1

A Basic Framework for Explanations in
Argumentation
AnneMarie Borg and Floris Bex
Abstract—We discuss explanations for formal (abstract and structured) argumentation – the question whether and why a certain
argument or claim can be accepted (or not) under various extension-based semantics. We introduce a flexible framework, which can
act as the basis for many different types of explanations. For example, we can have simple or comprehensive explanations in terms ofarguments for or against a claim, arguments that (indirectly) defend a claim, the evidence (knowledge base) that supports or isincompatible with a claim, and so on. We show how different types of explanations can be captured in our basic framework, discuss areal-life application and formally compare our framework to existing work.
Index Terms—Artificial intelligence, knowledge representation formalisms and methods, nonmonotonic reasoning and belief revision

F

1

I NTRODUCTIONR

ECENTLY , explainable AI (XAI) has received much attention, mostly directed at new techniques for explaining
decisions of (subsymbolic) machine learning algorithms [1].
However, explanations also play an important role in (symbolic) knowledge-based systems [2]. Argumentation is one
research area in symbolic AI that is frequently mentioned
in relation to XAI. For example, arguments can be used
to provide reasons for or against decisions [2], [3]. The
focus can also be on the argumentation itself, where it isexplained whether and why a certain argument or claim
can be accepted under certain semantics for computational
argumentation [4], [5], [6], [7]. It is the latter type of explanations we are interested in.
Two central concepts in argumentation are abstract argumentation frameworks [8] – sets of arguments and theattack relations between them – and structured or logicalargumentation frameworks (e.g., [9]) – where arguments areconstructed from a knowledge base and a set of rules andthe attack relation is based on the individual elements in thearguments. For both abstract and structured argumentation
frameworks we can determine extensions, sets of argumentsthat can collectively be considered as acceptable, under
different semantics [8]. In XAI terms [10], this is a global
explanation – what can we conclude from the model asa whole? However, as argumentation is being applied inreal-life AI systems with lay-users, we would rather have
simpler, more compact explanations for the acceptability of
individual arguments – a local explanation for a particulardecision or conclusion. We noticed the need for such explanations when deploying an argumentation system at theDutch National Police, which assists citizens in filing online
reports and complaints [11], [12].
We propose a basic framework for explanations in struc-••

Both authors are with the Department of Information and Computing
Science, Utrecht University, The Netherlands.
E-mail: {a.borg, f.j.bex}@uu.nl
Floris Bex is also with the Department of Law, Technology, Markets, andSociety, Tilburg University, The Netherlands.

Manuscript received August 5, 2020.

tured and abstract argumentation, with which explanations for (non-)accepted arguments and (sub-)conclusions
can be generated. Though some work on explanations forargumentation-based conclusions exists in the literature ([4],[5], [6], [7], Section 5), our framework is generic in that theunderlying argumentation framework does not have to be
adjusted and the definitions are semantics-independent –for example, the explanations based on the new semantics
of Fan and Toni [4] are a special case of our framework.
The framework is also flexible, as the contents of explanations can be varied. For example, rather than returning
all defending or attacking arguments, we can return only
those that can defend themselves, or the ones that directly
attack an argument. Furthermore, we are the first to use thestructure of arguments for explanations: not just arguments
for a conclusion, but also elements of these arguments (e.g.,premises or rules) can be returned as an explanation.

2

P RELIMINARIESAn abstract argumentation framework (AF) [8] is a pair
AF = hArgs, Atti, where Args is a set of arguments andAtt ⊆ Args × Args is an attack relation on these arguments.
An AF can be viewed as a directed graph, in which thenodes represent arguments and the arrows represent attacks
between arguments.
Example 1. Consider the AF AF 1 = hArgs1 , Att1 i whereArgs1 = {A1 , A2 , A3 , A4 } and Att1 = {(A2 , A1 ), (A3 , A2 ),(A3 , A4 ), (A4 , A3 )}.
Given an AF AF , Dung-style semantics [8] can be applied to it, to determine what combinations of arguments(called extensions) can collectively be accepted.
Definition 1. Let AF = hArgs, Atti be an AF, S ⊆ Args a setof arguments and let A ∈ Args. Then:•••

S attacks A if there is an A0 ∈ S such that (A0 , A) ∈Att;
S defends A if S attacks every attacker of A;
S is conflict-free if there are no A1 , A2 ∈ S such that
(A1 , A2 ) ∈ Att; and2•

S is admissible if it is conflict-free and it defends all of
its elements.

ASPIC+

2.1

The grounded extension (grd) is the minimal (withrespect to ⊆) complete extension;
A preferred extension (prf ) is a maximal (with respectto ⊆) complete extension; andA stable extension (stb) is a complete extension that
attacks every argument not in it.

We investigate explanations for a well-known approach tostructured argumentation: ASPIC+ [9], which allows for two
types of premises – axioms that cannot be questioned andordinary premises that can be questioned – and two types ofrules – strict rules that cannot be questioned and defeasible
rules. We choose ASPIC+ as the structured argumentation
approach in this paper since it allows to vary the form of the
explanations in many ways (see Section 4). The definitionsin this section are based on [9].

Extsem (AF) denotes the set of all the extensions of AF
under the semantics sem ∈ {cmp, grd, prf, stb}.

