On Explanation of Propositional Logic-based Argumentation System

Teeradaj Racharak and Satoshi Tojo

School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa, Japan

Keywords:

Deductive Argumentation, Argumentation System, Explainable Artiﬁcial Intelligence, Natural Deduction.

Abstract:

We present a characterization about argumentation and proof in logic. Indeed, we show that proof for a

claim α from a set of premises Φ can be deemed as a structured form of an argument for that claim. Due to the

expressivity of classical propositional logic (PL), this work considers that the knowledge-base is represented in

PL, in which the semantics and proof systems for individual arguments are studied and utilized. We show that

natural deduction (ND) can be used as a basis of proof for an argument and also for modeling counterarguments

in the form of canonical undercut. We reveal that ND does not merely enable for the construction of arguments

but also paves the way naturally for a human-understandable form of argumentative reasoning. Finally, we

demonstrate that our approach gives the feasibility of developing explainable artiﬁcial intelligence systems

that can offer human-friendly explanations to the users.

1 INTRODUCTION

Argumentation is an important aspect of human in-

telligence. Humans always search for pros and cons

of arguments as well as their consequences when at-

tempting to understand a facing situation for mak-

ing decisions. This argumentative reasoning can be

formalized by utilizing a logical language for the

premises and an appropriate consequence relation

for showing that claims logically follow from the

premises (a.k.a. logic-based argumentation).

There are a number of proposals for logic-based

formalization of argumentation (cf. (Besnard and

Hunter, 2018; Ches

nevar et al., 2000; Vreeswijk

and Prakken, 2001) for the existing literature).

These works allow the representation of arguments

for claims, the representation of counterarguments

against them, and the relationships between the argu-

ments. Despite the diversity, an argument in logic-

based argumentation is commonly deﬁned as a pair

of which the ﬁrst item is a set of formulae that proves

the second item (i.e. a logical formula). There have

been several investigation of and success with the use

of proof techniques in logic. For instance, (Prakken

and Sartor, 1997) developed proof procedures to ﬁnd

acceptable arguments in Dung’s semantics from a de-

feasible logic knowledge-base. As an example in

propositional logic knowledge-base, (Efstathiou and

Hunter, 2011) proposed to generate arguments and

counterarguments using the resolution principle and

connected graph (Kowalski, 1975; Kowalski, 1979).

Unfortunately, these approaches do not concretely of-

fer computational content of an argument in a form

that is understandable by naive users. This is a vital

aspect of developing explainable artiﬁcial intelligence

systems; reasoners should provide understandable ex-

planations in order to facilitate the process of evolv-

ing the theory between explainers and explainees (i.e.

a group of people who receive the explanations).

Here, we suppose that a knowledge-base ∆ is rep-

resented by classical propositional logic (PL); thereby

proof theories in PL are investigated for construc-

tion of arguments and counterarguments from ∆. For-

mally, ﬁnding an argument for claim α involves seek-

ing for a consistent subset Φ of ∆ which can logically

derive α, i.e., one can prove the validity of α from Φ

using some proof systems. This basically amounts to

investigate well-established proof theories of the base

logic towards arguments’ construction. Figure 1 (a)

depicts an example of applying the resolution tech-

nique (as adopted in (Efstathiou and Hunter, 2011)) to

prove that q is a valid consequence from {p, ¬p ∨q},

in which {} denotes the empty clause.

Researchers have put an essentially great deal

of effort into development of structured argumenta-

tion framework, but addressing understandable com-

puted content of argumentation models have received

less attention. Nevertheless, we might utilize proce-

dures which offer to generate an adequate explana-

tion for a developed argument. Considering Figure

Racharak, T. and Tojo, S.

On Explanation of Propositional Logic-based Argumentation System.

DOI: 10.5220/0010318103230332

In Proceedings of the 13th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2021) - Volume 2, pages 323-332

ISBN: 978-989-758-484-8

323

Figure 1: (a) A resolution proof for claim q given assumptions p and ¬p ∨ q; (b) A corresponding explanation of the proof.

1 (b) as a motivating example, one may observe that

an argument developed from the resolution carry too

much information than the need for a meaningful ex-

planation of why ‘q’ given assumptions ‘p implies q’

and ‘q’. While there exist more diverse procedures in

logic (e.g. analytic tableaux (Smullyan, 1995) and se-

quent calculus (Takeuti, 2013)), this work argues for

using natural deduction (ND) (Gentzen, 1935), taken

as a mean to identify an argument’s structure from

proof. We demonstrate that the pattern represented by

ND is close to what humans can perceive as an ar-

gument drawing a conclusion from any conjunction

that it contains. We elaborate upon our formalization

based on ND proof in Section 3.

