This paper provides an interpretation of abstract argumentation semantics in terms of structured bipolar argumentation. While abstract semantics evaluate which arguments to accept, we understand arguments to consist of various sentences that form premises and conclusions and evaluate both which arguments and which sentences to accept. Analysing the correspondence between evaluating arguments and evaluating sentences allows us to gain a new perspective on abstract argumentation semantics. Namely, while preferred semantics aims to maximise the set of accepted arguments, this does not always correspond to maximising the set of accepted sentences. We then detail the conditions under which the two do coincide. Thus, we can explain preferred semantics from the sentence perspective. We define structured bipolar argumentation frameworks (SBAFs) to implement both argument-based and sentence-based semantics that are admissibility-based. This takes into account the idea, common in philosophical and linguistic approaches, that it is sometimes rational to reject defended arguments. This, then, forms the basis on which we analyse abstract argumentation semantics.
When we engage in debates or argumentations, we argue about certain claims. For instance, we might debate proposals for new immigration policies, whether nuclear energy is safe, or elements of our world-views. When we argue for or against these claims, we put forth further claims that come together to form arguments. If these claims are in turn attacked or require support, we argue by introducing even more claims to the debate. While most analyses focus on the arguments we use, ultimately, we are interested in the claims we argue about. That is, we want to know whether we should accept a new immigration policy, that nuclear energy is safe, or the controversial elements of our world-views. If, during the debate, sub-discussions on how we justify our stances to these claims came up, the same goes for those.
Computational argumentation answers the question “Given all the arguments in a debate, what
should you believe?” by providing diferent ways of modelling debates that each come with their
range of semantics [
3rd International Workshop on Argumentation for eXplainable AI (ArgXAI@ECAI) 2025 $ michael.mueller@unifr.ch (M. A. Müller)
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structured [
Example 1. Consider the following situation, loosely inspired by [
It might be intuitive for agents to indicate which individual sentences they accept without thinking about the arguments directly. For instance, we might accept the sentences : that Clara is cited in a newspaper article, : that the violin is a Stradivarius, and : that it is owned by Hilary (see Example 2 for a full translation). Is {, , } an acceptable set of sentences? In the terms we introduce in Section 3.2, we can understand this as a weakly adequate language extension and it corresponds to the argument extension {2, 3}. Language extensions can also be used to explain argument extensions in that in this example the rejection of 1 is explained by the agent doubting that Alex claimed that the violin is a Stradivarius, even though the agent accepts that it is indeed one and the argument is unattacked and thus defended.
The idea of rejecting arguments based on doubting one of its sentences also illustrates how informal
approaches often evaluate arguments on the level of their sentences. For people confronted with
arguments, it is often more intuitive to think about what sentences they accept rather than which
arguments [
tation (Section 2) and then introduce SBAFs in Section 3. Then we define both argument and language semantics for SBAFs (Section 3.2). Afterwards, in the remaining part we detail how our semantics compare to those of abstract argumentation (Section 4), which allows us to interpret them in terms of acceptable sets of sentences.
Definition 1 (Abstract Argumentation Framework). An abstract argumentation framework (AF) is a tuple = ⟨, →⟩ where is a finite set of arguments and → ⊆ × an attack relation.
We write → in case (, ) ∈ → and generalise to sets of arguments, i.e. → in case → for some ∈ and → in case → for some ∈ . Further, we write ̸→ if (, ) ̸∈ →. For two sets of arguments, → ′ means → for some ∈ and ∈ ′. A set of arguments ⊆ defends an argument ∈ if ∀ ∈ : → =⇒ → .
The semantics are as usual. Let = ⟨, →⟩ be an AF. An extension ⊆ is called conflict-free if ∀, ∈ : ̸→ , admissible if it is conflict-free and defends all its arguments, complete if it is admissible and it contains all arguments it defends, preferred if it is ⊆ -maximal among admissible extensions.
Definition 2 (Language). A language ℒ = ⟨, , ⟩ consists of a non-empty set of sentences , a (partial) incompatibility function : → 2, and a (partial) naming function : 2 × → .