Definition 4. An argumentation system is a tuple AS =hL, R, ni, where:

An admissible set that contains all the arguments that it
defends is a complete extension (cmp).
•••

Where AF = hArgs, Atti is an AF, sem a semantics andExtsem (AF) 6= ∅, it is said thatTA ∈ Args is skeptically[resp. credulously] accepted if A ∈ Extsem (AF) [resp. A ∈S
Extsem (AF)]. These acceptability strategies are denotedby ∩ [resp. ∪]. A is said to be skeptically [resp. credulously]
non-accepted in AF if for some [resp. all] E ∈ Extsem (AF),A∈/ E . When these are arbitrary, result in the same or are
clear from the context, we will refer to accepted respectively
non-accepted arguments.
The notions of attack and defense can also be defined
between arguments:
Definition 2. Let AF = hArgs, Atti be an AF, A, B ∈ Args
and E ∈ Extsem (AF) for some sem. A can defend B directly
or indirectly: A directly defends B if there is some C ∈ Args
such that (C, B) ∈ Att and (A, C) ∈ Att, and A indirectly
defends B if A defends C ∈ Args and C defends B . It is said
that A defends B in E if A defends B and A ∈ E .
Similarly, A can attack B directly or indirectly: A directly
attacks B if (A, B) ∈ Att and A indirectly attacks B if A attacks
some C ∈ Args and C defends B .
Next we define two notions that will be used in the basic
definitions of explanations. The first, used for acceptance
explanations, denotes the set of arguments that defend
the argument A, while the last, used for non-acceptance
explanations, denotes the set of arguments that attack A andfor which there is no defense in the given extension.
Definition 3. Let AF = hArgs, Atti be an AF, A ∈ Args andE ∈ Extsem (AF) an extension for some semantics sem.
•••

DefBy(A) = {B ∈ Args | B defends A};
DefBy(A, E) = DefBy(A) ∩ E denotes the set ofarguments that defend A in E ;NotDef(A, E) = {B ∈ Args | B attacks A and E
does not attack B}, denotes the set of all attackers
of A for which no defense exists from E .

Example 2. In AF 1 (recall Example 1), example conflictfree sets are {A1 , A3 } and {A2 , A4 }. Extcmp (AF 1 ) = {∅,{A1 , A3 }, {A2 , A4 }}, while Extprf (AF 1 ) = Extstb (AF 1 ) ={{A1 , A3 }, {A2 , A4 }} and Extgrd (AF 1 ) = {∅}. None of thearguments in Args1 is skeptically accepted, while all of them
are credulously accepted for sem ∈ {cmp, prf, stb}.
Argument A3 directly attacks A4 , and attacks A2 both
directly and indirectly. A3 defends A1 directly against A2
and indirectly against A4 . Moreover, DefBy(A1 ) = {A3 },DefBy(A1 , {A1 , A3 }) = {A3 } and NotDef(A3 , {A2 , A4 }) ={A4 }.

•••

L is a propositional language closed under classicalnegation (¬), we denote ψ = −φ if ψ = ¬φ or φ =¬ψ .
R = Rs ∪ Rd is a set of strict (Rs ) and defeasible(Rd ) inference rules of the form φ1 , . . . , φn → φ resp.
φ1 , . . . , φn ⇒ φ, such that {φ1 , . . . , φn , φ} ⊆ L andRs ∩ Rd = ∅.
Where r ∈ R, Ant(r) = {φ1 , . . . , φn } are the antecedents of the rule and Cons(r) = φ is the consequentof the rule. Moreover, Rules(R, φ) = {r ∈ R |Cons(r) = φ}.
n : Rd → L is a naming convention for defeasible
rules.

A knowledge base in an argumentation system hL, R, ni is aset of formulas K ⊆ L which contains two disjoint subsets:
K = Kp ∪ Kn , the set of axioms Kn and the set of ordinary
premises Kp .
Arguments in ASPIC+ are constructed in an argumentation system from a knowledge base.
Definition 5. An argument A on the basis of a knowledge
base K in an argumentation system hL, R, ni is:1)

2)

φ if φ ∈ K, where Prem(A) = Sub(A) = {φ},Conc(A) = φ, Rules(a) = ∅ and TopRule(A) =undefined;
A1 , . . . , A n
ψ , where ∈ {→, ⇒}, if A1 , . . . , An
are arguments such that there exists a rule
Conc(A1 ), . . . , Conc(An )ψ in Rs if= → andin Rd if= ⇒.
Prem(A) = Prem(A1 ) ∪ . . . ∪ Prem(An ); Conc(A) =ψ ; Sub(A) = Sub(A1 ) ∪ . . . ∪ Sub(An ) ∪ {A};Rules(A) = Rules(A1 ) ∪ . . . ∪ Rules(An ) ∪{Conc(A1 ), . . . , Conc(An )ψ}; DefRules(A) ={r ∈ Rd | r ∈ Rules(A)}; TopRule(A) =Conc(A1 ), . . . , Conc(An )ψ.

The above notation can be generalized to sets.SFor example,
where S is a set of arguments Prem(S) = {Prem(A) |AS ∈ S}, Conc(S) = {Conc(A) | A ∈ S} and DefRules(S) ={DefRules(A) | A ∈ S}.
Example 3. AS2 = hL2 , R2 , ni is an argumentation system
where R2 = R2s ∪ R2d such that R2s = ∅, R2d = {d1 , . . . , d5 }(the application of these rules is shown in the arguments

3
2
below), let K2 = Kn
∪ Kp2 where Kn2 = {t} and Kp2 = {r}.
The following arguments can be constructed:

A1 : tDefinition 8. Let AF(AT) = hArgs, Atti for a semantics sem,
Extsem (AF) 6= ∅ and let some φ ∈ L. We define:B1 : rd•d

A2 : A1 ⇒3 ¬r

B2 : B1 ⇒2 pdd

A3 : A1 , A2 ⇒4 q

B3 : B1 ⇒5 ¬q•d

A4 : A3 ⇒1 p
We denote the set of arguments constructed from AS2
and K2 by Args2 . For A4 we have that Prem(A4 ) ={t}, Conc(A4 ) = p, Sub(A4 ) = {A1 , A2 , A3 , A4 } andRules(A4 ) = {d1 , d3 , d4 }. Furthermore, Rules(R2 , p) ={d1 , d2 }.

