2.1 Nelson’s constructive logic

The concept of constructive falsity was introduced into logic by Nelson (Reference Nelson1949) and it is often denoted as $\textbf{N3}$ . It was first axiomatized by Vorob’ev (Reference Vorob’ev1952), and later studied by Markov (Reference Markov1953), who related intuitionistic and strong negation, and by Rasiowa (Reference Rasiowa1969), who provided an algebraic characterization. Versions of constructive logic without the “explosive” axiom $\varphi \to (\sim\!\varphi \to \psi)$ are usually denoted as $\textbf{N4}$ and they are based on a four valued assignment for each world corresponding to the values unknown, (constructively) true, (constructively) false and inconsistent (or overdetermined). The logic $\textbf{N3}$ can be obtained by adding back the “explosive” axiom. We describe next a Kripke semantics for a version of $\textbf{N4}$ (Thomason Reference Thomason1969; Gurevich Reference Gurevich1977) with the falsity constant $\bot$ , which is denoted as $\textbf{N4}^\bot$ by Odintsov and Rybakov (Reference Odintsov and Rybakov2015). We follow here an approach with two forcing relations in the style of the work by Akama (Reference Akama1987). An alternative characterization using 2-valued assignments plus an involution has been described by Routley (Reference Routley1974).

Syntactically, we assume a logical language with a strong negation connective “ $\sim\!$ ”. That is, given some (possibly infinite) set of atoms At, a formula ${\varphi}$ is defined using the grammar:

\begin{equation*}{\varphi} \quad ::= \quad \bot \ \mid \ a \ \mid \ \sim\! {\varphi} \ \mid \ {\varphi} \wedge {\varphi} \ \mid \ {\varphi} \vee {\varphi} \ \mid \ {\varphi} \to {\varphi},\end{equation*}

with $a \in At$ . We use Greek letters ${\varphi}$ and ${\psi}$ and their variants to stand for propositional formulas. Intuitionistic negation is defined as ${\neg {\varphi} \underline{\underline{\mathrm{def}}} ({\varphi} \to \bot)}$ . We also define the derived operators ${{\varphi} \leftrightarrow {\psi} \underline{\underline{\mathrm{def}}} ({\varphi} \to {\psi}) \wedge ({\psi} \to {\varphi})}$ and ${\top \underline{\underline{\mathrm{def}}} \sim\! \bot}$ .

A Kripke frame ${\mathcal{F}} = {\langle W,\leq \rangle}$ is a pair where W is a nonempty set of worlds and $\leq$ is a partial order on W. A valuation ${V : W \longrightarrow 2^{At}}$ is a function mapping each world to a subset of atoms. A Nelson’s interpretation (N-interpretation) is a 3-tuple ${{\mathcal{I}} = {\langle {\mathcal{F}},V^+,V^- \rangle}}$ where ${{\mathcal{F}} = {\langle W,\leq \rangle}}$ is a Kripke frame and where both $V^+$ and $V^-$ are valuations satisfying, for every pair of worlds $w,w' \in W$ with $w \leq w'$ and every atom $a \in At$ , the following preservation properties:

1. (i) $V^+(w) \subseteq V^+(w')$ , and

2. (ii) $V^-(w) \subseteq V^-(w')$ .

Intuitively, $V^+$ represents our knowledge about constructive truth while $V^-$ represents our knowledge about constructive falsity. We say that ${\mathcal{I}}$ is consistent if, in addition, it satisfies:

1. (iii) $V^+(w) \cap V^-(w) = \varnothing$ for every world $w \in W$ .

Two forcing relations $\models^+$ and $\models^-$ are defined with respect to any N-interpretation ${{\mathcal{I}} = {\langle {\mathcal{F}},V^+,V^- \rangle}}$ , world $w \in W$ and atom $a \in At$ as follows:

\begin{eqnarray*}{\mathcal{I}}, w \models^+ a &\text{ iff }& a \in V^+(w),\\{\mathcal{I}}, w \models^- a &\text{ iff }& a \in V^-(w).\end{eqnarray*}

These two relations are extended to compounded formulas as follows:

\begin{eqnarray*}{\mathcal{I}}, w \not\models^+ \bot\\{\mathcal{I}}, w \models^+ \varphi_1 \wedge \varphi_2 &\text{ iff }& {\mathcal{I}}, w \models^+ \varphi_1 \text{ and } {\mathcal{I}}, w \models^+ \varphi_2\\{\mathcal{I}}, w \models^+ \varphi_1 \vee \varphi_2 &\text{ iff }& {\mathcal{I}}, w \models^+ \varphi_1 \text{ or } {\mathcal{I}}, w \models^+ \varphi_2\\{\mathcal{I}}, w \models^+ \varphi_1 \!\to\! \varphi_2 &\text{ iff }& \forall w'\!\geq\! w \ {\mathcal{I}}, w' \!\not\models^+\! \varphi_1 \text{ or } {\mathcal{I}}, w' \!\models^+\! \varphi_2\\{\mathcal{I}}, w \models^+ \sim\!\varphi &\text{ iff }& {\mathcal{I}}, w \models^-\varphi\\{\mathcal{I}}, w \models^- \bot\\{\mathcal{I}}, w \models^- \varphi_1 \wedge \varphi_2 &\text{ iff }& {\mathcal{I}}, w \models^- \varphi_1 \text{ or } {\mathcal{I}}, w \models^- \varphi_2\\{\mathcal{I}}, w \models^- \varphi_1 \vee \varphi_2 &\text{ iff }& {\mathcal{I}}, w \models^- \varphi_1 \text{ and } I, w \models^- \varphi_2\\{\mathcal{I}}, w \models^- \varphi_1 \!\to\! \varphi_2 &\text{ iff }& {\mathcal{I}}, w \models^+ \varphi_1 \text{ and } {\mathcal{I}}, w \models^- \varphi_2\\{\mathcal{I}}, w \models^- \sim\!\varphi &\text{ iff }& {\mathcal{I}}, w \models^+\varphi.\end{eqnarray*}

