Paraconsistent Labeling Semantics for Abstract Argumentation

: Dung’s abstract argumentation framework is a popular formalism in formal argumentation. The present work develops paraconsistent labeling semantics for abstract argumentation such that the incomplete and inconsistent information can be expressed


Introduction
Formal argumentation is an important research field in artificial intelligence (cf., e.g., [1][2][3][4]).Dung's theory of abstract argumentation framework is useful for dealing with disputes between agents.This framework is simple but has powerful expressiveness (cf.[5]).An argumentation framework (AF) is a pair (A, R) where A is a set of arguments and R a binary attack relation on A. For all a, b ∈ A, aRb means that the argument a attacks b.
There are various kinds of semantics such as grounded, complete, preferred, stable and semi-stable semantics defined for AF in the literature (cf., e.g., [5,6]).For the purpose of representing different kinds of information, AF has been extended to many forms such as AF with uncertain information [4,[7][8][9], bipolar AF [10][11][12], AF with preferences [13,14] and AF with constrains [15,16].In particular, there are two ways to extend AF to represent uncertain information: one is incomplete AF (iAF) where arguments and attacks may be uncertain [7,8], and the other is probabilistic AF (PrAF) where arguments and attacks are associated with probabilities [4,9].This paper uses AAF to deal with uncertain (incomplete) and paraconsistent information by introducing paraconsistent (four-valued) labeling semantics without utilizing probability.This differs from iAF in the following sense: the attack relation is certain, and the semantics is changed into a bilateral one such that an argument can accept (support) or reject a proposition.Thus, given an argument a and proposition φ, there are four possibilities: (C1) a accepts φ; (C2) a rejects φ; (C3) a both accepts and rejects φ; and (C4) a neither accepts nor rejects φ.Here, (C1) means that φ is certainly true for a; (C2) means that φ is certainly not true for a; (C3) means that φ is a contradictory information for a; and (C4) means that φ is irrelevant with a.
The paraconsistent approach given in the present paper puts uncertainties on the satisfaction relation but not on the attack relation.Dung said, "an argument is an abstract entity whose role is solely determined by its relations to other arguments.No special attention is paid to the internal structure of the arguments" [5].In our framework, an argument is an abstract entity whose role is determined both by its relations to other arguments and its relations to properties or propositions (supporting or rejecting).This is due to the fact that two arguments may attack the same arguments, but they are given based on different reasons.Belnap [17,18] and Dunn [19,20] proposed the four-valued semantics for modeling different states of information.In the present paper, the paraconsistent labeling semantics is used to deal with incomplete and paraconsistent information.After we present the bilateral semantics, we will introduce a sound and complete Hilbert-style axiomatic system.Finally, we will compare this kind of semantics with some relevant theories of abstract argumentation.Some logicians (cf., e.g., [21,22]) have considered paraconsistency both at the level of propositional symbol and at the level of accessibility relation.The present paper has considered paraconsistency only at the level of the propositional symbol, but we generalize four-valued paraconsistent modal logic in a different direction: we generalize a single accessibility relation to bi-directional accessibility relations.Indeed, we use a four-valued paraconsistent tense logic with a global modality.Hence, the logic in the present paper can talk about the relations of both attacking and being attacked.
The structure of this paper is as follows.Section 2 introduces the formal language and paraconsistent labeling semantics.Section 3 gives a Hilbert-style axiomatic system and proves its soundness and completeness.Section 4 gives the comparison between our semantics and other theories.Section 5 gives some concluding remarks and directions of future work.

Paraconsistent Labeling Semantics
This section introduces the formal language and the paraconsistent labeling semantics.The modal logic of abstract argumentation under this kind of semantics is introduced.Here, R(a) is the set of all arguments that a attacks, and Ȓ(a) is the set of all arguments that attack a.For every set of arguments X ⊆ A, the set of all arguments that arguments in X attack is defined as R[X] = a∈X R(a).The set of all arguments that attack arguments in X is defined as Ȓ[X] = a∈X Ȓ(a).For n ≥ 0, the set R n (a) of all arguments reachable from a in n-steps of attack is defined inductively by R 0 (a) = {a} and R n+1 = R[R n (a)].The set Ȓn (a) of all arguments that can reach a in n-steps of attack is defined inductively by Ȓ0 (a) = {a} and Ȓn+1 = Ȓ[ Ȓn (a)].
Let V = {p i : i ∈ ω} be a denumerable set of propositional variables.We assume that V is denumerable, and in practical scenarios, V can be finite.Let P (V) be the powerset of V.In the standard labeling in an AAF, a set of propositional variables is assigned to each argument.Here, we assign each argument a pair of sets of propositional variables, namely, each argument a takes an element in the product P (V) × P (V) as its label.For each argument a, the label l(a) consists of a pair of sets ⟨l + (a), l − (a)⟩ where l + (a) is the set of all propositional variables that a accepts, and l − (a) is the set of all propositional variables that a rejects.The two sets of propositional variables have four cases in the following diagram: The region (I) consists of propositional variables which a accepts; (II) consists of propositional variables which a rejects; (III) consists of propositional variables which a accepts and rejects; and (IV) consists of propositional variables which a neither accepts nor rejects.Now, we give the formal definition of labeling and model as follows.
Definition 2. Let F = (A, R) be an AAF.A labeling in F is the function l : A → P (V) × P (V).
A model is M = (F , l) where F is an AAF, and l is a labeling in F .For every model M = (A, R, l) and a ∈ A, the pair l(a) = ⟨l + (a), l − (a)⟩ is given the label of a.The label l(a) is Clearly, R(a) = {a, b, c} and Ȓ(c) = {a, b}.Note that aRa means a attacks itself, namely, a is contradictory.Moreover, {b, c} ⊆ R n (a) for all n ≥ 1.Let l be the labeling in F such that l + (a) = {q} and l − (a) = {q, r}.Then, l(a) is both inconsistent and incomplete.
Let a be an argument a in a model and φ a formula.We give the acceptance relation a |= + φ and the rejection relation a |= − φ to show the paraconsistent labeling semantics.
For every model M = (A, R, l) and φ ∈ Fm, the following hold: Proof.We only prove the following items, and the proof of others is omitted.
( ( Here, we have defined the paraconsistent modal logic PML which is the logic of argumentation under the paraconsistent labeling semantics.In the next section, we will give a Hilbert-style axiomatization of PML.

