf NoDefAgainst}(A,\mathcal{E})=\\{B\in{\sf Args}\mid B\allowbreak\text{
attacks }\allowbreak A\allowbreak\text{ and
}\allowbreak\mathcal{E}\allowbreak\text{ does not }\allowbreak\text{defend
}A\text{ against this }\allowbreak\text{attack}\\}$, denotes the set of all
(in)direct attackers of $A$ that are not defended by $\mathcal{E}$.

All three functions (i.e., ${\sf Defending}(A)$, ${\sf
Defending}(A,\mathcal{E})$ and ${\sf NoDefAgainst}(A,\mathcal{E})$) result in
relevant sets of arguments for $A$. In particular, all arguments in ${\sf
Defending}(A)$ and ${\sf Defending}(A,\mathcal{E})$ are defending-relevant and
all arguments in ${\sf NoDefAgainst}(A,\mathcal{E})$ are conflict-relevant.

######  Example 3 (Example 2 continued).

For $\mathcal{AF}_{1}$ from Example 1 we have that: ${\sf
Defending}(A)=\\{C,E\\}$, ${\sf Defending}(D)=\\{F\\}$, ${\sf
NoDefAgainst}(A,\mathcal{E}_{2})=\\{B,D,F\\}$ and ${\sf
NoDefAgainst}(A,\mathcal{E}_{3})=\\{B\\}$.

###  4.1 Acceptance Explanations

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF and
let $A\in{\sf Args}$. If $A$ is accepted w.r.t. a semantics ${\sf
Sem}\in\\{{\sf Grd},\allowbreak{\sf Cmp},\allowbreak{\sf Prf},\allowbreak{\sf
Sstb}\\}$ and an acceptance strategy $\star\in\\{\cap,\cup\\}$ then an
acceptance explanation can be requested. The explanation depends on the
acceptance strategy: for a skeptical reasoner the explanation has to account
for the acceptance of the argument in each ${\sf Sem}$-extension, while for a
credulous reasoner explaining the acceptance of the argument in one ${\sf
Sem}$-extension is sufficient.

######  Definition 5 (Argument acceptance explanation).

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF, let
$A\in{\sf Args}$ be an argument that is accepted given some semantics ${\sf
Sem}$ and an acceptance strategy ($\cap$ or $\cup$). Then:

  * •

${\sf Sem}{\sf Acc}^{\cap}(A)=\bigcup_{\mathcal{E}\in{\sf
Sem}(\mathcal{AF})}{\sf Defending}(A,\mathcal{E})$

  * •

${\sf Sem}{\sf Acc}^{\cup}(A)\in\\{{\sf
Defending}(A,\mathcal{E})\mid\mathcal{E}\in{\sf Sem}{\sf With}(A)\\}$.

The $\cap$-explanation returns all the arguments that defend $A$ in at least
one of the ${\sf Sem}$-extensions, while the $\cup$-explanation is a set of
arguments that defend $A$ in one ${\sf Sem}$-extension.

######  Example 4 (Example 3 continued).

In $\mathcal{AF}_{1}$ we have that: ${\sf Prf}{\sf Acc}^{\cup}(A)=\\{C,E\\}$;
and ${\sf Prf}{\sf Acc}^{\cup}(D)=\\{F\\}$.

Next we show some properties of the acceptance explanations. Proposition 1
shows that the defending arguments of an argument $A$ also defend the
arguments defended by $A$, while Proposition 2 shows that an explanation for
an argument is only empty when it is not attacked.

######  Proposition 1.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF,
$\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ for ${\sf Sem}\in\\{{\sf
Adm},\allowbreak{\sf Cmp},\allowbreak{\sf Grd},\allowbreak{\sf
Prf},\allowbreak{\sf Sstb}\\}$ and let $A,B\in{\sf Args}$. Then:

  * •

if $A\in{\sf Defending}(B,\mathcal{E})$, then ${\sf
Defending}(A,\mathcal{E})\subseteq{\sf Defending}(B,\mathcal{E})$;

  * •

if $A\in{\sf Defending}(B,\mathcal{E})$ and $B\in{\sf
Defending}(A,\mathcal{E})$, then ${\sf Defending}(A,\mathcal{E})={\sf
Defending}(B,\mathcal{E})$.

###### Proof.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF,
$\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ for ${\sf Sem}\in\\{{\sf Adm},{\sf
Cmp},{\sf Grd},{\sf Prf},{\sf Sstb}\\}$ and let $A,B\in{\sf Args}$. Suppose
that $A\in{\sf Defending}(B,\mathcal{E})$. By definition of ${\sf Defending}$
it follows that $A\in\mathcal{E}$. Let $C\in{\sf Args}$ such that $C\in{\sf
Defending}(A,\mathcal{E})$. Then there is some $D\in{\sf Args}$ such that
$(D,A)\in{\sf Att}$ and $C$ defends $A$ against this attack. However, since
$A$ defends $B$, it follows that $D$ attacks $B$ as well, from which it
follows that $C$ defends $B$ as well. Therefore $C\in{\sf
Defending}(B,\mathcal{E})$. The second item follows immediately. ∎

######  Proposition 2.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF and
let $A\in{\sf Args}$ be such that $A$ is accepted w.r.t. ${\sf Sem}\in\\{{\sf
Adm},\allowbreak{\sf Cmp},\allowbreak{\sf Grd},\allowbreak{\sf
Prf},\allowbreak{\sf Sstb}\\}$ and $\star\in\\{\cap,\cup\\}$. Then ${\sf
Sem}{\sf Acc}^{\star}(A)=\emptyset$ iff there is no $B\in{\sf Args}$ such that
$(B,A)\in{\sf Att}$.

###### Proof.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF and
let $A\in{\sf Args}$ be such that $A$ is accepted w.r.t. ${\sf Sem}\in\\{{\sf
Adm},\allowbreak{\sf Cmp},\allowbreak{\sf Grd},\allowbreak{\sf
Prf},\allowbreak{\sf Sstb}\\}$ and $\star\in\\{\cap,\cup\\}$.

$\Rightarrow\quad$ Suppose that ${\sf Sem}{\sf Acc}^{\star}(A)=\emptyset$.
Then for each $\mathcal{E}\in{\sf Sem}{\sf With}(A)$, ${\sf
Defending}(A,\mathcal{E})=\emptyset$. Hence there is no attacker of $A$ that
is defended by some argument from $\mathcal{E}$. Since $A\in\mathcal{E}$, $A$
is defended against its attackers. It follows that $A$ is not attacked at all.

$\Leftarrow\quad$ Now suppose that $A$ is not attacked. Then there is no
argument that defends $A$. Therefore, for any $\mathcal{E}\in{\sf Sem}{\sf
With}(A)$, ${\sf Defending}(A,\mathcal{E})=\emptyset$. It follows that ${\sf
Sem}{\sf Acc}^{\star}(A)=\emptyset$. ∎

###  4.2 Non-acceptance Explanations

In order to explain a contrast between an accepted and non-accepted argument,
we need non-acceptance explanations as well. Therefore, in this section, basic
definitions for explanations of non-accepted arguments are recalled. There are
again two types of explanations.

