In this paper we propose a general framework to provide a many-valued preferential interpretation of gradual argumentation semantics. The approach allows for conditional reasoning over arguments and boolean combination of arguments, with respect to some chosen gradual semantics, through the verification of graded (strict or defeasible) implications over a preferential interpretation. Argumentation is one of the major fields in non-monotonic reasoning (NMR) which has been shown to be very relevant for decision making and for explanation [1]. The relationships between preferential semantics of commonsense reasoning [2, 3, 4, 5] and argumentation semantics are very strong [6, 4]. While for Dung-style argumentation semantics and for Abstract Dialectical Frameworks, the relationships with conditional reasoning have been deeply investigated [7, 8, 9, 10], this is not the case for gradual argumentation [11, 12, 13, 14, 15, 16, 17]. The paper proposes a general approach to develop a preferential interpretation of an argumentation graph under a gradual semantics, provided some weak conditions on the domain of argument interpretation are satisfied. The approach allows for conditional reasoning over the argumentation graph, by formalizing conditional properties of the graph (with respect to the chosen semantics) in a many-valued logic with typicality: a many-valued propositional logic in which arguments play the role of propositional variables and in which a typicality operator is introduced, inspired by the typicality operator proposed in the Propositional Typicality Logic [18] and in Description Logics (DLs) with typicality [19]. The operator allows for the definition of 7th Workshop on Advances in Argumentation in Artificial Intelligence (AI 3), November 06-09, 2023, Rome, Italy * Corresponding author. † These authors contributed equally. " mario.alviano@unical.it (M. Alviano); laura.giordano@uniupo.it (L. Giordano); dtd@uniupo.it (D. Theseider Dupré) ~ https://alviano.net/ (M. Alviano); https://people.unipmn.it/laura.giordano/ (L. Giordano); https://people.unipmn.it/dtd/ (D. Theseider Dupré) 0000-0002-2052-2063 (M. Alviano); 0000-0001-9445-7770 (L. Giordano); 0000-0001-6798-4380 (D. Theseider Dupré)
conditional implications T(1) → 2, meaning that “normally argument 1 implies argument
2", in the sense that “in the typical situations where 1 holds, 2 also holds". The truth degree
of such implications can be determined with respect to a preferential interpretation defined from
a set of labellings of an argumentation graph, according to the chosen (gradual) argumentation
semantics. They correspond to conditional implications |∼ in the KLM approach [
More precisely, the paper considers graded implications of the form → ≥ , where and can be boolean combination of arguments possibly containing occurrences of the typicality operator. In particular, graded conditionals of the form T( ) → ≥ have the meaning that “normally argument implies argument with degree at least ". They are inspired by graded inclusion axioms in fuzzy DLs [21] and in weighted defeasible DLs knowledge bases [22].
The satisfiability of such implications in the multi-preferential interpretation of an argumentation graph (with respect to some given semantics ), exploits multiple preference relations < over labellings, each one associated with a boolean combination of arguments .
In [23] it has been shown that the satisfiability of a graded conditional T( ) → ≥ in a ifnite preferential interpretation can be decided in polynomial time in the product of the size of the interpretation and the size of the conditional formula. For well-founded preferences, the KLM postulates of a preferential consequence relation, reformulated for graded conditionals, can be proven to be satisfied by the conditionals which hold in the multi-preferential interpretation , for some choice of combination functions. In this paper, we consider an extension of the multi-preferential approach in [23] by lifting the well-foundedness restriction on the preference relations.
Given an argumentation graph and some gradual argumentation semantics , we define a preferential (many-valued) interpretation of the argumentation graph , with respect to the gradual semantics . We generalize the approach proposed in [24] for weighted argumentation graphs, without assuming a specific gradual semantics. In the following, we will consider both weighted and non-weighted argumentation graphs.
We follow Baroni, Rago and Toni [16, 25] (in their definition of a Quantitative Bipolar Argumentation Framework, QBAF) in the choice of the domain of argument interpretation, letting it to be a set , equipped with a preorder relation ≤ , an assumption which is considered general enough to include the domain of argument valuations in most gradual argumentation semantics. As usual, we let < iff ≤ and ̸≤ .
As in [16], we do not assume contains a minimum element and a maximum element. However,
if a minimum element and a maximum element belong to , we will denote them by 0 and 1
(or simply 0 and 1), respectively. If not, we will add the two elements 0 and 1 at the bottom
and top of the values in , respectively. We will also call the truth value set (or the truth
degree set). For instance, may be the unit interval [
For the definition of an argumentation graph, we consider the definition of edge-weighted QBAF by [26], for a generic domain . As we want to capture both weighted and non-weighted argumentation graphs, in the following, we will let the label of edges of the graph be +1 or − 1 to denote support and attack in the non-weighted case.