We assume to be symmetric, i.e. ∀, ∈ : ∈ ⇐⇒ ∈ . Additionally, we assume that (⟨{}, ⟩) = ∅.
The set of sentences can be any set of objects. All the structure we need to represent arguments
is given by the incompatibility and the naming functions. Incompatibility is used to model conflict
between sentences in that it associates each sentence with the set of sentences incompatible with it. It
is a symmetric notion of contrariness, cf. [
Example 2. Let us continue Example 1 and consider a language with = {, , , , , , , }, corresponding to: : “Alex says this violin is a Stradivarius”, : “This violin is a Stradivarius”, : “This violin is expensive”, : “Clara is cited in a newspaper article mentioning that Anne-Sophie owns this violin”, : “Clara says that Anne-Sophie owns this violin”, : “Anne-Sophie owns this violin”, : “Diego says Hilary owns this violin”, and : “Hilary owns this violin”, with ∈ and ∈ . We can also add (⟨{}, ⟩) : “This violin is a Stradivarius since Alex says so”.
As in abstract argumentation, we do not construct arguments. Rather, we take them as given, e.g. through a debate. This means that we do not need any logic or inference rules that determine from which sentences we can infer others. Formally, any combination of premises and conclusion could be an argument. In the following definition, we use as a function from arguments to sets of sentences in order to indicate the premises of an argument and as a function from arguments to sentences in order to indicate the conclusion of an argument. For an argument , () is its set of sentences and () is its conclusion.
Definition 3 (Arguments). An argument in a language ℒ = ⟨, , ⟩ is a tuple = ⟨ (), ()⟩ where () is a non-empty finite subset of and () ∈ .
We also define the set of sentences of an argument as () := () ∪ {()}. The set of sentences generalises to sets of arguments () = ⋃︀∈ ().
We say is a minimal argument for sentence ∈ if = ⟨{}, ⟩.
Example 3. In Example 2, we already saw one argument: 1 : ⟨{}, ⟩, which corresponds to “This violin
is a Stradivarius since Alex says so”. The other arguments are: 2 : ⟨{}, ⟩, 3 : ⟨{}, ⟩, 4 : ⟨{}, ⟩,
and 5 : ⟨{}, ⟩. An example of a minimal argument would be ⟨{}, ⟩, i.e. “This violin is a Stradivarius
since this is a Stradivarius”. It is an edge case of an argument as it does not contain any real inference step,
but they are useful to represent single sentences in frameworks (see e.g. [
Next, we denfie supports and attacks between arguments. Attacks are defined such that an argument
attacks another argument if its conclusion is incompatible with some part of it [
The support relation requires more explanation. We want to capture situations where accepting some
set of arguments commits you to accepting another: If you accept an argument 1 : ⟨{}, ⟩ and there
is an argument 2 : ⟨{}, ⟩, you should also accept 2 since you accept all its premises. When it comes
to arguments with more than one premise, only a set of arguments might force its acceptance. For
instance, accepting 1 does not force acceptance of 3 : ⟨{, }, ⟩. But if you also accept 4 : ⟨{}, ⟩,
then you should accept 3. Thus, we define support not as a binary relation between arguments but
as a relation between sets of arguments and arguments. With these, we capture a notion of premise
support [
Suppose there is also an argument 5 : ⟨{}, ⟩. Then, accepting 1 means accepting all premises of
5 and thus 5 should be accepted as well. This is a kind of premise-sharing-support, sometimes called
exhaustion [
Definition 4 (Support and Attack). Let , be arguments in language ℒ.
We say a set of arguments supports if () ⊆ ().
We say that attacks if () ∈ for some ∈ () or if () ∈ (). Example 4. Using the setting of Examples 1, 2 and 3, we have a support from 1 : ⟨{}, ⟩ to 2 : ⟨{}, ⟩ and a mutual rebut between 4 : ⟨{}, ⟩ and 5 : ⟨{}, ⟩.