######  Definition 6 (Argument non-acceptance explanation).

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF, let
$A\in{\sf Args}$ be an argument that is not accepted w.r.t. ${\sf Sem}$ and
$\star\in\\{\cap,\cup\\}$. Then:

$\displaystyle\bullet\ {\sf Sem}{\sf NotAcc}^{\cap}(A)$
$\displaystyle=\bigcup_{\mathcal{E}\in{\sf Sem}{\sf Without}(A)}{\sf
NoDefAgainst}(A,\mathcal{E})$ $\displaystyle\bullet\ {\sf Sem}{\sf
NotAcc}^{\cup}(A)$ $\displaystyle=\bigcup_{\mathcal{E}\in{\sf
Sem}(\mathcal{AF})}{\sf NoDefAgainst}(A,\mathcal{E}).$

Thus, a non-acceptance explanation contains all the arguments that attack $A$
and for which no defense exists in: some ${\sf Sem}$-extensions (for $\cap$)
of which $A$ is not a member; all ${\sf Sem}$-extensions (for $\cup$). That
for $\cap$ only some extensions have to be considered follows since $A$ is
skeptically non-accepted as soon as ${\sf Sem}{\sf Without}(A)\neq\emptyset$,
while $A$ is credulously non-accepted when ${\sf Sem}{\sf Without}(A)={\sf
Sem}(\mathcal{AF})$.

######  Example 5 (Example 3 continued).

Consider again $\mathcal{AF}_{1}$. We have that: ${\sf Prf}{\sf
NotAcc}^{\cap}(A)=\\{B,D,F\\}$ and ${\sf Prf}{\sf NotAcc}^{\cap}(D)=\\{E\\}$.

The next proposition, the counterpart of Proposition 2, shows that a non-
acceptance explanation is never empty.

######  Proposition 3.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF and
let $A\in{\sf Args}$ be such that $A$ is non-accepted w.r.t. ${\sf
Sem}\in\\{{\sf Cmp},{\sf Grd},{\sf Prf},{\sf Sstb}\\}$ and
$\star\in\\{\cap,\cup\\}$. Then ${\sf Sem}{\sf
NotAcc}^{\star}(A)\neq\emptyset$.

###### Proof.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ and $A\in{\sf
Args}$ be such that $A$ is non-accepted w.r.t. ${\sf Sem}\in\\{{\sf
Cmp},\allowbreak{\sf Grd},\allowbreak{\sf Prf},\allowbreak{\sf Sstb}\\}$ and
$\star\in\\{\cap,\cup\\}$. Assume that ${\sf Sem}{\sf
NotAcc}^{\star}(A)=\emptyset$, then there is no argument
$B\in\bigcup_{\mathcal{E}\in{\sf Sem}{\sf Without}(A)}{\sf
NoDefAgainst}(A,\mathcal{E})$. Hence, for each $\mathcal{E}\in{\sf Sem}{\sf
Without}(A)$, ${\sf NoDefAgainst}(A,\mathcal{E})=\emptyset$. It follows that
there is no $B\in{\sf Args}$ such that $(B,A)\in{\sf Att}$. But then, by the
completeness of $\mathcal{E}$ it follows that $A\in\mathcal{E}$. A
contradiction. Therefore ${\sf Sem}{\sf NotAcc}^{\star}(A)\neq\emptyset$. ∎

That the above proposition does not hold for ${\sf Sem}={\sf Adm}$ follows
since not every admissible extension contains all the arguments that it
defends. Take for example an AF with arguments $A$ and $B$ and no attacks
between them. Then $\\{B\\}$ is an admissible extension, thus $A$ is
skeptically non-accepted, yet ${\sf NoDefAgainst}(A,\\{B\\})=\emptyset$. In
fact: ${\sf Adm}{\sf NotAcc}^{\cap}(A)={\sf Adm}{\sf
NotAcc}^{\cap}(B)=\emptyset$.

For the non-acceptance counterpart of Proposition 1 note that $A\in{\sf
NoDefAgainst}(B,\mathcal{E})$ entails that $A$ (in)directly attacks $B$.
Hence, if $A$ is not accepted either, the arguments in ${\sf
NoDefAgainst}(A,\mathcal{E})$ (in)directly defend $B$. In Section 4.3 we study
how acceptance and non-acceptance are related.

###  4.3 Comparing Acceptance and Non-acceptance

When looking at Examples 4 and 5 for $A$ and $D$ one can observe that
acceptance and non-acceptance explanations are related. In this section we
formalize this observation. In particular, we show that non-acceptance
explanations contain the acceptance explanations of (1) the direct attackers;
(2) the directly attacked arguments; and (3) the indirectly attacked
arguments.

######  Proposition 4.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an
argumentation framework, let $\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ for some
${\sf Sem}\in\\{{\sf Adm},\allowbreak{\sf Cmp},\allowbreak{\sf
Grd},\allowbreak{\sf Prf},\allowbreak{\sf Sstb}\\}$ and let $A,\allowbreak
B_{1},\allowbreak\ldots,\allowbreak B_{n},\allowbreak
C_{1},\allowbreak\ldots,\allowbreak C_{k}\in{\sf Args}$ such that
$(B_{1},A),\ldots,\allowbreak(B_{n},A)\in{\sf Att}$ and $A$ indirectly attacks
$C_{1},\ldots,C_{k}$. Then:

  1. 1.

where $B_{1},\ldots,B_{m}\in\mathcal{E}$, $m\leq n$: ${\sf
NoDefAgainst}(A,\mathcal{E})\supseteq{\sf
Defending}(B_{1},\mathcal{E})\cup\ldots\cup{\sf
Defending}(B_{m},\mathcal{E})$;

  2. 2.

when $A\in\mathcal{E}$: ${\sf Defending}(A,\mathcal{E})\subseteq{\sf
NoDefAgainst}(B_{1},\mathcal{E})\allowbreak\cup\allowbreak\ldots\allowbreak\cup\allowbreak{\sf
NoDefAgainst}(B_{n},\allowbreak\mathcal{E})$;

  3. 3.

where $A\in\mathcal{E}$ and $C_{1},\ldots,C_{j}\notin\mathcal{E}$, $j\leq k$:
${\sf Defending}(A,\mathcal{E})\subseteq{\sf NoDefAgainst}(C_{i},\mathcal{E})$
for all $i\in\\{1,\ldots,j\\}$.

###### Proof.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF,
$\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ for some ${\sf Sem}\in\\{{\sf
Adm},{\sf Cmp},{\sf Grd},{\sf Prf},{\sf Sstb}\\}$ and $A\in{\sf Args}$.

  1. 1.

Let $B\in\mathcal{E}$ be such that $(B,A)\in{\sf Att}$. If ${\sf
Defending}(B,\mathcal{E})=\emptyset$, we are done, hence, let $C\in{\sf
Defending}(B,\mathcal{E})$. Then, by the proof of Proposition 2 there is some
$D\in{\sf Args}$ such that $(D,B)\in{\sf Att}$ and $C$ (in)directly attacks
$D$. Since $B$ attacks $A$, it follows that $D$ defends $A$ and that $C$
(in)directly attacks $A$. Since $C\in\mathcal{E}$, $\mathcal{E}$ does not
defend $A$ against the attack from $C$ and therefore $C\in{\sf
NoDefAgainst}(A,\mathcal{E})$.

  2. 2.

Let $C\in{\sf Defending}(A,\mathcal{E})$, then, by Proposition 2, $n\neq 0$.
Suppose that $C$ directly defends $A$, then there is a
$B_{i}\in\\{B_{1},\ldots,B_{n}\\}$ such that $(C,B_{i})\in{\sf Att}$. Since
$C\in\mathcal{E}$ it follows that $C\in{\sf
NoDefAgainst}(B_{i},\allowbreak\mathcal{E})$. Now suppose that $C$ indirectly
defends $A$. Then there are $D_{1},D_{2},\ldots,D_{k}\in{\sf Args}$, where $k$
is odd, such that
$(D_{1},B_{i}),(D_{2},D_{1}),\ldots(D_{k},D_{k-1}),\allowbreak(C,\allowbreak
D_{k})\in{\sf Att}$. Since $D_{k}$ defends $B_{i}$ and $C$ attacks $D_{k}$ it
follows that $C$ attacks $B_{i}$ as well. Hence $C\in{\sf
NoDefAgainst}(B_{i},\mathcal{E})$. Note that, for any $D\in{\sf
Defending}(A,\mathcal{E})$ a $B_{i}\in\\{B_{1},\ldots,B_{n}\\}$ exists. It
therefore follows that ${\sf Defending}(A,\mathcal{E})\subseteq{\sf
NoDefAgainst}(B_{1},\mathcal{E})\cup\ldots\cup{\sf
NoDefAgainst}(B_{n},\mathcal{E})$.

