=Paper=
{{Paper
|id=Vol-3733/paper13
|storemode=property
|title=On the Extension of Argumentation Logic
|pdfUrl=https://ceur-ws.org/Vol-3733/paper13.pdf
|volume=Vol-3733
|authors=Antonis Kakas,Paolo M. Mancarella
|dblpUrl=https://dblp.org/rec/conf/cilc/KakasM24
}}
==On the Extension of Argumentation Logic==
https://ceur-ws.org/Vol-3733/paper13.pdf
On the Extension of Argumentation Logic
Antonis Kakas1,† , Paolo M. Mancarella2,*,†
1
Department of Computer Science, University of Cyprus, Cyprus, antonis@ucy.ac.cy
2
Department of Computer Science, University of Pisa, Italy, paolo.mancarella@unipi.it
Abstract
This paper shows how Argumentation Logic can be further extended to cover more fully paraconsistent forms
of logical reasoning. The extension is based on the notion of non-acceptable self-defeating arguments as a
generalization of the Reductio ad Absurdum principle.
Keywords
Propositional Logic, Argumentation Logic, Para consistency, Reductio ad Absurdum
1. Motivation and Background
Argumentative inference relies on the central normative condition of the acceptability of a (set of)
argument(s). Informally, this condition states that “a (set of) argument(s) is acceptable only when it
defends against all its counter-arguments”. An acceptable argument thus forms a “case” that supports
satisfactorily its claim and hence the claim is a possible or credulous conclusion under the argumentative
reasoning. One way to formalise this notion of acceptability of arguments is via a recursive operator
that first defines the more general notion of relative acceptability, 𝐴𝑐𝑐(∆, ∆0 ), giving the acceptability
of a set of arguments ∆ with respect to a given set ∆0 and then projecting down to 𝐴𝑐𝑐(∆, {}) for the
semantics.
This acceptability semantics for argumentation was first proposed in the context of Logic Program-
ming [1, 2, 3] showing how it captures and extends the semantics of negation as failure. It was then
an easy matter to apply this to abstract argumentation [4]. More recently, it was shown [5] that this
type of recursive acceptability semantics for argumentation can be applied to formal logical reasoning
where arguments are sets of logical formulae, e.g., propositional formulae. In this, an acceptable case
of arguments corresponds to a set of formulae which can be enveloped in a model of the theory and
thus a credulous conclusion corresponds to a satisfiable formula. We can then show that such a form of
Argumentation Logic (AL) is logically equivalent to classical Propositional Logic (PL).
This equivalence holds only when reasoning under a set of given premises that are classically
consistent. When the premises are inconsistent, AL does not trivialize but smoothly extends PL into a
paraconsistent logic. Technically, AL does this by encompassing in its form of logical reasoning the
notion of proof by contradiction in a way that prevents this from using an inconsistency in one part of
the theory to derive any conclusion that may be “unrelated” to the inconsistency. In argumentation,
Reduction ad absurdum is captured through the notion of non-acceptability of arguments, namely
the contrary notion of acceptability of arguments. Non-acceptable arguments are “self-defeating”
arguments. Informally, such an argument is one that either forms a counter-argument to itself or that
it is a counter-argument to an argument that it necessarily needs in order to defend against some
counter-argument to it. In other words, a non-acceptable or self-defeating argument invalidates its
possible case of support by rendering the set of arguments in the case incompatible with each other.
In this paper, we will explore further the notion of non-acceptable arguments and study how this can
give in the AL reformulation of PL new acceptable sets of arguments (under inconsistent premises) that
were not recognized as such before. The main idea is that we can extend the notion of acceptability
of a set of arguments ∆ by requiring that any counter-argument 𝐴 against this is either, as before,
CILC 2024: 39th Italian Conference on Computational Logic, June 26-28, 2024, Rome, Italy
*
Corresponding author.
†
These authors contributed equally.