When arbitrary or clear from the context, the superscript∪∩
will be omitted (e.g., |∼grd as |∼grd and |∼grd coincide).
Example 5 (Example 4 continued). For AF(AT2 )hArgs2 , Att2 i we have that:1)

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

2)

Definition 6. An argument A attacks an argument B iff A
undercuts, rebuts or undermines B , where:•••

A undercuts B (on B 0 ) iff Conc(A) = −n(r) for someB 0 ∈ Sub(B) such that B 0 ’s top rule r is defeasible,
it denies a rule;
A rebuts B (on B 0 ) iff Conc(A) = −φ for someB 0 ∈ Sub(B) of the form B100 , . . . , Bn00 ⇒ φ, it denies
a conclusion;
A undermines B (on φ) iff Conc(A) = −φ for someφ ∈ Prem(B) \ Kn , it denies a premise.

Argumentation theories and their corresponding Dungstyle argumentation frameworks can now be defined.

∪

Credulous entailment: AT |∼sem φ iff for some E ∈Extsem (AF) there is an argument A ∈ E with
Conc(A) = φ, it is said that φ is credulously accepted;∩
Skeptical entailment: AT |∼sem φ iff for each E ∈Extsem (AF) there is some A ∈ E such that
Conc(A) = φ, it is said that φ is skeptically accepted.

3)
4)

=

∩

AT2 |6∼grd φ and AT2 |6∼sem φ for φ ∈ {q, ¬q, r, ¬r},and sem ∈ {cmp, prf, stb}; while∪
AT2 |∼sem φ for any φ ∈ {p, q, ¬q, r, ¬r, t} and sem ∈{cmp, prf, stb};∩
AT2 |∼grd t and AT2 |∼sem t for sem ∈ {cmp, prf, stb};and∩AT2 |∼sem p for sem ∈ {prf, stb} but AT2 |6∼grd p.

This follows since each argument from Args2 is part of atleast one extension, but only A1 is part of every extension.
The last item follows since each sem-extension of AF(AT2 )contains either A4 or B2 for sem ∈ {prf, stb}.
2.2

Necessary Notation

This notation is meant to keep the definitions of explanations in Section 3 general and short.

Definition 7. An argumentation theory is a pair AT =hAS, Ki, where AS is an argumentation system and K isa knowledge base.
A structured argumentation framework (SAF) defined by anargumentation theory AT is a pair AF(AT) = hArgs, Atti,
where Args is the set of all arguments constructed from AT
and (A, B) ∈ Att iff A attacks B according to Definition 6.

Notation 1. Let AF = hArgs, Atti be an AF, A ∈ Args andS ⊆ Args. Then, for some sem ∈ {grd, cmp, prf, stb}:

Dung-style semantics, as in Definition 1, can be applied
to SAFs in the same way as they are applied in AFs.

The set of arguments that can be used to explain theacceptance of a formula differs depending on the acceptancestrategy. For this the following notation will be applied.

Example 4. (Example 3 continued) Consider the argumentation theory AT2 = hAS2 , K2 i. Figure 1 contains the graphicalrepresentation of AF(AT2 ) = hArgs2 , Att2 i. In this framework there are no undercuts, all the attacks from A2 areunderminers and all the other attacks are rebuts.

••

Notation 2. Let AF(AT) = hArgs, Atti be an SAF, φ ∈ L andlet sem ∈ {grd, cmp, prf, stb}. Then:
•
•

A1

A2

A3

A4

•

B1

B2

B3

Fig. 1. Graphical representation of AF (AT2 ).

Then:
Extgrd (AF(AT2 ))={A1 };
andExtsem (AF(AT2 )) = {{A1 , A2 , A3 , A4 }, {A1 , B1 , B2 , B3 }},for sem ∈ {prf, stb}.
Entailment relations, induced by the structured argumentation framework and a semantics, are defined by:

Esem
A = {E ∈ Extsem (AF) | A ∈ E} denotes the set ofsem-extensions of AF which contain A;
Esem= {E ∈ Extsem (AF) | A ∈/ E} denotes the set ofAZ
sem-extensions of AF which do not contain A.

Argsφ = {A ∈ Args | Conc(A) = φ} denotes the setof all arguments of AF(
S AT) with conclusion φ;
sem,∪
Argsφ
= {A ∈ Extsem (AF(AT)) | Conc(A) =φ} denotes the set of all arguments of AF(AT)with conclusion φ that are part of at least one semextension (i.e.,( that are credulously∩accepted);∅
if AT |6∼sem φ
sem,∩
Argsφ
=
sem,∪
Argsφ
otherwise
sem,∪is the same as Argsφ
if φ is skeptically acceptedand ∅ if it is not skeptically accepted.

Example 6. (Example 4 continued) Whenever Argssem,∩
6= ∅,p
there is no difference between ∪ and ∩. But Argsq =Argsqsem,∪ = {A3 } while Argsqsem,∩ = ∅ for sem ∈{cmp, prf, stb}.
Next it is defined what it means for two formulas to beconnected in an argumentation system.

4

Definition 9. Let AS = hL, R, ni be an argumentation
system. Then, φ is connected to ψ if φ = ψ , or:••there is some r ∈ R with Cons(r) = ψ and φ ∈Ant(r);
there is some γ ∈ L such that φ is connected to γ and
γ is connected to ψ .

The set of all connected formulas of ψ is denoted by:•Connected(ψ) = {φ ∈ L | φ is connected to ψ}.