  3. 3.

Let $A\in\mathcal{E}$ and suppose that $C_{1},\ldots,C_{j}\notin\mathcal{E}$
for some $j\leq k$. By assumption $A$ indirectly attacks $C_{i}$ for all
$i\in\\{1,\ldots,j\\}$ and since $A\in\mathcal{E}$, $C_{i}$ is not defended
against this attack by $A$. Therefore $A\in{\sf
NoDefAgainst}(C_{i},\mathcal{E})$. Note that any $D\in{\sf
Defending}(A,\mathcal{E})$ defends $A$ and therefore indirectly attacks
$C_{i}$ as well. It therefore follows that $D\in{\sf
NoDefAgainst}(C_{i},\mathcal{E})$ and hence ${\sf
Defending}(A,\mathcal{E})\subseteq{\sf NoDefAgainst}(C_{i},\mathcal{E})$ for
all $i\in\\{1,\ldots,j\\}$. ∎

To see that ${\sf NoDefAgainst}(A,\mathcal{E})\not\subseteq{\sf
Defending}(B_{1},\mathcal{E})\cup\ldots\cup{\sf
Defending}(B_{n},\mathcal{E})$, take a look at the following example.
Intuitively this is the case since, in terms of labeling semantics [2], an
argument can be in the extension, attacked by the extension (i.e., out) or
attacked by an argument that is not in or out (i.e., undecided).

######  Example 6.

Let $\mathcal{AF}_{2}=\left\langle{\sf Args}_{2},{\sf Att}_{2}\right\rangle$,
as shown in Figure 2. There are two preferred extensions: ${\sf
Prf}(\mathcal{AF}_{3})=\\{\\{A\\},\allowbreak\\{B\\}\\}$. Here we have that
${\sf NoDefAgainst}(C,\\{B\\})=\\{B,D,E,F\\}$ but only $(B,C)\in{\sf Att}$
such that $B\in\mathcal{E}$, for which: ${\sf Defending}(B,\\{B\\})=\\{B\\}$.

$A$$B$$C$$D$$E$$F$ Figure 2: Graphical representations of the AF
$\mathcal{AF}_{2}$.

Knowing how acceptance and non-acceptance explanations are related is useful
in the context of contrastive explanations, where explanations are not only
about the requested argument, but about arguments that are conflicting with
the requested argument as well.

##  5 Contrastive Explanations

A contrastive explanation explains $A$ by explaining _why $A$ rather than
$B$_. Important in contrastive explanations is that the difference between
fact (i.e., $A$) and foil (i.e., $B$) is highlighted. In this paper we assume
that fact and foil are not always compatible: it cannot be the case that both
$A$ and $B$ are skeptically accepted. Intuitively, we make this assumption
since otherwise there is no contrastive question for fact and foil (i.e., _why
both $A$ and $B$_ is not contrastive).

In the context of formal argumentation contrastive explanations are modeled by
comparing the elements of the basic explanations that explain the acceptance
[resp. non-acceptance] of the fact and, at the same time, explain the non-
acceptance [resp. acceptance] of the foil.

######  Definition 7 (Contrastive explanations).

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF, let
$A\in{\sf Args}$ (the fact) and let ${\sf S}\subseteq{\sf Args}$ (a set of
foils) such that there is no $\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ in which
$A,B\in\mathcal{E}$ for all $B\in{\sf S}$. Moreover, $\star\in\\{\cap,\cup\\}$
and ${\dagger}=\cap$ if $\star=\cup$ and ${\dagger}=\cup$ if $\star=\cap$ and
${\sf sem}$. Contrastive explanations are then defined as in Figure 3.

$\displaystyle{\sf Sem}{\sf Cont}^{\star}(A,{\sf S})=$
$\displaystyle\begin{cases}{\sf Sem}{\sf Acc}^{\star}(A)\cap\bigcup_{B\in{\sf
S}}{\sf Sem}{\sf NotAcc}^{\dagger}(B)&\text{ if }{\sf Sem}{\sf
Acc}^{\star}(A)\cap\bigcup_{B\in{\sf S}}{\sf Sem}{\sf
NotAcc}^{\dagger}(B)\neq\emptyset\\\ \left\langle{\sf Sem}{\sf
Acc}^{\star}(A),\bigcup_{B\in{\sf S}}{\sf Sem}{\sf
NotAcc}^{\dagger}(B)\right\rangle&\text{ otherwise}.\end{cases}$
$\displaystyle{\sf Sem}{\sf ContN}^{\star}(A,{\sf S})=$
$\displaystyle\begin{cases}{\sf Sem}{\sf
NotAcc}^{\star}(A)\cap\bigcup_{B\in{\sf S}}{\sf Sem}{\sf
Acc}^{\dagger}(B)&\text{ if }{\sf Sem}{\sf
NotAcc}^{\star}(A)\cap\bigcup_{B\in{\sf S}}{\sf Sem}{\sf
Acc}^{\dagger}(B)\neq\emptyset\\\ \left\langle{\sf Sem}{\sf
NotAcc}^{\star}(A),\bigcup_{B\in{\sf S}}{\sf Sem}{\sf
Acc}^{\dagger}(B)\right\rangle&\text{ otherwise}.\end{cases}$ Figure 3:
Definition of contrastive explanations (see Definition 7).

In words, when there are arguments that cause the fact to be accepted [resp.
non-accepted] and the foil to be non-accepted [resp. accepted], the
contrastive explanation is the set of such arguments, this is the first case.
If there are no common causes for the acceptance [resp. non-acceptance] of the
fact and the non-acceptance [resp. acceptance] of the foil, the contrastive
explanation is a pair of the respective explanations, the second case.

######  Example 7 (Examples 4 and 5 continued).

For the framework $\mathcal{AF}_{1}$ we have that: ${\sf Prf}{\sf
Cont}^{\cup}(A,D)=\\{E\\}$, ${\sf Prf}{\sf Cont}^{\cup}(A,B)=\\{C\\}$ and
${\sf Prf}{\sf Cont}^{\cup}(A,\\{B,D\\})=\\{C,E\\}$ while ${\sf Prf}{\sf
ContN}^{\cap}(A,B)=\\{B\\}$ and ${\sf Prf}{\sf ContN}^{\cap}(A,D)=\\{F\\}$.

Thus, while the acceptance of $A$ can generally be explained by $C$ and $E$
(recall Example 4) when compared to the non-acceptance of $D$ [resp. $B$] the
acceptance of $A$ is explained by $E$ [resp. $C$] alone.

One could consider contrastive explanations more meaningful when these return
a set, rather than a pair. This is the case since then there are arguments
that influence both the acceptance [resp. non-acceptance] of the fact and the
non-acceptance [resp. acceptance] of the foil. The next proposition shows that
in most cases the explanation is a set. Only when the accepted argument is not
attacked at all or fact and foil are not conflict-relevant is the intersection
empty.