It is worth mentioning that current studies on

logic-based argumentation have mostly concerned on

exploiting logic for modeling structured argumenta-

tion such as (Besnard and Hunter, 2018); however,

how it contributes to the development of explainable

artiﬁcial intelligence (XAI) systems is not fully in-

vestigated. Thus, this work aims at bridging this gap

between argumentation and its applications on XAI.

The contribution in this paper is that we introduce

an approach to modeling arguments based on ND cal-

culus towards the development of XAI systems. Our

approach offers three main advantages: (1) explicit

information used to build up arguments, (2) a trans-

parent connection between the supports and the claim

corresponding to the consequence relation, and (3)

obvious translation for generating a human-friendly

argument from the proposed formalization. The use

of ND supports explanation generation from the com-

puted deductive arguments; and also, the use of ar-

gumentative proof procedure coincides with everyday

explanations used by humans (cf. Section 4). We

review the basic elements in (Dung’s) abstract argu-

mentation and classical propositional logic including

the natural deduction in Section 2. Section 5 relates

our approach to others. Finally, Section 6 provides a

conclusion and discussion of future directions.

2 PRELIMINARIES

2.1 Abstract Argumentation

Abstract argumentation (AA) provides a good start-

ing point for formalizing argumentation in human rea-

soning. In (Dung, 1995), an AA framework is a pair

hA, R i of which A represents a set of arguments and

R ⊆ A × A represents attack between arguments. Ar-

guments may attack each other and thereby their sta-

tuses are subject to an evaluation. Semantics for AA

returns sets of arguments called extensions, which are

conﬂict-free and defend themselves against attacks.

Formally, a set S ⊆ A of arguments is conﬂict-

free iff there are no arguments A, B ∈ S such that

(A, B) ∈ R . Moreover, S defends A ∈ A iff, for any

argument B ∈ A, (B, A) ∈ R implies an existence of

C ∈ S such that (C, B) ∈ R . A conﬂict-free set S is

admissible iff each argument A ∈ S is defended by

S. These conﬂict-freeness and admissibility proper-

ties form the basis of all AA semantics as follows.

Let Defended(S)

= {A | S defends A} be a function

which yields a set of arguments defended by a cer-

tain set. Then, set S is a complete extension iff S is

conﬂict-free and S = Defended(S); set S is a grounded

extension iff it is the minimal complete extension

(w.r.t. set inclusion); set S is a preferred extension iff

it is a maximal complete extension (w.r.t. set inclu-

sion); and set S is a stable extension iff S is conﬂict-

free and S attacks every argument which is not in S.

2.2 Propositional Logic and Proof

In AA, the structure and meaning of arguments and

attacks are abstract. On the one hand, the abstract

deﬁnition enables to study properties which are in-

dependent of any speciﬁc aspects (Baroni and Gia-

comin, 2009). On the other hand, this generality fea-

tures a limited expressivity and can be hardly adopted

to model practical target situations. To ﬁll out this

gap, less abstract formalisms were considered, deal-

ing in particular with the construction of arguments

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324

and the conditions for an argument to attack another

e.g. ASPIC

(Modgil and Prakken, 2014), DeLP

(Garc

ıa and Simari, 2004), and assumption-based ar-

gumentation (ABA) (Dung et al., 2009). There have

also been some investigation of logic-based argumen-

tation in which well-established logical frameworks

are used to model arguments and the relationship be-

tween them (cf. Section 5). Since the expressivity of

classical propositional logic (PL) allows for model-

ing sufﬁcient knowledge-base, this work focuses on a

knowledge-base expressible in PL. Some preliminary

deﬁnitions are provided below for self-containment.

Let L be a PL language obtained from a given set

of atoms with ¬, ∧, ∨, and → as connectives. Though

all PL formulae can be formulated by using only ¬

and ∧, we also include others to simplify the presen-

tation of our approach. We also assume that L con-

tains the special symbol ⊥ representing inconsistency.

To show that a sentence is derivable (or provable), we

use the following natural deduction (ND) rules for any

φ, ψ ∈ L (Van Dalen, 2004):

φ ψ

φ ∧ ψ

(∧I)

φ ∧ ψ

(∧E)

φ ∧ ψ

(∧E)

φ ¬φ

⊥

(¬E)

⊥

(⊥)

[¬φ]

⊥

(RAA)

[φ]

φ → ψ

(→ I)

φ → ψ φ

(→ E)

φ ∨ ψ

(∨I)

φ ∨ ψ

(∨I)

φ ∨ ψ

[φ]

[ψ]

(∨E)

Derived consequences result from applying these

ND rules in sequence and we denote Φ ` φ if φ is

derivable from Φ. This search can be performed in

the forward direction, from Φ to φ, in the backward

direction, from φ to Φ, or even from both directions

concurrently (Ferrari and Fiorentini, 2015). Our deﬁ-

nition of argument (cf. Section 3) insists on the back-

ward generation of arguments by applying ND rules

with formulae in a knowledge-base.