An N-interpretation is said to be an N-model of a formula $\varphi$ , in symbols ${\mathcal{I}} \models^+ \varphi$ , iff ${\mathcal{I}} , w \models^+ \varphi$ for every $w \in W$ . It is said to be N-model of a theory $\Gamma$ , in symbols also ${{\mathcal{I}} \models^+ \Gamma}$ , iff it is an N-model of all its formulas ${{\mathcal{I}} \models^+ \varphi}$ . A formula $\varphi$ is said to be a consequence of a theory $\Gamma$ iff every model of $\Gamma$ is also a model of $\varphi$ , that is ${\mathcal{I}} \models^+ \varphi$ for every ${\mathcal{I}} \models^+ \Gamma$ . This formalization characterizes $\textbf{N4}$ while a restriction to consistent N-interpretation s would characterize $\textbf{N3}$ . As mentioned above, $\textbf{N4}$ is “somehow” paraconsistent in the sense that a formula $\varphi$ and its strongly negated counterpart $\sim\!\varphi$ may simultaneously be consequences of some theory: for instance, we have that ${\{a, \sim\! a\}} \models^+ a$ and ${\{a, \sim\! a\}} \models^+ \sim\! a$ . Intuitively, these two forcing relations determine the four values above mentioned: a formula $\varphi$ satisfying ${\mathcal{I}} \not\models^+ \varphi$ and ${\mathcal{I}} \not\models^- \varphi$ is understood as unknown. If it satisfies ${\mathcal{I}} \models^+ \varphi$ and ${\mathcal{I}} \not\models^- \varphi$ , is understood as true. False if ${\mathcal{I}} \not\models^+ \varphi$ and ${\mathcal{I}} \models^- \varphi$ , and inconsistent if ${\mathcal{I}} \models^+ \varphi$ and ${\mathcal{I}} \models^- \varphi$ .

2.2 Logic programming, equilibrium logic, and here-and-there Nelson’s models

In order to accommodate logic programming conventions, we will indistinctly write ${\varphi \leftarrow \psi}$ instead of ${\psi \to \varphi}$ when describing logic programs. An explicit literal is either an atom $a \in At$ or an atom preceded by strong negation $\sim\! a$ . A literal is either an explicit literal l or an explicit literal preceded by intuitionistic negation $\neg l$ . A literal that contains intuitionistic negation is called negative. Otherwise, it is called positive. A rule is a formula of the form $H \leftarrow B$ where H is a disjunction of atoms and B is a conjunction of literals. A logic program $\Pi$ is a set of rules.

Given some set of explicit literals ${\mathbf{T}}$ and some formula $\varphi$ , we write ${{\mathbf{T}} \models^+ \varphi}$ when ${\langle {\mathcal{F}},V^+,V^- \rangle} \models^+ \varphi$ holds for the Kripke frame ${\mathcal{F}}$ with a unique world w and valuations: $V^+(w) = {\mathbf{T}} \cap At$ and $V^-(w) = {\{\ { a }\ \big|\ { \sim\! a \in {\mathbf{T}}}\ \}}$ . A set of explicit literals ${\mathbf{T}}$ is said to be closed under $\Pi$ if ${\mathbf{T}} \models^+ H \leftarrow B$ for every rule $H \leftarrow B$ in $\Pi$ .

Next, we recall the notions of reduct and answer set (Gelfond and Lifschitz Reference Gelfond and Lifschitz1991):

1. i. Remove all rules with $\neg l$ in the body s.t. $l \in {\mathbf{T}}$ ,

2. ii. Remove all negative literals for the remaining rules.

Set ${\mathbf{T}}$ is a stable model of $\Pi$ if ${\mathbf{T}}$ is a $\subseteq$ -minimal closed set under $\Pi$ .

For characterizing logic programs in constructive logic, we are only interested in a particular kind of N-interpretation s over Here-and-There (HT) frames. These frames are of the form ${\mathcal{F}_{HT}} = {\langle {\{h,t\}}, \leq \rangle}$ where $\leq$ is a partial order satisfying $h \leq t$ . We refer to N-interpretation s with an HT-frame as HT-interpretation s. A HT-model is an N-model which is also a HT-interpretation. We use the generic terms interpretation (resp. model) for both HT and N-interpretations (resp. models) when it is clear by the context. At first sight, it may look that restricting ourselves to HT frames is an oversimplification. However, once the closed world assumption is added to intuitionistic logic, this logic can be replaced without loss of generality by any proper intermediate logic (Osorio et al. Reference Osorio, PÉrez and Arrazola2005; Cabalar et al. Reference Cabalar, Fandinno, FariÑas del Cerro, Pearce and Valverde2017).