######  Proposition 5.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF and
$A,B\in{\sf Args}$. Then ${\sf Sem}{\sf Acc}^{\star}(A)\cap{\sf Sem}{\sf
NotAcc}^{\dagger}(B)=\emptyset$ implies that ${\sf Sem}{\sf
Acc}^{\star}(A)=\emptyset$; or $A$ is not conflict-relevant for $B$.

###### Proof.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF and
let $A,B\in{\sf Args}$. That ${\sf Sem}{\sf Acc}^{\star}(A)\cap{\sf Sem}{\sf
NotAcc}^{\dagger}(B)=\emptyset$ when ${\sf Sem}{\sf Acc}^{\star}(A)=\emptyset$
follows immediately. Recall from Proposition 3 that ${\sf Sem}{\sf
NotAcc}^{\dagger}(B)\neq\emptyset$. Suppose that $A$ is conflict-relevant for
$B$ and, without loss of generality, that ${\sf Sem}{\sf
Acc}^{\star}(A)\neq\emptyset$. Since ${\sf Sem}{\sf Acc}^{\star}(A)$ is
requested and, by assumption ${\sf Sem}{\sf Acc}^{\star}(A)\neq\emptyset$,
there is some $\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ such that
$A\in\mathcal{E}$ and ${\sf Defending}(A,\mathcal{E})\neq\emptyset$. If
$(A,B)\in{\sf Att}$, by Proposition 4.1 ${\sf
Defending}(A,\mathcal{E})\subseteq{\sf NoDefAgainst}(B,\mathcal{E})$. If $A$
indirectly attacks $B$, then by Proposition 4.3 it follows that ${\sf
Defending}(A,\mathcal{E})\subseteq{\sf NoDefAgainst}(B,\mathcal{E})$ as well.
Since ${\sf Defending}(A,\mathcal{E})\neq\emptyset$ we have that ${\sf
Sem}{\sf Acc}^{\star}(A)\cap{\sf NotAcc}^{\dagger}(B)\neq\emptyset$. ∎

In view of the above result, the following conditions are introduced on the
fact and foil. By requiring these conditions to hold, meaningful contrastive
explanations can be obtained. For this let $\mathcal{AF}=\left\langle{\sf
Args},{\sf Att}\right\rangle$ be an AF and let $\\{A\\}\cup{\sf
S}\subseteq{\sf Args}$. Then ${\sf Sem}{\sf Cont}^{\star}(A,{\sf S})$ [resp.
${\sf Sem}{\sf ContN}^{\star}(A,{\sf S})$] can be requested when, for each
$B\in{\sf S}$:

  * •

$A$ is at least credulously accepted [resp. skeptically non-accepted] and $B$
is at least skeptically non-accepted [resp. credulously accepted];

  * •

for each $\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ it never holds that
$\\{A,B\\}\subseteq\mathcal{E}$;

  * •

either $A$ is conflict-relevant for $B$ or $B$ is conflict-relevant for $A$.

These conditions ensure that fact and foil are incompatible, but still
relevant for each other: it is explained what makes the fact accepted [resp.
non-accepted] and, simultaneously causes the foil to be non-accepted [resp.
accepted]. This prevents contrastive explanations for arguments that are not
related or conflicting. These conditions are not exhaustive, depending on,
e.g., the application, a user might wish to enforce further conditions on fact
or foil.

###  5.1 Non-explicit Foil

When humans request a (contrastive) explanation the foil is sometimes left
implicit, yet the expected explanation does not provide all reasons for the
fact happening, but should rather explain the difference between fact and
foil. While humans are able to detect the foil based on, e.g., context, this
is a challenge for AI systems, including argumentation. In particular, it is
impossible to provide one strategy, since different applications entail
different foils. For example, if argumentation is applied to determine a yes
or no answer (e.g., whether one qualifies for a loan), the foil would be _not
A_. However, if argumentation is applied to choose one element from a larger
set (e.g., a medical diagnosis), the foil might be any member of that set.

Since in the definition of contrastive explanations it is necessary to provide
a foil, a way to determine the foil is required. This is where one of the
advantages of formal argumentation comes in: the explicit nature of conflicts
between arguments makes that the foil or a set of foils can be constructed
from an AF. Since the relation between arguments is only determined by the
attack relation in our setting, it is impossible to distinguish between
attackers. To illustrate the possibilities, in the remainder of the paper the
foil will consist of all directly attacking arguments.

######  Definition 8.

Let $\mathcal{AF}=\langle{\sf Args},\allowbreak{\sf Att}\rangle$ be an AF and
let $A\in{\sf Args}$. Then: ${\sf Foil}(A)=\\{B\in{\sf Args}\mid B\text{
directly attacks }A\\}$.

######  Example 8.

For $\mathcal{AF}_{1}$ we have that: ${\sf Foil}(A)=\\{B,D\\}$; ${\sf
Foil}(B)=\\{C\\}$ and ${\sf Foil}(D)=\\{E\\}$.

Note that, for our running example, the explanations with implicit foil do not
change:

######  Example 9 (Examples 7 and 8 continued).

For the AF $\mathcal{AF}_{1}$: ${\sf Prf}{\sf Cont}^{\cup}(A,{\sf
Foil}(A))=\\{C,E\\}$ and ${\sf Prf}{\sf ContN}^{\cap}(D,{\sf
Foil}(D))=\\{E\\}$.

In what follows it will be assumed that ${\sf Foil}(A)\neq\emptyset$, for fact
$A$, i.e., that a foil exists. Note that, by Definition 8, for any AF
$\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ and $A\in{\sf
Args}$, ${\sf Foil}(A)=\emptyset$ iff there is no $B\in{\sf Args}$ such that
$(B,A)\in{\sf Att}$. Hence, any argument without a foil is not attacked at
all. In such a case a non-acceptance explanation is not applicable and, by
Proposition 2, the acceptance explanation is empty. Therefore, this
requirement does not restrict our results.

The next proposition shows that the obtained contrastive explanations are
meaningful when the first condition of the applicability of contrastive
explanations is fulfilled and the foil is defined as in Definition 8.

######  Proposition 6.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF, let
$A\in{\sf Args}$ be such that ${\sf Foil}(A)\neq\emptyset$ and ${\sf
Sem}\in\\{{\sf Adm},{\sf Cmp},{\sf Grd},{\sf Prf},{\sf Sstb}\\}$. Then a
contrastive acceptance [resp. non-acceptance] explanation can be requested for
$A$, when $A$ is at least credulously accepted [resp. skeptically non-
accepted] and for all $B\in{\sf Foil}(A)$, $B$ is at least skeptically non-
accepted [resp. credulously accepted].

###### Proof.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF, let
$A\in{\sf Args}$ be such that ${\sf Foil}(A)\neq\emptyset$ and ${\sf
Sem}\in\\{{\sf Adm},\allowbreak{\sf Cmp},\allowbreak{\sf Grd},\allowbreak{\sf
Prf},\allowbreak{\sf Sstb}\\}$. To show that:

  1. 1.

for each $\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ it is never the case that
$\\{A,B\\}\subseteq\mathcal{E}$: by definition, ${\sf Foil}(A)=\\{B\in{\sf
Args}\mid(B,A)\in{\sf Att}\\}$ and hence, since each $\mathcal{E}\in{\sf
Sem}(\mathcal{AF})$ is conflict-free, it is never the case that
$\\{A,B\\}\subseteq\mathcal{E}$.

  2. 2.

either $A$ is conflict-relevant for $B$ or $B$ is conflict-relevant for $A$:
by definition, $(B,A)\in{\sf Att}$ and hence $B$ is conflict-relevant for $A$.
∎

In view of the above proposition we obtain the following corollary from
Propositions 5 and 6.