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
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Workshop ISSN 1613-0073
Proceedings
defended against explicitly by some other argument, or by recognizing that 𝐴 is by itself non-acceptable
or self-defeating. In this second possibility a counter-argument is dealt with by showing, in analogy
to proof by contradiction, that it is by itself invalid and hence it cannot affect the acceptability of ∆.
Whereas in the previous work in [5] this was used only for the limiting case of non-acceptability of a
self-attacking counter-argument, in this paper we will show how more complex forms of self-defeating
non-acceptable arguments can be identified and used to “neutralize” the effect of such arguments when
they appear as counter-arguments to other arguments.
Section 2 reviews the acceptability semantics for general abstract argumentation frameworks under
which the classical Propositional Logic is reformulated as an Argumentation logic. Section 3 defines the
proposed general extension of the acceptability semantics. Section 4 applies the general theory to the
specific case of AL as a reformulation of PL and shows how this extends the existing definition of AL.
Section 5 concludes with a discussion of future work (further possible extensions of AL, e.g. for directly
inconsistent premises, where we just extend suitably the defense relation).
2. Acceptability semantics of Argumentation
Let us briefly review the area of Abstract Argumentation and its semantics [4, 6] as developed and used
in the area of Artificial Intelligence. In abstract argumentation we are not interested in the internal
structure of arguments but only in their relative properties. An abstract argumentation framework is
defined as follows.
Definition 1. [Abstract Argumentation Framework]
An abstract argumentation framework is a triple, ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩, where
• 𝐴𝑟𝑔 is a set (of arguments)
• 𝐴𝑡𝑡 is a binary (partial) relation on 𝐴𝑟𝑔 (attack relation)
• 𝒟𝑒𝑓 is a binary (partial) relation on 𝐴𝑟𝑔 (defense relation)
Given 𝐴, ∆, 𝐷 ⊆ 𝐴𝑟𝑔, we say that 𝐴 attacks ∆ (written 𝐴 ⇝ ∆) iff there exists 𝑎 ∈ 𝐴 and 𝑏 ∈ ∆
such that (𝑎, 𝑏) ∈ 𝐴𝑡𝑡 and that 𝐷 defends against 𝐴 (written 𝐷 ↠ 𝐴) iff (𝑑, 𝑐) ∈ 𝒟𝑒𝑓 for some
𝑑 ∈ 𝐷 and 𝑐 ∈ 𝐴.
This definition differs from the usual classical definition used in the area of abstract argumentation,
where an argumentation framework is simply a tuple 𝐴𝐹 𝑐 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡𝑐 ⟩, consisting of arguments and
an attack relation 𝐴𝑡𝑡𝑐 between arguments. In the triple argumentation frameworks defined above,
we expand the classic attack relation 𝐴𝑡𝑡𝑐 into its two relations, i.e. whenever (𝑎, 𝑏) ∈ 𝐴𝑡𝑡𝑐 then
(𝑎, 𝑏) ∈ 𝐴𝑡𝑡 and (𝑎, 𝑏) ∈ 𝒟𝑒𝑓 . This transformation makes explicit the two properties captured by the
classic attack relation, namely when (𝑎, 𝑏) ∈ 𝐴𝑡𝑡𝑐 then (i) the argument 𝑎 can form a counter-argument
to 𝑏 and (ii) 𝑎 can defend against 𝑏 when this is a counter-argument (to any argument).
A typical realization of a triple argumentation framework in some language, ℒ, for constructing and
comparing arguments is given by: (1) 𝑎 is in conflict in ℒ with 𝑏 for (𝑎, 𝑏) ∈ 𝐴𝑡𝑡 to hold and (2) 𝑎 is at
least as strong in ℒ as 𝑏 for (𝑎, 𝑏) ∈ 𝒟𝑒𝑓 to hold. In such realizations, the attack relation is symmetric
and the defense relation is a subset of the attack relation. The detailed study of these links is beyond
the scope of this paper.