In explanations for formulas for which no argument
exists the following notation will be used:
Notation 3. Let AF(AT) = hArgs, Atti be an SAF and let
φ ∈ L be such that there is no argument for it in Args. Then:S•
NoArgAnt(φ) = {ψ | ψ ∈{Ant(r) | r ∈Rules(R, φ)} and @A ∈ Args s.t. Conc(A) = ψ} denotes the set of formulas in antecedents of rules for
φ for which no argument exists.
•
NoArgPrem(φ) = {ψ∈ Connected(φ) |Rules(R, ψ) = ∅ and ψ ∈/ K} denotes the set of
formulas that are connected to φ but that are not partof K and for which no rules exist.
Intuitively, NoArgAnt determines the formulas for whicharguments are missing in order for an argument for φ to beavailable, while NoArgPrem determines the formulas thatare not derivable from AF(AT) (neither from K nor as aconclusion of some rule) and which could be part of thederivation of an argument for φ.
Example 7. Consider AS2 from Example 3, but let K20 = Kp2
2(i.e., Kn= ∅). It follows that the arguments A1 , A2 , A3 andA4 no longer exist. Thus there is no argument for ¬r nor
for q (though there is still an argument for p: B2 ). We have
that: NoArgAnt(q) = {t, ¬r}, Connected(q) = {t, ¬r} andNoArgPrem(q) = {t}.

3BASIC E XPLANATIONS

We now define basic explanations in terms of two functions.
D determines the depth of the explanation, how “far away”
we should look when considering attacking and defendingarguments as explanations. F determines the form of the
explanation, whether we want, for example, an argument
as an explanation or only its premises. A formal definitionof these functions is not provided since domain (F) andcodomain (D and F) are not fixed. We will sometimes use thesuperscriptes acc and na to denote the function used in thecontext of acceptance [resp. non-acceptance] explanations.
3.1

Basic Explanations for AcceptanceWe define two types of acceptance explanations, where ∩explanations provide all the reasons why an argument orformula can be accepted by a skeptical reasoner, while∪-explanations provide one reason why an argument orformula can be accepted by a credulous reasoner. For the
purpose of this section let Dacc (A, S) = DefBy(A, S) andFacc (T) = id(T) = T (i.e., id(S) = S for any set S).

3.1.1 Explanations for Accepted Arguments
An argument explanation for an accepted argument A consists of the arguments that defend it, depending on the extensions considered according to the acceptability strategy.
Definition 10 (Argument explanation). Let AF =hArgs, Atti be an AF and let A ∈ Args be an accepted
argument, given some sem ∈ {cmp, grd, prf, stb} and an
acceptance strategy (∩ or ∪). Then:[
Acc∩
Dacc (A, E);
sem (A) =E∈Extsem (AF )∪Accsem (A) ∈ {Dacc (A, E) | E ∈ Esem
A }.

Acc∩
sem (A) provides for each sem-extension E the arguments that defend A in E , and Acc∪
sem (A) the argumentsthat defend A in one of the sem-extensions.
Example 8 (Example 2 continued). Recall AF 1 =hArgs1 , Att1 i. We have that:•Acc∪
prf (A2 ) = {A4 };•Acc∪
prf (A3 ) = {A3 }.

3.1.2 Explanations for Accepted Formulas
In structured argumentation explanations for the acceptanceof a formula φ can be requested, in addition to argument
explanations. For φ to be accepted, at least one argument forφ must exist. Therefore, the existence of such an argument
is part of the explanation as well.
Definition 11 (Formula explanation). Let AF(AT) =?
hArgs, Atti be an SAF and let φ ∈ L be such that AT |∼sem φ,for sem ∈ {cmp, grd, prf, stb} and ? ∈ {∩, ∪}. Here S =
sem,∩
sem,∪
Argsφ
, A ∈ Argsφand SA ∈ {Dacc (A, E) | E ∈ Esem
A }:

+
*
[ [
acc
Acc∩
(S), Facc 
Dacc (B, E) ;sem (φ) = F
B∈S E∈Esem
B

*

Acc∪
sem (φ) =

+
acc

F

acc

(A), F(SA ) .

The first part of the explanation denotes arguments for φ(recall Notation 2) – all arguments in the case of Acc∩
sem (φ)and one argument in the case of Acc∪
(φ)
.
The
secondsem
part of the explanation is similar to the set of arguments
in an argument explanation, although now the function F isapplied to it. This makes it possible to change the form of the
explanation (e.g., premises instead of arguments). The maindifference with argument explanations is that more than one
argument for φ may be considered in the ∩-explanation. The(skeptical) ∩-explanation again takes all extensions in Esem
B
into account to determine the arguments that defend B ,while for the (credulous) ∪-explanation again the defendingarguments for A from just one extension in Esem
A are taken.
Example 9. (Example 5 continued) Consider the SAF∩
AF(AT2 ) for AT2 = hAS2 , K2 i. Recall that AT2 |∼prf p, hence:•Acc∩
prf (p) = h{A4 , B2 }, {A2 , A3 , B1 }i.

For other formulas the Acc∩
sem -explanation does not apply,
since none of these are skeptically accepted. However:•Acc∪
prf (q) = h{A3 }, {A2 , A3 }i;•Acc∪
prf (¬q) = h{B3 }, {B1 , B3 }i.