######  Corollary 1.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF, let
$A\in{\sf Args}$ be such that ${\sf Foil}(A)\neq\emptyset$ and ${\sf
Sem}\in\\{{\sf Adm},\allowbreak{\sf Cmp},\allowbreak{\sf Grd},\allowbreak{\sf
Prf},\allowbreak{\sf Sstb}\\}$. Then:

  * •

${\sf Sem}{\sf Cont}^{\star}(A,{\sf Foil}(A))$ is never of the form
$\left\langle{\sf Sem}{\sf Acc}^{\star}(A),\bigcup_{B\in{\sf Foil}(A)}{\sf
Sem}{\sf NotAcc}^{\dagger}(B)\right\rangle$;

  * •

${\sf Sem}{\sf ContN}^{\star}(A,{\sf Foil}(A))=\left\langle{\sf Sem}{\sf
NotAcc}^{\star}(A),\allowbreak\bigcup_{B\in{\sf Foil}(A)}{\sf Sem}{\sf
Acc}^{\dagger}(B)\right\rangle$ iff ${\sf Sem}{\sf
Acc}^{\dagger}(B)=\emptyset$ for all $B\in{\sf Foil}(A)$.

Thus, when the foil is determined as in Definition 8, non-acceptance
contrastive explanations are pairs if and only if the fact is only attacked by
non-attacked arguments.

##  6 Contrastive Explanations in Structured Argumentation

Since many approaches to structured argumentation result in an abstract
argumentation framework (see e.g., [4]), the basic and contrastive
explanations as well as the results in this paper are applicable to such
approaches as well. However, like in [7], the structure of the arguments
within any approach to structured argumentation, makes it possible to refine
the explanations. For this we take ASPIC+ [23].222For the sake of simplicity
and conciseness we take classical negation (denoted by $\neg$) as the
contrariness function and we do not considered preferences in this paper.

#### ASPIC+

In ASPIC+, an _argumentation system_
$\text{AS}=\left\langle\mathcal{L},\mathcal{R},n\right\rangle$ consisting of a
propositional language $\mathcal{L}$, a set of rules
$\mathcal{R}=\mathcal{R}_{s}\cup\mathcal{R}_{d}$ (of the form
$r=\phi_{1},\ldots,\phi_{n}\rightarrow\psi$ for strict rules
($r\in\mathcal{R}_{s}$) and $r=\phi_{1}\ldots,\phi_{n}\Rightarrow\psi$ for
defeasible rules ($r\in\mathcal{R}_{d}$)) and the naming convention $n$ for
defeasible rules and the _knowledge base_
$\mathcal{K}=\mathcal{K}_{n}\cup\mathcal{K}_{p}$ (containing the disjoint sets
of axioms ($\mathcal{K}_{n}$) and ordinary premises ($\mathcal{K}_{p}$)) form
an _argumentation theory_
$\text{AT}=\left\langle\text{AS},\mathcal{K}\right\rangle$, within which
arguments can be constructed:

######  Definition 9.

An _argument_ $A$ on the basis of a knowledge base $\mathcal{K}$ in an
argumentation system $\left\langle\mathcal{L},\mathcal{R},n\right\rangle$ is:

  1. 1.

$\phi$ if $\phi\in\mathcal{K}$, with ${\sf Prem}(A)={\sf Sub}(A)=\\{\phi\\}$,
${\sf Conc}(A)=\phi$ and ${\sf TopRule}(A)=\text{undefined}$;

  2. 2.

$A_{1},\ldots,A_{n}\rightarrow/\Rightarrow\psi$ if $A_{1},\ldots,A_{n}$ are
arguments such that there exists a strict/defeasible rule ${\sf
Conc}(A_{1}),\allowbreak\ldots,{\sf Conc}(A_{n})\rightarrow/\Rightarrow\psi$
in $\mathcal{R}_{s}/\mathcal{R}_{d}$.

${\sf Prem}(A)={\sf Prem}(A_{1})\cup\ldots\cup{\sf Prem}(A_{n})$; ${\sf
Conc}(A)=\psi$; ${\sf Sub}(A)={\sf Sub}(A_{1})\cup\ldots\cup{\sf
Sub}(A_{n})\cup\\{A\\}$; ${\sf TopRule}(A)={\sf Conc}(A_{1}),\ldots,{\sf
Conc}(A_{n})\rightarrow/\Rightarrow\psi$.

For a set of argument ${\sf S}$: ${\sf Prem}({\sf S})=\\{{\sf Prem}(A)\mid
A\in{\sf S}\\}$.

Attacks on an argument are based on the rules and premises applied in the
construction of that argument.

######  Definition 10.

An argument $A$ _attacks_ an argument $B$ iff $A$ _undercuts_ , _rebuts_ or
_undermines_ $B$, where:333We let $\phi=-\psi$ if $\phi=\neg\psi$ or
$\psi=\neg\phi$.

  * •

$A$ _undercuts_ $B$ (on $B^{\prime}$) iff ${\sf Conc}(A)=-n(r)$ for some
$B^{\prime}\in{\sf Sub}(B)$ such that $r={\sf
TopRule}(B^{\prime})\in\mathcal{R}_{d}$;

  * •

$A$ _rebuts_ $B$ (on $B^{\prime}$) iff ${\sf Conc}(A)=-\phi$ for some
$B^{\prime}\in{\sf Sub}(B)$ of the form
$B^{\prime\prime}_{1},\ldots,B^{\prime\prime}_{n}\Rightarrow\phi$;

  * •

$A$ _undermines_ $B$ (on $\phi$) iff ${\sf Conc}(A)=-\phi$ for some
$\phi\in{\sf Prem}(B)\setminus\mathcal{K}_{n}$.

_Abstract argumentation frameworks_ can be derived from argumentation
theories: $\mathcal{AF}(\text{AT})=\left\langle{\sf Args},{\sf
Att}\right\rangle$, where ${\sf Args}$ is the set of all arguments constructed
from the argumentation theory AT and $(A,B)\in{\sf Att}$ iff $A$ attacks $B$.

Dung-style semantics can be applied to such AFs. We denote: ${\sf All}{\sf
Args}(\phi)=\\{A\in{\sf Args}\mid{\sf Conc}(A)=\phi\\}$, ${\sf Sem}{\sf
Accept}(\phi)={\sf All}{\sf Args}(\phi)\cap\bigcup{\sf Sem}(\mathcal{AF})$,
${\sf Sem}{\sf With}(\phi)=\bigcup\\{{\sf Sem}{\sf With}(A)\mid A\in{\sf
Args}_{\phi}\\}$, ${\sf Sem}{\sf Without}(\phi)=\bigcap\\{{\sf Sem}{\sf
Without}(A)\mid A\in{\sf Args}_{\phi}\\}$.

