$ASPIC ^?$ and the Postulates of Non-interference and Crash-Resistance

Abstract

We introduce an interrogation mark ? in $ASPIC ^+$ languages as a plausibility operator to enhance any defeasible conclusion does not have the same status as an irrefutable one. The resulting framework, dubbed $ASPIC ^?$, is tailored to make a distinction between strong inconsistencies and weak inconsistencies. The aim is to avoid the former and to tolerate the latter. This means the extensions obtained from the $ASPIC ^?$ framework are free of strong conflicts, but tolerant to weak conflicts. Then, in the current study, we show $ASPIC ^?$ satisfy reasonable properties. In particular, we focus on the property that a conflict between two arguments should not interfere with the acceptability of other unrelated arguments. With this purpose in mind, we prove under which conditions the important principles of Non-interference and Crash-Resistance hold in $ASPIC ^?$.

1 Introduction

As noticed in [1], contradictions can be considered under the mantle of many points of views: as a consequence of the only correct description of a contradictory world, as a temporary state of our knowledge, as the outcome of a particular language which we have chosen to describe the world, as the result of conflicting observational criteria, as the superposition of world-views, or as the result from the best theories available at a given moment. Indeed, in [2], it is argued that inconsistency is a natural companion to defeasible methods of reasoning and that paraconsistency (the property of a logic admitting non-trivial inconsistent theories) should play a role in the formalisation of these methods. In fact, they introduced an interrogation mark ? as a plausibility operator to enhance any defeasible conclusion do not have the same status as an irrefutable one, obtained from deduction.

Inspired by these ideas, we will present the $ASPIC ^?$ framework by extending the $ASPIC ^+$ framework [3], one of the most important formalisms to represent and reason with structured argumentation. In $ASPIC ^?$ (as well as in $ASPIC ^+$), we identify two types of rules: strict (irrefutable) and defeasable. Unlike $ASPIC ^+$, the distinguishing characteristic of $ASPIC ^?$ is the conclusion of any defeasible rule will be a plausible (?-suffixed) formula $\phi ?$. The intended meaning is the conclusion of $\phi ?$ will not necessarily prevent the conclusion of $\lnot \phi ?$; it is required an argument with conclusion $\lnot \phi$, which can only be obtained from a strict rule, to attack the conclusion $\lnot \phi ?$ of an argument. Thus, to produce a strong conflict between the conclusions of the arguments, at least in one of them, the conclusion should be a ?-free formula obtained via a strict rule.

In $ASPIC ^?$, strong inconsistencies as in $\left\{ \phi , \lnot \phi \right\}$ (or $\left\{ \phi ?, \lnot \phi \right\}$) are distinguished from those weak inconsistencies as in $\left\{ \phi ?, \lnot \phi ?\right\}$: the first should be avoided; the second can be tolerated. This means the (weak) conflict between $\phi ?$ and $\lnot \phi ?$ can be accommodated in the same extension. Hence, the extensions in $ASPIC ^?$ will be free of strong conflicts, but tolerant to weak conflicts.

Given that much current work on structured argumentation [3,4,5] combines strict and defeasible inference rules, unexpected results can arise when two arguments based on defeasible rules have contradictory conclusions. This is particularly critical (see [6]) if the strict inference rules include the Ex Falso principle (that an inconsistent set implies anything), because for any formula $\phi$, an argument concluding $\lnot \phi$ can be constructed from these two arguments. As consequence, any other argument is potentially under threat!

In order to solve this problem for $ASPIC ^+$, Wu [7, 8] requires that in each argument, the set of conclusions of all its sub-arguments are classically consistent. Another approach was taken in [6], in which they replace classical logic as the source for strict rules by the (weaker) paraconsistent logic presented in [9] to invalidate the Ex Falso principle as a valid strict inference rule.

Here we will also exploit how to avoid the application of the Ex Falso principle in $ASPIC ^?$ by combining these two solutions: 1) as in [6], we resort to paraconsistent reasoning to tolerate conflicts; our differential is we tolerate only weak conflicts. 2) as in [7, 8], we require for each argument, the set of conclusions of all its sub-arguments are consistent; our differential is that we eliminate only those arguments whose sets of conclusions lead to a strong conflict.

Then, we show $ASPIC ^?$ satisfies reasonable properties. In particular, we focus on the property that a conflict between two arguments should not interfere with the acceptability of other unrelated arguments. With this purpose in mind, we prove under which conditions the important principles of Non-interference and Crash-Resistance [10] hold in $ASPIC ^?$.

The rest of the paper is organised as follows: in Sect. 2, $ASPIC ^?$ framework is presented. Then, we introduce the corresponding argumentation framework with two kinds of defeats (strong and weak) and its semantics. Section 3 is focused on proving the satisfaction of the principles of Non-interference and Crash-Resistance. Finally, we summarise our contributions and future developments.

2 The $ASPIC ^?$ Framework

An Abstract Argumentation Framework $AF$ [11] is a pair $(\mathcal A, \mathcal D)$ in which $\mathcal A$ is a set of arguments and $\mathcal D \subseteq \mathcal A \times \mathcal A$ is a relation of defeat. An argument A defeats $\mathcal B$ if $(A,B) \in \mathcal D$. The $ASPIC ^+$ framework [3, 12] gives structure to the arguments and defeat relation in an $AF$. In this section, we introduce in $ASPIC ^+$ languages an interrogation mark ? as a plausibility operator to enhance defeasible conclusions do not have the same status as those irrefutable. The resulting framework, $ASPIC ^?$, is tailored to distinguish strong inconsistencies from weak inconsistencies. The aim is to avoid the former and to tolerate the latter. We start by defining the argumentation systems specified by $ASPIC ^?$:

Definition 1

(Argumentation System). An argumentation system is a tuple $AS = (\mathcal L,^-,\mathcal R,n)$, in which

• $\mathcal L= \mathcal L^* \cup \mathcal L^?$ is a logical language with a unary negation symbol $\lnot$ and a unary plausibility symbol ? such that

  • $\mathcal L^*$ is a ?-free logical language with a unary negation symbol.

  • $\mathcal L^? = \left\{ \phi ? \mid \phi \in \mathcal L^*\right\}$.

• $^-$ is a function from $\mathcal L$ to $2^{\mathcal L}$, such that

  • $\varphi$ is a contrary of $\psi$ if $\varphi \in \overline{\psi }$, $\psi \not \in \overline{\varphi }$;

  • $\varphi$ is a contradictory of $\psi$ (denoted by $\varphi = -\psi$), if $\varphi \in \overline{\psi }$, $\psi \in \overline{\varphi }$;

• $\mathcal R= \mathcal R_s\cup \mathcal R_d$ is a set of strict ($\mathcal R_s$) and defeasible ($\mathcal R_d$) inference rules of the form $\phi _1, \ldots , \phi _n \rightarrow \phi$ and $\phi _1, \ldots , \phi _n \Rightarrow \psi ?$ respectively (in which $\phi _1, \ldots , \phi _n, \phi$ are meta-variables ranging over wff in $\mathcal L$ and $\psi$ is a meta-variable ranging over wff in $\mathcal L^*$), and $\mathcal R_s\cap \mathcal R_d= \emptyset$.

• n is a partial function such that $n : \mathcal R_d\longrightarrow \mathcal L$.

For any formula $\phi \in \mathcal L^*$, we say $\psi \in - \phi$ if $\psi = \lnot \phi$ or $\psi = \lnot \phi ?$ or $\phi = \lnot \psi$ or ($\phi = \lnot \gamma$ and $\psi = \gamma ?$); we say $\psi \in - \phi ?$ if $\psi = \lnot \phi$ or $\phi = \lnot \psi$.

Intuitively, contraries can be used to model well-known constructs like negation as failure in logic programming. Note for any $\phi \in \mathcal L^*$, $\phi$ and $\phi ?$ are contradictories of $\lnot \phi$; whilst, only $\phi$ is a contradictory of $\lnot \phi ?$. This means $\phi ?$ is not a contradictory of $\lnot \phi ?$. A set as $\left\{ \phi , - \phi \right\}$ (or $\left\{ \phi ?, - \phi ?\right\}$) is intended to represent a strong inconsistency, and $\left\{ \phi ?, \lnot \phi ?\right\}$ is intended to represent a weak inconsistency. We will refer to these two kinds of inconsistencies (strong and weak) as epistemic inconsistencies or simply inconsistencies.

It is also required a knowledge base to provide premises for the arguments.

Definition 2

(Knowledge Base). A knowledge base in an argumentation system $AS = (\mathcal L,^-,\mathcal R, n)$ is a set $\mathcal K\subseteq \mathcal L$ consisting of two disjoint subsets $\mathcal K_n$ (the axioms) and $\mathcal K_p$ (the ordinary premises).

Axioms are certain knowledge and cannot be attacked, whilst, ordinary premises are uncertain and can be attacked. Now we can define an argumentation theory:

Definition 3

An argumentation theory $( AS , \mathcal K)$ is a pair in which $AS$ is an argumentation system and $\mathcal K$ is a knowledge base in $AS$.