Hilbert-Style Axiomatic System
In this section, we shall present a Hilbert-style axiomatic system S for the logic PML of abstract argumentation.We shall prove the soundness and completeness of S. Let CL be the classical propositional logic in the language {¬, ∨}.
A proof of a formula φ in S a finite sequence of formulas φ 1 , . . ., φ n such that φ n = φ and each φ i is either an axiom or derived from previous formulas by an inference rule.A formula φ is provable (or a theorem) in S (notation: ⊢ S φ) if there exists a proof of φ in S. A formula φ is derivable from a set of formulas Γ in S (notation: Here, ∆ is the conjunction of all formulas in ∆.In particular, ∅ = ⊤.
Lemma 2. For all formulas φ, ψ 1 and ) is obtained from φ by substituting ψ 2 for one or more occurrences of ψ 1 in φ.
Hence, ⊢ S □φ ∧ ♢ψ → ♢(φ ∧ ψ) by (A1).Similarly, ⊢ S ■φ ∧ ♦ψ → ♦(φ ∧ ψ) and ⊢ S Uφ ∧ Eψ → E(φ ∧ ψ).Now, we prove the completeness of S by the canonical model method.First, we define deductive filters and use them to define the canonical model.Definition 7. A nonempty set of formulas F is an S-deductive filter if φ ∈ F and ⊢ S φ → ψ imply ψ ∈ F. An S-deductive filter F is proper if ⊥ ̸ ∈ F. The set of all S-deductive filters is denoted by F (S).A proper S-deductive filter F is prime if φ ∨ ψ ∈ F implies φ ∈ F or ψ ∈ F. The set of all prime S-deductive filters is denoted by F p (S).

For a nonempty set of formulas
(2) If φ ̸ ⊢ S ψ, then there exists G ∈ F p (S) such that φ ∈ G and ψ ̸ ∈ G.
Proof.For (1), assume that F ∈ F (S) and ψ ̸ ∈ F. Consider the set By Zorn's lemma, there exists a ⊆-maximal element G in X.Clearly, F ⊆ G and ψ ̸ ∈ G.It suffices to show that G is prime.For a contradiction, suppose that The canonical model for S is defined as the model M S = (F S , l S ) where l S is a labeling given by l The canonical AAF F S consists of two relations R S and R E , and so it is not an AAF in the sense of Definition 2. Thus, the proof of completeness is not standard.Using the following Lemma 7, we obtain a submodel M S (F) generated by a point F ∈ W S and so it will be a model in the sense of Definition 4.
Lemma 8. Let φ ∈ Fm and G ∈ W F .The following hold: (1 Proof.We prove ( 1) and ( 2) simultaneously by induction on the the complexity of φ.
(2) ♦φ ∈ F if and only if φ ∈ G for some G ∈ F p (S) such that GR S F. ( Proof.(1) The right-to-left direction follows from the definition of R S .Assume that ♢φ ∈ F.