The semantics of an abstract argumentation framework is defined via subsets of arguments that
satisfy an acceptability property, 𝒜𝑐𝑐(∆, ∆0 ), whose informal meaning is that the set of arguments
∆ is acceptable in the context of a given set of arguments ∆0 , only when ∆ can defend against all its
counter-arguments. The precise definition of the acceptability property is given as follows.
Definition 2. [Acceptability property]
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be an abstract argumentation framework and ∆, ∆0 ⊆ 𝐴𝑟𝑔. Then:
• 𝒜𝑐𝑐(∆, ∆0 ) iff
– ∆ ⊆ ∆0 , or
– for any 𝐴 ⊆ 𝐴𝑟𝑔 such that 𝐴 ⇝ ∆:
∘ 𝐴 ̸⊆ ∆ ∪ ∆0 , and
∘ there exists 𝐷 ⊆ 𝐴𝑟𝑔 such that 𝐷 ↠ 𝐴 and 𝒜𝑐𝑐(𝐷, ∆ ∪ ∆0 )
We can thus see that for ∆ to be acceptable in the context of ∆0 all its counter-arguments must be
defended by arguments that are themselves acceptable in the extended context of ∆ ∪ ∆0 . Extending
the context in this way means that (a chosen set of arguments) ∆ can contribute to its own defense.
Formally, the acceptability property is defined through the least fixed point of an associated monotonic
operator on the binary Cartesian product of sets of arguments
ℛ = 2𝐴𝑟𝑔 × 2𝐴𝑟𝑔
Definition 3. [Acceptability operator]
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be an abstract argumentation framework. The acceptability operator 𝒜 :
ℛ → ℛ is defined as follows. Given 𝑟 ∈ ℛ and ∆, ∆0 ⊆ 𝐴𝑟𝑔, (∆, ∆0 ) ∈ 𝒜(𝑟) iff:
• ∆ ⊆ ∆0 , or
• for any 𝐴 ⊆ 𝐴𝑟𝑔 such that 𝐴 ⇝ ∆:
– 𝐴 ̸⊆ ∆ ∪ ∆0 , and
– there exists 𝐷 ⊆ 𝐴𝑟𝑔 such that 𝐷 ↠ 𝐴 and (𝐷, ∆ ∪ ∆0 ) ∈ 𝑟
We denote by 𝒜𝑓 𝑖𝑥 the least fixed point of this operator. Then the semantics of an argumentation
framework is given through the subsets of arguments ∆ that are acceptable with respect to the empty
set of arguments, i.e. such that (∆, {}) ∈ 𝒜𝑓 𝑖𝑥 holds. We say that such sets of arguments are acceptable.
Example 1.
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be the abstract argumentation framework where
• 𝐴𝑟𝑔 = {𝑎, 𝑏}
• 𝐴𝑡𝑡 = {(𝑎, 𝑏), (𝑏, 𝑎)}
• 𝒟𝑒𝑓 = {(𝑏, 𝑎)}
In this framework its two arguments attack each other but only argument 𝑏 is able to defend against its
counter-argument of 𝑎, e.g., because 𝑏 is stronger than 𝑎. We can then see that the set {𝑏} is acceptable
whereas the set {𝑎} is not acceptable as it cannot defend against its counter-argument 𝐴 = {𝑏}. Instead, if
the defense relation contained also (𝑎, 𝑏), e.g., when the two arguments are of equal strength, then both
{𝑎} and {𝑏} would be acceptable sets of arguments.
To illustrate the basic idea of the problem for the need of extending the semantics of argumentation
let us consider the following example.
Example 2. [Motivating Example 1]
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be the abstract argumentation framework where
• 𝐴𝑟𝑔 = {𝑎, 𝑏}
• 𝐴𝑡𝑡 = {(𝑎, 𝑎), (𝑎, 𝑏)}
• 𝒟𝑒𝑓 = {}
The argument set {𝑏} is not acceptable as it cannot defend against its attack by {𝑎}: there are no arguments
that could be used as a defense. Nevertheless, {𝑎} is itself self-attacking and hence we would expect that it
is not necessary to find an explicit way of defending against it. Thus we would want the argument set {𝑏}
to be acceptable.