In $ASPIC ^?$, arguments are constructed recursively from an argumentation theory by the successive application of construction rules:

Definition 4

(Argument). An argument A on the basis of an argumentation theory $( AS , \mathcal K)$ and an argumentation system $(\mathcal L,^-, \mathcal R, n)$ is

• 1. $\phi$ if $\phi \in \mathcal K$ with $\mathtt {Prem}(A) = \left\{ \phi \right\}$, $\mathtt {Conc}(A) = \phi$, $\mathtt {Sub}(A) = \left\{ \phi \right\}$, $\mathtt {DefR}(A) = \emptyset$, $\mathtt {Rules}(A) = \emptyset$, $\mathtt {TopRule}(A) = \textit{undefined}$.

• 2. $A_1, \ldots , A_n \rightarrow \psi$ if $A_1, \ldots , A_n$ are arguments s.t. there is a strict rule $\mathtt {Conc}(A_1), \ldots , \mathtt {Conc}(A_n) \rightarrow \psi \in \mathcal R_s$; $\mathtt {Prem}(A) = \mathtt {Prem}(A_1) \cup \cdots \cup \mathtt {Prem}(A_n)$; $\mathtt {Conc}(A) = \psi$; $\mathtt {Sub}(A) = \mathtt {Sub}(A_1) \cup \cdots \cup \mathtt {Sub}(A_n) \cup \left\{ A\right\}$; $\mathtt {Rules}(A) = \mathtt {Rules}(A_1) \cup \cdots \cup \mathtt {Rules}(A_n) \cup \left\{ \mathtt {Conc}(A_1), \ldots , \mathtt {Conc}(A_n) \rightarrow \psi \right\}$; $\mathtt {TopRule}(A) = \mathtt {Conc}(A_1), \ldots , \mathtt {Conc}(A_n) \rightarrow \psi$.

• 3. $A_1, \ldots , A_n \Rightarrow \psi ?$ if $A_1, \ldots , A_n$ are arguments such that there exists a defeasible rule $\mathtt {Conc}(A_1), \ldots , \mathtt {Conc}(A_n) \Rightarrow \psi ? \in \mathcal R_d$; $\mathtt {Prem}(A) = \mathtt {Prem}(A_1) \cup \cdots \cup \mathtt {Prem}(A_n)$; $\mathtt {Conc}(A) = \psi ?$; $\mathtt {Sub}(A) = \mathtt {Sub}(A_1) \cup \cdots \cup \mathtt {Sub}(A_n) \cup \left\{ A\right\}$; $\mathtt {Rules}(A) = \mathtt {Rules}(A_1) \cup \cdots \cup \mathtt {Rules}(A_n) \cup \left\{ \mathtt {Conc}(A_1), \ldots , \mathtt {Conc}(A_n) \Rightarrow \psi ?\right\}$; $\mathtt {TopRule}(A) = \mathtt {Conc}(A_1), \ldots , \mathtt {Conc}(A_n) \Rightarrow \psi ?$.

For any argument A we define $\mathtt {Prem}_n(A) = \mathtt {Prem}(A) \cap \mathcal K_n$; $\mathtt {Prem}_p(A) = \mathtt {Prem}(A) \cap \mathcal K_p$; $\mathtt {DefR}(A) = \left\{ r \in \mathcal R_d\mid r \in \mathtt {Rules}(A)\right\}$ and $\mathtt {StR}(A) = \left\{ r \in \mathcal R_s\mid r \in \mathtt {Rules}(A)\right\}$.

Example 1

Consider the argumentation system $AS = (\mathcal L,^-,\mathcal R,n)$, in which

• $\mathcal L= \mathcal L^* \cup \mathcal L^?$ with $\mathcal L^* = \{a,b,f,w, \lnot a, \lnot b, \lnot f, \lnot w, {\sim a}, {\sim b}, {\sim f}, {\sim w}, {\sim \lnot a}, {\sim \lnot b}, {\sim \lnot f}, {\sim \lnot w}\}$. The symbols $\lnot$ and $\sim$ respectively denote strong and weak negation.

• For any $\phi \in \mathcal L^*$ and any $\psi \in \mathcal L$, (1) $\phi \in \overline{\psi }$ iff (a) $\psi = \lnot \phi$ or $\psi = \lnot \phi ?$ or $\phi = \lnot \psi$ or ($\phi = \lnot \gamma$ and $\psi = \gamma ?$); or (b) $\psi = {\sim \phi }$ or ($\psi = {\sim \phi ?}$. (2) $\phi ? \in \overline{\psi }$ iff (a) $\psi = \lnot \phi$ or $\phi = \lnot \psi$; or (b) $\psi = {\sim \phi }$.

• $\mathcal R_s= \left\{ \lnot f \rightarrow \lnot w; b \rightarrow a\right\}$ and $\mathcal R_d= \left\{ a \Rightarrow \lnot f?; b, {\sim \lnot w} \Rightarrow w?; \lnot f? \Rightarrow \lnot w?\right\}$.

Let $\mathcal K$ be the knowledge base such that $\mathcal K_n= \emptyset$ and $\mathcal K_p= \left\{ b, {\sim \lnot w}\right\}$. The arguments defined on the basis of $\mathcal K$ and $AS$ are $A_1 = [b]$, $A_2 = [{\sim \lnot w}]$, $A_3 = [A_1 \rightarrow a]$, $A_4 = [A_3 \Rightarrow \lnot f?]$, $A_5 = [A_1, A_2 \Rightarrow w?]$ and $A_6 = [A_4 \Rightarrow \lnot w?]$.

An argument A is for $\phi$ if $\mathtt {Conc}(A) = \phi$; it is strict if $\mathtt {DefR}(A) = \emptyset$; defeasible if $\mathtt {DefR}(A) \ne \emptyset$; firm if $\mathtt {Prem}(A) \subseteq \mathcal K_n$; plausible if $\mathtt {Prem}(A) \cap \mathcal K_p \ne \emptyset$. An argument is fallible if it is defeasible or plausible and infallible otherwise. We write $S \vdash \phi$ if there is a strict argument for $\phi$ with all premises taken from S, and if there is a defeasible argument for $\phi$ with all premises taken from S. The next definition will be repeatedly employed in Sect. 3:

Definition 5

Let $AT = ( AS , \mathcal K)$ be an argumentation theory with argumentation system $AS = (\mathcal L,^-, \mathcal R, n)$. For a formula $\phi \in \mathcal L$, we define $\texttt {Atoms}(\phi ) = \left\{ a \mid a \textit{ is an atom occurring in } \phi \right\}$. For a set $\mathcal F \subseteq \mathcal L$ of formulas in $\mathcal L$, we define $\texttt {Atoms}(\mathcal F) = \bigcup _{\phi \in \mathcal F} \texttt {Atoms}(\phi )$; furthermore, for a set of atoms $\mathfrak A$, $\mathcal F_{\mid \mathfrak A} = \left\{ \phi \in \mathcal F \mid \phi \textit{ contains only atoms in } \mathfrak A\right\}$. For a strict rule $s = \phi _1, \ldots , \phi _n \rightarrow \varPsi$, we define $\texttt {Atoms}(s) = \texttt {Atoms}(\left\{ \phi _1, \ldots , \phi _n, \psi \right\} )$. For a defeasible rule $d = \phi _1, \ldots , \phi _n \Rightarrow \varPsi$, we define $\texttt {Atoms}(d) = \texttt {Atoms}(\left\{ \phi _1, \ldots , \phi _n, \psi \right\} )$. For a set $\mathcal S = \left\{ s_1, \ldots , s_n\right\}$ of strict rules, we define $\texttt {Atoms}(\mathcal S) = \texttt {Atoms}(s_1) \cup \cdots \cup \texttt {Atoms}(s_n)$. For a set $\mathcal D = \left\{ d_1, \ldots , d_n\right\}$ of defeasible rules, we define $\texttt {Atoms}(\mathcal D) = \texttt {Atoms}(d_1) \cup \cdots \cup \texttt {Atoms}(d_n)$. For an argumentation system $AS = (\mathcal L,^-, \mathcal R, n)$, we define $\texttt {Atoms}( AS )= \texttt {Atoms}(\mathcal R_d) \cup \texttt {Atoms}(\left\{ n(r) \mid r \in \mathcal R_d\textit{ and } n(r) \textit{ is defined}\right\} )$, in which $\mathcal R_d\subseteq \mathcal R$ is the set of defeasible rules in $\mathcal R$. For an argumentation theory $AT = ( AS , \mathcal K)$, we define $\texttt {Atoms}( AT ) = \texttt {Atoms}( AS ) \cup \texttt {Atoms}(\mathcal K)$. For an argument A, we define $\texttt {Atoms}(A) = \texttt {Atoms}(\mathtt {StR}(A)) \cup \texttt {Atoms}(\mathtt {DefR}(A))$. Finally, for a set $\mathcal A= \left\{ A_1, \ldots , A_n\right\}$ of arguments, we define $\texttt {Atoms}(\mathcal A) = \texttt {Atoms}(A_1) \cup \cdots \cup \texttt {Atoms}(A_n)$.