Given any HT-interpretation, ${\mathcal{I}} = {\langle {\mathcal{F}_{HT}},V^+,V^- \rangle}$ , we define four sets of atoms as follows:

\begin{eqnarray*}H_{\mathcal{I}}^+ &\underline{\underline{\text{def}}}& V^+(h)\\H_{\mathcal{I}}^- &\underline{\underline{\text{def}}}& V^-(h)T_{\mathcal{I}}^+ &\underline{\underline{\text{def}}}& V^+(t)\\T_{\mathcal{I}}^- &\underline{\underline{\text{def}}}& V^-(t)\end{eqnarray*}

These sets of atoms correspond to the atoms verified at each corresponding world and valuation. Every HT-interpretation ${\mathcal{I}}$ is fully determined by these four sets. We will omit the subscript and write, for instance, $H^+$ instead of $H^+_{\mathcal{I}}$ when ${\mathcal{I}}$ is clear from the context. Furthermore, any HT-interpretations can be succinctly rewritten as a pair ${\mathcal{I}} = {\langle {\mathbf{H}},{\mathbf{T}} \rangle}$ where ${\mathbf{H}} = H^+ \cup \sim\! H^-$ and ${\mathbf{T}} = T^+ \cup \sim\! T^-$ are sets of literals. ^{Footnote 1} Note that, by the preservation properties of N-interpretations, we have that ${{\mathbf{H}} \subseteq {\mathbf{T}}}$ . We say that an HT-interpretation ${\mathcal{I}} = {\langle {\mathbf{H}},{\mathbf{T}} \rangle}$ is total iff ${\mathbf{H}} = {\mathbf{T}}$ . Given HT-interpretations ${\mathcal{I}} = {\langle {\mathbf{H}},{\mathbf{T}} \rangle}$ and ${\mathcal{I}}' = {\langle {\mathbf{H}}',{\mathbf{T}}' \rangle}$ , we write ${\mathcal{I}} \leq {\mathcal{I}}'$ iff ${\mathbf{H}} \subseteq {\mathbf{H}}'$ and ${\mathbf{T}} = {\mathbf{T}}'$ . As usual, we write ${\mathcal{I}} < {\mathcal{I}}'$ iff ${\mathcal{I}} \leq {\mathcal{I}}'$ and ${\mathcal{I}} \neq {\mathcal{I}}'$ .

Next, we introduce the definition of equilibrium model (Pearce Reference Pearce1996).

Interestingly, consistent equilibrium models precisely capture the answer set of a logic program. The following is a rephrase of Proposition 2 by Pearce (Reference Pearce1996) using our notation.

More in general, it has been shown by Odintsov and Pearce (Reference Odintsov and Pearce2005) that the (possible nonconsistent) equilibrium models of a logic program capture its paraconsistent answer sets (Sakama and Inoue Reference Sakama and Inoue1995).

The following propositions characterizes some interesting properties of HT and strong negation that will be useful through the paper ^{Footnote 2} :

1. 1. $I, w \models^+ \varphi$ implies $I, t \models^+ \varphi$ , and

2. 2. $I, w \models^- \varphi$ implies $I, t \models^- \varphi$ .

1. i. ${\mathcal{I}}, w \models^+ \neg\varphi$ iff ${\mathcal{I}}, t \not\models^+ \varphi$ , and

2. ii. ${\mathcal{I}}, w \models^+ \neg\neg\varphi$ iff ${\mathcal{I}}, t \models^+ \varphi$ , and

3. iii. ${\mathcal{I}}, w \models^+ \neg\neg\neg\varphi$ iff ${\mathcal{I}}, w \models^+ \neg\varphi$ , and

4. iv. ${\mathcal{I}}, w \models^- \neg\varphi$ iff ${\mathcal{I}}, w \models^- \sim\!\varphi$ .

2.3 Abstract argumentation frameworks

Since their introduction, the syntax of AFs have been extended in different ways. One of these extensions, usually called SETAFs, consists in generalizing the notion of binary attacks to collective attacks such that a set of arguments B attacks some argument a (Nielsen and Parsons 2007). Another such extension, usually called Bipolar AFs (BAFs), consists in frameworks with a second positive relation called support (Karacapilidis and Papadias Reference Karacapilidis and Papadias2001; Verheij Reference Verheij2003a; Amgoud et al. Reference Amgoud, Cayrol and Lagasquie-Schiex2004). In particular, Verheij (Reference Verheij2003b) introduced the idea that, in AFs, arguments are considered as prima-facie justified statements, which can be considered true until proved otherwise, that is, until they are defeated. This allows introducing a second class of ordinary arguments, which cannot be considered true unless get supported by the prima-facie ones. Later, Polberg and Oren (Reference Polberg and Oren2014) developed this idea by introducing Evidence-Based AFs (EBAFs), an extension of SETAFs (and, this, of AFs) which incorporates the notions of support and prima-facie arguments. Next we introduce an equivalent definition by Cayrol et al. (Reference Cayrol, Fandinno, FariÑas del Cerro and Lagasquie-Schiex2018), which is closer to the logic formulation we pursue here.