We let a (weighted) argumentation graph to be a quadruple = ⟨, ℛ, 0, ⟩, where is a set of arguments, ℛ ⊆ × a set of edges, 0 : → assigns a base score of arguments, and : ℛ → R is a weight function assigning a positive or negative weight to edges. An example of weighted argumentation graph is in Figure 1, where the base score is not represented.
A pair (, ) ∈ ℛ is regarded as a support of argument to argument when the weight (, ) is positive and as an attack of argument to argument when (, ) is negative. In case the graph is non-weighted, we let (, ) = − 1 of attacks and (, ) = +1 for supports.
Bipolar argumentation has been studied in the literature [27, 16, 25, 26] through different
frameworks. We refer to the Quantitative Bipolar Argumentation Framework (QBAF) by Baroni,
Rago and Toni [16, 25] for a classification and the properties of gradual semantics, when the
argumentation graph is non-weighted, and to Potyka’s work [26] for the framework of
edgeweighted QBAFs and its properties. The properties of edge-weighted argumentation graphs with
weights in [
Whatever semantics is considered for an argumentation graph , we will assume that identifies a set Σ of labellings of the graph over a domain of argument valuation . A labelling of over is a function : → , which assigns to each argument an acceptability degree (or a strength) in the domain of argument valuation . In some cases, we may omit the base score 0, and consider the set of labellings Σ of a graph for all the possible choices of the base score, or for a subset of them. In the following we will assume that, whatever the concrete definition of a semantics might be, the semantics of can be regarded, abstractly, as a pair (, Σ ): a domain of argument valuation and a set of labellings Σ over the domain. Example 1 ([24]). As an example, in the -coherent semantics for weighted argumentation graphs, in the finitely-valued case, for = with = 5, the graph in Figure 1 has 36 labellings, while, for = 9, has 100 labellings. For instance, = (0, 4/5, 3/5, 2/5, 2/5, 3/5) (meaning that (1) = 0, (2) = 4/5, and so on) is a labelling for = 5.
In the following, we introduce a propositional language to represent boolean combination of arguments and a many-valued semantics for it over the domain of argument valuation. Then, we extend the language with a typicality operator, to introduce defeasible implications over boolean combinations of arguments and define a (multi-)preferential interpretation associated with the argumentation graph and a set of labellings Σ .
Given an argumentation graph = ⟨, ℛ, 0, ⟩, let ℒ be a propositional language whose set of propositional variables is the set of arguments . We assume that the language ℒ contains the connectives ∧, ∨, ¬ and →, and that formulas are defined inductively, as usual. Formulas built from the propositional variables in correspond to a boolean combination of arguments (denoted , , ), which are considered, for instance, by Hunter et al. [28] in their epistemic approach to probabilistic argumentation.
We consider a many-valued semantics for boolean combination of arguments, with as the
truth degree set. Let ⊗ , ⊕ , ▷ and ⊖ be the truth degree functions in for the connectives ∧,
∨, ¬ and → (respectively). When is [
A labelling : → of graph , assigning to each argument ∈ a truth degree in , can be regarded as a many-valued valuation. A valuation can be inductively extended to all propositional formulas of ℒ as follows: ( ∧ ) = ( ) ⊗ ( ), ( ∨ ) = ( ) ⊕ ( ), ( → ) = ( ) ▷ ( ), and (¬ ) = ⊖ ( ). Based on the choice of the combination functions, a labelling uniquely assigns a truth degree to any boolean combination of arguments. We will assume that the false argument ⊥ and the true argument ⊤ are formulas of ℒ and that (⊥) = 0 and (⊤) = 1, for all labellings .
In this section, given an argumentation graph and a semantics (, Σ ) of , we aim at defining a preferential interpretation of the graph. We first introduce a preference relation on the set of labellings Σ , associated to any boolean combination of arguments .
Definition 1. Given a set of labellings Σ , for each boolean combination of arguments , we define a preference relation < on Σ , as follows: for , ′ ∈ Σ , < ′ iff ′( ) < ( ).
Labelling is preferred to ′ with respect to an argument (or a boolean combination of arguments) when is more plausible than ′ for argument , that is, when the degree of truth of in is greater than the degree of truth of in ′. The preference relation < is a strict partial order relation on Σ .
When the set Σ of labellings of a graph in an argumentation semantics is infinite, the preference relations < (and < ) are not guaranteed to be well-founded, as there may be infinitely-descending chains of labellings.
Let us define the preferential interpretation of a graph with respect to a set of labellings. Definition 2. Given an argumentation graph , a gradual semantics with domain of argument valuation , and the set of labellings Σ of wrt , we let the preferential interpretation of wrt to be the triple = (, Σ , {< }).