Let $AT = ( AS , \mathcal K)$ be an argumentation theory with $AS = (\mathcal L, ^-, \mathcal R, n)$, and $\mathcal A$ the set of all arguments constructed from $\mathcal K$ in $AS$. Assume $\mathcal K= (\mathcal K_n, \mathcal K_p)$, such that $\mathcal K_n= \left\{ a,b,c\right\}$ and $\mathcal K_p= \emptyset$. Consider $\mathcal R= \left\{ a \rightarrow d, b \Rightarrow e\right\}$. The resulting arguments are $A = [a]$, $B=[b]$, $C=[c]$, $D=[A \rightarrow d]$, and $E=[B \Rightarrow e]$. We have $\texttt {Atoms}(D) = \left\{ a, d\right\}$, $\texttt {Atoms}( AT ) = \left\{ a, b, c, e\right\}$. Note those atoms occurring only in the strict rules (as d) are not considered as atoms in $\texttt {Atoms}( AT )$.

2.1 Attacks and Defeats

In $ASPIC ^?$ arguments are related to each other by attacks (as in $ASPIC ^+$) and by weak attacks:

Definition 6

(Attacks). Consider the arguments A and B. We say A attacks B iff A undercuts, undermines and rebuts B, in which

• A undercuts B (on $B'$) iff $\mathtt {Conc}(A) \in \overline{n(r)}$ for some $B' \in \mathtt {Sub}(B)$ such that $B'$’s top rule r is defeasible.

• A undermines B (on $\phi$) iff $\mathtt {Conc}(A) \in \overline{\phi }$ and $\phi \in \mathtt {Prem}_p(B)$. In such a case, A contrary-undermines B iff $\mathtt {Conc}(A)$ is a contrary of $\phi$.

• A rebuts B (on $B'$) iff $\mathtt {Conc}(A) \in \overline{\phi ?}$ for some $B' \in \mathtt {Sub}(B)$ of the form $B_1'',\ldots ,B_n'' \Rightarrow \phi ?$. In such a case, A contrary-rebuts B iff $\mathtt {Conc}(A)$ is a contrary of $\phi ?$.

We say A weakly attacks B iff A weakly undermines or weakly rebuts B, in which

• A weakly undermines B (on $\phi ?$ (resp. $\lnot \phi ?$)) iff $\mathtt {Conc}(A) = \lnot \phi ?$ (resp. $\mathtt {Conc}(A) = \phi ?$) for an ordinary premise $\phi ?$ (resp. $\lnot \phi ?$) of B.

• A weakly rebuts B (on $B'$) iff $\mathtt {Conc}(A) = \lnot \phi ?$ (resp. $\mathtt {Conc}(A) = \phi ?$) for some $B' \in \mathtt {Sub}(B)$ of the form $B_1'',\ldots ,B_n'' \Rightarrow \phi ?$ (resp. $B_1'',\ldots ,B_n'' \Rightarrow \lnot \phi ?$).

Example 2

Recalling Example 1, we have $A_5$ weakly rebuts $A_6$ and $A_6$ weakly rebuts $A_5$. Besides, $A_6$ contrary-undermines $A_2$ and $A_5$ on ${\sim \lnot w}$. If in addition, one had the argument $A_7 = [A_4 \rightarrow \lnot w?]$, then $A_7$ (like $A_6$) would weakly rebut $A_5$ on $A_5$; however, $A_7$ (unlike $A_6$) would not be weakly rebutted by $A_5$.

Definition 7

($SAF$). A structured argumentation framework $SAF$ defined by an argumentation theory $AT = ( AS , \mathcal K)$ is a tuple $\langle \mathcal A, \mathcal C, \mathcal C', \preceq \rangle$, in which

• $\mathcal A$ is the set of all arguments A constructed from $\mathcal K$ in $AS$ such that it satisfies Definition 4 and $\texttt {Atoms}(A) \subseteq \texttt {Atoms}( AT )$;

• $(X, Y ) \in \mathcal C$ iff X attacks Y and $(X, Y ) \in \mathcal C'$ iff X weakly attacks Y;

• $\preceq$ is a preference ordering on $\mathcal A$.

The restriction $\texttt {Atoms}(A) \subseteq \texttt {Atoms}( AT )$ is to avoid including in the $SAF$ arguments built from strict rules without relation to $\texttt {Atoms}( AT )$. Next, we define the corresponding defeat relation:

Definition 8

(Defeat). [3] Let $A,B \in \mathcal A$ and A attacks B. If A undercut, contrary-rebut or contrary-undermine attacks B on $B'$ then A is said to preference-independent attack B on $B'$; otherwise A is said to preference-dependent attack B on $B'$. A defeats B iff for some $B'$ either A preference-independent attacks B on $B'$ or A preference-dependent attacks B on $B'$ and $A \not \prec B'$.

As observed in the previous definition, a preference-dependent attack from one argument to another only succeeds (as a defeat) if the attacked argument is not stronger than the attacking argument. Thus, if an argument A preference-dependent attacks B and B is preferred over A, then the attack of A to B does not succeed, and B is not defeated by A.

2.2 Abstract Argumentation Frameworks with Two Kinds of Defeats

As $SAF$s have two kinds of attacks, the associated abstract argumentation frameworks have to couple with two kinds of defeats:

Definition 9

(Argumentation frameworks with two kinds of defeats). An abstract argumentation framework with two kinds of defeats ($AF _2$) corresponding to a $SAF = \langle \mathcal A, \mathcal C, \mathcal C', \preceq \rangle$ is a tuple $(\mathcal A, \mathcal D, \mathcal D')$ such that $\mathcal D = \{(X,Y) \in \mathcal C \mid X \textit{ defeats } Y \}$ and $\mathcal D' = \left\{ (X,Y) \in \mathcal C' \mid X \not \prec Y\right\}$. For $A \in \mathcal A$, we define

$$ A^+ = \left\{ B \in \mathcal A\mid (A,B) \in \mathcal D \cup \mathcal D'\right\} \textit{ and } A^- = \left\{ B \in \mathcal A\mid (B,A) \in \mathcal D \cup \mathcal D'\right\} . $$

Given a $SAF SA$ defined by an argumentation theory $AT$ and an $AF _2\, AF$ corresponding to $SA$, we will refer to $AF$ as the resulting $AF _2$ from $AT$.

Example 3

(Example 2 continued). Let $\preceq = \left\{ (A_6,A_2)\right\}$, i.e., $A_6 \prec A_2$ be a preference ordering on $\mathcal A = \left\{ A_1, A_2, A_3, A_4, A_5, A_6\right\}$. In the $SAF (\mathcal A, \mathcal C, \mathcal C', \preceq )$ defined by $AT$, we have $\mathcal C = \left\{ (A_6,A_2)\right\}$ and $\mathcal C' = \left\{ (A_5,A_6), (A_6,A_5)\right\}$. As $(A_6,A_2)$ is a preference independent attack, we obtain $\mathcal D = \mathcal C$ and $\mathcal D' = \mathcal C'$.

Traditional approaches to argumentation semantics ensure arguments attacking each other are not tolerated in the same set, which is said to be conflict free. In $ASPIC ^?$, we distinguish a strong conflict from a weak conflict. The aim is to avoid strong conflicts, which are carried over by the defeat relation $\mathcal D$, and to tolerate weak conflicts, which are carried over by the relation $\mathcal D'$. We say the resulting set is compatible:

Definition 10

(Compatible sets). Let $AF = (\mathcal A, \mathcal D, \mathcal D')$ be an $AF _2$ and $S \subseteq \mathcal A$. A set S is compatible (in $AF$) if $\forall A \in S$, $\not \exists B \in S$ such that $(B,A) \in \mathcal D$.

Compatible sets do not contain the arguments A and B if A attacks B (or vice versa); however they can accommodate weak attacks between their members. This means in Example 3, the set $\left\{ A_5,A_6\right\}$ is compatible, but $\left\{ A_2, A_6\right\}$ is not. Now we are entitled to define semantics to deal with compatible sets:

Definition 11

(Semantics). Let $AF = (\mathcal A, \mathcal D, \mathcal D')$ be an $AF _2$ and $S \subseteq \mathcal A$ be a compatible set of arguments. Then $X \in \mathcal A$ is acceptable with respect to S iff

• $\forall Y \in \mathcal A$ such that $(Y,X) \in \mathcal D$ : $\exists Z \in S$ such that $(Z, Y ) \in \mathcal D$ and

• $\forall Y \in \mathcal A$ such that $(Y,X) \in \mathcal D'$ : $\exists Z \in S$ such that $(Z, Y ) \in \mathcal D \cup \mathcal D'$.