The notion of acceptability is extended by requiring not only defense against all attacking arguments, but also support from some prima-facie arguments. Furthermore, the defense can be provided not only by defeating all attacking sets of arguments, but also by denying the necessary support for some of the non-prima-facie arguments of these attacks.

1. 1. a is defeated w.r.t. E iff there is some $B \subseteq E$ s.t. $(B,a) \in \mathbf{R}_a$ , Def(E) will denote the set of arguments that are defeated w.r.t. E.

2. 2. a is supported w.r.t. E iff either $a \in \mathbf{P}$ or there is some $B \subseteq E \setminus{\{a\}}$ whose elements are supported w.r.t. $E \setminus{\{a\}}$ and such that $(B,a) \in \mathbf{R}_s$ ,

3. 3. a is supportable w.r.t. E iff it is supported w.r.t. $\mathbf{A} \setminus {Def}({E})$ ,

4. 4. a is unacceptable w.r.t. E iff it is either defeated or not supportable,

5. 5. a is acceptable w.r.t. E iff it is supported and, for every $(B,a) \in \mathbf{R}_a$ , there is $b \in B$ such that b is unacceptable w.r.t. E

Sup(E) (resp. UnAcc(E) and Acc(E)) will denote the set of arguments that are supported (resp. unacceptable and acceptable) w.r.t. E.

Then, semantics are defined as follows:

1. 1. self-supporting iff $E \subseteq {Sup}({E})$ ,

2. 2. conflict-free iff $E \!\cap\! {Def}({E}) \!=\! \varnothing$ ,

3. 3. admissible iff it is conflict-free and $E \subseteq {Acc}({E})$ ,

4. 4. complete iff it is conflict-free and $E = {Acc}({E})$ ,

5. 5. preferred iff it is a $\subseteq$ -maximal admissible set,

6. 6. stable iff $E = \mathbf{A} \setminus {Un}{Acc}({E})$ .

SETAFs can be seen as special cases where the set of supports is empty and all arguments are prima-facie. In this sense, we write $\mathbf{SF}= {\langle \mathbf{A},\mathbf{R}_a \rangle}$ instead $\mathbf{EF} = {\langle \mathbf{A},\mathbf{R}_a,\varnothing,\mathbf{A} \rangle}$ . Furthermore, in their turn, AFs can be seen as a special case of SETAFs where all attacks have singleton sources. In such case, we just write ${\mathbf{AF}= {\langle \mathbf{A},\mathbf{R} \rangle}}$ instead $\mathbf{SF}= {\langle \mathbf{A},\mathbf{R}_a \rangle}$ , where $\mathbf{R} = {\{\ { (b,a) }\ \big|\ { ({\{b\}},a) \in \mathbf{R}_a }\ \}}$ For this kind of frameworks, the respective notions of conflict-free (resp. admissible, complete, preferred or stable) coincide with those being defined by Nielsen and Parsons (2007) and Dung (Reference Dung1995), respectively.

To illustrate the notions of support and prima-facie arguments, consider the well-known Tweety example:

1. 1. birds (normally) can fly,

2. 2. penguins are birds,

3. 3. penguins cannot fly and

4. 4. Tweety is a penguin.

We can formalize this by the following graph:

where pT, bT and fT respectively stand for “Tweety is a penguin”, “Tweety is a bird” and “Tweety can fly.” Double arrows represent support while simple ones represent attacks. Furthermore, circles with solid border represent prima-facie arguments while dashed border ones represent ordinary ones. That is, “Tweety is a penguin” is considered a prima-facie argument that supports that “Tweety is a bird” which, in its turn, supports that “Tweety can fly.” The latter is then considered also prima-facie, that is, true unless proven otherwise. Note that “Tweety is a penguin” also attacks that “Tweety can fly”, so the latter cannot be accepted as true. Formally, this corresponds to the framework $\mathbf{EF}_1 ={\langle \mathbf{A},\mathbf{R}_a,\mathbf{R}_s,\mathbf{P} \rangle}$ with $\mathbf{R}_a = {\{ ({\{pT\}},fT) \}}$ and $\mathbf{R}_s = {\{ ({\{pT\}},bT), \, ({\{bT\}},fT) \}}$ and $\mathbf{P} = {\{ pT \}}$ whose unique admissible, complete, preferred and stable extension is ${\{pT , bT\}}$ . In other words, we conclude that “Tweety cannot fly.” Note that “Tweety is a penguin” provides conflicting evidence for whether it can fly or not. In EBAFs, this is solved by giving priority to the attack relation, so “Tweety cannot fly” is inferred.