5

3.2

Basic Explanations for Non-Acceptance

Similar to acceptance explanations, there are two types of
non-acceptance explanations: ∩-explanations for why anargument or formula is not accepted in some extensions (i.e.,
is not skeptically accepted), and ∪-explanations for why
an argument or formula is not accepted in all extensions(i.e., is not credulously accepted). For this let Dna (A, S) =NotDef(A, S) and Fna (T) = id(T) = T.

formulas which are neither part of the knowledge base nor
the consequent of a rule).
Example 11. (Example 7 continued) Consider again AS2
from Example 3, with the knowledge base K20 from Example 7 (i.e., K20 = K2 \ {t}). There are no arguments for ¬r
and q :••

3.2.1

Explanations for Non-Accepted Arguments

In any Dung-style semantics based on the complete semantics, an argument is not accepted if it is attacked and it is notdefended by an accepted argument. Hence, intuitively, theexplanation for the non-acceptance of an argument is the setof arguments for which no defense exists.
Definition 12 (Non-acceptance argument explanation). Let
AF = hArgs, Atti be an AF and let A ∈ Args be an argumentthat is not accepted, given some sem ∈ {cmp, grd, prf, stb}and some ? ∈ {∩, ∪}. Then:[
NotAcc∩Dna (A, E);
sem (A) =E∈EsemA

NotAcc∪sem (A) =[
AThis follows since, although there is a rule for q (i.e., d4 ∈R2d ) [resp. for ¬r (i.e., d3 ∈ R2d )], there is some ψ ∈ Ant(d4 )[resp. ψ ∈ Ant(d3 )] (i.e., ψ = t [resp. ψ = ¬r]) such that
there is no argument for t [resp. ¬r] in AF(AT2 ) and whenlooking at the missing premises to derive q [resp. ¬r] theformula t, necessary for d3 is found.
Like for non-acceptance argument explanations, if anargument for φ exists but it is not accepted, there has tobe an attacker for which there is no defense.
Definition 14 (Non-acceptance formula explanation). Let
AF(AT) = hArgs, Atti be an SAF and let φ ∈ L be such?
that AT |6∼sem φ, given some sem ∈ {cmp, grd, prf, stb} and? ∈ {∩, ∪}. Here, Sφ = Argsφ .

Dna (A, E).

E∈Extsem (AF )So the non-acceptance argument explanation contains
all the arguments in Args that attack A and for which no
defense exists in: some sem-extensions (for ∩) of which A
is not a member; all sem-extensions (for ∪). That for ∩ only
some extensions have to be considered follows since A isskeptically non-accepted as soon as Esem
6= ∅, while A is
A
Z

=
Ext
credulously non-accepted when Esemsem (AF).
AZExample 10. (Example 5 continued) Recall AF(AT2 ) shownin Figure 1. Then:•

NotAcc∪grd (A3 ) = {B1 , B3 };•NotAcc∪prf (B3 ) = {A2 , A3 }.

3.2.2

NotDer(¬r) = h{d3 }, {t}, {t}i;
NotDer(q) = h{d4 }, {t, ¬r}, {t}i.

Explanations for Non-Accepted Formulas

The non-acceptance of a formula φ can have two causes:
either there is no argument for φ at all (i.e., it is notderivable) or all arguments for φ are attacked. In the firstcase φ is not part of the knowledge base K. Moreover, if
there are rules with φ as consequent, for each rule there is atleast one antecedent for which no argument exists.
Definition 13 (Non-derivability explanation). Let AF(AT)be an SAF and let φ be some non-derivable formula. Then:

NotDer(φ) = hRules(R, φ),NoArgAnt(φ), NoArgPrem(φ)i .
The idea is that the explanation points out the gaps
in the argumentation theory: the missing knowledge baseelements and/or missing rules. If there are rules for φ these
are collected in the first part of the explanation, the second
part contains the missing antecedents of these rules (if there
would be arguments for all antecedents, there would be anargument for φ) and the third part contains the formulasthat are connected to φ but for which no rule exists (i.e.,

+



*

NotAcc∩sem (φ) =

na

na 

F (Sφ ), F

[

[

na

D (A, E)A∈Sφ E∈Esem
6A



*

NotAcc∪sem (φ) =na

F (Sφ ), F

na 

+
[

[

na

D (A, E)  .

A∈Sφ E∈Extsem (AF )These explanations consist of the existing arguments for
φ and the arguments for which no defense exists from E
under Dna . Similar to non-acceptance argument explanations, for ∩ only the extensions without any argument forφ have to be considered, while for ∪ all extensions have tobe accounted for. By assumption Sφ 6= ∅, since otherwise
the explanation for the non-acceptance of φ would be its
non-derivability.
Example 12. (Example 9 continued) Consider again
AF(AT2 ). Recall that all arguments are credulously accepted, we do however have:

4•

NotAcc∪prf (q) = h{A3 }, {B1 , B3 }i;•

NotAcc∪prf (¬q) = h{B3 }, {A2 , A3 }i.

VARYING D AND F

This section proposes several variations for D and F, themain purpose of which is to show the flexibility of thebasic framework. We focus on notions of defense, which aresuitable for the completeness-based semantics in this paper.
For, for example, naive semantics, one might want to base
D on conflicts instead. In Section 4.4 these variations are
discussed in the context of a real-life application.
4.1

Notions of Defense

We start by only considering the arguments that defend
themselves against all attacks.