Example 2.1. For a knowledge-base ∆

= {b →

a, c → b, c ∧ b}, where a, b, c represents ‘avoid

steroids’, ‘get vaccine against hepatitis B’, and ‘plan

to visit Africa’, respectively. In the following, we

show that ∆ ` a:

b → a

c ∧ b

Naturally, reading this deduction tree from top

to bottom corresponds to the following explanation;

noted that one can also read the tree from bottom to

top to obtain a similar natural language sentence:

1. It is assumed that c and b;

2. So is b from #1;

3. It is also assumed that b implies a;

4. So is a from #2 and #3.

Hence, we show that assuming ‘b implies a’ and ‘c

and b’ derives ‘a’.

3 NDSA FRAMEWORK: ND FOR

STRUCTURED

ARGUMENTATION

Observe that a derivation in Example 2.1 corresponds

to querying ‘should we avoid steroids and why if so?’.

Hence, it is quite natural to deem that ND for formula

α represents a logical argument for claim α supported

by a corresponding set of premises. Indeed, reading a

ND tree from top to bottom (also, from bottom to top)

yields an interpretation of a logical argument, allow-

ing to extract naturally a human-friendly explanation

as to why the claim α is so.

Deﬁnition 3.1 (ND Argument). Given a PL

knowledge-base ∆, an argument for claim α sup-

ported by Φ ⊆ ∆ (denoted by hΦ, αi) is a ND proof

tree such that α is derivable (backwards) from α to Φ

and ¬α is not derivable from Φ.

Set Φ is called supports or assumptions; and also,

α is called a claim of an argument. Observe that Deﬁ-

nition 3.1 imposes consistency constraint to avoid the

construction of illogical arguments (such as via ex

falso quodlibet).

Unlike several other work, e.g. those of (Besnard

and Hunter, 2018) and (Garc

ıa and Simari, 2004), we

do not impose the restriction that the support of an

argument be minimal. For instance, the same conse-

quence as in Example 2.1 can be derived; however

more verbose, as follows:

b → a

c → b

c ∧ b

This ND proof tree corresponds to the following log-

ical argument for explaining why we should avoid

steroids (a) given the assumptions (reading from top

to bottom of the ND proof tree):

1. We know that c and b by our assumptions;

On Explanation of Propositional Logic-based Argumentation System

325

2. So is c from #1;

3. We also know that c implies b by our assumptions;

4. So is b from #2 and #3;

5. We also know that b implies a by our assumptions;

6. So is a from #4 and #5.

Though the above argument is not minimal, it is

also relevant in the sense that their supports contribute

to deducing the conclusion. Minimal checking is one

way to ensure relevancy but may come at a compu-

tational cost. Nonetheless, our arguments are guar-

anteed to be relevant without imposing on minimality

due to the backward generation of ND proof trees.

It is worth mentioning that applying ND is advan-

tageous for us since the hypotheses appear only on

top layers of a deduction tree, that sufﬁces our prime

goal of yielding human-friendly arguments. In com-

parison to other formalisms, a Hilbert-style axiomati-

zation requires us to supply many axioms in the midst

of a proof tree. As for the analytic tableau method,

we need to show our goal to prove ﬁrst on the top line,

that is against our objective. Gentzen’s sequent calcu-

lus (Kleene et al., 1952) might be the most polished

style of deduction; however, each sequent becomes a

long and messy sequence of formulae and is thus dif-

ﬁcult for proof’s visualization.

Given two arguments, it is possible to compare

which argument is more general than one another.

The following deﬁnition captures this relation be-

tween two arguments from a knowledge-base.

Deﬁnition 3.2. An argument hΦ, αi is comparable to

and more concise than an argument hΨ, βi iff Φ ⊂ Ψ

and α ≡ β.

From the above deﬁnition, one can say that argu-

ment h{b → a, c ∧ b}, ai is comparable to and more

concise than argument h{b → a, c → b, c ∧ b}, ai.

Equipping argumentation into knowledge-base

reasoning enables to deal with existence of inconsis-

tent premises; derived conclusions of a knowledge-

base are the claims of arguments in a concerned ex-

tension. Since a knowledge-base may be inconsistent,

logical arguments constructed from the knowledge-

base may be conﬂicting with each other. To deﬁne

counterarguments, we consider the notion classical

direct undercut (Besnard and Hunter, 2018), which

is largely applied in the literature.

Deﬁnition 3.3. Let A

= hΦ, αi and B

= hΨ, βi be

arguments. Then, we say that argument A attacks ar-

gument B iff ∃φ ∈ Ψ such that α ≡ ¬φ.

It is worth noticing that if an argument is attacked

by another argument, then other arguments which are

less concise than it will also be attacked by the same

counterargument. Hence, it is redundant to account

for less concise arguments in argumentative reason-

ing and is omitted to show in our running examples.