######  Example 10.

Let $\text{AT}_{3}=\left\langle\text{AS}_{3},\mathcal{K}_{3}\right\rangle$,
where $\text{AS}_{3}=\left\langle\mathcal{L},\mathcal{R},n\right\rangle$ where
$\mathcal{R}$ and
$\mathcal{K}_{3}=\mathcal{K}^{3}_{p}=\\{\textit{cf},\textit{rc},\textit{sa},\textit{ka},\textit{kp},\textit{rr}\\}$
are such that the set of arguments ${\sf Args}_{3}$ that can be constructed
from $\text{AT}_{3}$ is:

$\displaystyle A_{1}:\ $ cf $\displaystyle A_{2}:\textit{rc}$ $\displaystyle
A_{3}:\textit{sa}$ $\displaystyle A_{4}:\ $ ka $\displaystyle
A_{5}:\textit{kp}$ $\displaystyle A_{6}:\textit{rr}$ $\displaystyle B_{1}:\ $
$\displaystyle A_{1}\overset{d_{1}}{\Rightarrow}\textit{iw}$ $\displaystyle
B_{2}:A_{2}\overset{d_{2}}{\Rightarrow}\neg n(d_{1})$ $\displaystyle
B_{3}:A_{5}\overset{d_{5}}{\Rightarrow}\neg\textit{rc}$ $\displaystyle B_{4}:\
$ $\displaystyle B_{1},A_{3}\overset{d_{3}}{\Rightarrow}m$ $\displaystyle
B_{5}:A_{4}\overset{d_{4}}{\Rightarrow}\neg n(d_{3})$ $\displaystyle
B_{6}:A_{6}\overset{d_{6}}{\Rightarrow}\neg\textit{ka}.$

See Figure 4 for a graphical representation of the corresponding argumentation
framework $\mathcal{AF}(\text{AT}_{3})$ and Section 7 for a realistic
instantiation of this example. Note that ${\sf
Grd}(\mathcal{AF}(\text{AT}_{3}))=\\{A_{1},A_{3},A_{5},A_{6}\\}$ and ${\sf
Sem}(\mathcal{AF}(\text{AT}_{3}))=\\{\\{A_{1},\allowbreak A_{2},\allowbreak
A_{3},\allowbreak A_{4},\allowbreak A_{5},\allowbreak A_{6},\allowbreak
B_{2},\allowbreak B_{5}\\},\allowbreak\\{A_{1},\allowbreak A_{2},\allowbreak
A_{3},\allowbreak A_{5},\allowbreak A_{6},\allowbreak B_{2},\allowbreak
B_{6}\\},\allowbreak\\{A_{1},\allowbreak A_{3},\allowbreak A_{5},\allowbreak
A_{4},\allowbreak A_{6},\allowbreak B_{1},\allowbreak B_{3},\allowbreak
B_{5}\\},\allowbreak\\{A_{1},\allowbreak A_{3},\allowbreak A_{5},\allowbreak
A_{6},\allowbreak B_{2},\allowbreak B_{3},\allowbreak B_{4},\allowbreak
B_{6}\\}\\}$ for ${\sf Sem}\in\\{{\sf Prf},\allowbreak{\sf Stb}\\}$.

$A_{1}$$A_{2}$$A_{3}$$A_{4}$$A_{5}$$A_{6}$$B_{1}$$B_{2}$$B_{3}$$B_{4}$$B_{5}$$B_{6}$
Figure 4: Graphical representation of the AF $\mathcal{AF}(\text{AT}_{3})$.

#### Basic Explanations for ASPIC+

In abstract argumentation the arguments are abstract entities, however, in
ASPIC+ the structure of the arguments is known and can be used in the
explanations. To this end we use the function, $\mathbb{F}$, which determines
the content of an explanation (e.g., explanations can consist of arguments
($\mathbb{F}={\sf id}$) or of the premises of those arguments
($\mathbb{F}={\sf Prem}$)).444See [7] for additional variations for
$\mathbb{F}$. Formula explanations differ in two ways from the explanations in
Section 4: the function $\mathbb{F}$ is applied; and the arguments for $\phi$
have to be considered (e.g., all accepted arguments for $\phi$ for
$\cap$-acceptance and an accepted argument for $\phi$ for $\cup$-acceptance).
The basic explanations from Section 4 for formulas are defined by:

######  Definition 11 (Basic formula explanations).

Let $\phi\in\mathcal{L}$ be a formula and
$\mathcal{AF}(\text{AT})=\left\langle{\sf Args},{\sf Att}\right\rangle$ be
based on AT. Suppose that $\phi\in\mathcal{L}$ is accepted w.r.t. ${\sf Sem}$
and $\cap$ or $\cup$. Then:

$\displaystyle\bullet\ $ $\displaystyle{\sf Sem}{\sf Acc}^{\cap}(\phi)\hskip
14.22636pt=\hskip 14.22636pt\mathbb{F}\left(\bigcup_{A\in{\sf Sem}{\sf
Accept}(\phi)}\bigcup_{\mathcal{E}\in{\sf Sem}(\mathcal{AF})}{\sf
Defending}(A,\mathcal{E})\right);$ $\displaystyle\bullet\ $ $\displaystyle{\sf
Sem}{\sf Acc}^{\cup}(\phi)\hskip 14.22636pt\in\hskip
14.22636pt\left\\{\mathbb{F}({\sf Defending}(A,\mathcal{E}))\mid A\in{\sf
Sem}{\sf Accept}(\phi),\mathcal{E}\in{\sf Sem}{\sf With}(A)\right\\}.$

Suppose now that $\phi$ is non-accepted w.r.t. ${\sf Sem}$ and $\cap$ or
$\cup$:

$\displaystyle\bullet\ $ $\displaystyle{\sf Sem}{\sf
NotAcc}^{\cap}(\phi)\hskip 14.22636pt=\hskip
14.22636pt\mathbb{F}\left(\bigcup_{A\in{\sf All}{\sf
Args}(\phi)}\bigcup_{\mathcal{E}\in{\sf Sem}{\sf Without}(A)}{\sf
NoDefAgainst}(A,\mathcal{E})\right);$ $\displaystyle\bullet\ $
$\displaystyle{\sf Sem}{\sf NotAcc}^{\cup}(\phi)\hskip 14.22636pt=\hskip
14.22636pt\mathbb{F}\left(\bigcup_{A\in{\sf All}{\sf
Args}(\phi)}\bigcup_{\mathcal{E}\in{\sf Sem}(\mathcal{AF})}{\sf
NoDefAgainst}(A,\mathcal{E})\right).$

That for the $\cap$-non-acceptance explanation all arguments for $\phi$ have
to be accounted for follows since it might be the case that an explanation
does not contain one particular argument for $\phi$ but it does contain
another.

######  Example 11 (Example 10 continued).

For the AF $\mathcal{AF}(\text{AT}_{3})$ based on $\text{AT}_{3}$ from Example
10:

  * •

${\sf Prf}{\sf Acc}^{\cup}(m)=\\{B_{3},B_{6}\\}$ for $\mathbb{F}={\sf id}$ and
${\sf Prf}{\sf Acc}^{\cup}(m)=\\{\textit{kp},\textit{rr}\\}$ for
$\mathbb{F}={\sf Prem}$;

  * •

${\sf Prf}{\sf Acc}^{\cup}(\neg n(d_{1}))=\\{A_{2}\\}$ for $\mathbb{F}={\sf
id}$ and ${\sf Prf}{\sf Acc}^{\cup}(\neg n(d_{1}))=\\{\textit{rc}\\}$ for
$\mathbb{F}={\sf Prem}$;

  * •

${\sf Prf}{\sf NotAcc}^{\cap}(m)=\\{A_{2},A_{4},B_{2},B_{5}\\}$ for
$\mathbb{F}={\sf id}$ and ${\sf Prf}{\sf
NotAcc}^{\cap}(m)=\\{\textit{rc},\textit{ka}\\}$ for $\mathbb{F}={\sf Prem}$.