6

Definition 15. Let AF = hArgs, Atti be an AF, A, B ∈ Args
and let E ∈ Extsem (AF) for some semantics sem. Then:

FinalDef(A, E) = {B ∈ DefBy(A, E) | ∀C ∈ Args s.t. (C, B)[∈ Att, (B, C) ∈ Att} ∪ {DefBy(B, E) | B ∈ DefBy(A, E),∀C ∈ DefBy(B, E), DefBy(C, E) = DefBy(B, E) and @D∈ DefBy(B, E) s.t. ∀E ∈ Args s.t. (E, D) ∈ Att, (D, E) ∈ Att}
denotes the set of arguments that defend A in E and that are
not attacked at all, defend themselves against any attacker
or are part of an even cycle that is not attacked.
Intuitively this means that these arguments that defend
A do not need other arguments to be defended and, given
E , can be considered as safe to be accepted. To see whyeven cycles should be regarded, take a look at the following
example:Example 13. (Figure 2(a)) Note that here Extgrd (AF 3 ) =∅, while Extsem (AF 3 ) = {{A, D, F, H}, {A, D, F, I}, {B,C, E, H}, {B, C, E, I}} for sem ∈ {prf, stb}. Let E ={A, D, F, H}. Then FinalDef(F, E) = {A, D, H}. This follows since H defends itself against the attack from I and{A, D} is part of an even cycle that is not attacked. If evencycles would not be covered by FinalDef , the defense of theattack (E, F ) would not be accounted for.
Another option is to consider only the arguments that
directly defend the considered argument.
Definition 16. Let AF = hArgs, Atti be an AF, A, B ∈ Args
and let E ∈ Extsem (AF) for some semantics sem. Then:
DirDef(A, E) = {B ∈ E | B directly defends A}, denotes
the set of arguments in E that directly defend A.
One reason for looking at direct conflicts might be that
direct conflicts are often more clear from the context thanindirect conflicts.
Example 14. (Figure 2(c)). Here Extsem (AF 4 ) = {{A1 , A3 ,A5 }} for any sem ∈ {grd, cmp, prf, stb}. Moreover:•••

Acc(A1 ) = {A3 , A5 } for D = DefBy;
Acc(A1 ) = {A5 } for D = FinalDef ; andAcc(A1 ) = {A3 } for D = DirDef .

This minimal example can be seen as a discussion in theform of a sequence of arguments attacking and defending
the topic A1 . When at the end an explanation for theacceptance of A1 is requested: DefBy returns all argumentsthat defend A1 ; FinalDef returns the last argument that was
put forward, which is uncontested; and DirDef returns theargument against the direct attacker of the topic.

Definition 17. Let AF = hArgs, Atti be an AF, A, B ∈ Args
and let E ∈ Extsem (AF) be an extension for some semantics sem. Then: NoDir(A, E) = {B ∈ NotDef(A, E) |A does not (in)directly attack B} denotes the set of arguments that attack A for which no defense exists in E andwhich are not attacked by A itself.
Intuitively, the members of NoDir(A, E) attack A but inorder to defend A against the attack another argument thanA itself is necessary.
Example 16. Let AF 5 = h{A, B}, {(A, B), (B, A)}i. Here
Extprf (AF 5 ) = {{A}, {B}}, NotAcc∩ (A) = {B} for
D = NotDef but NotAcc∩
prf (A) = ∅ for D = NoDir
since by accepting A, A can indeed be concluded.
Now let AF 05 as in Figure 2(d). Then Extprf (AF 05 ) ={{A, D}, {B, C}, {B, D}}, NotAcc∩
prf (A) = {B, C} for D =NotDef and NotAcc∩
(A)
=
{C}
for D = NoDir, since inprf
order to defend A, just accepting A is not enough, D isneeded to defend against the attack from C .
Example 17. (Example 12 continued) Consider AF(AT2 )from Example 3. Then, for Facc = id and Dna = NoDir:

•

acc
Acc∩
=
prf (p) = h{A4 , B2 }, {A2 , A3 , B1 }i, for D
DirDef ;Acc∩= h{A4 , B2 }, {A2 , B1 }i, for Dacc =prf (p)FinalDef .

In the case of non-acceptance explanations, D was defined as the set of all attacking arguments against which
no defense exists. The next definition considers only those
attackers that A does not (in)directly attack itself.

NotAcc∩prf (q) = h{A3 }, {B1 }i;•

NotAcc∩prf (¬q) = h{B3 }, {A2 }i.

4.2

Element Explanations

In structured argumentation, one can provide full arguments as the explanation (e.g., F = id), but the structureof the arguments provides other possibilities as well.
Definition 18. Let AF(AT) = hArgs, Atti be an SAF
and S ⊆ Args a set of formulas. Then AntTop(S) ={Ant(TopRule(A)) | A ∈ S} denotes the set of antecedentsof the top rule of all arguments in S.
The above definition, combined with the introduced
notation in Definition 5, provides some ideas of how F canbe defined. For example, explanations in terms of premises
explain the conclusion in terms of knowledge base items.
The notion AntTop provides explanations in terms of closely
related information and the rule with which the conclusion
is derived from that information.
Example 18. (Examples 9 and 12 continued) Consider
AF(AT2 ) from Example 3. Then, for Dacc = DefBy and
Dna = NotDef :Example 15. (Example 9 continued) Consider AF(AT2 ).
Then, for Facc = id:
•

•

4.3

•

acc
Acc∩
= Prem;prf (p) = h{t, r}, {t, r}i for F

•

acc
Acc∩
= AntTop;prf (p) = h{q, r}, {t, ¬r}i for F•

na
NotAcc∪= Prem;prf (q) = h{t}, {r}i for F•

na
NotAcc∪= AntTop.
prf (q) = h{¬r, t}, {r}i for F

Comparing the Size of Explanations

When choosing a definition for D and F the size of theresulting explanation might be one of the considerations.
While for F this depends on the argumentation framework(e.g., an argument might have many premises or the top
rule might have only one antecedent), for D the size of thedifferent definitions can be compared. We will apply ≤ tothe size of the sets, i.e., S1 ≤ S2 denotes |S1 | ≤ |S2 |.

7

F

G

B1

B3

A1

A6

A5

B4

B2

A2

A4

A3

B
C1
A

D

E
H

I

C
(a) AF 3 , Example 13.

A1

A2

A3

A4

(b) AF(AT6 ), Example 19.