This idea that arguments that are themselves self-attacking or more generally, as we shall see below
in this paper, self-defeating was first studied in the context of the argumentation-based semantics of
Negation as Failure in Logic Programming (see [1, 2, 3] and references therein). These ideas were then
lifted to the case of Abstract Argumentation in [4] showing their general applicability.
Both in the case of Logic Programming and Abstract Argumentation the approach is based on first
formulating a relative notion of acceptability of arguments, as we have reviewed above in Definitions 2
and 3, from which we then extract or project to notions of absolute acceptability of arguments (recently
a semantically equivalent reformulation of relative acceptability was proposed in [7]). This approach
to extending the semantics of argumentation via a relative notion of acceptability is indirectly related
to a variety of other approaches [8, 9, 10] most of which are studies of how to address odd loops in
the attack relation of an argumentation framework. In particular, in [10] where, under the labeling
semantics of argumentation, arguments can be labeled IN, OUT or UNDECIDED, there is a close link
between this later label of UNDECIDED and the notion of self-defeated arguments under the relative
acceptability semantics.
2.1. Propositional Logic as Argumentation Logic
An important application of the relative acceptability semantics is that of the reformulation of classical
Propositional Logic in terms of argumentation [5]. We will briefly review this reformulation and
its paraconsistent extension of Argumentation Logic as a realization of the abstract argumentation
framework and its acceptability semantics.
Definition 4. [Argumentation Logic Framework]
We denote by ⊢𝑀 𝑅𝐴 the Natural Deduction direct derivation relation of propositional logic modulo
Reduction ad Absudrum (MRA), i.e. without the proof rule of Reduction ad Absudrum.
Let 𝑇 be a propositional theory. The argumentation logic framework corresponding to 𝑇 is the triple
𝐴𝐹 𝑇 = ⟨𝐴𝑟𝑔𝑠, 𝐴𝑡𝑡, 𝐷𝑒𝑓 ⟩ with:
• 𝐴𝑟𝑔 = {Σ | Σ is a finite set of propositional sentences}
• given ∆, Γ ∈ 𝐴𝑟𝑔, with ∆ ̸= {}, (Γ, ∆) ∈ 𝐴𝑡𝑡 iff 𝑇 ∪ Γ ∪ ∆ ⊢𝑀 𝑅𝐴 ⊥
• given ∆ ∈ 𝐴𝑟𝑔, ({𝜑}, ∆) ∈ 𝐷𝑒𝑓 , where 𝜑 is the complement of some sentence 𝜑 ∈ ∆ and
({}, ∆) ∈ 𝐷𝑒𝑓 whenever 𝑇 ∪ ∆ ⊢𝑀 𝑅𝐴 ⊥.
We see that the attack relation is symmetric, i.e. arguments are always counter-arguments of each
other when together they are directly inconsistent in the context of the given premises 𝑇 . The defense
relation essentially expresses the fact that any argument can be defended against by undermining one
of its premises. In logical terms, the defense relation expresses the property that for any formula 𝜑 we
are free to choose this or its complement. The second part of the defense relation expresses the fact
that if an argument is self-inconsistent with respect to the given premises, then this can be trivially
defended against by the “safe” empty argument (which in turn can not be attacked). We will see below
that when we extend the acceptability semantics, this second part of the defense relation will not need
to be stated explicitly at this level, but will be captured at the extended acceptability semantic level.
We will denote by 𝒜ℒ𝑓 𝑖𝑥 (or simply by 𝒜ℒ) the least fixpoint of the corresponding operator 𝒜 in
the general abstract argumentation frameworks as above in definition 3. We then have a logical corre-
spondence between propositional logic (for classically consistent premises 𝑇 ) and the argumentation
acceptability semantics [5]. For any formula 𝜑: 𝜑 is acceptable, i.e., ({𝜑}, {}) ∈ 𝒜ℒ𝑓 𝑖𝑥 if and only
if there is a model of 𝑇 in which 𝜑 is true. Furthermore, for classically inconsistent premises which
are directly consistent, i.e. consistent under the restricted derivation of ⊢𝑀 𝑅𝐴 , the argumentation
semantics does not trivialize but smoothly extends the propositional deductive semantics into such
cases of inconsistent premises.
The full technical details of these results can be found in [5]. For the purposes of this paper it is
important to point out that the results rest on the correspondence between proofs via Reductio ad
Absurdum and the non-acceptability of formulae, namely that for any formula 𝜑: ({𝜑}, {}) ̸∈ 𝒜ℒ𝑓 𝑖𝑥
holds if and only if there exists in Natural Deduction a restricted1 Reductio ad Absurdum proof for the
complement of 𝜑, i.e. for 𝜑. Hence, if a posited formula 𝜑 is shown via the restricted form of Reductio
ad Absurdum to lead to an inconsistency, then the argument set {𝜑} can not be acceptable. This means
that some argument, 𝐴, that attacks {𝜑}, i.e. it is directly inconsistent with it, cannot be defended
against by some set of formulae 𝐷 that is acceptable in the context of {𝜑}. In other words, the argument
{𝜑} defeats its possible defenses, it is self-defeating.
3. Non-acceptable Arguments
In this section, we will examine further the nature of non-acceptable arguments and the relative
defeatedness of such arguments in the context of a given set of arguments. In particular, we will be
interested in a sub-case of non-acceptability that relates to arguments that are defeated in their own
context.
Let us return to abstract argumentation and consider the following example.
Example 3. [Motivating Example 2]
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be the abstract argumentation framework where
• 𝐴𝑟𝑔 = {𝑎, 𝑏}
• 𝐴𝑡𝑡 = {(𝑎, 𝑏)}
• 𝒟𝑒𝑓 = {}
Argument set {𝑏} is attacked by argument set {𝑎}. Trivially then, ({𝑏}, {𝑎}) ̸∈ 𝒜𝑓 𝑖𝑥 , i.e. {𝑏} is non-
acceptable in the context of {𝑎}, as {𝑏} is attacked by an argument that belongs to the context . We will
also say that {𝑏} is defeated in the context of {𝑎}. Similarly, if we have that (𝑏, 𝑎) ∈ 𝐴𝑡𝑡, i.e. {𝑏}
also attacks {𝑎}, then {𝑎} is defeated in the context of {𝑏}. Note also that these particular contextual
defeat cases hold irrespective of what is contained in the defense relation. Hence even if we had the tuple
(𝑏, 𝑎) ∈ 𝒟𝑒𝑓 it would still be the case that {𝑏} is non-acceptable or defeated in the context of {𝑎}.
Let us return to the first motivating example.
Example 4. [Motivating Example 1 cont.]
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be the abstract argumentation framework where
• 𝐴𝑟𝑔 = {𝑎, 𝑏}
• 𝐴𝑡𝑡 = {(𝑎, 𝑎), (𝑎, 𝑏)}
• 𝒟𝑒𝑓 = {}
The argument set {𝑎} is self-attacking and hence it is non-acceptable or defeated in its own context. We
consider this argument as a self-defeating argument exactly because it contains an (one of its) attack.
Note that the property of {𝑎} being self-defeating is not affected by the argument {𝑏}.
Recognizing this property of self-defeatness of arguments, it is reasonable to require that other
arguments, whose counter-arguments are such self-defeated arguments, are acceptable (wrt {}). For
instance, in the above example it is reasonable to accept that argument {𝑏} is acceptable (in the context
of {}), because its only attack is “self-defeating”. In other words, it is not necessary to explicitly defend
against such a self-defeating counter-argument, as this attack is an argument that invalidates itself.