We have explicitly associated the preference relations < to the set of labellings Σ of the graph, although preference are induced by the labellings in Σ . Often, we will simply write or , rather than (and (, Σ , {< }) rather then (, Σ , {< })).
Language ℒ
T is obtained by extending language ℒ with a unary typicality operator T.
Intuitively, “a sentence of the form T( ) is understood to refer to the typical situations in which
holds" [18]. The typicality operator allows for the formulation of conditional implications (or
defeasible implications) of the form T( ) → whose meaning is that "normally, if then ", or
"in the typical situations when holds, also holds". They correspond to conditional
implications |∼ of KLM preferential logics [
→ is called strict. In the language ℒT, we allow for general implications and may contain occurrences of the typicality operator. The interpretation of a typicality → , where formula T( ) is defined with respect to a preferential interpretation = (, Σ , {< }). valuation of a propositional formula T( ) in is defined as follows:
Given a preferential interpretation = (, Σ , {< }), and a labelling ∈ Σ , the (T( )) = ︂{ ( ) 0 otherwise if ∈ < (Σ) (1) where < (Σ) =
{ : ∈ Σ and ∄ ′ ∈ Σ s.t. ′ < }.
When (T( )) > 0, is a labelling assigning a maximal degree of acceptability to argument in , i.e., it maximizes the acceptability of argument , among all the labellings in . As we lifted the requirement that preferences < are well-founded, the set < (Σ) might be empty.
Given a preferential interpretation = (, Σ , {< }), we can now define the satisfiability in of a graded implication, having form → ≥ or → ≤ , with and in and and boolean combination of arguments. We first define the truth degree of an implication a preferential interpretation as follows: → wrt under a semantics , the truth degree of an implication
Given a preferential interpretation = (, Σ , {< }) of an argumentation graph → wrt. is defined as: (
→ ) = ∈Σ( ( ) ▷ ( )).
We can now define the satisfiability of a graded implication in an interpretation . or it is not (i.e., ̸|= graded implications, such as
Given a preferential interpretation = (, Σ , {< }) of an argumentation graph wrt. , satisfies a graded implication satisfies a graded implication → ≤ (written |= → ≤ ) iff (
→ ) ≤ . → ≥ (written |= → ≥ ) iff ( → ) ≥ ; Notice that the valuation of a graded implication (e.g., → ≥ ) in a preferential interpretation is two-valued, that is, either the graded implication is satisfied in (i.e., |= → ≥ ) → ≥ ). Hence, it is natural to consider boolean combinations of (T(1) → 2 ∧ 3 ≤ 0.7) ∧ (T(3) → 4) ≥ 0.6) → (T(1) → 4) ≥ and define their satisfiability in an interpretation in the obvious way, based on the semantics of classical propositional logic.
The preferential interpretation can be used to validate properties of interest of an argumentation graph , expressed by graded implications (including strict or defeasible implications or their boolean combination) based on the semantics . For instance, the boolean combination of graded conditionals above allows to verify to validate whether the graded conditional (T(1) → 4) ≥ 0.6) holds, for all the labellings of the graph (in the semantics ) satisfying the graded conditionals (T(1) → 2 ∧ 3 ≤ 0.7) and (T(3) → 4) ≥ 0.6) .
When the preferential interpretation is finite (i.e., it contains a finite set of labellings), the satisfiability of graded implications (and their boolean combinations) can be verified by model checking over the preferential interpretation . In case there are infinitely many labellings of the graph in the semantics , approximations of the semantics over a finite domain can be considered for proving properties of the argumentation graph. As a proof of concept, in [24] we have developed an ASP approach for defeasible reasoning over an argumentation graph under the -coherent semantics in the finitely-valued case.
In [7] Weydert has proposed one of the first approaches for combining abstract argumentation with a conditional semantics. He has studied “how to interpret abstract argumentation frameworks by instantiating the arguments and characterizing the attacks with suitable sets of conditionals describing constraints over ranking models”. In doing this, he has exploited the JZ-evaluation semantics, which is based on system JZ [30].
For Abstract Dialectical Frameworks (ADFs) [8], the correspondence between ADFs and Nonmonotonic Conditional Logics has been studied in [9] with respect to the two-valued models, the stable, the preferred semantics and the grounded semantics of ADFs.
In [10] Ordinal Conditional Functions (OCFs) are interpreted and formalized for Abstract Argumentation, by developing a framework that allows to rank sets of arguments with respect to their plausibility. An attack from argument a to argument b is interpreted as the conditional relationship, “if a is acceptable then b should not be acceptable". Based on this interpretation, an OCF inspired by System Z ranking function is defined.
Our approach does not commit to a specific gradual argumentation semantics, and aims at providing a preferential conditional interpretation for a large class of gradual argumentation semantics. In this paper we focus on the gradual case, based on a many-valued logic.