Following this idea, we are now ready to instantiate

an abstract argumentation framework from a (possi-

bly inconsistent) PL knowledge-base.

Figure 2 illustrates an instantiated abstract argu-

mentation framework, in which each triangle repre-

sents an argument corresponding to each ND deriva-

tion. At each argument, the bottom part represents the

claim, the top part denotes its supports, and arrows

represent argument-counterargument relationship.

The proposed approach gives a straightforward

way for instantiating an abstract argumentation

framework from a PL knowledge-base. However, ap-

plying these deﬁnitions may cause inﬁnite construc-

tion of arguments; thereby causing the attack rela-

tion among them to be also inﬁnite. For instance,

if an instantiated abstract argumentation framework

contains argument h{a}, ai, it also means that the in-

stantiated framework contains arguments h{a}, a ∨bi,

h{a}, a ∨ b ∨ ci, and so on using the ∨I-ND rule. This

kind of inﬁnite abstract argumentation frameworks

may be hardly used and requires a special treatment

to deal with the inﬁnite construction. In the follow-

ing, we consider a core of an argumentation frame-

work (Amgoud et al., 2011) which can be identiﬁed

by the following notions.

Deﬁnition 3.4 (Structural Equivalence of Argu-

ments). Arguments A

= hΦ, αi and B

= hΨ, βi are

structurally equivalent iff Φ = Ψ and α ≡ β.

Deﬁnition 3.5. Let F

= hA

, R

i and F

= hA, R i

represent different AA frameworks. Then, F

is a

core of F iff A

and R

are ﬁnite; and, for any A ∈ A,

there exists A

∈ A

such that A

and A are structurally

equivalent and A

satisﬁes the following conditions:

• For any argument B ∈ A such that (B, A) ∈ R ,

there also exists argument B

∈ A

such that

, A

) ∈ R

, and

• For any argument B ∈ A such that (A, B) ∈ R ,

there also exists argument B

∈ A

such that

, B

) ∈ R

When multiple arguments are structurally equiva-

lent, it is adequate to choose exactly one argument of

them in an instantiated abstract argumentation frame-

work. Restricted our attention on structurally equiva-

lent arguments, one can identify a core of an argumen-

tation framework. For instance, it can be shown that a

core of an instantiated abstract argumentation frame-

work in Example 2.1 contains three arguments as in

Figure 3 in which A

denotes a derivation for c from

c ∧ b, A

denotes a derivation for b from c ∧ b, and

denotes a derivation for a from c ∧ b together with

b → a. At implementation, this imposition can be re-

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

326

Figure 2: An instantiated abstract argumentation framework.

duced to checking whether α ∧¬β and β∧ ¬α are un-

satisﬁable by any Boolean satisﬁability (SAT) solver,

although checking if two logical formulae are equiv-

alent may come with a computational cost. Some

more heuristic techniques for performance improve-

ment may also be investigated and are remained as

our future tasks.

Figure 3: A core of an instantiated abstract argumentation

framework.

To complete our earlier deﬁnitions of argument

and attack, we formally give a deﬁnition of our natural

deduction-based structured argumentation (NDSA)

framework as follows.

Deﬁnition 3.6 (NDSA). A NDSA framework is a

triple hL, ∆, `

i, where L is a PL language, ∆ is

a knowledge-base modeled based upon language L,

and `

is a consequence relation represented by the

natural deduction proof calculus.

Proposition 3.1. Let F

= hA, R i be an AA frame-

work built according to NDSA. Then, F

= hA

, R

is a core of F if it is constrained from F as follows:

• For each set A ⊆ A of structurally equivalent

arguments, only one argument A of A is non-

deterministically included in A

and all attacks

of arguments A \ A are excluded from hA, R i to

yield hA

, R

i .

Proof. We show that the above construction yields

a core of any argumentation framework hA, R i. Fix

any A ⊆ A of structurally equivalent arguments, we

show if an argument A of A is non-deterministically

in A

, then the following conditions hold:

• (Condition 1) Fix any argument B ∈ A \ A such

that (B, A) ∈ R , we show that there exists an ar-

gument B

∈ A

such that (B

, A) ∈ R

. Since A

is a set of structurally equivalent arguments, then

(B, A) ∈ R

. This means B

= B. Therefore, this

condition trivially holds.

• (Condition 2) Fix any argument B ∈ A \ A such

that (A, B) ∈ R , we show that there exists an ar-

gument A

∈ A

such that (A

, B) ∈ R

. Since A is

a set of structurally equivalent arguments and A is

a singleton, then we know A

= A. Therefore, this

condition also trivially holds.

The above proposition provides us an algorithmic

procedure to indicate a core of an abstract argumenta-

tion framework. In the following, we illustrates an-

other (but less trivial) example about identifying a

core of an abstract argumentation framework. It also

highlights that the proposed framework can be used

in multi-agent reasoning, in which all agents possess

their own consistent set of the knowledge (but, their

integration is inconsistent). For instance, it often oc-

curs that witnesses in jurisdictions may hold differ-

ent consistent sets of beliefs but integration among

those beliefs turns to be inconsistent. This shows that

NDSA allows to represent an argumentation dialogue

and to detect the argument-counterargument interac-

tion between each agent’s utterance.

Example 3.1. Let ∆

= {a, a → b} and ∆

{¬b, a → b} represent different knowledge-bases pos-

sessed by Agents A and B, respectively, in which a de-

notes ‘plan to visit Africa’ and b denotes ‘get vaccine

against hepatitis B’. For the integrated knowledge-

base ∆

= ∆

∪ ∆

, it can be shown that a core of an

AA framework built from ∆ according to NDSA is

presented as in Figure 4.

Noted that, in the ﬁgure, A

represents a deriva-

tion for h{a}, ai, A

represents a derivation for

h{¬b}, ¬bi, A

represents a derivation for h{a →

b}, a → bi, A

represents a derivation for h{a, ¬b}, a∧

¬bi, A

represents a derivation for h{¬b, a → b}, ¬ai,

and A

represents a derivation for h{a, a → b}, bi.

4 ACCEPTABILITY AND

EXPLANATIONS OF NDSA

Since NDSA instantiates an AA framework from a

knowledge-base, all semantics for determining the

‘acceptability’ of arguments in AA also apply to ND

arguments. As a common approach in logic-based ar-

gumentation, consequences of a knowledge-base are

claims of those arguments in a concerned extension.

Deﬁnition 4.1. Let hL, ∆, `

i be a NDSA frame-

work and ext(F ) be an extension of an AA frame-

On Explanation of Propositional Logic-based Argumentation System

327

Figure 4: A core of an instantiated abstract argumentation framework.

work F built from the NDSA. Then, a set of conse-

quences from ∆ w.r.t. ext(F ) (denoted by Con

ext(F )

)

is deﬁned as: Con

ext(F )

= {α | hΦ, αi ∈ ext(F )}.

For instance, considering Example 3.1, applying

stable semantics (cf. Subsection 2.1) yields three ex-

tensions: {A

, A

}, {A

, A

}, and {A

, A

Hence, there are three sets of accepted consequences

w.r.t. the stable semantics from this knowledge-base:

{a, a → b, b}, {¬b, a → b, ¬a}, and {a, ¬b, a ∧ ¬b}.

Proposition 4.1. Let F

= hA

, R

i and F

= hA, R i

represents AA frameworks. If F

is a core of F , then

the following holds for any concerned extension:

• For any α ∈ Con

ext(F

)

, there exists β ∈ Con

ext(F )

such that α ≡ β; and

• For any β ∈ Con

ext(F )

, there exists α ∈ Con

ext(F

)

such that α ≡ β.

Proof.

• (Condition 1) Fix any α ∈ Con

ext(F

)

, we show

that there exists β ∈ Con

ext(F )

such that α ≡ β.

By assumption, we know that, for any A ∈ A,

there exists A

∈ A

such that A

and A are struc-

turally equivalent. Therefore, this condition triv-

ially holds by Deﬁnitions 3.4 – 3.5.

• (Condition 2) Fix any β ∈ Con

ext(F )

, we show

that there exists α ∈ Con

ext(F

)

such that α ≡ β.

By assumption, we know that, for any A ∈ A,

there exists A

∈ A

such that A

and A are struc-

turally equivalent. Therefore, this condition triv-

ially holds by Deﬁnitions 3.4 – 3.5.

Notice that incorporating the computation of ac-

ceptability in AA together with ND in PL knowledge-

base naturally represents formal reasoning used by

humans, as (Gentzen, 1935)

and (Dung, 1995) aimed

at this. Furthermore, separating levels of proof of-

fers non-monotonic behavior for PL (which is a

monotonic logic) since adding/removing arguments

can surpass the acceptance of some arguments and

“First, I wished to construct a formalism that comes as

close as possible to actual reasoning. Thus, arose a calculus

of natural deduction”, quoted from (Gentzen, 1935).

thereby the corresponding claims are prevailed. Due

to the well-investigated computational models of ar-

gumentation and ND proof in PL, we can outline an

implementation of our proposed approach as shown

in Algorithm 1. This algorithm ﬁnds all acceptable

arguments w.r.t. a concerned extension.

Algorithm 1 : Finding accepted arguments in a NDSA

framework hL, ∆, `

1: input: a knowledge-base ∆, an AA semantics s

2: output: sets of acceptable arguments w.r.t. s

3: function ACCEPTEDARGUMENTS(∆, s)

4: Let G be an empty directed graph.

5: G

= Construction of an abstract argumenta-

tion framework based on Deﬁnitions 3.1 – 3.3.

6: Indicate a core of G based on Proposition 3.1.

7: Remove irrelevant arguments and attack rela-

tions (that are not part of the core) in G.

8: exts

= Sets of acceptable arguments w.r.t. the

semantics s in G.

9: return exts

10: end function

Regarding the tasks of providing their explana-

tions, NDSA considers two-level interpretation on a

core of an argumentation framework:

1. Explain why arguments are acceptable w.r.t. AA

semantics;

2. Explain why accepted arguments are logically de-

rived based on ND.

These two levels correspond to macro-scoping and

micro-scoping explanations, respectively, for an ac-

cepted argument. The ﬁrst level interpretation can be

viewed as a debate between two ﬁctitious agents (i.e.

the proponent and the opponent) arguing why argu-

ments in an extension should be accepted. On the

other hand, for the second level interpretation, one can

speciﬁcally zoom into a ND derivation of an argument

as a basis for serving an explanation.

The following ND deduction proof exempliﬁes

this intuition by explaining why argument A

(in Ex-

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

328

ample 3.1) for claim ‘get vaccine against hepatitis B’

is derivable in the ﬁrst stable extension:

a a → b

This inferential step enables to yield the following ex-

planations to the users. As aforementioned, it corre-

sponds to the micro-scoping level of explanation:

1. Given that ‘plan to visit Africa’ (a) and ‘planning

to visit Africa implies to get vaccine against hep-

atitis B’ (a → b);

2. Hence, ‘get vaccine against hepatitis B’ (b).

We discuss more about the generation process of

explanations on our two levels of explanations (cf.

Subsection 4.1) and the evaluation of explanations in

an aspect of good explanations based upon the view-

point of human cognition in social science (cf. Sub-

section 4.2) in the following subsections.

4.1 Generation of Explanations for

Claims of NDSA Arguments

This subsection brieﬂy introduces the idea of provid-

ing explanations from NDSA, which basically com-

prise two forms of explanation as follows.

4.1.1 Dialogical Explanations

This explanation corresponds to the macro-scoping

interpretation of our proposed framework. Indeed,

given an abstract argumentation framework hA, R i

instantiated from NDSA, one can explain the outcome

of a concerned extension dialogically by reinterpret-

ing a dispute tree T of an argument A ∈ A, which can

be constructed by the following procedure:

1. Every node of T is of the form [L : B] where L is

either proponent (P) or opponent (O) and B ∈ A,

2. The root node of T is always labeled by [P : A],

3. For every node [P : B] of T with B ∈ A, and for

every C ∈ A with (C, B) ∈ R , there exists a child

of [P : B] which is labeled by [O : C],

4. For every node [O : B] of T with B ∈ A, there

exists exactly one child of [O : B] which is labeled

by [P : C] with (C, B) ∈ R ,

5. There are no other nodes in T except #1 – #4.

The proponent wins if he/she can counter-attack

against every attacking argument by the opponent.

The set of all arguments belonging to the proponent

nodes in T is called the defence set of T (Dung et al.,

2006). This defence set represents a reason for why a

certain claim should be accepted.

Note that a branch in a dispute tree may be either

ﬁnite or inﬁnite. A ﬁnite branch represents a win-

ning sequence of arguments that ends with an argu-

ment by the proponent in which the opponent is un-

able to attack. An inﬁnite branch represents a winning

sequence of arguments that the proponent counter-

attacks every attack of the opponent ad inﬁnitum.

Several studies (Dung et al., 2006; Modgil and Cami-

nada, 2009) have put an essentially great deal of effort

into investigation of its winning strategies in order to

help determining the membership of arguments in an

extension of any abstract argumentation framework.

Deﬁnition 4.2. Let hA, R i be an abstract argumenta-

tion framework. A dispute tree T for A ∈ A is admis-

sible iff no argument labels both P and O;

Theorem 4.1 ((Dung et al., 2009)). Let hA, R i be an

abstract argumentation framework. We know:

1. If T is an admissible dispute tree for an argument

A ∈ A, then the defence set of T is admissible;

2. If A ∈ S for an admissible set S ⊆ A, then there ex-

ists an admissible dispute tree for A with defence

set S

such that S

⊆ S and S

is admissible;

This work does not focus on the strategies; but rather,

we employ this notion for explanation in a contrafac-

tual argumentative situation for the consequences de-

rived by NDSA (cf. Deﬁnition 4.1).