We define $f_ AF (S) = \left\{ A \in \mathcal A\mid A \text { is acceptable w.r.t. } S\right\}$. For a compatible set S in $AF$, we say 1) S is an admissible set of $AF$ iff $S \subseteq F_ AF (S)$; 2) S is a complete extension of $AF$ iff $f_ AF (S) = S$; 3) S is a preferred extension of $AF$ iff it is a set inclusion maximal complete extension of $AF$; 4) S is the grounded extension iff it is the set inclusion minimal complete extension of $AF$; 5) S is a stable extension iff S is complete extension of $AF$ and $\forall Y \not \in S$, $\exists X \in S$ s.t. $(X, Y ) \in \mathcal D \cup \mathcal D'$. 6) S is a semi-stable extension iff it is a complete extension of $AF$ such that there is no complete extension $S_1$ of $AF$ in which $S \cup S^+ \subset S_1 \cup S_1^+$.

Notice for an argument X to be acceptable w.r.t. S, if $(Y,X) \in \mathcal D$, there should exist an argument $Z \in S$ such that $(Z,Y) \in \mathcal D$, i.e., a weak defeat as $(Z,Y) \in \mathcal D'$ is not robust enough to defend a defeat as $(Y,X) \in \mathcal D$. Otherwise, X defends a weak defeat $(Y,X) \in \mathcal D'$ if $\exists Z \in S$ such that Z (weak) defeats Y.

Example 4

(Example 3 continued).

Regarding the $AF _2$ constructed in Example 3, we obtain

• Complete Extensions: $\left\{ A_1, A_3, A_4\right\}$, $\left\{ A_1, A_3, A_4, A_5\right\}$, $\left\{ A_1, A_3, A_4, A_6\right\}$, $\{A_1,A_3, A_4, A_5, A_6 \}$;

• Grounded Extension: $\left\{ A_1, A_3, A_4\right\}$;

• Preferred Extension: $\left\{ A_1, A_3, A_4, A_5, A_6\right\}$

• Stable/Semi-stable Extensions: $\left\{ A_1, A_3, A_4, A_6\right\}$, $\left\{ A_1, A_3, A_4, A_5, A_6\right\}$

3 The Postulates of Non-interference and Crash-Resistance

In this section, we show under which conditions, $ASPIC ^?$ satisfies the property that a conflict between two arguments should not interfere with the acceptability of other unrelated arguments. Let us illustrate it via the following example:

Example 5

[6] Let $\mathcal R_d= \left\{ p \Rightarrow q; r \Rightarrow \lnot q; t \Rightarrow s \right\}$ be a set of defeasible rules in $ASPIC ^+$ (their conclusions are not ?-suffixed formulas), $\mathcal K_p= \emptyset$ and $\mathcal K_n= \left\{ p, r, t\right\}$, while $\mathcal R_s$ consists of all propositionally valid inferences. The corresponding $AF$ includes the arguments $A_1 = [p]$, $A_2 = [ A_1 \Rightarrow q]$, $B_1 = [r]$, $B_2 = [B_1 \Rightarrow \lnot q]$, $C = [A_2, B_2 \rightarrow \lnot s]$, $D_1 = [t]$ and $D_2 = [D_1 \Rightarrow s]$. We have C defeats $D_2$ if $C \not \prec D_2$. This is problematic as s can be any formula. Hence, any defeasible argument unrelated to $A_2$ or $B_2$ can, depending on $\preceq$, be defeated by C owing to the explosiveness of classical logic as the source for $\mathcal R_s$.

This property is guaranteed by proving the postulates (originally conceived in [10]) of Non-interference (Definition 22) and Crash-Resistance (Definition 25) hold in $ASPIC ^?$. Unlike [7, 8], however, we will not eliminate all arguments whose set of conclusions of all its sub-arguments is contradictory, but only those whose set of conclusions contains strong contradictions as $\left\{ \phi , \lnot \phi \right\}$, $\left\{ \phi , \lnot \phi ?\right\}$ or $\left\{ \lnot \phi , \phi ?\right\}$. Weak contradictions as $\left\{ \phi ?, \lnot \phi ?\right\}$ will not lead to the elimination of the argument (see Definition 14). We proceed by introducing several definitions and lemmas before proving the satisfaction of these postulates:

Definition 12

(Consistency). Let $\mathcal A= \left\{ A_1, \ldots , A_n\right\}$ be a set of arguments on the basis of an argumentation theory $( AS , \mathcal K)$ and an argumentation system $AS = (\mathcal L,^-, \mathcal R, n)$. An argument $A \in \mathcal A$ is inconsistent iff $\forall \phi \in \mathcal L$, it holds $\left\{ \mathtt {Conc}(A') \mid A' \in \mathtt {Sub}(A)\right\} \vdash \phi$. Otherwise, A is consistent. The set $\mathcal A$ is inconsistent if $\forall \phi \in \mathcal L$, it holds $\mathtt {Concs}(\mathtt {Sub}(A_1)) \cup \ldots \cup \mathtt {Concs}(\mathtt {Sub}(A_n)) \vdash \phi$. Otherwise $\mathcal A$ is consistent.

A strict rule as $\phi _1, \ldots , \phi _n \rightarrow \psi$ represents that if $\phi _1, \ldots , \phi _n$ hold, then without exception it holds that $\psi$. It has a very general meaning; the unique restriction we will impose to prove our results is that we will assume throughout this section that every strict rule in an argumentation system is reasonable:

Definition 13

(Reasonable strict rules). Let $( AS , \mathcal K)$ be an argumentation theory and $AS = (\mathcal L,^-, \mathcal R_s\cup \mathcal R_d, n)$ be an argumentation system. A strict rule $\phi _1, \ldots , \phi _n \rightarrow \psi \in \mathcal R_s$ is reasonable iff 1) for each $\phi _i$ ($1 \le i \le n$) it holds $\texttt {Atoms}(\phi _i) \subseteq \texttt {Atoms}(\psi )$ or 2) $\forall \psi \in \mathcal L$, it holds $\left\{ \phi _1, \ldots , \phi _n\right\} \vdash \psi$.

A strict rule $\phi _1, \ldots , \phi _n \rightarrow \psi$ is reasonable if $\forall \phi _i$ ($1 \le i \le n$), each atom in $\phi _i$ is also in $\psi$ or $\left\{ \phi _1, \ldots , \phi _n\right\}$ is inconsistent. Reasonable strict rules are very usual in many propositional logics. Now we define an inconsistency cleaned $AF _2$:

Definition 14

(Inconsistency-cleaned $AF _2$). Let $\langle \mathcal A, \mathcal D, \mathcal D' \rangle$ be a $AF _2$ resulting from an argumentation theory $AT$. We define $\mathcal A_c = \{ A \in \mathcal A\mid A \text { is consistent} \}$, $\mathcal D_c = \mathcal D \cap (\mathcal A_c \times \mathcal A_c)$ and $\mathcal D_c' = \mathcal D' \cap (\mathcal A_c \times \mathcal A_c)$. We refer to $(\mathcal A_c, \mathcal D_c, \mathcal D_c')$ as the inconsistency cleaned $AF _2$ resulting from an argumentation theory $AT$.

By inconsistency-cleaned version of the $ASPIC ^?$ system, we mean the $ASPIC ^?$ system from which the inconsistency-cleaned $AF _2$ is constructed. The next concept is important to simplify the proofs of the results we will show in this section:

Definition 15

(Flat arguments). Let A be an argument on the basis of an argumentation theory $AT$. We say that A is flat iff $\mathtt {TopRule}(A)$ is strict, $A = [A_1, \ldots , A_n \rightarrow \alpha ]$ and $\forall A_i$ ($1 \le i \le n$), one of the following conditions holds:

• 1. $\mathtt {TopRule}(A_i)$ is defeasible or

• 2. $\mathtt {Rules}(A_i) = \emptyset$.

Flat arguments have strict top rule and every of its strict subarguments comes from the set of premises. Henceforth, for any $AF _2\langle \mathcal A, \mathcal D, \mathcal D' \rangle$ resulting from argumentation theory $AT$, we will assume without loss of generality every $A \in \mathcal A$ with a strict top rule is flat. In order to prove some results in this section, we will need to identify the set of conclusions associated with a set of arguments:

Definition 16

Let $\mathcal A$ be a set of arguments whose structure complies with Definition 4. We define $\mathtt {Concs}(\mathcal A) = \left\{ \mathtt {Conc}(A) \mid A \in \mathcal A\right\}$.

Next, we define the consequence function $Cn _ sem$, such that $Cn _ sem ( AT )$ is a set of sets of conclusions under the argumentation semantics $sem$.