3.1 A conservative extension of logic programming

Let us now consider the language formed with the set of logical connectives

\begin{equation*}\mathcal{C}_{LP} \underline{\underline{\mathrm{def}}} {\{ \bot, \sim\!, \wedge, \vee, {\Rightarrow}, {not}\, \}}.\end{equation*}

In other words, a $\mathcal{C}_{LP}$ -formula ${\varphi}$ is defined using the following grammar:

\begin{equation*}{\varphi} \quad ::= \quad \bot \ \mid \ a \ \mid \ \sim\! {\varphi} \ \mid \ {not}\, \varphi \ \mid \ {\varphi} \wedge {\varphi} \ \mid \ {\varphi} \vee {\varphi} \ \mid \ {\varphi} {\Rightarrow} {\varphi},\end{equation*}

with $a \in At$ being an atom. A $\mathcal{C}_{LP}$ -literal is either an explicit literal l or is default negation ${not}\, l$ . A $\mathcal{C}_{LP}$ -rule is a formula of the form $H {\Leftarrow} B$ where H is a disjunction of atoms and B is a conjunction of $\mathcal{C}_{LP}$ -literals. $\mathcal{C}_{LP}$ -theories and $\mathcal{C}_{LP}$ -programs are respectively defined as sets of $\mathcal{C}_{LP}$ -formulas and $\mathcal{C}_{LP}$ -rules. The definition of an answer set is applied straightforwardly as in Definition 1. Given any theory $\mathcal{C}_{LP}$ -theory $\Gamma$ , by $\mathcal{C}_{N}({\Gamma})$ we denote the result of

1. 1. replacing every occurrence of ${\Rightarrow}$ by $\to$ and

2. 2. and every occurrence of ${not}\, $ by $\neg$ .

Then, the following results follow directly from Propositions 4 and 5:

In other words, the equilibrium models semantics are a conservative extension of the answer set semantics. The following example shows the usual representation of the Tweety scenario in this logic (an alternative representation using contradictory evidence will be discussed in the Discussion section).

Example 5 (Ex. 1 continued) Consider again the Tweety scenario. The following logic program $P_9$ is a usual way of representing this scenario in LP:

(4) \begin{eqnarray}&{flyTweety} &{\Leftarrow}& {birdTweety} \wedge {not}\, \sim\!{flyTweety,} \end{eqnarray}

(5) \begin{eqnarray}&{birdTweety} &{\Leftarrow}& {penguinTweety,}\end{eqnarray}

(6) \begin{eqnarray}\sim\!\,&{flyTweety} &{\Leftarrow}& {penguinTweety}\end{eqnarray}

\begin{eqnarray}&{{penguinTweety,}} \notag\end{eqnarray}

where rule (4) formalizes the statement “birds normally can fly.” This is achieved by considering $\sim\!{flyTweety}$ as an exception to this rule. It can be checked that $P_{9}$ has a unique equilibrium model ${\mathcal{I}}_{9}$ , which is consistent, and which satisfies ${\mathcal{I}}_{9} \not\models {flyTweety}$ and ${\mathcal{I}}_{9} \models {not}\, {flyTweety}$ . In other words, Tweety cannot fly.

Example 6 (Ex. 3 continued) Consider now the theory obtained by replacing formulas $a {\Rightarrow} c$ and $b {\Rightarrow} d$ in $\Gamma_2$ by the following two formulas:

\begin{eqnarray*}{not}\, e \wedge a &{\Rightarrow}& c&{not}\, e \wedge b &{\Rightarrow}& d.\end{eqnarray*}

Let $\Gamma_10$ be such theory. It is easy to see that neither $\Gamma_10$ nor $\mathcal{C}_{N}({\Gamma_10})$ monotonically entail c nor d. This is due to the fact that the negation of e is not monotonically entailed: $\Gamma_10 \not\models^+ {not}\, e$ and $\mathcal{C}_{N}({\Gamma_10}) \not\models^+ \neg e$ . On the other hand, the negation of e is non-monotonically entailed in both cases: $\Gamma_10 \models {not}\, e$ and $\mathcal{C}_{N}({\Gamma_10}) \models \neg e$ . Note that both $\Gamma_10$ and $\mathcal{C}_{N}({\Gamma_10})$ have a unique equilibrium model, ${\mathcal{I}}_{10} = {\langle {\mathbf{T}},{\mathbf{T}} \rangle}$ and ${\mathcal{I}}_{10}' = {\langle {\mathbf{T}}',{\mathbf{T}}' \rangle}$ with ${\mathbf{T}} = {\{a,b ,\sim\! b, c\}}$ and ${\mathbf{T}}' = {\{a,b, \sim\! b, c, d\}}$ , respectively, and in both cases we have ${\mathcal{I}}_{10} \models {not}\, e$ and ${\mathcal{I}}_{10}' \models \neg e$ . As a result, we get that both theories cautiously entail c. However, as happened in Example 3, only $\mathcal{C}_{N}({\Gamma_10})$ cautiously entails d, because the unique evidence for d comes from b for which we have inconsistent evidence. This behavior is different from paraconsistent answer sets (Sakama and Inoue Reference Sakama and Inoue1995; Odintsov and Pearce Reference Odintsov and Pearce2005). As pointed out by Sakama and Inoue (Reference Sakama and Inoue1995), the truth of d is less credible than the truth of c, since d is derived through the contradictory fact b. In order to distinguish such two facts Sakama and Inoue (Reference Sakama and Inoue1995), also define suspicious answer sets which do not consider d as true. ^{Footnote 4}

This example also helps us to illustrate the strengthened closed world assumption principle CW. On the one hand, we have that $\Gamma_10 \models {not}\, e$ holds because there is no evidence for e. On the other hand, we have that $\Gamma_10 \models {not}\, b$ holds because we have contradictory evidence for b. Moreover, we have that $\Gamma_10 \models {not}\, d$ holds because the only evidence we have for d is based on the contradictory evidence for b.