Intuitively, this form explains as a debate between

a proponent P seeking to establish the acceptance of

an argument in an extension and an opponent O seek-

ing to withdraw such acceptance. For example, one

can unfold the debate for the acceptance of A

in a sta-

ble extension {A

, A

} in Example 3.1 as follows

(cf. Figure 5). First, P moves argument A

repre-

senting the acceptability of A

. Then, O puts forward

argument A

representing the attack on a support of

. Then, P has to counter-argue O’s argument by

putting forward argument A

as it defends A

. Next,

O puts forward argument A

representing the attack

on a support of A

. It is in turn provided that this ar-

gument is counter-attacked by the same argument A

of P. We note that the same arguments A

, A

put for-

ward by O can also counter-attack A

of P; however,

this attack and counter-attack relationship represents

two winning sequences of arguments that the propo-

nent counterattacks every attack of the opponent ad

inﬁnitum. We can handle this situation of an inﬁnite

branch by disabling its repetition on the dispute tree

(cf. the dot lines in Figure 5). Since the arguments

forwarded by P are unattacked by O, these sequences

of argument moves indicate the acceptance of argu-

ment A

. Dialogical explanations for other arguments

in the extension can be obtained similarly.

On Explanation of Propositional Logic-based Argumentation System

329

Figure 5: A dispute tree for argument h{a}, ai w.r.t. the NDSA framework in Example 3.1.

4.1.2 Logical Explanations

This explanation corresponds to the micro-scoping

interpretation of our proposed framework. As we

demonstrate earlier, the computational content of ND

can facilitate to generate human-friendly logical ex-

planations on theorem hood in a theory. It is worth

noting that other proof systems may also be used as a

basis to explanation generation; nonetheless, we ob-

serve that they may carry more information and ex-

press less intuitively to human interpretation. The

idea to unfold explanations from a ND derivation is

as follows. We trace on a ND proof tree from top

to bottom (toward the conclusion of the proof) by a

procedure that generates text corresponding to each

inference step and tries not to repeat conjunctive par-

ticles (e.g. if – then, thus, hence, etc.). After that, we

put together phrases derived from each subproof.

4.2 Evaluation of Explanations for

Claims of NDSA Arguments

To evaluate the explanations generated from NDSA,

we compare and relate with major ﬁndings on charac-

teristics of good explanations in philosophy, cognitive

psychology/science, and social psychology. Early on,

most works devoted for explainable artiﬁcial intelli-

gence use only the researchers’ intuition of what con-

stitutes a good explanation (Miller, 2019) and over-

look the important insights from these research ﬁelds.

We ﬁll in this gap by reviewing relevant work and

demonstrating that NDSA can support XAI systems

to develop human-like explanations, i.e. contrastive

explanations (Subsubsection 4.2.1) and selected ex-

planations (Subsubsection 4.2.2). In other words, an

important outcome of this work is to draw a closer

connection between argumentation theory and its ap-

plications towards the development of XAI systems.

4.2.1 Contrastive Explanations

Humans always seek in response to particular coun-

terfactual cases (Hilton, 1990; Lipton, 1990). Re-

search in related ﬁelds especially the social science

shows that people do not explain the causes for an

event per se, but rather explain the cause of an event

relative to some other event that does not occur; that

is, an explanation is of the form ‘why α rather than

β?’, in which α represents the ﬁnal conclusion and β

represents an opposite outcome.

It is worth observing that our dialogical explana-

tions are inherently contrastive by the construction;

that is, to explain that α must occur, the explainer (the

proponent) deliberates that a counterfactual β is not

necessarily the case to the explainee (the opponent) by

ﬁnding a counterargument to β. For instance, given a

question “why should we plan to visit Africa?”, the

explainer may try to answer “why should not we plan

to visit Africa?”. To answer this negated question,

the explainer can answer it by showing that “not visit-

ing Afriva (¬a)” is not necessarily the case, i.e., with

an answer “because we plan to visit Africa (a) and if

we plan to visit Africa, then we get vaccine against

hepatitis B (a → b)”. This indicates that our explana-

tions satisfy this form of good explanations by natu-

rally reinterpreting from the dialogical explanation.

4.2.2 Selected Explanations (w.r.t. a Context)

Research results on social psychology and cognitive

science show that humans rarely expect an explana-

tion to consist of both an actual and complete cause of

a decision. But rather, they select one or two causes

from a (possibly inﬁnite) number of causes to be the

explanation for the explainee (Hilton, 2017). Indeed,

while a decision may have many argumentative claim-

backings, often the explainee cares only about a small

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

330

subset (relevant to the context). That is, the explainer

selects a subset of the possible explanations (based on

different criteria), and the explainer and the explainee

may interact and argue about these explanations.

It is worth observing that the generation of dia-

logical explanations can be tailored to selected expla-

nations if the context of the explainee is given. For

that, we consider only a branch in a dispute tree for

the argument in which the selected branch coincides

with the criteria speciﬁed in the context; the context

may mean a concerned formula indicated by the ex-

plainee, the length of a considering branch of the dis-

pute tree, and etc. For instance, according to Figure

5, there might be two possible selected explanations

to be transmitted to a receiver of the explanation for

why argument h{a}, ai as the dispute tree has two

branches. This indicates that our forms of explanation

also enable to create selected explanations by manip-

ulating on the generation of dialogical explanations.