Definition 17

Let $\mathfrak {AT}$ be the set of all argumentation theories that can be constructed from a language $\mathcal L$. Let $AT \in \mathfrak {AT}$ be an argumentation theory and $AF = (\mathcal A, \mathcal D, \mathcal D')$ be the resulting $AF _2$ from $AT$. We define $Cn _{ sem } : \mathfrak {AT} \rightarrow 2^{2^{\mathtt {Concs}(\mathcal A)}}$ is a function s.t. $Cn _{ sem }( AT ) = \{\mathtt {Concs}(E) \mid E \subseteq \mathcal A\textit{ is an extension} \textit{of } AF _2 \textit{ under semantics } sem \}$, where $sem \in \{\textit{complete, grounded, preferred,} \textit{stable, semi-stable}\}$. We will use $Cn _{\textit{c}}( AT )$ as the shortening of $Cn _{ complete }( AT )$. For a set $\mathfrak A$ of propositional atoms, by $Cn _{ sem }( AT )_{\mid \mathfrak A}$, we mean the set $\left\{ \mathcal F_{\mid \mathfrak A} \mid \mathcal F \in Cn _{ sem }( AT )\right\}$.

In order to define the postulate for non-interference, we need to specify what the union of two argumentation theories looks like:

Definition 18

(Union of argumentation theories). Let $AT _1 = ( AS _1, \mathcal K_1)$ and $AT _2 = ( AS _2, \mathcal K_2)$ be argumentation theories s.t. $\mathcal K_1 = \mathcal K_{n_1} \cup \mathcal K_{p_1}$, $\mathcal K_2 = \mathcal K_{n_2} \cup \mathcal K_{p_2}$, $AS _1 = (\mathcal L,^-, \mathcal R_{s_1} \cup \mathcal R_{d_1}, n_1)$ and $AS _2 = (\mathcal L,^-, \mathcal R_{s_2} \cup \mathcal R_{d_2}, n_2)$. Besides, we assume $n_1(r) = n_2(r)$ for any $r \in \mathcal R_{d_1} \cup \mathcal R_{d_2}$. We define $AT _1 \cup AT _2$ is an argumentation theory $AT = ( AS , \mathcal K)$ s.t. $\mathcal K= \mathcal K_n\cup \mathcal K_p$ with $\mathcal K_n= \mathcal K_{n_1} \cup \mathcal K_{n_2}$, and $\mathcal K_p= \mathcal K_{p_1} \cup \mathcal K_{p_2}$; $AS = (\mathcal L,^-, \mathcal R_s \cup \mathcal R_d, n)$ with $\mathcal R_s = \mathcal R_{s_1} \cup \mathcal R_{s_2}$, $\mathcal R_d = \mathcal R_{d_1} \cup \mathcal R_{d_2}$ and $n(r) = n_1(r)$ if $r \in \mathcal R_{d_1}$; otherwise, $n(r) = n_2(r)$.

Other very important notion to guarantee our results in this section is that of syntactically disjoint argumentation theories; it will be employed to characterise non-interference (Definition 22) and contamination (Definition 24).

Definition 19

(Syntactically disjoint argumentation theories). Let $AT _1$ and $AT _2$ be argumentation theories. We say $AT _1$ and $AT _2$ are syntactically disjoint when $\texttt {Atoms}( AT _1) \cap \texttt {Atoms}( AT _2) = \emptyset$.

The depth of an argument will be employed in the proof of Lemma 1:

Definition 20

(Depth of an argument). Let A be an argument whose structure complies with Definition 4. The depth of A, denoted by $\texttt {depth}(A)$, is 1 if $\mathtt {Rules}(A) = \emptyset$ or else $\texttt {depth}(A) = 1 + max \{\texttt {depth}(A') \mid A' \in \mathtt {Sub}(A) \}$.

The following essential lemma states that for every argument A such that $\mathtt {Conc}(A) \subseteq \texttt {Atoms}(AF)$ there exists an argument $A'$ with the same conclusion, $\texttt {Atoms}(A') \subseteq \texttt {Atoms}(AF)$, and is not more vulnerable than A.

Lemma 1

Let $AS _1 = (\mathcal L,^-, \mathcal R_s\cup \mathcal R_{d_1}, n_1)$ and $AS _2 = (\mathcal L,^-, \mathcal R_s\cup \mathcal R_{d_2}, n_2)$ be argumentation systems s.t. $\mathcal R_s$ is a set of strict rules and $\mathcal R_{d_1}$ and $\mathcal R_{d_2}$ are sets of defeasible rules. Let $AT = AT _1 \cup AT _2$ be an argumentation theory where $AT _1 = ( AS _1, \mathcal K_1)$ and $AT _2 = ( AS _2, \mathcal K_2)$ are syntactically disjoint, and $AF = (\mathcal A, \mathcal D, \mathcal D')$ and $AF _1 = (\mathcal A_1, \mathcal D_1, \mathcal D_1')$ be respectively the inconsistency cleaned $AF _2$s resulting from $AT$ and $AT _1$. For each argument $C \in \mathcal A$ such that $\mathtt {Conc}(C) \subseteq \texttt {Atoms}( AT _1)$, $\exists C' \in \mathcal A_1$ with $\mathtt {Conc}(C') = \mathtt {Conc}(C)$, $C'^+ \subseteq C^+$ and $C'^- \subseteq C^-$.

Proof

Let $\mathcal K_1 = \mathcal K_{n_1} \cup \mathcal K_{p_1}$ and $\mathcal K_2 = \mathcal K_{n_2} \cup \mathcal K_{p_2}$. We will prove by induction on $\texttt {depth}(C)$ that $\forall C \in \mathcal A$, where $\texttt {Atoms}(\mathtt {Conc}(C)) \subseteq \texttt {Atoms}( AT _1)$, $\exists C' \in \mathcal A_1$ such that $\mathtt {Conc}(C') = \mathtt {Conc}(C)$ and (1) $\mathtt {Concs}(\mathtt {Sub}(C')) \subseteq \mathtt {Concs}(\mathtt {Sub}(C))$ and (2) $\mathtt {DefR}(C') \subseteq \mathtt {DefR}(C)$. Note $C'^+ = C^+$ follows directly from $\mathtt {Conc}(C') = \mathtt {Conc}(C)$; condition (1) guarantees $C'$ is consistent as C is consistent; condition (2) suffices to show $C'^- \subseteq C^-$ as $AS _1$ and $AS _2$ share the same set $\mathcal R_s$ of strict rules:

Suppose $\texttt {depth}(C) = 1$. There are two possibilities: $C \in \mathcal K_{n_1}$ or $C \in \mathcal K_{p_1}$. Thus, for $C' = C$, $C'^- = C^-$ and $\mathtt {Conc}(C') = \mathtt {Conc}(C)$. Now assume conditions (1) and (2) hold for any argument C with $\texttt {depth}(C) \le k$. We will show for any argument C with $\texttt {depth}(C) = k + 1$ they also hold. There are two possibilities:

• $\mathtt {TopRule}(C) \in \mathcal R_d= \mathcal R_{d_1} \cup \mathcal R_{d_2}$. Note C is of the form $C_1, \ldots , C_n \Rightarrow \mathtt {Conc}(C)$. As $AT _1$ and $AT _2$ are syntactically disjoint and $\texttt {Atoms}(\mathtt {Conc}(C)) \subseteq \texttt {Atoms}( AT _1)$, $\mathtt {TopRule}(C) \in \mathcal R_{d_1}$. It follows that for each $i \in \left\{ 1, \ldots , n\right\}$, $\texttt {Atoms}(\mathtt {Conc}(C_i)) \subseteq \texttt {Atoms}(AT_1)$. As $\texttt {depth}(C_i) \le k$, by induction hypothesis, there exists $C'_i \in \mathcal A_1$ such that $\mathtt {Conc}(C'_i) = \mathtt {Conc}(C_i)$, $\mathtt {DefR}(C'_i) \subseteq \mathtt {DefR}(C'_i)$, and $\mathtt {Concs}(\mathtt {Sub}(C'_i)) \subseteq \mathtt {Concs}(\mathtt {Sub}(C_i))$. Applying $\mathtt {TopRule}(C)$ we construct $C' = C'_1, \ldots , C'_n \Rightarrow \mathtt {Conc}(C)$ from $AT_1$. Now we show that $C'$ satisfies the requested properties.

  • $C' \in \mathcal A_1$ since $\texttt {Atoms}(C') = \texttt {Atoms}(\mathtt {TopRule}(C)) \cup \bigcup ^n_{i=1} \texttt {Atoms}(C'_i) \subseteq \texttt {Atoms}( AT _1)$.

  • $\mathtt {Conc}(C') = \mathtt {Conc}(C)$ since C and $C'$ share the same top rule.

  • $\mathtt {DefR}(C') = \left\{ \left\{ \mathtt {DefR}(C)\right\} \cup \bigcup ^n_{i=1} \mathtt {DefR}(C'_i)\right\} \subseteq$ $\left\{ \left\{ \mathtt {TopRule}(C)\right\} \cup \bigcup ^n_{i=1} \mathtt {DefR}(C_i)\right\} = \mathtt {DefR}(C)$.