A5

B

A

C

D

(d) AF 05 , Example 16.

(c) AF 4 , Example 14.
Fig. 2. Graphical representations of the AFs in Section 4.

Proposition 1. Let AF = hArgs, Atti be an AF, let A ∈ Args
and let E ∈ Extsem (AF) be an extension for it. Where ∈ {≤
, ⊆}:
1)
2)
3)

and where R6 is such that the following arguments can be6derived from K6 = Kn
= {p, w, su, re}:B1 : pB2 : w
B3 : su
B4 : reDirDef(A, E)  DefBy(A, E);
FinalDef(A, E)  DefBy(A, E);
NoDir(A, E)  NotDef(A, E).

This follows since DirDef(A, E) and FinalDef(A, E) arealways subsets of DefBy(A, E) and, similarly, NoDir(A, E) isalways a subset of NotDef(A, E). Indeed, Acc∩
prf (p) is both≤- and ⊆-smaller for Dacc = DirDef than for Dacc = DefBy(see Example 15). Similarly, NotAcc∩
prf (q), is ≤- and ⊆smaller for Dna = NoDir than for Dna = NotDef (seeExample 17).

Applying the Basic Framework

One of the inspirations for this paper is an argumentationbased system in use by the Dutch National Police, whichassists citizens who might have been the victim of internet
trade fraud (e.g., malicious web shops or traders) in filing acriminal report [11], [12]. From this report basic observations
such as ‘money was paid by the complainant to the counterparty’ or ‘no package was delivered to the complainant’are collected, and these observations are used as premises inlegal arguments to infer whether or not the report concerns
a possible case of fraud. This conclusion is then provided
to the complainant who filed the report. The system isbased on ASPIC+ [9], with axioms (the observations) anddefeasible rules (based on Dutch law concerning fraud), andall attacks are rebuts. The next example illustrates such anargumentation framework.
Example 19. Let AS6 = hL6 , R6 , ni be an argumentation system, where L6 contains the propositions p (the complainantpaid), w (the wrong package arrived), fk (the product is fake),su (the product looks suspicious), re (counterparty states thatthe product is real), cd (the complainant delivered), cpd (thecounterparty delivered) and f (it is fraud) and their negations

A4 : A3 ⇒ ¬cpd
A5 : B4 ⇒ ¬fk
A6 : C1 , A4 ⇒ f

Figure 2(b) shows the corresponding SAF AF(AT6 ). The
preferred extensions of AF(AT6 ), only mentioning the A arguments, are {A1 , A2 , A3 }, {A1 , A2 , A5 }, {A1 , A3 , A4 } and{A3 , A4 , A6 }. So none of A1 , . . . , A6 is skeptically accepted
and all are credulously accepted. Take conclusion f, whereE = {A3 , A4 , A6 , B1 , B2 , B3 , B4 , C1 }. Then:
•
•

4.4

C1 : B1 ⇒ cd
A1 : B2 ⇒ ¬f
A2 : B2 ⇒ cpd
A3 : B3 ⇒ fk

•
•
•

acc
Acc∪
= id andprf (f) = h{A6 }, {A3 , A4 , A6 }i for Facc
D ∈ {DefBy, DirDef};acc= Prem andAcc∪
prf (f) = h{p, su}, {p, su}i for F
Dacc ∈ {DefBy, DirDef};acc= AntTop andAcc∪
prf (f) = h{cd, ¬cpd}, {su}i for F
acc
D = FinalDef ;na
NotAcc∪= idprf (¬f) = h{A1 }, {A3 , A4 , A6 }i for F
acc
and D = NotDef ;na
NotAcc∪= id andprf (¬f) = h{A1 }, {A3 , A4 }i for F
acc
D = NoDir.

Looking at the different possibilities for F, we see thatinstead of the full arguments we can also return just thepremises (observations) of the supporting arguments, so ‘fbecause p and su’. This is what the police system currently
does. The reasoning behind this is that citizens understandthese more factual observations better than more legal concepts such as delivering under a contract. On the other
hand, for the public prosecutor involved in the processingof complaints, an explanation in legal terms – ‘f because cd
and ¬cpd’ (based on AntTop) – might make more sense.
For D there are also different options. For example,
FinalDef returns arguments that do not need other arguments to defend them. That A3 is such an argument w.r.t.
A6 means that this argument A3 for fk is the ‘main reason’we accept f , that is, without A3 the conclusion f will neverbe accepted. With NoDir, no directly conflicting arguments

8

are given (e.g., A6 which directly conflicts with A1 ). Thisavoids explanations such as ‘(the argument for) ¬f is not
accepted because (there is an argument for) f ’.
4.5

Overview

In this section we have considered variations for the functions D and F. Acceptance explanations can be given interms of all the defending arguments (D = DefBy), thearguments that need no further defense (D = FinalDef ), andarguments that defend against direct conflicts (D = DirDef ).
Non-acceptance explanations can be given in terms of all
the attackers for which no defense exists (D = NotDef )and those arguments that need to be defended by anotherargument (D = NoDir). In a structured setting (e.g., in
ASPIC+ ), the form of each of these explanations can bevaried. We discussed sets of arguments (F = id), sets ofpremises/observations (F = Prem) and sets of antecedentsof the rule that was last used (F = AntTop).