4.1 Dung’s argumentation frameworks

Now, let us formalize the notion of attack introduced in AT, by defining the following connective:

(7) \begin{gather}\varphi_1 {\leadsto} \varphi_2 \ \ \underline{\underline{\mathrm{def}}} \ \ \varphi_1 {\Rightarrow} \sim\!\varphi_2.\end{gather}

Here, we identify the acceptability of $\varphi_1$ with having a consistent proof of it, or in other words, as having a proof of the truth of $\varphi_1$ and not having a proof of its falsity. Then, (7) states that the acceptability of $\varphi_1$ allows to construct a proof of the falsity of $\varphi_2$ . In this sense, we identify a proof of the falsity of $\varphi_2$ with $\varphi_2$ being defeated.

1. i. ${\mathcal{I}} \models \varphi_1$ and ${\mathcal{I}} \models^+ \varphi_1 {\leadsto} \varphi_2$ imply ${\mathcal{I}} \models^- \varphi_2$

Using the language $\mathcal{C}_{AF} = {\{{\leadsto}\}}$ , we can translate any AF as follows:

Definition 6 Given some framework $\mathbf{AF}= {\langle \mathbf{A},\mathbf{R} \rangle}$ , we define the theory:

(8) \begin{eqnarray}{ l ,C, l}\mathcal{C}_{AF}({\mathbf{AF}}) &\underline{\underline{\mathrm{def}}}& \mathbf{A} \cup {\{\ { a {\leadsto} b }\ \big|\ { (a,b) \in \mathbf{R} }\ \}}. \end{eqnarray}

In addition, we assign a corresponding set of arguments ${E_{\mathcal{I}}} \underline{\underline{\mathrm{def}}} {\{\ {\! a \in \mathbf{A} \!}\ \big|\ {\! {\mathcal{I}} \models a \!}\ \}}$ to every interpretation ${\mathcal{I}}$ .

Translation $\mathcal{C}_{AF}({\cdot})$ applies the notion of attack introduced in AT to translate an AF into a logical theory. The strengthened close world assumption CW is used to retrieve the arguments ${E_{\mathcal{I}}}$ corresponding to each stable model ${\mathcal{I}}$ of the logical theory obtained from this translation.

Then, we have that $\mathcal{C}_{AF}({\mathbf{AF}_11})$ is the theory containing the following two attacks:

\begin{gather*}a {\leadsto} b b {\leadsto} c,\end{gather*}

plus the facts ${\{a,b,c\}}$ .

1. i. f a is defeated w.r.t. ${E_{\mathcal{I}}}$ , then ${\mathcal{I}} \models^+ \sim\! a$

2. ii. ${E_{\mathcal{I}}}$ is conflict-free.

If, in addition, ${\mathcal{I}}$ is an $\leq$ -minimal model, then

• (iii) a is defeated w.r.t. ${E_{\mathcal{I}}}$ iff ${\mathcal{I}} \models^+ \sim\! a$ .

In fact, we can generalize this correspondence between the stable extensions and the equilibrium models to any argumentation framework as stated by the following theorem:

1. i. if ${\mathcal{I}}$ is an equilibrium model of $\mathcal{C}_{AF}({\mathbf{AF}})$ , then ${E_{\mathcal{I}}}$ is a stable extension of $\mathbf{AF}$ ,

2. ii. if E is a stable extension of $\mathbf{AF}$ and ${\mathcal{I}}$ is a total interpretation such that $T_{\mathcal{I}}^+ \!=\! \mathbf{A}$ and $T_{\mathcal{I}}^- \!=\! {Def}({E})$ , then ${\mathcal{I}}$ is an equilibrium model of $\mathcal{C}_{AF}({\mathbf{AF}})$ .

Proof sketch. ^{Footnote 5} First, note that condition (i) follows directly from (iii) in Proposition 9 and the facts that (a) equilibrium models are $\leq$ -minimal models and (b) ${E_{\mathcal{I}}}$ is a stable extension iff ${E_{\mathcal{I}}}$ are exactly the non-defeated arguments w.r.t. ${E_{\mathcal{I}}}$ . To show (ii), it is easy to see that ${E_{\mathcal{I}}}$ being a stable extension implies that ${\mathcal{I}}$ is a model of $\mathcal{C}_{AF}({\mathbf{AF}})$ . Hence, to show that ${\mathcal{I}}$ is an equilibrium model what remains is to prove that any ${\mathcal{J}} < {\mathcal{I}}$ is not a model of $\mathcal{C}_{AF}({\mathbf{AF}})$ . Any such ${\mathcal{J}}$ must satisfy $H_{\mathcal{J}}^+ = H_{\mathcal{I}}^+ = \mathbf{A}$ and $H_{\mathcal{J}}^- \subset H_{\mathcal{I}}^- = T_{\mathcal{I}}^- = {Def}({E})$ . Therefore, there is some defeated argument such that $a \notin H_{\mathcal{J}}^-$ and some defeating attack $(b,a) \in \mathbf{R}_a$ such that $b \in E = H_{\mathcal{I}}^+ \setminus T_{\mathcal{I}}^- = H_{\mathcal{J}}^+ \setminus T_{\mathcal{J}}^-$ . This implies that $b {\leadsto} a \in \mathcal{C}_{AF}({\mathbf{AF}})$ and ${\mathcal{J}} \models b$ which, in its turn, implies that $a \in H_{\mathcal{J}}^-$ . This is a contradiction and, consequently, ${\mathcal{I}}$ is an equilibrium model.