Definition 5 (Structured Bipolar Argumentation Framework). A structured bipolar argumentation framework (SBAF) is a tuple ℬ = ⟨ℒ, , →, →·⟩ where is a finite set of arguments in language ℒ and →, →·are the corresponding attack and support relations.
We write →· in case (, ) ∈ →·and use the same notational conventions as with attacks. Example 5. The following example is based on ∈ , ∈ , ∈ (6), and ∈ (and accordingly ∈ ).
4 : ⟨{}, ⟩ 5 : ⟨{, }, ⟩
6 : ⟨{}, ⟩
Definition 6 (Saturated SBAFs). An SBAF ℬ is called saturated (resp. strongly saturated) if ∀ ∈ () .. ∃ ∈ () ∩ , there is a minimal argument for or (resp. and) for in , and ∀ ∈ () .. ∈ () for some ∈ , there is a minimal argument for in .
The SBAF in Example 5 is not saturated, but could be made so by adding, for instance, ⟨{}, ⟩ and ⟨{}, ⟩. To make it strongly saturated, ⟨{}, ⟩ and ⟨{}, ⟩ would also have to be added.
and to evaluate both acceptable sets of arguments and acceptable sets of sentences.
is not forced to accept all defended arguments, but their rejection is limited by support. A simple way
of doing this is to start with admissible semantics and add a condition that makes sure that extensions
are closed under support (cf. [
already accept undercutting information for it. In such a case, you accept that the inference in the argument does not hold and as such you can reject it, even if you accept all its premises.
While both options can be seen as reasonable, in this paper we opt for a weak interpretation where you are never forced to accept undefended arguments. In some sense, this interpretation allows you to conclude that the inference of the supported but undefended argument must be faulty without having explicit undercutting information.
Definition 7 (Weakly coherent Argument Extensions). A weakly coherent argument extension in an SBAF ℬ = ⟨ℒ, , →, →·⟩ is an admissible extension that satisfies: Weak Support-Closure: ∀ ∈ : if supports , does not contain undercutting information for , and defends , then ∈ .
Example 6. Consider the SBAF of example 5. The extension {1, 2, 3, 4, 6} is weakly coherent.
sets of sentences, we first require that they do not contain incompatible sentences. We call a set of sentences compatible if ∀, ∈ : ̸∈ . But apart from this condition, it is the arguments that limit the choice of sentences we can accept—this is the point of arguing. For this we need to know to which arguments you are committed to when accepting some sentences and vice-versa.
Given an argument extension, we can simply say that the corresponding language extension consists of all sentences of accepted arguments. When we start with a language extension, however, it is less clear how the accepted sentences translate to accepted arguments. For instance, perhaps it is possible to accept all premises and the conclusion of an argument without accepting the argument itself—one can simply disbelief that the premises support the conclusion. As with weak support-closure, we assume that this is possible in case the argument in question is undefended. This gives a translation from language extensions to argument extensions that is in line with weakly coherent argument extensions.
Given a set of sentences in an SBAF ℬ, we define its weak argument set which collects together
all the arguments in ℬ which should be accepted. As we only require accepting defended arguments,
we define it using a fixpoint construction analogous to that in abstract argumentation for complete
extensions [
ℬ() := { ∈ | ∈ () and defends }.
With this, we can define the weak argument set.
Definition 9 (Weak Argument Set). If is compatible, then its initial set, (), is defined as the largest admissible subset of { ∈ | () ⊆ and () ∩ = ∅}. The weak argument set of a compatible , (), then is the least fixpoint of that contains ().
ℬ
We need to make sure that the weak argument set contains (), as otherwise the least fixpoint of would sometimes be too small. For instance, with two arguments 1 : ⟨{}, ⟩ and 2 : ⟨{}, ⟩ with ∈ and ∈ , the least fixpoint for the language extension = {, } would be empty. However, we want it to contain 1 as all its sentences are accepted and it defends itself.
Proposition 1. The weak argument set is well-defined.
Example 7. Consider again the SBAF of example 5 and take the language extension = {, , , , , , }. For its weak argument set, we have: () = {1, 2, 3, 4} and () = {1, 2, 3, 4, 6}.
Consider also ′ = {, , , , , } with (′) = {1, 2, 3, 5}. While both and ′ end up accepting all of 1, 2, and 3, thus supporting both 4 and 5, they difer in which of them they accept. On the argument level, the situation looks symmetrical, but using the sentence perspective, we can explain this diference through the prior commitment of to the conclusion of 4 and the prior commitment of ′ to that of 5.
Proposition 2. For a compatible language extension , () is admissible.
Definition 10 (Adequate Language Extensions). A weakly adequate language extension in an SBAF ℬ = ⟨ℒ, , →, →·⟩ is a compatible set of sentences ⊆ () that satisfies sentence-closure. Sentence-Closure: ∀ ∈ () : () ⊆ .
Example 8. We can now fully analyse Example 1. The language is provided in Example 2. We get the following SBAF. We can now see that {3} is weakly coherent, and that {, , } is weakly adequate.
Example 9. Consider an SBAF with two arguments 1 : ⟨{}, ⟩ and 2 : ⟨{}, ⟩ with ∈ . Then, {1, 2} is a weakly coherent argument extension, but its set of sentences is not compatible and hence not weakly adequate.
Proposition 3 (Direct correspondence for weak extensions). Let ℬ = ⟨ℒ, , →, →·⟩ be a saturated SBAF. Then for every weakly adequate language extension , its weak argument set () is a weakly coherent argument extension.
Also, for every weakly coherent argument extension , its set of sentences () is a weakly adequate language extension.
Proof. Let be a weakly adequate language extension. We check all conditions of ().
Weak Support-Closure: Assume for some ∈ that () supports , i.e. () ⊆ (()), does not contain undercutting information, i.e. () ∩ (()) = ∅, and defends . We need to show that ∈ (). First note that (()) ⊆ , as Lemma ?? shows that () = () and all sentences occuring in arguments of () are already contained in . Thus we have () ⊆ . Further, we can show that () ∩ = ∅. Suppose for a contradiction that there is some ∈ () ∩ . By saturatedness of ℬ, there is a minimal argument for , for which we know that → . Since () defends , we know that there is some ∈ () such that → . By being a minimal argument, we know that () ∈ , and we further know by sentence-closure that () ∈ . But this contradicts compatibility of , since also ∈ . Thus we conclude that () ∩ = ∅. Now we can use that () is a fixpoint of to conclude that, since () also defends , we have ∈ () as desired.
Now let be a weakly coherent argument extension. We check all conditions for (). Compatibility: Suppose there are , ∈ () such that ∈ . Then we have arguments , ∈ such that ∈ () and ∈ (). Further, by saturatedness of ℬ, there is w.l.o.g. a minimal argument for (note that → ). Now, we need to additionally show that defends . Note that since ∈ (), any attack on is also an attack on . And since defends , thus also defends . Weak support-closure then gives ∈ . But since → , this contradicts conflict-freeness of . Thus we conclude that () is compatible.
Defence: We show that = (()), from which defence follows directly. By Lemma ??, we know that () = , thus it sufices to show that = () (since then () will itself be the smallest fixpoint containing it). ⊆: ⊇:
Take any ∈ . Then we have () ⊆ () . Further, we know that defends and ℬ is saturated, thus by Lemma ??, we know that () ∩ () = ∅. This gives us that ∈ { ∈ | () ⊆ () and () ∩ () = ∅}. But note that we have just now shown that ⊆ { ∈ | () ⊆ () and () ∩ () = ∅}, and since is admissible, we can directly infer that ⊆ (()) , as the latter is the largest admissible subset of { ∈ | () ⊆ () and () ∩ () = ∅}.
Take any ∈ (()). Then we have that () ⊆ () and ()∩() = ∅. If we can show that defends , then weak support-closure gives us the desired ∈ . Thus take any attacker of . There are two cases. (1) () ∈ for some ∈ (). Since () ⊆ () , there is some argument ∈ such that ∈ (). But then → , and since defends , we also have → . That is, defends against . (2) () ∈ (). By saturatedness of ℬ, there is a minimal argument for (). Note that → and since is defended by (()), there is some argument ∈ (()) such that → . This gives us in particular () ∈ () and since is a minimal argument, we also have () ∈ (). Further, () ∈ () gives us some argument ∈ such that () ∈ (). Recall that incompatibility is symmetric, thus → . Finally, since defends , we have → , that is, defends against .
In sum, defends and weak support-closure gives us ∈ as desired.
Sentence-Closure: Take any ∈ (()). Since (()) ⊆ , we have ∈ and thus () ⊆ ()) as desired.
Observation 1. Complete extensions are weakly coherent and a weakly coherent extension is ⊆ -maximal if it is preferred.
correspond to an agent who accepts as many arguments as possible, what kind of extensions do we get from an agent who accepts as many sentences as possible? We first define confident extensions to capture this idea.
Definition 11 (Confident Extensions). Let ℬ be an SBAF. A language extension ⊆ () is called a confident weakly adequate language extension if it is ⊆ -maximal amongst weakly adequate language extensions.
An argument extensions ⊆ is called a confident weakly coherent argument extension if it is weakly coherent and there exists a confident weakly adequate language extension ⊆ () such that = ().
Example 10. In the SBAF of example 5, we have the following confident weakly coherent extensions: {1, 2, 3, 5, 7}, and {1, 2, 3, 4, 6}.
Proposition 4 (Indirect correspondence for confident extensions). Let ℬ be a saturated SBAF. Then for every confident weakly adequate language extension, its weak argument set is confident weakly coherent.
Also, for every confident weakly coherent argument extension , there exists a confident weakly adequate language extension such that () = .
Example 11. In the following SBAF (with ∈ ), ∅ is confident weakly coherent, based on {, } being confident weakly adequate, but it is not preferred.
frameworks, where many minimal arguments are added, that maximising arguments and maximising sentences coincide. The correspondence between confident weakly coherent extensions and preferred extensions also shows that, in strongly saturated SBAFs, the former does not take support into account, as the latter can be calculated in pure attack-frameworks.
Proposition 5. Let ℬ = ⟨ℒ, , →, →·⟩ be a saturated SBAF. Then any preferred extension ⊆ is confident weakly coherent.
Proof. Let be a preferred extension. Since it is then also complete, Observation 1 gives us that it is weakly coherent. It thus remains to show that there exists a confident weakly adequate language extension such that = ().
Consider the set { ⊆ () | = () and is weakly adequate}. Note that it is nonempty, since () is weakly adequate and (()) = (Proposition 3). Recall that we only consider finite frameworks, thus there exists a ⊆ -maximal element ′ that is weakly adequate and (′) = . It remains to show that ′ is also ⊆ -maximal amongst weakly adequate language extensions.
Take any weakly adequate ′′ such that ′ ⊊ ′′. Since ′ is ⊆-maximal amongst weakly adequate extensions with () = , we know that (′′) ̸= . We first show that ⊊ { ∈ | () ⊆ ′′ and ()∩′′ = ∅}. Take any ∈ . Then clearly, () ⊆ () ⊆ ′ ⊊ ′′. Now suppose there is some ∈ () ∩ ′′. Then, by saturatedness, there exists a minimal argument for . Note that → and since is defended, there is some ∈ such that → , that is, () ∈ . But since () ⊆ () ⊆ ′ ⊆ ′′, this contradicts compatibility of ′′. Hence, () ∩ ′′ = ∅ and we can note that ⊊ { ∈ | () ⊆ ′′ and () ∩ ′′ = ∅}. Since is admissible, this gives ⊆ ( ′′) ⊆ (′′).
In sum, ⊊ (′′), but since (′′) is admissible (Proposition 3), this contradicts that is preferred. Hence, there exists not weakly adequate ′′ such that ′ ⊊ ′′ and we conclude that is confident weakly coherent.
Proposition 6. Let ℬ = ⟨ℒ, , →, →·⟩ be a strongly saturated SBAF. Then any confident weakly coherent argument extension ⊆ is preferred.
Proof. Let be weakly coherent and suppose it is not preferred. We show that is not confident. We need to show that for any ⊆ () such that () = , there exists an adequate language extension ′ such that ⊊ ′.
Since is admissible, there exists a preferred extension ′ such that ⊊ ′. By Proposition 5, we know that ′ is confident weakly coherent. Hence, there exists a confident weakly adequate language extension ′ such that (′) = ′.
Now take any ⊆ () such that () = . We show that ⊊ ′. Take any ∈ ()∖′. Then ̸∈ ′ and since ′ is confident, we know that ′ ∪ {} is not weakly adequate. We can show that ′ ∪ {} is not compatible. Suppose it is. Then (′ ∪ {}) is admissible (Proposition 2), and we know that ′ ̸⊆ (′ ∪ {}) (since otherwise either ′ ∪ {} would be weakly adequate, if ′ = ′ ∪ {}, or ′ not preferred, if ⊊ ′ ∪ {}). This lets us take an argument ∈ ′∖(′ ∪ {}) and we know that ∈ () (since otherwise ∈ (′ ∪ {})). By saturatedness of ℬ, we then have a minimal argument for . Note that → and since ′ is defended, there is some ∈ ′ such that () ∈ . But note that () ⊆ () = ( (′)) ⊆ ′ (see proof of Proposition 3), contradicting compatibility of ′ ∪ {}. Thus, we know that ′ ∪ {} cannot be compatible. But then there is some ∈ ′ such that ∈ . By strong saturatedness, there exists a minimal argument for . Since it is a minimal argument, {} is admissible and hence ∈ (′) ⊆ ′. Since → , we also have ′ → . But since ⊆ ′, we know that ̸∈ (since otherwise ′ would not be conflict-free). Finally, since is a minimal argument for , we know that ̸∈ (since otherwise ∈ () = ). In sum, ⊆ ′.
It remains to note that ̸= ′, since otherwise = () = (′) = ′. Thus ⊊ ′ and is not confident. This establishes the result, since we know that any preferred ⊆ is weakly coherent.
The notion of support we use is related to that of deductive support [
rejection of unattacked arguments, namely if they are not supported by evidence. Thus, this approach
implements a notion of doubt that is based on a notion of evidential support. However, this notion of
support is very diferent from ours as it requires all accepted arguments to be supported. Evidential
support is a version of necessary support [
such as ABA [
bipolar argumentation frameworks (pBAFs) [
Example 12. Consider the following SBAF with ∈ , ∈ , and ∈ (3).
Argument 1 defends itself against the closed set {2, 3} because it attacks 3, but it is not defended against 2. Accordingly, {1} is not weakly coherent, but it would be Γ-admissible.
claim and provide semantics that work purely on the level of claims, similarly as our language extensions. They also compare the strategies of maximising accepted claims and maximising accepted arguments, and they conclude that these strategies are identical in well-formed CAFs, meaning that arguments with the same claim attack the same arguments. As SBAFs are well-formed by definition, their result difers from ours, as confident weakly coherent extensions and preferred extensions do not coincide in general. One crucial diference between our approaches is that CAFs only take into account the conclusions of arguments while SBAFs also consider their premises.
Argumentation in practice has many features and aspects, which in turn lead to many ways of modelling it. In this paper, we followed ideas from informal approaches to argumentation and developed an approach to structured bipolar argumentation. This approach takes the language perspective seriously and allows evaluation of both argument and language extensions. While both capture similar ideas, they give diferent result in general and only agree in saturated SBAFs. The language perspective further allows us to compare the strategies of maximising accepted arguments and maximising accepted sentences and we detail in what situations the two strategies coincide, namely in strongly saturated
[
dation for Autonomous Agents and Multiagent Systems, 2022, pp. 1319 – 1327. [39] F. H. van Eemeren, B. Garssen, B. Meufels, Fallacies and judgments of reasonableness: Empirical research concerning the pragma-dialectical discussion rules, Springer, 2009.