5 COMPARISON WITH THE

STATE OF THE ART WORK

This section compares our approach described in this

paper with existing work on logic-based argumenta-

tion where the knowledge-base is formulated by PL.

(Efstathiou and Hunter, 2011) adopt the idea of

connection graph (Kowalski, 1975; Kowalski, 1979)

to model arguments from a knowledge-base ∆ of

clauses where each claim is a literal. At a high level,

each node in a connection graph represents a clause

in ∆; and, each edge connects nodes φ, ψ if there is a

disjunct in b ∈ φ with its complement being a disjunct

¬b ∈ ψ. To ﬁnd an argument for claim α, the authors

considered the set of complements of the disjuncts of

α together with ∆. Then, for any clause φ in the graph,

if there is a disjunct b ∈ φ and there are no edges con-

necting φ, ψ where the complement of b is a disjunct

in ψ, then the clause φ is deleted together with edges

involving φ. This process of deletion is continued un-

til no more clauses can be identiﬁed for deletion. If

the resulting graph is non-empty, then it contains a

set of formulae that entails α. Though the idea of

connection graph can be used to ﬁnd arguments, our

proposed approach can model logical arguments more

naturally where connections between the claim and its

supporting premises are visualized explicitly; thereby

our approach enables to extract explanation as more

human-friendly arguments for non-technical users.

(Dung et al., 2009) introduced a more concrete

framework for instantiating abstract argumentation

called assumption-based argumentation, in which ar-

guments can be constructed deductively from infer-

ence rules and the notion of attack is deﬁned based

on the contrary of an argument’s claim. It is worth

observing that the authors’ proposal was also similar

to ours in a sense that their arguments were modeled

by applying modus ponens on inferential knowledge-

base whereas our approach models arguments by ap-

plying ND rules to prove a claim (i.e. ND proof trees

as logical arguments). The fact that we do not only fo-

cus on just one rule enables to model deductive argu-

ments in a way that can be understood by naive users.

(Kakas et al., 2014) developed argumentation

logic (AL), which can be viewed as an extension of

PL. AL re-interpreted and extended PL to deal with

inconsistency by modeling arguments from a set of

PL formulae. Attack between two arguments in AL

was deﬁned based on a proof for inconsistency (in-

spired by Reductio ad Absurdum in ND) between

two sets of PL formulae representing arguments. As

for entailment, an argument is said to hold if it can

be successfully defended and it cannot be success-

fully objected against. AL does not follow conven-

tional semantics in AA; hence, the framework may

infer different results to ours. For instance, let ∆

{α, β, α ∧ β, ¬α ∨ ¬β}, the knowledge-base does not

entail α in AL; however, our approach ﬂavors ND for

modeling arguments from sets of PL formulae and

evaluates the acceptability of arguments based on se-

mantics in AA. For example, α may be inferred from

∆ if stable semantics is used or otherwise if grounded

semantics is considered.

6 CONCLUSION AND FUTURE

DIRECTIONS

This work presents an approach to a logic-based ar-

gumentation framework for reasoning with an (incon-

sistent) PL knowledge-base, especially in multi-agent

reasoning in which each agent holds different set of

(mutually inconsistent) knowledge, with an aim at the

introduction of human-friendly explanations in argu-

mentative reasoning. We show that good explanations

investigated in cognitive science and social psychol-

ogy can be formalized as a NDSA framework to de-

velop an explainable artiﬁcial intelligence system.

Our approach exploits two main aspects of for-

malisms: (1) the naturalness of natural deduction and

(2) argumentative semantics in AA. First, we utilize

natural deduction proof in PL for ﬁnding valid argu-

ments in a knowledge-base. While we are inspired

by ND for modeling arguments from the knowledge-

base, our approach can also be applied with other

proof systems even though verbose explanations may

be generated due to the proof procedure (cf. Figure

On Explanation of Propositional Logic-based Argumentation System

331

1). Indeed, a derivation for a formula is re-interpreted

as an argument supporting that formula; and also, ar-

guments supporting the contrary of the premises are

seen as its attack. Second, when modeled arguments

are in conﬂict, the notion of acceptability and se-

mantics in AA are used to handle inconsistency. We

believe that the computational content which brings

together these two formalisms can generate human-

friendly explanations on theorem hood in a theory.

It is widely accepted by now that answers of an

intelligent systems should be able to explain for why

to the users. Therefore, in the future, we would like to

extend this idea for other logics (e.g. description logic

and modal logic) and develop argumentation-based

reasoning engines that offer human-friendly explana-

tions to naive users for applying on real-world appli-

cations such as legal reasoning and ontology merging.

ACKNOWLEDGEMENTS

The authors would like to thank anonymous reviewers

for valuable comments. This study was supported by

JSPS KAKENHI Grant Number 17H02258.

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