The above example shows a simple (and limiting) case of a non-acceptable self-defeating argument.
More complex forms of such arguments exist, as it is illustrated in the next example.
1
This restriction requires that the posited hypothesis must be necessary for its inconsistency to be derived (see [5]). For
classically consistent premises 𝑇 such a restricted proof always exists when a non-restricted ordinary Reductio ad Absurdum
proof exists for the same posited hypothesis.
Example 5. [Motivating Example 3]
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be the abstract argumentation framework where
• 𝐴𝑟𝑔 = {𝑎, 𝑏, 𝑎1 , 𝑑1 }
• 𝐴𝑡𝑡 = {(𝑎, 𝑏), (𝑎1 , 𝑎), (𝑎, 𝑑1 ), (𝑑1 , 𝑎1 )}
• 𝒟𝑒𝑓 = {(𝑑1 , 𝑎1 )}
Argument 𝑎 is attacked by 𝑎1 which can only be defended against by argument 𝑑1 . But 𝑎 attacks this
defence of 𝑑1 , i.e. 𝑑1 is defeated in the context of 𝑎. Hence, as in the example above, 𝑎 is non-acceptable and
we can consider it as self-defeating, but now in an indirect way, because 𝑎 renders its necessary defending
argument(s) non-acceptable or defeated in its own context. If we thus accept that 𝑎 is self-defeating then
again, as in the example above, we would want that argument 𝑏, which is attacked only by 𝑎, to be an
acceptable argument.
These more complex forms of self-defeated arguments arise from the recursive nature of non-
acceptability given by negating the recursive definition of acceptability.
Proposition 1. [Non-acceptability]
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be an abstract argumentation framework and ∆, ∆0 ⊆ 𝐴𝑟𝑔. Let
𝑛𝑜𝑛_𝐴𝑐𝑐(∆, ∆0 ) denote the statement (∆, ∆0 ) ̸∈ 𝒜𝑓 𝑖𝑥 . Then the following holds directly from the
definition of acceptability:
• 𝑛𝑜𝑛_𝐴𝑐𝑐(∆, ∆0 ) iff ∆ ̸⊆ ∆0 and
∃𝐴 ⊆ 𝐴𝑟𝑔 such that 𝐴 ⇝ ∆ and
∗ 𝐴 ⊆ ∆ ∪ ∆0 , or
∗ ∀𝐷 ⊆ 𝐴𝑟𝑔 s.t. 𝐷 ↠ 𝐴: 𝑛𝑜𝑛_𝐴𝑐𝑐(𝐷, ∆ ∪ ∆0 ).
Hence, a general non-acceptable argument 𝐴 such that (𝐴, {}) ̸∈ 𝒜𝑓 𝑖𝑥 holds, is one where, when we
collect recursively the defenses against one of its counter-arguments and recursively the further defenses
against attacks of the earlier defenses, we end up with a collection of defenses that is self-attacking.
Definition 5. [Self-defeating sets of arguments]
Let 𝐴, 𝐴′ ⊆ 𝐴𝑟𝑔. We say that 𝐴 is defeated in the context of 𝐴′ iff 𝑛𝑜𝑛_𝐴𝑐𝑐(𝐴, 𝐴′ ) holds. When a set is
defeated in the context of the empty set, i.e. 𝑛𝑜𝑛_𝐴𝑐𝑐(𝐴, {}) holds, we say that 𝐴 is self-defeating and
denote this by 𝒮𝒟(𝐴).
Proposition 2.
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be an abstract argumentation framework.
(i) If (𝑎, 𝑏) ∈ 𝐴𝑡𝑡, then 𝑛𝑜𝑛_𝐴𝑐𝑐({𝑏}, {𝑎}) holds;
(ii) If ∆ is self-attacking then it is self-defeating, i.e. 𝒮𝒟(𝐴) holds.