Like for arguments (recall Propositions 2 and 3) we have the following result:

######  Proposition 7.

Let $\mathcal{AF}(\text{AT})=\left\langle{\sf Args},{\sf Att}\right\rangle$ be
an AF based on AT, let $\phi\in\mathcal{L}$, $\star\in\\{\cap,\cup\\}$, ${\sf
Sem}\in\\{{\sf Adm},\allowbreak{\sf Cmp},\allowbreak{\sf Grd},\allowbreak{\sf
Prf},\allowbreak{\sf Sstb}\\}$ and $\mathbb{F}$ be such that $\mathbb{F}({\sf
S})\neq\emptyset$ when ${\sf S}\neq\emptyset$:

  * •

when $\phi$ is accepted w.r.t. $\star$ and ${\sf Sem}$, ${\sf Sem}{\sf
Acc}^{\star}(\phi)=\emptyset$ iff there is no $B\in{\sf Args}$ such that
$(B,C)\in{\sf Att}$ for any $C\in{\sf Sem}{\sf Accept}(\phi)$;

  * •

when $\phi$ is non-accepted w.r.t. $\star$ and ${\sf Sem}$, then ${\sf
Sem}{\sf NotAcc}^{\star}(\phi)\neq\emptyset$.

This proposition shows that an acceptance explanation for a formula $\phi$ is
only empty when no argument for $\phi$ is attacked, while a non-acceptance
explanation for $\phi$ is never empty. The proof of the proposition is similar
to the proofs of Propositions 2 and 3 and left to the reader.

#### Contrastive Explanations for ASPIC+

With these basic explanations for formulas, contrastive explanations can be
defined for formulas in a similar way as for arguments (recall Figure 3),
where the fact is now a formula and the foil is a set of formulas. When the
foil is not explicit, we can define, for example, ${\sf
Foil}(\phi)=\\{-\phi\mid-\phi\in{\sf Concs}({\sf Args})\\}$.

In order to obtain meaningful contrastive explanations in the structured
setting, we redefine the conditions on the application of contrastive
explanations in the context of ASPIC+. Let $\mathcal{AF}(\text{AT})$ be an AF
based on AT, let $\\{\phi\\}\cup{\sf S}\subseteq{\sf Concs}({\sf Args})$. Then
${\sf Sem}{\sf Cont}^{\star}(\phi,{\sf S})$ [resp. ${\sf Sem}{\sf
ContN}^{\star}(\phi,{\sf S})$] can be requested when for each $\psi\in{\sf
S}$:

  * •

$\phi$ is at least credulously accepted (i.e., ${\sf Sem}{\sf
With}(\phi)\neq\emptyset$) [resp. skeptically non-accepted (i.e., ${\sf
Sem}{\sf Without}(\phi)\neq\emptyset$)] and $\psi$ is at least skeptically
non-accepted [resp. credulously accepted];

  * •

for each $\mathcal{E}\in{\sf Sem}(\mathcal{AF})$ it is never the case that
$\\{\phi,\psi\\}\subseteq{\sf Concs}(\mathcal{E})$;

  * •

either some $A\in{\sf All}{\sf Args}(\phi)$ is conflict-relevant for some
$B\in{\sf All}{\sf Args}(\psi)$ or some $B\in{\sf All}{\sf Args}(\psi)$ is
conflict-relevant for some $A\in{\sf All}{\sf Args}(\phi)$.

######  Example 12 (Example 11 continued).

For $\mathcal{AF}(\text{AT}_{4})$, where $\mathbb{F}={\sf Prem}$ we have that:

  * •

${\sf Prf}{\sf Cont}^{\cup}(m,\neg n(d_{1}))=\\{\textit{kp}\\}$;

  * •

${\sf Prf}{\sf Cont}^{\cup}(m,\neg n(d_{3}))=\\{\textit{rr}\\}$; and

  * •

${\sf Prf}{\sf ContN}^{\cap}(m,\neg n(d_{1}))=\\{\textit{rc}\\}$.

Next we turn to the formula counterparts of the results from Section 5. First
the counterpart of Proposition 5 on empty contrastive explanations.

######  Proposition 8.

Let $\mathcal{AF}(\text{AT})=\left\langle{\sf Args},{\sf Att}\right\rangle$ be
an AF based on AT and let $\phi,\psi\in\mathcal{L}$ such that
$\phi,\psi\in{\sf Concs}({\sf Args})$, it holds that ${\sf Sem}{\sf
Acc}^{\star}(\phi)\cap{\sf Sem}{\sf NotAcc}^{\dagger}(\psi)=\emptyset$ implies
that ${\sf Sem}{\sf Acc}^{\star}(\phi)=\emptyset$; or for each $A\in{\sf
All}{\sf Args}(\phi)$ and each $B\in{\sf All}{\sf Args}(\psi)$, $A$ is not
conflict-relevant for $B$.

###### Proof.

Let $\mathcal{AF}(\text{AT})=\left\langle{\sf Args},{\sf Att}\right\rangle$ be
an AF based on AT and let $\phi,\psi\in\mathcal{L}$ such that
$\phi,\psi\in{\sf Concs}({\sf Args})$. That ${\sf Sem}{\sf
Acc}^{\star}(\phi)\cap{\sf Sem}{\sf NotAcc}^{\dagger}(\psi)=\emptyset$ when
${\sf Sem}{\sf Acc}^{\star}(\phi)=\emptyset$ follows immediately. Recall from
Proposition 7 that it always holds that ${\sf Sem}{\sf
NotAcc}^{\dagger}(\psi)\neq\emptyset$. From Proposition 5 it is known that
${\sf Sem}{\sf Acc}^{\star}(A)\cap{\sf Sem}{\sf
NotAcc}^{\dagger}(B)=\emptyset$ if $A$ is not conflict-relevant for $B$.
Hence, if for each $A\in{\sf All}{\sf Args}(\phi)$ and each $B\in{\sf All}{\sf
Args}(\psi)$ $A$ is not conflict-relevant for $B$ then, by the definitions of
the basic explanations for formulas, ${\sf Sem}{\sf Acc}^{\star}(\phi)\cap{\sf
Sem}{\sf NotAcc}^{\dagger}(\psi)=\emptyset$. ∎

Recall that for Proposition 6 as well as for Corollary 1 it was assumed that a
foil exists. We assume this for the formula counterparts of these results as
well.

######  Proposition 9.

Let $\mathcal{AF}(\text{AT})=\left\langle{\sf Args},{\sf Att}\right\rangle$ be
an AF based on AT, let $\phi\in\mathcal{L}$ be such that ${\sf
Foil}(\phi)\neq\emptyset$ and ${\sf Sem}\in\\{{\sf Adm},{\sf Cmp},{\sf
Grd},{\sf Prf},{\sf Sstb}\\}$. Then: a contrastive acceptance [resp. non-
acceptance] explanation can be requested for $\phi$, when $\phi$ is at least
credulously accepted [resp. skeptically non-accepted] and for all $\psi\in{\sf
Foil}(\phi)$, $\psi$ is at least skeptically non-accepted [resp. credulously
accepted].

###### Proof.

Let $\mathcal{AF}(\text{AT})=\left\langle{\sf Args},{\sf Att}\right\rangle$ be
an AF based on AT, let $\phi\in\mathcal{L}$ be such that ${\sf
Foil}(\phi)\neq\emptyset$ and ${\sf Sem}\in\\{{\sf Adm},\allowbreak{\sf
Cmp},\allowbreak{\sf Grd},\allowbreak{\sf Prf},\allowbreak{\sf Sstb}\\}$. To
show that:

  1. 1.

for each $\mathcal{E}\in{\sf Sem}(\mathcal{AF}(\text{AT}))$ it is never the
case that $\\{\phi,\psi\\}\subseteq{\sf Concs}(\mathcal{E})$: since
$\psi=-\phi$, every argument for $\psi$ rebuts every arguments for $\phi$ and
vice versa. Let $A\in{\sf All}{\sf Args}(\psi)$ and $B\in{\sf All}{\sf
Args}(\phi)$, then $(A,B),(B,A)\in{\sf Att}$. Since $\mathcal{E}\in{\sf
Sem}(\mathcal{AF}(\text{AT}))$ is always conflict-free, it follows that
$\\{\phi,\allowbreak\psi\\}\not\subseteq{\sf Concs}(\mathcal{E})$.