Discussion
In an AAF, consider different arguments a and b which have the same relations to other arguments.The following Example 2 gives such a scenario.The problem is how a and b can be distinguished from each other.The semantics given by Dung [5] can not tell the difference between arguments a and b since an argument is completely determined by its relations to other arguments.However, using the semantics presented in this paper, we can achieve this.For an argument a, it has not only the attack relationship to other arguments, for example, the argument b attacks a and a attacks c, but also can accept or reject some properties.For example, a accepts the property p and rejects q.Thus, from this semantical perspective, an argument is an abstract entity whose role is determined both by its relations to other arguments and its semantical relations to propositions.
Example 2. Marry and John both oppose presidential candidate A. However, they oppose him for different reasons.Marry opposes them because she thinks that he is not honest; and John opposes them because he thinks that his policies can not support economic growth.
The two arguments in Example 2 (Marry and John's arguments) both attack the argument that presidential candidate A should be elected, but they are different arguments.Using the labeling semantics, a formula φ can be interpreted as a property of arguments or a set of all arguments accepting φ.The formula ∼φ is interpreted as a set of all arguments rejecting φ.The formula ♢φ is interpreted as a set of all arguments which attack some arguments supporting φ.The formula □φ is interpreted as a set of all arguments which attack only arguments supporting φ.The formulas ♦φ and ■φ are interpreted similarly.The formula Eφ is interpreted as that there is an argument accepting φ, and Uφ as that all arguments accept φ.By the labeling semantics, arguments are distinguished by properties.
Furthermore, many arguments are based on incomplete and inconsistent information.Dung [5] said that "all forms of reasoning with incomplete information rest on the simple intuitive idea that a defeasible statement can be believed only in the absence of any evidence to the contrary which is very much like the principle of argumentation.".Since information is usually not only incomplete but also inconsistent based on multiple information sources, the labeling semantics given in the present paper works naturally since it introduces the acceptance and rejection semantical relations.Grossi [23,24] introduced the doing argumentation theory to study abstract argumentation.In this framework, an argument a belonging to I(p) in a given model means that a has the property p or that p is true of a. Thus, either a has the property p or a does not have p, but not both.This approach is essentially bivalent.In our paraconsistent labeling semantics, we have the following four cases: (1) a accepts p (p is true of a).
(2) a rejects p (p is false of a).
(3) a both accepts and rejects p (p is both true and false of a).(4) a neither accepts nor rejects p (p is neither true nor false of a).
Grossi's doing argumentation theory can not express the inconsistency or lacking information of arguments.For example, the arguments which attack an argument supporting φ and rejecting φ can be expressed by the formula ♢(φ ∧ ∼φ) under our labeling semantics.
Caminada [25] proposed an alternative labeling semantics for abstract argumentation.In their framework, every argument is labeled (or be valued) by in, out or undec, which means, respectively, the argument is accepted, rejected and illegally undec.
A Caminada Labeling is a function λ from the set of all arguments to the set of labels (or propositional variables), in, out and undec, satisfying the following conditions.Caminada's labeling demands consistency, i.e., an argument a is rejected if it is attacked by an accepted argument b.It also demands maximal acceptance, i.e., an argument a which is not attacked by no argument is necessarily accepted.However, in our paraconsistent approach, neither consistency nor maximal acceptance is assumed.

Concluding Remarks and Future Work
This paper develops a paraconsistent labeling semantics for abstract argumentation.Then, we introduces the paraconsistent modal logic PML and its Hilbert-style axiomatic system S.The system S is shown to be sound and complete with respect to the paraconsistent labeling semantics.Moreover, we have compared paraconsistent labeling semantics with other semantical approaches to the abstract argumentation.
The present paper leaves some interesting directions which are worth exploring in further work.One direction is that paraconsistency at the level of accessibility relation may be generalized in two different ways: one is the four-valued accessibility relation, and the other is the birelational frame semantics.In the latter way, we have the attack relation and support relation so that an argument can both attack and support the same arguments.Some logicians have given a sort of four-valued paraconsistent modal logic based on birelational frame semantics, but the resulting logic is not a temporal logic (cf., e.g., [26]).
Another approach is using the modal hybrid logic that can enrich the expressiveness of language on structures.We may add names of arguments to the language so that we can talk about properties of arguments in a direct way.Some logicians have considered this kind of modal hybrid logic (cf., e.g., [27]), but the used logic is not a temporal one.
The last approach is the study of measures of inconsistency in abstract argumentation (AAF) that can enable us to compare in a direct way two different AAFs with respect of the number of inconsistencies.For example, the arguments from a debate among the candidates consist of an argumentation; then, the less inconsistency there is in the argumentation, the clearer the candidate's policy.Measures of inconsistency in the context of argumentation theory are worth exploring further; although, this kind of work has been done in other contexts (cf., e.g., [27][28][29]).
Funding: This research was funded by the Chinese National Funding of Social Sciences (Grant no.18ZDA033).

Definition 1 .
An abstract argumentation framework (AAF) is a pair F = (A, R) where A ̸ = ∅ is a set of arguments, and R is a binary relation on A which is called the attack relation.The inverse of R is defined as Ȓ = {⟨a, b⟩ : bRa}.For every argument a ∈ A, we define R(a) = {b ∈ A : aRb} and Ȓ(a) = {b ∈ A : bRa}.

Definition 4 .
Let M = (A, R, l) be a model, a ∈ A and φ ∈ Fm.The acceptance relation M, a |= + φ and rejection relation M, a |= − φ are defined simultaneously by induction as follows: M, a |= + p if and only if

Lemma 6 .
The following hold in S: