1 Introduction
Formal argumentation theory [4] is nonmonotonic in the sense that when new arguments are added, some arguments may change their status. In this rapport, we show that preferred, stable, complete and grounded semantics satisfy attack removal monotonicity. This means that if an attack from to is removed, the status of cannot worsen, e.g. if was skeptically accepted, it cannot become rejected. Note that result we prove in the present document is the proof of Proposition 1 and Conjecture 1 of the recent paper by Amgoud et al. [2].
2 Formal setting
Let us introduce the formal setting. Let be an infinite set of all possible arguments. An argumentation graph is defined as follows:
Definition 1 (Argumentation Graph).
An argumentation graph is an ordered tuple , where is a finite subset of and . Let be the universe of all argumentation graphs built on .
If we say that attacks . The set of attackers of an argument is defined as . For a set of attacks , we define .
A set of arguments is conflict free if it does not contain two arguments and such that . A set of arguments defends itself iff for every argument , for every argument , if attacks , there exists such that attacks . Let be a conflictfree set.

is a complete extension iff it defends all its elements and contains all arguments it defends.

is a preferred extension iff it is a maximal (w.r.t. ) complete extension.

is a stable extension iff it attacks any .

is a grounded extension iff it is a minimal (w.r.t. set ) complete extension.
Let denote the set of all extensions of under semantics , where and (resp. ) stands for stable (resp. preferred, grounded, complete) semantics. Once extensions are computed, acceptability degrees are assigned to arguments. We follow the definition by Amgoud et al. [2].
Definition 2.
For every , if , then , otherwise,

iff .

iff s.t , .

iff and s.t and .

iff and s.t and .
3 Attack removal monotonicity
The principle we study, i.e. attack removal monotonicity, was introduced by Amgoud et al. [2], in Definition 6. The principle was simply called monotonicity. We prefer to use a more specific name, i.e. attack removal monotonicity, in order to distinguish this principle from other principles in the literature that are also called monotonicity, e.g. [1].
Definition 3 (Attack removal monotonicity).
A semantics satisfies attack removal monotonicity iff for every argumentation graph , , , it holds that .
4 Labellings
To prove that preferred, stable, complete and grounded semantics satisfy this principle, we first introduce the basic concepts behind the labeling [3].
Definition 4 (Labeling).
A labeling is a total function which assigns to each argument of every argumentation graph a value in the set . A labeling is complete iff ,

if , ,

if such that ,

otherwise.
Let be the set of all complete labellings of .
Let be an argumentation graph. For a set of arguments , we define the corresponding labelling as follows:

, ,

such that and , ,

such that and , .
For a labeling of , we define .
It was shown by Caminada [3] that the set of arguments labeled within a complete labeling forms a complete extension of the argumentation graph. Similarly, every complete extension can be labeled by a complete labeling. Correspondences between complete labellings and the three other semantics (grounded, preferred, stable) have also been shown. We denote by (respectively , ) the set of labellings corresponding to stable (respectively preferred, grounded) extension(s) of .
5 Two lemmas
Lemma 1.
Let . Let be such that , and . For any , for any , if , then .
Proof.
Let , and . Let be such that , and let . Let and . Assume that and . From Table 1 (in [5]), where ^{1}^{1}1For , we denote by the argumentation graph .. We construct thus a series of argumentation graphs , where for each , , and we repeat the previous reasoning in order to obtain that . Note that . Hence, . ∎
Lemma 2.
Let . Let be such that , and . For any , for any , if , then .
Proof.
Let and . Let be such that , , , and . Assume that and . From Table 2 (in [5]), where . We construct thus a series of argumentation graphs , where for each , , and we repeat the previous reasoning in order to obtain that . Note that . Hence, . ∎
6 The main result
Proposition 1.
Stable, preferred, complete and grounded semantics satisfy attack removal monotonicity.
Proof.
Let . Let be such that , , and . We put , where . Let us show that .
 Case 1.

. This means that .

(the case is possible only for stable semantics). By definition .

. From [4], , and hence .
Since by definition, , then .

 Case 2.

. Hence, and coincide, trivial.
 Case 3.

.
i) Assume that . Since by definition, , then .
ii) Assume now that . Thus, by definition . From now on, we proceed per semantics.
 Stable semantics:
 Grounded semantics:

 Complete and preferred semantics:

with .

Assume that . By definition, there exists such that . From Lemma 1, . Then, .

Assume that . By definition of , is attacked by at least one complete/preferred extension . Thus, for , it holds that . From Lemma 1, . Assume now that . There are three cases:

. This is impossible since , .

. This means that such that . From Lemma 2, . This would mean that . This contradicts the assumption .

. This is impossible since and meaning that is attacked by at least one complete/preferred extension.


∎
7 Appendix: the proof of the same result under an alternative definition of degrees
The choice of Definition 2 to assign the degree to all the arguments in case when there are no extensions can be disputable. We show that our main result still holds under an alternative definition of degrees.
Let us present an alternative definition of degrees, where all the arguments are assigned the degree in case there are no extensions. The rest of the following definition coincides with Definition 2.
Definition 5.
For every , if , then , otherwise,

iff .

iff s.t , .

iff and s.t and .

iff and s.t and .
Proposition 2.
Stable, preferred, complete and grounded semantics satisfy attack removal monotonicity under the alternative definition of acceptability degree (Definition 5).
Proof.
We only need to consider stable semantics, since the proof for the other semantics is identical to the proof of Proposition 1. Let . Let be such that , let . We define , where . Let us show that .
 Case 1.

If , the proof is over.
 Case 2.

Let . Let be a stable labelling of such that . From Lemma 1, is a stable labelling of . Hence .
 Case 3.

Let . It might be that there are no stable extensions of or that there is a stable extension of . In both cases, we proceed by the proof by reductio ad absurdum. Aiming at the contradiction, let us suppose . This means that has at least one stable extension. Let be a stable labelling of such that . From Lemma 2, is a stable labelling of . Hence, belongs to at least one extension of . Contradiction.
∎
References
 [1] Leila Amgoud and Jonathan BenNaim. Axiomatic foundations of acceptability semantics. In Proceedings of the 15th International Conference Principles of Knowledge Representation and Reasoning, KR’16, pages 2–11, 2016.

[2]
Leila Amgoud, Jonathan BenNaim, and Srdjan Vesic.
Measuring the intensity of attacks in argumentation graphs with
shapley value.
In
Proceedings of the 26th International Joint Conference on Artificial Intelligence, IJCAI’17
, pages 63–69. AAAI Press, 2017.  [3] M. Caminada. On the issue of reinstatement in argumentation. In Proceedings of the 10th European Conference on Logics in Artificial Intelligence JELIA’06, pages 111–123, 2006.

[4]
Phan Minh Dung.
On the Acceptability of Arguments and its Fundamental Role in NonMonotonic Reasoning, Logic Programming and nPerson Games.
Artificial Intelligence, 77:321–357, 1995.  [5] Tjitze Rienstra, Chiaki Sakama, and Leendert W. N. van der Torre. Persistence and monotony properties of argumentation semantics. In Third International Workshop on Theory and Applications of Formal Argumentation, TAFA’15, pages 211–225, 2015.
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