5R ELATED W ORK

Fan and Toni [4] define relevant explanations for a single
topic argument in the form of a new related admissibility
semantics, and show how explanations can be derived from
related admissible sets for abstract argumentation and ABA.
A set of arguments is called related admissible if it isadmissible and each argument in it defends the topic. An
explanation for an argument A (called here RA-explanation to
avoid confusion) is then defined as a related admissible setof arguments with topic A. In the next proposition we show
how RA-explanations can be expressed in our framework.
Proposition 2. Let AF = hArgs, Atti be an AF and let A ∈Args. Then {DefBy(A, E) | E ∈ Eadm
A } is the set of all RAexplanations for A.
Proof. Let AF = hArgs, Atti be an AF and let A ∈ Args.
Suppose that Eadm
6= ∅. Let S ∈ {DefBy(A, E) | E ∈ EadmAA },we first show that S is related admissible:
S defends A. This follows immediately by the definitionof S = DefBy(A, E).
S is admissible. Note that S ⊆ E for some E ∈ Eadm
A ,therefore S is conflict-free. Suppose that there is some B ∈ Ssuch that B is not defended against an attack from C ∈Args. By definition of DefBy, C (in)directly attacks A. Since
A, B ∈ E , there is some D ∈ E such that D defends A
and B against the attack from C . By assumption, D ∈/ S.
A contradiction with the definition of DefBy. Therefore S
defends all of its arguments and is thus admissible.
Now suppose that there is some S0 which is an RAexplanation for A but S0 ∈/ {DefBy(A, E) | E ∈ Eadm
A }.
By definition of related admissible sets A ∈ S0 , S0 ∈ Eadm
A
and for each B ∈ S0 , B = A or B defends A in S0 ,thus B ∈ DefBy(A, E), a contradiction. Hence any RAexplanation for A is in {DefBy(A, E) | E ∈ Eadm
A }.
This shows that any Acc∪
adm -explanation is an RAexplanation and that therefore our framework is a moregeneral version of [4].
Garcı́a et al. [6] study explanations for abstract argumentation and DELP. Explanations for a claim are definedas triples of dialectical trees that provide a warrant forthe claim, dialectical trees that provide a warrant for thecontrary of the claim, and dialectical trees for the claim andits contrary that provide no warrant. This means, on the
one hand, that explanations might contain many arguments
and, on the other hand, that the receiver of the explanation
is expected to understand argumentation and dialecticaltrees. With real-life applications in mind, we believe that
explanations that rely less on the underlying argumentation
framework and that can be adjusted to the application
would be more useful. Therefore, in our framework an
explanation consists of a set of (parts of) arguments, that
could be embedded in a natural language sentence to bepresented to a user, as suggested in Section 4.4.
Explanations for non-accepted arguments in abstract
argumentation are studied in [5], [7], both of which focus on
the structure of the AF and credulous non-acceptance underadmissible semantics. Note that we consider skeptical andcredulous non-acceptance for several Dung-style semantics.
In [5] an explanation consists of either a set of arguments
or a set of attacks, the removal of which would make theargument admissible. In structured argumentation it is notalways possible to remove exactly one argument (or attack).
In the AF of Figure 1, A3 would become skeptically acceptable for any semantics, if B1 would be removed. However,when looking at the underlying argumentation theory (recall Example 3), when B1 is removed, the arguments B2 andB3 do no longer exist and thus ¬q is no longer a credulous
conclusion. Therefore, in this paper the basic definition for
non-accepted arguments is defined in terms of the arguments for which no defense exist and no suggestion is madehow to change the argumentation framework in order to
get the considered argument accepted. In [7], explanations
are sub-frameworks, such that the considered argument iscredulously non-accepted in that sub-framework and anyof its super-frameworks. Though a note was added on theapplicability of such explanations in a structured setting,this is not formally investigated in that paper.
Summarizing, our basic framework is (formally) shown
to be more general, more flexible and specifically adjustable
to the receiver of the explanation. Furthermore, none of
the above-mentioned works consider the structure of thearguments when providing explanations.

6

C ONCLUSIONS AND F UTURE W ORK

We have introduced a generic, flexible basic framework for
explanations in structured and abstract argumentation. With
this framework, specialized local explanations for the (non)acceptance of arguments can be given, taking into accountcredulous and skeptical reasoners.
In future work, we plan to extend our framework withpreferences – although showing preferences is sometimes
considered less effective when providing explanations [3],the (non-)acceptance of arguments very often depends directly on them, making a preference the direct reason for(not) accepting an argument.
Given our basic framework, we will further study how
our explanations formally relate to acceptance strategies
and different semantics, and investigate the necessity andsufficiency of arguments and how to implement this in
explanations.

9

Aside from formal investigations, we also want to lookat how findings from the social sciences on what good
explanations are (see e.g., [1], [3]) can be integrated, andhow different types of explanations are evaluated by human users. Important in this respect is that explanations
are contrastive: while people may ask why A?, they often
mean why A rather than B ?, where A is called the fact and
B is called the foil. The goal is then to explain as muchof the differences between fact and foil as possible. Oneof the challenges for an AI system is that the foil is notalways explicit. We plan to study contrastive explanations
within our framework by combining acceptance and nonacceptance and the knowledge of conflicting arguments andcontraries in the case of an implicit foil.

AnneMarie Borg is a postdoctoral researcher in the Police-lab AI at
Utrecht University. Her research interests include formal argumentationand logic. She received her PhD from Ruhr University Bochum in 2019.
Contact her at Politielab AI, Departement Informatica, Pincetonplein 5,
3508 TB Utrecht, The Netherlands, a.borg@uu.nl.

ACKNOWLEDGMENTS
This research has been partly funded by the Dutch Ministryof Justice and the Dutch National Police.

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Floris Bex is scientific director of the Police-lab AI at Utrecht University,and professor of Data Science and the judiciary at Tilburg University.
His interests include argumentation and AI & Law. He received his PhDfrom the University of Groningen in 2009. Contact him at Politielab
AI, Departement Informatica, Pincetonplein 5, 3508 TB Utrecht, The
Netherlands, f.j.bex@uu.nl.