Theorem 2 captures the relation between the stable extensions of an AF and its translation into a logical theory. As mentioned above, this relation relies on the reasoning principles AT and CW: An $\mathbf{AF}= {\langle \mathbf{A},\mathbf{R} \rangle}$ is translated into a logical theory $\mathcal{C}_{AF}({\mathbf{AF}})$ using the notion of attack introduced in AT. The stable extension ${E_{\mathcal{I}}}$ of this AF is then retrieved from the equilibrium model ${\mathcal{I}}$ of $\mathcal{C}_{AF}({\mathbf{AF}})$ using the CW principle.

4.2 Set attack argumentation frameworks

We may also extend the results of the previous section to SETAFs using the language $\mathcal{C}_{SF} = {\{{\leadsto}, \wedge\}}$ and a similar translation.

Definition 7 Given some finitary set attack framework $\mathbf{SF}= {\langle \mathbf{A},\mathbf{R}_a \rangle}$ , we define

(9) \begin{eqnarray}{ l ,C, l}\Gamma_{\mathbf{R}_a} &\underline{\underline{\mathrm{def}}}& {\Big\{\ { \bigwedge A {\leadsto} b }\ \ \Big|\ \ { (A,b) \in \mathbf{R}_a}\ \Big\}}, \end{eqnarray}

and $\mathcal{C}_{SF}({\mathbf{SF}}) \underline{\underline{\mathrm{def}}} \mathbf{A} \cup \Gamma_{\mathbf{R}_a}$ .

Similar to Definition 6, translation $\mathcal{C}_{SF}({\cdot})$ applies the notion of attack introduced in AT to translate an AF into a logical theory. In this case, the set of attacking arguments becomes a conjuntion in the antecedent of the attack connective.

1. i. if ${\mathcal{I}}$ is an equilibrium model of $\mathcal{C}_{SF}({\mathbf{SF}})$ , then ${E_{\mathcal{I}}}$ is a stable extension of $\mathbf{SF}$ ,

2. ii. if E is a stable extension of $\mathbf{SF}$ and ${\mathcal{I}}$ is a total interpretation such that $T_{\mathcal{I}}^+ \!=\! \mathbf{A}$ and $T_{\mathcal{I}}^- \!=\! {Def}({E})$ , then ${\mathcal{I}}$ is an equilibrium model of $\mathcal{C}_{SF}({\mathbf{SF}})$ .

Proof sketch. The proof follows as in Theorem 2 by noting that any interpretation ${\mathcal{I}}$ and set of arguments B satisfy: $B \subseteq {E_{\mathcal{I}}}$ iff ${\mathcal{I}} \models b$ for all $b \in B$ iff ${\mathcal{I}} \models \bigwedge B$ .

4.3 Argumentation frameworks with evidence-based support

Let us now extend the language of SETAFs with the LP implication (1), in other words, we consider the language possessing the following set of connectives $\mathcal{C}_{EF} = {\{{\leadsto},\wedge,{\Rightarrow}\}}$ , so that we can translate any EBAF as follows:

Definition 8 Given any finitary evidence-based framework $\mathbf{EF}={\langle \mathbf{A},\mathbf{R}_a,\mathbf{R}_s,\mathbf{P} \rangle}$ , we define its corresponding theory as: $\mathcal{C}_{EF}({\mathbf{EF}}) \,\underline{\underline{\mathrm{def}}}\, \mathbf{P} \cup \Gamma_{\mathbf{R}_a} \cup \Gamma_{\mathbf{R}_s}$ with

(10) \begin{eqnarray}{ l ,C, l}\Gamma_{\mathbf{R}_s} &\underline{\underline{\mathrm{def}}}& {\Big\{\ { \bigwedge A {\Rightarrow} b }\ \ \Big|\ \ { (A,b) \in \mathbf{R}_s}\ \Big\}}, \end{eqnarray}

and $\Gamma_{\mathbf{R}_a}$ as stated in (9).

Note that, in contrast with AFs and SETAFs, the theory corresponding to an EBAFs do not contain all arguments as atoms, but only those that are prima-facie $\mathbf{P}$ . This reflects the fact that in EBAFs not all arguments can be accepted, but only those that are prima-facie or are supported by those prima-facie. Supports are represented using the LP implication ${\Rightarrow}$ and supported arguments are captured by the positive evaluation of each interpretation $H_{\mathcal{I}}^+$ . The following result extends Proposition 9 to EBAFs including the relation between supported arguments and models.