  2. 2.

either an argument for $\phi$ is conflict-relevant for an argument for $\psi$
or an argument for $\psi$ is conflict-relevant for an argument for $\phi$:
this follows since $(A,B)\in{\sf Att}$ and $(B,A)\in{\sf Att}$ for any
$A\in{\sf All}{\sf Args}(\psi)$ and $B\in{\sf All}{\sf Args}(\phi)$.∎

This shows that for ASPIC+, when the foil is defined by the negation of the
fact and the first condition for the applicability is fulfilled, the other
conditions are fulfilled as well. From Propositions 8 and 9 we obtain the
following corollary.

######  Corollary 2.

Let $\mathcal{AF}=\left\langle{\sf Args},{\sf Att}\right\rangle$ be an AF
based on AT, let $\phi\in\mathcal{L}$ be such that ${\sf
Foil}(\phi)\neq\emptyset$ and ${\sf Sem}\in\\{{\sf Adm},{\sf Cmp},{\sf
Grd},{\sf Prf},{\sf Sstb}\\}$. Then:

  * •

it is never the case that ${\sf Sem}{\sf Cont}^{\star}(\phi,{\sf Foil}(\phi))$
is of the form $\langle{\sf Sem}{\sf
Acc}^{\star}(\phi),\allowbreak\bigcup_{\psi\in{\sf Foil}(\phi)}{\sf Sem}{\sf
NotAcc}^{\dagger}(\psi)\rangle$;

  * •

${\sf Sem}{\sf ContN}^{\star}(\phi,{\sf Foil}(\phi))=\langle{\sf Sem}{\sf
NotAcc}^{\star}(\phi),\allowbreak\bigcup_{\psi\in{\sf Foil}(\phi)}{\sf
Sem}{\sf Acc}^{\dagger}(\psi)\rangle$ iff ${\sf Sem}{\sf
Acc}^{\dagger}(\psi)=\emptyset$ for all $\psi\in{\sf Foil}(\phi)$.

Like in the case of arguments, when the foil is determined by the negation of
the fact, the explanation ${\sf Cont}$ is never a pair and the explanation
${\sf ContN}$ is only a pair when all arguments for the formulas of the foil
are not attacked at all.

##  7 Applying Contrastive Explanations

At the Dutch National Police several argumentation-based applications have
been implemented [6]. These applications are aimed at assisting the police at
working through high volume tasks, leaving more time for tasks that require
human attention. For example, there is an application that helps Dutch
citizens to file a complaint on online trade fraud and can then determine
whether it is a case of fraud [22]. Another application helps the police to
identify malafide webshops [21]. As for any AI application, these applications
should be able to provide an explanation for the derived decision. In this
section we illustrate how the proposed contrastive explanations from this
paper can be applied in a webshop classification case.555In this paper we work
with an adjusted version of the actual system to make the conclusions more
interesting from an argumentative perspective and since the system cannot be
published.

Suppose that a complaint has been filed (cf) about a webshop, that it is
malafide ($m$). Usually, when a complaint is filed, an investigation into the
webshop is done (iw, rule $d_{1}$) and, when an investigation is done and it
is found that the url of the webshop is suspicious (sa), the webshop is found
to be malafide (rule $d_{3}$). Now, a complaint can be retracted (rc) in which
case $d_{1}$ is not applicable (rule $d_{2}$), unless the owner of the webshop
is known by the police (kp, rule $d_{5}$). Similarly, if the address is
registered at the chamber of commerce (ka) then rule $d_{3}$ is not applicable
(rule $d_{4}$), unless the registration was recently retracted (rr, rule
$d_{6}$).

This case is represented in $\mathcal{AF}(\text{AT}_{3})$ from Example 10 and
the graphical representation can be found in Figure 4. The basic explanations
for $m$ were discussed in Example 11:

  * •

The webshop is malafide: $m$ can be $\cup$-accepted since the owner of the
webshop is known by the police (kp) and the registration at the chamber of
commerce was recently retracted (rr), from which it follows that no exceptions
could be derived.

  * •

The webshop is not malafide: $m$ can be $\cap$-non-accepted since the
complaint was retracted (rc) and the url of the webshop is registered (ka).

These basic explanations are exhaustive: all the reasons why the webshop is
(not) malafide are provided. For applications with more arguments, this may
result in more reasons within one explanation. With our contrastive
explanations, the explanation can focus on an explicit contrastive question.
For example (recall Example 12), the contrast between the webshop being
malafide and the applicability of rule $d_{1}$ (if a complaint is filed the
webshop is investigated):

  * •

The webshop is malafide rather than that there is an exception to rule
$d_{1}$, since the owner is known by the police (kp).

  * •

The webshop is not malafide and an exception to rule $d_{1}$ applies, since
the complaint was retracted (rc).

Thus, the contrastive explanations are better tailored to one question and
result in smaller explanations than the basic explanations for
(non-)acceptance from [7].

##  8 Conclusion and Discussion

In this paper we have studied contrastive explanations for conclusions of
arguments in formal argumentation. For this we have employed a basic framework
for explanations introduced in [7] that can be applied on top of an AF as
introduced in [10] and which allows for explanations of both accepted and non-
accepted arguments. The result is a general approach to derive contrastive
explanations from AFs generated from an abstract setting. In Section 6 we have
refined these explanations such that these can also be applied to an
ASPIC+-setting [23]. Finally, we have discussed an example of how these
explanations can be applied in the argumentation-based systems employed by the
Dutch National Police. To the best of our knowledge this is the first
investigation into contrastive local explanations for conclusions derived from
both abstract and structured argumentation.

One of the challenges in the literature on contrastive explanations for
conclusions derived by some AI system is that the foil is not always explicit.
Since an AF comes with a clear notion of conflict (i.e., the attack relation
between the arguments of the framework), we were able to introduce a way to
find the foil of an argument or formula when it is not provided explicitly.
This gives the use of argumentation to explain decisions an advantage over
other approaches to AI that do not have such a notion of conflict.

For this paper we chose to highlight a particular form of contrast in the
context of formal argumentation, i.e., the difference between an accepted
[resp. non-accepted] argument and a (set of) non-accepted [resp. accepted]
argument(s). One could interpret our approach as modeling the alternative
contrastive explanations from [18]. There are however other forms of contrast
that could be highlighted in a contrastive explanation as well. For example,
in terms of accepted arguments, one could compare two accepted but different
arguments and return as the explanation the difference in their acceptance
explanation; explain why an argument is $\cup$-accepted but not
$\cap$-accepted; or compare different (but similar) AFs and explain why in one
AF an argument is accepted and in another it is non-accepted.

In future work we plan to further investigate different forms of
contrastiveness and model these in the context of explanations for
argumentation-based conclusions. We will also integrate further findings from
the social sciences on how humans request, generate, select, interpret and
evaluate explanations [19, 25, 24]. Moreover, we will study how research on
argumentation that aims to model other aspects of human reasoning can be
applied in our study of explanations (e.g., the use of preferences [3] or the
instantiation with a specific logic, see e.g., [5]).

### Acknowledgements

This research has been partly funded by the Dutch Ministry of Justice and the
Dutch National Police. We thank Daphne Odekerken for her help with designing
the examples and the anonymous reviewers for their valuable comments.

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