  • $\mathtt {Concs}(\mathtt {Sub}(C')) = \left\{ \left\{ \mathtt {Conc}(C)\right\} \cup \bigcup ^n_{i=1} \mathtt {Concs}(\mathtt {Sub}(C'_i))\right\} \subseteq$ $\left\{ \left\{ \mathtt {Conc}(C)\right\} \cup \bigcup ^n_{i=1} \mathtt {Concs}(\mathtt {Sub}(C_i))\right\} = \mathtt {Concs}(\mathtt {Sub}(C))$.

• $\mathtt {TopRule}(C) \in \mathcal R_s$. Note C is of the form $C_1, \ldots , C_n \rightarrow \mathtt {Conc}(C)$. As we have assumed C is flat, we can partition arguments $C_i$ into two sets $\mathfrak C_p \cup \mathfrak C_d = \left\{ 1, \ldots , n\right\}$, in which $i \in \mathfrak C_p$ iff $\mathtt {Conc}(C_i) \in \mathcal K_1 \cup \mathcal K_2$ and $\mathtt {TopRule}(C_i) = \textit{undefined}$, and $i \in \mathfrak C_d$ iff $\mathtt {TopRule}(C_i) \in \mathcal R_{d_1} \cup \mathcal R_{d_2}$. Since $AT _1$ and $AT _2$ are syntactically disjoint, for $i \in \mathfrak C_p$, $\texttt {Atoms}(\mathtt {Conc}(C_i)) \subseteq \texttt {Atoms}(AT_1)$ or $\texttt {Atoms}(\mathtt {Conc}(C_i)) \subseteq \texttt {Atoms}(AT_2)$, and for $i \in \mathfrak C_d$, $\texttt {Atoms}(\mathtt {TopRule}(C_i)) \subseteq \texttt {Atoms}(AT_1)$ or $\texttt {Atoms}(\mathtt {TopRule}(C_i)) \subseteq \texttt {Atoms}(AT_2)$. We can partition the subarguments of C into two disjoint sets $\mathfrak C_1$ and $\mathfrak C_2$ such that for $i \in \mathfrak C_1$, $\mathtt {Conc}(C_i) \subseteq \texttt {Atoms}(AT_1)$ and for $i \in \mathfrak C_2$, $\mathtt {Conc}(C_i) \subseteq \texttt {Atoms}(AT_2)$. Let $\mathfrak C_p \cap \mathfrak C_1 = \left\{ p_1, \ldots , p_k\right\}$, $\mathfrak C_d \cap \mathfrak C_1 = \left\{ d_1, \ldots , d_m\right\}$, and $\mathfrak C_2 = \left\{ b_1, \ldots , b_j\right\}$. For each $d_i \in \mathfrak C_d \cap \mathfrak C_1$, $\texttt {depth}(C_{d_i}) \le k$. By the induction hypothesis, for each $d_i \in \mathfrak C_d \cap \mathfrak C_1$, exists $C'_{d_i} \in \mathcal A_1$ s.t $\mathtt {Conc}(C'_{d_i}) = \mathtt {Conc}(C_{d_i})$, and $\mathtt {DefR}(C'_{d_i}) \subseteq \mathtt {DefR}(C_{d_i})$. Note that $\mathtt {Conc}(C_{p_1}), \ldots , \mathtt {Conc}(C_{p_k})$, $\mathtt {Conc}(C'_{d_1}), \ldots , \mathtt {Conc}(C'_{d_m}), \mathtt {Conc}(C_{b_1}),\ldots , \mathtt {Conc}(C_{b_j}) \rightarrow \mathtt {Conc}(C)$ corresponds to $\mathtt {TopRule}(C)$.

  Let $T = \texttt {Atoms}(\mathtt {Conc}(C_{b_1})) \cup \ldots \cup \texttt {Atoms}(\mathtt {Conc}(C_{b_j}))$. As $T \cap \texttt {Atoms}(\mathtt {Conc}(C))= \emptyset$, $\mathtt {Conc}(C_{p_1}),\ldots , \mathtt {Conc}(C_{p_n}), \mathtt {Conc}(C'_{d_1}), \ldots , \mathtt {Conc}(C'_{d_m}) \rightarrow \mathtt {Conc}(C) \in \mathcal R_s$; otherwise $\{\mathtt {Conc}(C_{b_1}), \ldots , \mathtt {Conc}(C_{b_j})\}$ is inconsistent (Definition 13), which cannot be true as C is consistent. Thus, we can construct an argument $C' = \mathtt {Conc}(C_{p_1}), \ldots , \mathtt {Conc}(C_{p_n}),\mathtt {Conc}(C'_{d_1}), \ldots , \mathtt {Conc}(C'_{d_m}) \rightarrow \mathtt {Conc}(C)$ from $AT_1$ s.t. $\mathtt {Conc}(C') = \mathtt {Conc}(C)$. Now we show $C'$ satisfies the requested properties.

  • $C' \in \mathcal A_1$ since $\texttt {Atoms}(C') = \texttt {Atoms}(\mathtt {TopRule}(C)) \cup \bigcup _{(1 \le i \le n)} \texttt {Atoms}(C'_i) \subseteq \texttt {Atoms}( AT _1)$.

  • $\mathtt {Conc}(C') = \mathtt {Conc}(C)$ since C and $C'$ share the same top rule.

  • In the construction of $C'$ we resort to $C_i$ for each $i \in \mathcal C_1 \cap \mathfrak C_p$ and for each $j \in \mathcal C_1 \cap \mathfrak C_d$, we resort to $C'_j$, obtained by induction hypothesis, s.t. $\mathtt {DefR}(C'_j) \subseteq \mathtt {DefR}(C_j)$ and $\mathtt {Concs}(\mathtt {Sub}(C_j')) \subseteq \mathtt {Concs}(\mathtt {Sub}(C_j))$. It follows $\mathtt {DefR}(C') \subseteq \mathtt {DefR}(C)$ and $\mathtt {Concs}(\mathtt {Sub}(C')) \subseteq \mathtt {Concs}(\mathtt {Sub}(C))$.    $\square$

The notion of defense expresses when an argument C defends B from A:

Definition 21

(Defense). Let $AF = (\mathcal A, \mathcal D, \mathcal D')$ be an $AF _2$ and $A, B, C \in \mathcal A$ such that $(A,B) \in \mathcal D \cup \mathcal D'$. An argument C defends B from A, denoted by $df (C,B,A)$, when 1) if $(A,B) \in \mathcal D$, then $(C,A) \in \mathcal D$; 2) if $(A,B) \in \mathcal D'$, then $(C,A) \in \mathcal D$ or $(C,A) \in \mathcal D'$.

In addition to the fundamental result obtained by Lemma 1, we will use Lemmas 2, 3, 5 and 6 to prove Theorem 1, which is one of our main results. With the following lemma we establish a connection between complete extensions of $AF$ with the arguments in $\mathcal A_1$:

Lemma 2

Let $AT = AT _1 \cup AT _2$ for syntactically disjoint argumentation theories $AT _1$ and $AT _2$, and $AF = (\mathcal A, \mathcal D, \mathcal D')$ and $AF _1 = (\mathcal A_1, \mathcal D_1, \mathcal D'_1)$ be respectively the resulting inconsistency-cleaned $AF _2$s from $AT$ and $AT _1$. For any complete extension $E \subseteq \mathcal A$ of $AF$, $\mathtt {Concs}(E \cap \mathcal A_1) = \mathtt {Concs}(E)_{\mid \texttt {Atoms}( AT _1)}$.

Proof

• If $\phi \in \mathtt {Concs}(E \cap \mathcal A_1)$, then $\exists A \in E \cap \mathcal A_1$ such that $\mathtt {Conc}(A) = \phi$. It follows $A \in E$ and $A \in \mathcal A_1$, and so $\phi \in \mathtt {Concs}(E)$ and $\phi \in \mathtt {Concs}(\mathcal A_1)$. As $\texttt {Atoms}(\mathtt {Conc}(\phi )) \subseteq \texttt {Atoms}( AT _1)$, it holds $\phi \in \mathtt {Concs}(E)_{\mid \texttt {Atoms}( AT _1)}$.

• If $\phi \in \mathtt {Concs}(E)_{\mid \texttt {Atoms}( AT _1)}$, then $\exists A \in E$ s.t. $\mathtt {Conc}(A) = \phi$ and $\texttt {Atoms}(\phi ) \subseteq \texttt {Atoms}( AT _1)$. From Lemma 1, $\exists A' \in \mathcal A_1$ such that $\mathtt {Conc}(A') = \phi$ and $A'^- \subseteq A^-$. As E is a complete extension of $AF$ and $A \in E$, it must be $A' \in E$, and so $A' \in E \cap \mathcal A_1$. Thus, $\phi \in \mathtt {Concs}(E \cap \mathcal A_1)$.    $\square$

Lemmas 3 and 4 will be used as intermediate steps in the demonstration of Lemma 5. Lemma 3 expresses the function $f_{ AF _1}$ associated with the subframework $AF _1$ in terms of the function $f_ AF$ associated with $AF$:

Lemma 3

Let $AT = AT _1 \cup AT _2$ for syntactically disjoint argumentation theories $AT _1$ and $AT _2$, and $AF = (\mathcal A, \mathcal D, \mathcal D')$ and $AF _1 = (\mathcal A_1, \mathcal D_1, \mathcal D'_1)$ be respectively the resulting inconsistency-cleaned $AF _2$s from $AT$ and $AT _1$. For any set of arguments $S \subseteq \mathcal A_1$, $f_{ AF _1}(S) = f_{ AF }(S) \cap \mathcal A_1$.