1. i. if a is supported w.r.t. ${E_{\mathcal{I}}}$ , then ${\mathcal{I}} \models^+ a$ ,

2. ii. if a is defeated w.r.t. ${E_{\mathcal{I}}}$ , then ${\mathcal{I}} \models^+ \sim\! a$ ,

3. iii. ${E_{\mathcal{I}}}$ is conflict-free.

If, in addition, ${\mathcal{I}}$ is an $\leq$ -minimal HT-model, then

1. i. a is supported w.r.t. ${E_{\mathcal{I}}}$ iff ${\mathcal{I}} \models^+ a$ ,

2. ii. a is defeated w.r.t. ${E_{\mathcal{I}}}$ iff ${\mathcal{I}} \models^+ \sim\! a$ ,

3. iii. ${E_{\mathcal{I}}}$ is self-supporting.

Example 9 (Ex. 1 continued) Consider now framework $\mathbf{EF}$ representing the Tweety scenario.

(11) \begin{eqnarray}{birdTweety} &{\Rightarrow}& {flyTweety,} \end{eqnarray}

(12) \begin{eqnarray}{penguinTweety} &{\Rightarrow}& {birdTweety,}\end{eqnarray}

(13) \begin{eqnarray}{penguinTweety} &{\leadsto}& {flyTweety} \end{eqnarray}

\begin{eqnarray}&&{penguinTweety.} \notag&\end{eqnarray}

As mentioned in Example 1, framework $\textbf{EF}_{1}$ has a unique stable extension

\begin{equation*}{\{{penguinTweety},\, {birdTweety}\}},\end{equation*}

which does not include the argument flyTweety. In other words, Tweety cannot fly. Interestingly, $\mathcal{C}_{SF}({\textbf{EF}_{1}})$ has also a unique equilibrium model ${\mathcal{I}}_{12} = {\langle {\mathbf{T}}_{12},{\mathbf{T}}_{12} \rangle}$ where ${\mathbf{T}}_{12}$ stands for the set:

\begin{gather*}{\{ {penguinTweety},\, {birdTweety},\, {flyTweety},\, \sim\!{flyTweety}\}}.\end{gather*}

This equilibrium model precisely satisfies the two arguments in that stable extension: ${\mathcal{I}}_{12} \models {penguinTweety}$ and ${\mathcal{I}}_{12} \models {birdTweety}$ . Note that we get ${\mathcal{I}}_{12} \not\models {flyTweety}$ from the fact that ${\mathcal{I}}_{12} \models^+ \sim\!{flyTweety}$ . In fact, this correspondence holds for any EBAF as shown by Theorem 4 below. Though more technically complex, the proof of Theorem 4 is similar that those of Theorems 2 and 3. In particular, it is necessary to prove the following relation between equilibrium models and supportable arguments:

1. i. a is supportable w.r.t. ${E_{\mathcal{I}}}$ iff ${\mathcal{I}} \models^+ a$ .

In contrast with the results for supported arguments stated in Proposition 10, this property does not hold for arbitrary $\leq$ -minimal models. This fact can be illustrated by considering a simple $\mathbf{EF}_13$ such that $\mathcal{C}_{EF}({\textbf{EF}_{13}}) = {\{ a,\, a {\Rightarrow} b \}}$ . Let ${\mathcal{I}}_{13} = {\langle {\mathbf{H}}_{13},{\mathbf{T}}_{13} \rangle}$ be some interpretation with ${\mathbf{H}}_{13} = {\{ a \}}$ and ${\mathbf{T}}_{13} = {\{ a, \sim\! a \}}$ . It is easy to see that ${\mathcal{I}}_{13}$ is a $\leq$ -minimal model of $\mathcal{C}_{EF}({\textbf{EF}_{13}})$ , though it is not an equilibrium model (because it is not a total interpretation). It can also be checked that a is not defeated and, consequently, that b is supportable w.r.t. $E_{{\mathcal{I}}_{13}} = \varnothing$ . On the other hand, the unique equilibrium model of $\mathcal{C}_{EF}({\textbf{EF}_{13}})$ is ${\mathcal{J}}_{13} = {\langle {\mathbf{H}}'_{13},{\mathbf{T}}'_{13} \rangle}$ with ${\mathbf{H}}'_{13} = {\{ a, b \}}$ and ${\mathbf{T}}'_{13} = {\{ a, b \}}$ . Here, both a and b are supportable (and supported) w.r.t. $E_{{\mathcal{J}}_{13}} = {\{a,b\}}$ .

The following result shows that, indeed, this correspondence holds for any EBAF:

1. i. if ${\mathcal{I}}$ is an equilibrium model of $\mathcal{C}_{EF}({\mathbf{EF}})$ , then ${E_{\mathcal{I}}}$ is a stable extension of $\mathbf{EF}$ ,

2. ii. if E is a stable extension of $\mathbf{EF}$ and ${\mathcal{I}}$ is a total interpretation such that $T_{\mathcal{I}}^+ \!=\! {Sup}({E})$ and $T_{\mathcal{I}}^- \!=\! {Def}({E})$ , then ${\mathcal{I}}$ is an equilibrium model of $\mathcal{C}_{EF}({\mathbf{EF}})$ .