In [31, 32] an approach is presented which regards a weighted argumentation graph as a weighted conditional knowledge base in a fuzzy defeasible Description Logic. In this approach, a pair of arguments (, ) ∈ ℛ with weight (representing an attack or a support), corresponds to a conditional implication T() ⊑ with weight . Based on this correspondence, some semantics for weighted knowledge bases with typicality [33] have inspired some argumentation semantics [31], and vice-versa. In particular, in [24] we have developed an ASP approach for defeasible reasoning over an argumentation graph under the -coherent semantics in the finitelyvalued case. In this paper, we have generalized the approach beyond the -coherent semantics, to deal with a large class of gradual semantics.
In this paper, we have developed a general framework to define a many-valued preferential interpretation of an argumentation graph, with respect to a gradual argumentation semantics. The approach allows for graded (strict and conditional) implications involving arguments and boolean combination of arguments (with typicality) to be evaluated in the preferential interpretation of the argumentation graph, which can be constructed based on a given gradual argumentation semantics . When the preferential interpretation is finite, the validation of graded conditionals can be done by model-checking over interpretation .
In [23] we have shown that graded conditionals T( ) → ≥ 1, which are satisfied in , satisfy the postulates of a preferential consequence relation [20] (suitable reformulated in this setting), for some choice of combination functions. Whether such properties are satisfied by the semantics considered in this paper, which does not require the preference relations to be well-founded, will be a subject of future work.
The definition of a preferential interpretation associated with an argumentation graph
and a gradual semantics also sets the ground for the definition of a probabilistic interpretation
for gradual semantics with domain of argument valuation in the unit real interval [
Laura Giordano and Daniele Theseider Dupré were partially supported by the Università del Piemonte Orientale, and by INDAM-GNCS Project 2022 “ LESLIE: LogichE non-claSsiche per tooL Intelligenti ed Explainable". Mario Alviano was partially supported by Italian Ministry of Research (MUR) under PNRR project FAIR “Future AI Research”, CUP H23C22000860006 and by LAIA lab (part of the SILA labs). [7] E. Weydert, On the plausibility of abstract arguments, in: Proc. Symbolic and Quantitative Approaches to Reasoning with Uncertainty - 12th European Conference, ECSQARU 2013, Utrecht, The Netherlands, July 8-10, 2013, volume 7958 of LNCS, Springer, 2013, pp. 522–533. [8] G. Brewka, H. Strass, S. Ellmauthaler, J. P. Wallner, S. Woltran, Abstract dialectical frameworks revisited, in: IJCAI 2013, Proceedings of the 23rd International Joint Conference on Artificial Intelligence, Beijing, China, August 3-9, 2013, IJCAI/AAAI, 2013, pp. 803–809. [9] J. Heyninck, G. Kern-Isberner, M. Thimm, On the correspondence between abstract dialectical frameworks and nonmonotonic conditional logics, in: Proceedings of the ThirtyThird International Florida Artificial Intelligence Research Society Conference, May 17-20, 2020, AAAI Press, 2020, pp. 575–580. [10] K. Skiba, M. Thimm, Ordinal conditional functions for abstract argumentation, in: COMMA 2022, Cardiff, Wales, UK, 14-16 September 2022, volume 353 of Frontiers in Artificial Intelligence and Applications, IOS Press, 2022, pp. 308–319. [11] C. Cayrol, M. Lagasquie-Schiex, Graduality in argumentation, J. Artif. Intell. Res. 23 (2005) 245–297. doi:10.1613/jair.1411. [12] J. Janssen, M. D. Cock, D. Vermeir, Fuzzy argumentation frameworks., in: IPMU 2008, 2008, pp. 513–520. [13] P. E. Dunne, A. Hunter, P. McBurney, S. Parsons, M. J. Wooldridge, Weighted argument systems: Basic definitions, algorithms, and complexity results, Artif. Intell. 175 (2011) 457–486. [14] S. Egilmez, J. G. Martins, J. Leite, Extending social abstract argumentation with votes on attacks, in: TAFA 2013, Beijing, China, Aug. 3-5, LNCS 8306, Springer, 2013, pp. 16–31. [15] L. Amgoud, J. Ben-Naim, D. Doder, S. Vesic, Acceptability semantics for weighted argumentation frameworks, in: IJCAI 2017, Melbourne, Australia, 2017, pp. 56–62. [16] P. Baroni, A. Rago, F. Toni, How many properties do we need for gradual argumentation?, in: Proc. AAAI 2018, New Orleans, Louisiana, USA, February 2-7, 2018, pp. 1736–1743. [17] L. Amgoud, D. Doder, Gradual semantics accounting for varied-strength attacks, in:
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