Proof

• $f_{ AF _1}(S) \subseteq f_{ AF }(S) \cap \mathcal A_1$. If $A \in f_{ AF _1}(S)$, then $A \in \mathcal A_1$. It remains to prove $A \in f_{ AF }(S)$: let $B \in \mathcal A$ s.t. $(B,A) \in \mathcal D \cup \mathcal D'$. It means $\texttt {Atoms}(\mathtt {Conc}(B)) \subseteq \texttt {Atoms}( AT _1)$. By Lemma 1, $\exists B' \in \mathcal A_1$ s.t. $\mathtt {Conc}(B') = \mathtt {Conc}(B)$ and $B'^- \subseteq B^-$. Thus, $(B',A) \in \mathcal D \cup \mathcal D'$. As $A \in f_{ AF _1}(S)$, $\exists C \in S$ s.t. $df (C,A,B')$. From $B'^- \subseteq B^-$, $(C,B) \in \mathcal D \cup \mathcal D'$. It follows $A \in f_{ AF }(S)$. Thus $A \in f_{ AF }(S) \cap \mathcal A_1$.

• $f_{ AF }(S) \cap \mathcal A_1 \subseteq f_{ AF _1}(S)$. Let $A \in f_{ AF }(S) \cap \mathcal A_1$. It means that $\forall B \in \mathcal A$ such that $(B,A) \in \mathcal D \cup \mathcal D'$, $\exists C \in S$ such that $df (C,A,B)$. As $\mathcal D_1 \cup \mathcal D'_1 \subseteq \mathcal D \cup \mathcal D'$, it follows $\forall B \in \mathcal A_1$ such that $(B,A) \in \mathcal D_1 \cup \mathcal D_1'$, $\exists C \in S$ such that $df (C,A,B)$. Then $A \in f_{ AF _1}(S)$.    $\square$

By Lemma 4 (employed in Lemma 5), if an argument A is attacked by a set of arguments that also attack arguments in a complete extension E, then $A \in E$.

Lemma 4

Let $AF = (\mathcal A, \mathcal D, \mathcal D')$ be the $AF _2$ resulting from an argumentation theory $AT$. Let $AF _c = (\mathcal A_c, \mathcal D_c, \mathcal D'_c)$ be the inconsistency-cleaned $AF _2$. Let E be a complete extension of $AF _c$, $S \subseteq E$ and $A \in \mathcal A$. If A is consistent and $A^- \subseteq S^-$, then $A \in E$.

Proof

Suppose A is consistent. It follows $A \in \mathcal A_c$. As E is a complete extension of $AF$, $\forall B \in S$, $B \in f_{ AF }(E)$. As $A^- \subseteq S^-$, it follows $A \in f_{ AF }(E) = E$.    $\square$

Lemmas 5 assures the complete extensions of $AF$ when restricted to the arguments of its subframework $AF _1$ is a complete extension of $AF _1$:

Lemma 5

Let $AT = AT _1 \cup AT _2$ for syntactically disjoint argumentation theories $AT _1$ and $AT _2$, and $AF = (\mathcal A, \mathcal D, \mathcal D')$ and $AF _1 = (\mathcal A_1, \mathcal D_1, \mathcal D'_1)$ be the inconsistency-cleaned $AF _2$s resulting from $AT$ and $AT _1$ respectively. If E is a complete extension of $AF$, then $E \cap \mathcal A_1$ is a complete extension of $AF _1$.

Proof

Let E be a complete extension of $AF$. Assume $E' = E \cap \mathcal A_1$. We will prove that $E'$ is a complete extension of $AF _1$. Note $E'$ is compatible in $AF _1$, since $E' \subseteq E$, E is compatible in $AF$ and there are no new defeats in $AF _1$. Now we will show $E' = f_{ AF _1}(E')$:

• $E' \subseteq f_{ AF _1}(E')$. Let $A \in E'$. As also $A \in E$ and E is a complete extension of $AF$, it means $\forall B \in \mathcal A_1$ such that $(B,A) \in \mathcal D_1 \cup \mathcal D_1'$, $\exists C \in E$ such that $df (C,A,B)$. As $B \in \mathcal A_1$, $\texttt {Atoms}(\mathtt {Conc}(C)) \subseteq \texttt {Atoms}( AT _1)$. By Lemma 1, $\exists C' \in \mathcal A_1$, such that $\mathtt {Conc}(C') = \mathtt {Conc}(C)$ and $C'^- \subseteq C^-$. From Lemma 4, $C' \in E$, and so $C' \in E'$. Thus, $A \in f_{ AF _1}(E')$.

• $f_{ AF _1}(E') \subseteq E'$. As f is a monotony function, and E is a complete extension of $AF$, $f_{ AF }(E') \subseteq f_{ AF }(E) = E$, and so $f_{ AF }(E') \cap \mathcal A_1 \subseteq E \cap \mathcal A_1$. From Lemma 3, $f_{ AF _1}(E') = f_{ AF }(E') \cap \mathcal A_1$. But then, we obtain $f_{ AF _1}(E') \subseteq E \cap \mathcal A_1$. Thus, $f_{ AF _1}(E') \subseteq E'$.    $\square$

Lemma 6 assures the admissible sets in $AF _1$ are also admissible sets in $AF$:

Lemma 6

Let $AT = AT _1 \cup AT _2$ for syntactically disjoint argumentation theories $AT _1$ and $AT _2$, and $AF = (\mathcal A, \mathcal D, \mathcal D')$ and $AF _1 = (\mathcal A_1, \mathcal D_1, \mathcal D'_1)$ be respectively the resulting inconsistency-cleaned $AF _2$s from $AT$ and $AT _1$. Let $S \subseteq \mathcal A_1$. If S is an admissible set in $AF _1$, then S is an admissible set if $AF$.

Proof

As S is a compatible set in $AF _1$, S is also a compatible set in $AF$, since $\mathcal A_1 \subseteq \mathcal A$. It remains to show that $S \subseteq f_{ AF }(S)$. Assume $B \in \mathcal A$ such that for some $A \in S$, $(B,A) \in \mathcal D \cup D'$. It means $\texttt {Atoms}(\mathtt {Conc}(B)) \subseteq \texttt {Atoms}( AT _1)$. From Lemma 1, $\exists B' \in \mathcal A_1$ such that $\mathtt {Conc}(B') = \mathtt {Conc}(B)$ and $B'^- \subseteq B^-$. Note $(B',A) \in \mathcal D \cup D'$ in $AF _1$. As S is admissible in $AF _1$, from Definition 11, $\exists C \in S$ s.t. $df(C,A,B')$. Given $B'^- \subseteq B^-$, $(C,B) \in \mathcal D \cup \mathcal D'$. It follows $S \subseteq f_{ AF }(S)$.    $\square$

In the sequel, we formally define the concept of non-interference.

Definition 22

(Non-interference). The $ASPIC ^?$ system satisfies non-interference under a semantics $sem$ iff for every syntactically disjoint argumentation theories $AT _1$ and $AT _2$, it holds $Cn _{ sem }( AT _1 \cup AT _2)_{\mid \texttt {Atoms}( AT _1)} = Cn _{ sem }( AT _1)$.

Non-interference means that, for disjoint argumentation theories $AT _1$ and $AT _2$, $AT _1$ does not influence the outcome with respect to the language of $AT _2$.

Theorem 1

The inconsistency-cleaned version of the $ASPIC ^?$ system satisfies non-interference under complete semantics.

Proof

Let $AT = AT _1 \cup AT _2$ for syntactically disjoint argumentation theories $AT _1$ and $AT _2$ and $AF = (\mathcal A, \mathcal D, \mathcal D')$ and $AF _1 = (\mathcal A_1, \mathcal D_1, \mathcal D'_1)$ be respectively the resulting inconsistency-cleaned $AF _2$s from $AT$ and $AT _1$. Let $\mathfrak S_1 = \left\{ B_1, \ldots , B_n\right\}$ and $\mathfrak S_2 = \left\{ S_1, \ldots , S_m\right\}$ be the set of complete extensions of $AF$ and $AF _1$ respectively. We will prove that $L = R$, in which $L = Cn _{\textit{c}}( AT )_{\mid \texttt {Atoms}( AT _1)} = \left\{ \mathtt {Concs}(B_1)_{\mid \texttt {Atoms}( AT _1)}, \ldots , \mathtt {Concs}(B_n)_{\mid \texttt {Atoms}( AT _1)}\right\}$, and $R = Cn _{\textit{c}}( AT _1)_{\mid \texttt {Atoms}( AT _1)}= \left\{ \mathtt {Concs}(S_1), \ldots , \mathtt {Concs}(S_m)\right\}$.

From Lemma 2, $L = \left\{ \mathtt {Concs}(B_1 \cap \mathcal A_1), \ldots , \mathtt {Concs}(B_n \cap \mathcal A_1)\right\}$. For each complete extension B of $AF$, $B \cap \mathcal A_1$ is a complete extension of $AF _1$ (Lemma 5). It remains to prove for any $S \in \mathfrak S_2$, $\exists B \in \mathfrak S_1$ with $B \cap \mathcal A_1 = S$. If S is a complete extension of $AF _1$ ($S = f_{ AF _1}(S)$), S is an admissible set in $AF$ (Lemma 6), i.e., $S \subseteq f_{ AF }(S)$. Then $B = \bigcup ^{\infty }_{n=1} f^n_{ AF }(S)$ is a complete extension of $AF$ as the least fixed point of $f_{ AF }$ contains S. We will prove that $B \cap \mathcal A_1 = S$. Intersecting both sides of $B = \bigcup ^{\infty }_{n=1} f^n_{ AF }(S)$ with $\mathcal A_1$, and applying Lemma 3, we get $B \cap \mathcal A_1 = (\bigcup ^{\infty }_{n=1} f^n_{ AF }(S)) \cap \mathcal A_1 = \bigcup ^{\infty }_{n=1} f^n_{ AF }(S) \cap \mathcal A_1 = \bigcup ^{\infty }_{n=1} f^n_{ AF _1}(S) = \bigcup ^{\infty }_{n=1} S = S$.    $\square$

$ASPIC ^?$ is non trivial under semantics $sem$ if the conclusions of an argumentation theory are never fully determined by the atoms.

Definition 23

(Non-trivial). The $ASPIC ^?$ system is non-trivial under semantics $sem$ iff for each nonempty set $\mathfrak A$ of atoms, there are argumentation theories $AT _1$ and $AT _2$ such that $\texttt {Atoms}( AT _1) = \texttt {Atoms}( AT _2)$ and $Cn _{ sem }( AT _1)_{\mid \mathfrak A} \ne Cn _{ sem }( AT _2)_{\mid \mathfrak A}$.

In the following theorem, we show that the inconsistency-cleaned version of the $ASPIC ^?$ system satisfies non-triviality under complete semantics:

Theorem 2

The inconsistency-cleaned version of the $ASPIC ^?$ system satisfies non-triviality under complete semantics.

Proof

Let $\mathfrak A = \left\{ a_1, \ldots , a_n\right\} (n \ge 1)$ a set of atoms. We will show that there are two inconsistency-cleaned argumentation frameworks $AF _1 = (\mathcal A_1,\mathcal D_1, \mathcal D_1')$ and $AF _2 = (\mathcal A_2,\mathcal D_2, \mathcal D'_2)$ resulting from $AT _1$ and $AT _2$ respectively such that $\texttt {Atoms}( AT _1)= \texttt {Atoms}( AT _2)$ and $Cn _{\textit{c}}( AT _1)_{\mid \mathfrak A} \ne Cn _{\textit{c}}( AT _2)_{\mid \mathfrak A}$. Let $\mathcal K_{n_1} = \mathcal K_{n_2} = \mathcal K_{p_2} = \emptyset$, $\mathcal K_{p_1} = \left\{ a_1, \ldots , a_n\right\}$, $\mathcal R_{d_2} = \left\{ a_1 \Rightarrow a_1?; \ldots ; a_n \Rightarrow a_n? \right\}$, $\mathcal R_{s_1} = \mathcal R_{s_2} = \mathcal R_{d_1} = \emptyset$. Thus, $Cn _{\textit{c}}( AT _1) = \left\{ \left\{ a_1, \ldots , a_n\right\} \right\}$ and $Cn _{\textit{c}}( AT _2) = \left\{ \emptyset \right\}$, and so $Cn _{\textit{c}}( AT _1) = Cn _{\textit{c}}( AT _1)_{\mid \mathfrak A} \ne Cn _{\textit{c}}( AT _2)_{\mid \mathfrak A} = Cn _{\textit{c}}( AT _2)$.   $\square$

An argumentation theory $AT _1$ is contaminating when any other unrelated argumentation theory $AT _2$ becomes irrelevant when merged with $AT _1$:

Definition 24

(Contamination). An argumentation theory $AT _1$ is contaminating under a semantics $sem$ iff for every argumentation theory $AT _2$ s.t. $AT _1$ and $AT _2$ are syntactically disjoint, it holds $Cn _{ sem }( AT _1) = Cn _{ sem }( AT _1 \cup AT _2)$.

Crash-resistance is strongly related to the concept of contamination:

Definition 25

(Crash-resistance). We say that $ASPIC ^?$ under a semantics $sem$ satisfies crash-resistance iff there does not exists an argumentation theory $AT$ that is contaminating under $sem$.

The intuition behind crash-resistance is that one wants to avoid local problems having global effects.

Theorem 3

If $ASPIC ^?$ satisfies non-interference and non-triviality under complete semantics, then it also satisfies crash-resistance under complete semantics.

Proof

(1) By absurd suppose the $ASPIC ^?$ does not satisfy crash-resistance. Then there exists an argumentation theory $AT _1$ that is contaminating and $\texttt {Atoms}( AT _1) \subset \mathfrak A$. Let $\mathfrak B = \mathfrak A \backslash \texttt {Atoms}( AT _1)$. (2) By assumption $ASPIC ^?$ is non-trivial. Thus, there are argumentation theories $AT _2$ and $AT _3$ such that $\texttt {Atoms}( AT _2) = \texttt {Atoms}( AT _3) \subseteq \mathfrak B$ and $Cn _\textit{c}( AT _2)_{\mid \mathfrak B} \not = Cn _\textit{c}( AT _3)_{\mid \mathfrak B}$. Note that both $AT _2$ and $AT _3$ are syntactically disjoint from $AT _1$. (3) By assumption $ASPIC ^?$ satisfies non-interference, from which follows $Cn _\textit{c}( AT _2)_{\mid \mathfrak B} = Cn _\textit{c}( AT _2 \cup AT _1)_{\mid \mathfrak B}$ and $Cn _\textit{c}( AT _3 \cup AT _1)_{\mid \mathfrak B} = Cn _\textit{c}( AT _3)_{\mid \mathfrak B}$. (4) Given $AT _1$ is contaminating, $Cn _\textit{c}( AT _1 \cup AT _2)_{\mid \mathfrak B} = Cn _\textit{c}( AT _1)_{\mid \mathfrak B} = Cn _\textit{c}( AT _1 \cup AT _3)_{\mid \mathfrak B}$. From (3) and (4), it follows that $Cn _\textit{c}( AT _2)_{\mid \mathfrak B} = Cn _\textit{c}( AT _3)_{\mid \mathfrak B}$. It is an absurd as from (2) we have $Cn _\textit{c}( AT _2)_{\mid \mathfrak B} \not = Cn _\textit{c}( AT _3)_{\mid \mathfrak B}$.   $\square$

Theorem 4

The inconsistency-cleaned version of the $ASPIC ^?$ system satisfies crash-resistance under complete semantics.

Proof

It follows from Theorems 2, 1 and 3.

4 Conclusion and Future Works

In this work, we defined an argumentation framework, dubbed $ASPIC ^?$, by introducing in $ASPIC ^+$ [3] an interrogation mark ? as a plausibility operator to enhance any defeasible conclusion does not have the same status than an irrefutable one: In $ASPIC ^?$, any defeasible rule have the form $\phi _1, \ldots , \phi _n \Rightarrow \phi ?$.

As in [2], we distinguish strong contradictions from weak ones. We avoid the former and tolerate the latter. Then, we showed in $ASPIC ^?$ conflicting arguments does not interfere with the acceptability of unrelated arguments. This is proved by combining solutions found in [6] and in [7, 8] to show the postulates of Non-interference and Crash-Resistance hold in inconsistency-cleaned $ASPIC ^?$: 1) as in [6], we resort to paraconsistent reasoning to tolerate conflicts; our differential is we tolerate only weak conflicts. 2) as in [7, 8], we require for each argument, the set of conclusions of all its sub-arguments are consistent; our differential is that we eliminate only those arguments whose sets of conclusions lead to a strong conflict. Thus, our work paves the way to investigate in the context of structured argumentation alternative solutions to satisfy the postulates of Non-interference and Crash-resistance without having to delete all inconsistent arguments.

In the future we will study other ways to satisfy these postulates and which monotonic paraconsistent logics can be used as source of strict rules to avoid contaminating argumentation theories. We will also exploit the relation between $ASPIC ^?$ and extended logic programas with paraconsistent semantics [13].