Proof.
(i) {𝑎} ⇝ {𝑏} and {𝑎} ⊆ {𝑏} ∪ {𝑎}. Hence 𝑛𝑜𝑛_𝐴𝑐𝑐({𝑏}, {𝑎}).
(ii) Obvious, since ∆ ⇝ ∆ and ∆ ⊆ ∆ ∪ {}.
4. Extended Acceptability semantics
The extension of the notion of acceptability of arguments follows the simple idea that counter-arguments
that are non-acceptable or self-defeating can be dealt with without the need to explicitly defend against
them. It is sufficient to recognize that such attacks are self-defeating.
Definition 6. [Extended Acceptability]
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be an abstract argumentation framework and ∆, ∆0 ⊆ 𝐴𝑟𝑔. Then a set of
arguments ∆ is acceptable in the context of ∆0 , denoted by 𝒜𝑐𝑐+ (∆, ∆0 ), when the following holds:
𝒜𝑐𝑐+ (∆, ∆0 ) iff
– ∆ ⊆ ∆0 , or
– for any 𝐴 ⊆ 𝐴𝑟𝑔 such that 𝐴 ⇝ ∆:
∗ 𝐴 ̸⊆ ∆ ∪ ∆0 , and
∗ (𝐴, {}) ̸∈ 𝒜𝑓 𝑖𝑥 , or there exists 𝐷 ⊆ 𝐴𝑟𝑔 such that 𝐷 ↠ 𝐴 and (𝐷, ∆ ∪ ∆0 ) ∈ 𝒜𝑓 𝑖𝑥
Proposition 3.
Let 𝐴𝐹 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ be an abstract argumentation framework and ∆, ∆0 ⊆ 𝐴𝑟𝑔. Then
(∆, ∆0 ) ∈ 𝒜𝑓 𝑖𝑥 =⇒ 𝒜𝑐𝑐+ (∆, ∆0 )
Proof. Straightforward by the definition of 𝒜𝑓 𝑖𝑥 and 𝒜𝑐𝑐+ .
Example 6. [Motivating Example 1 Revisited]
In both of these examples ({𝑏}, {}) does not belong to 𝒜𝑓 𝑖𝑥 , i.e. the argument set {𝑏} is not acceptable.
However, 𝒜𝑐𝑐+ ({𝑏}, {}) holds because the only (minimal) attack against {𝑏}, namely the set {𝑎}, is
self-defeating. Hence the argument set {𝑏} is acceptable in the extended semantics.
4.1. AL+ : Extended Argumentation Logic
We will now revisit the reformulation of classical Propositional Logic and its paraconsistent extension
by Argumentation Logic as a realization of the abstract argumentation framework. We will apply the
extended acceptability semantics of the general Definition 6 to the case of Argumentation Logic. This
will give an extended form of argumentation logic that takes more fully into account the presence
of non-acceptable arguments. In effect, this will give a generalized use of the principle of proof by
contradiction under inconsistent premises.
Definition 7. [Extended Argumentation Logic]
Let 𝐴𝐹 𝑇 = ⟨𝐴𝑟𝑔𝑠, 𝐴𝑡𝑡, 𝐷𝑒𝑓 ⟩ be the argumentation logic framework corresponding to a (directly
consistent) propositional theory 𝑇 . The extended argumentation logic, 𝒜ℒ+ , is given by:
𝒜ℒ+ (∆, {}) holds iff for any 𝐴 ⊆ 𝐴𝑟𝑔 such that 𝐴 ⇝ ∆:
• 𝐴 ̸⊆ ∆, and
• (𝐴, {}) ̸∈ 𝒜ℒ, or ∃ 𝐷 ⊆ 𝐴𝑟𝑔 such that 𝐷 ↠ 𝐴 and (𝐷, ∆) ∈ 𝒜ℒ
Hence a set of formulae is acceptable in 𝒜ℒ+ either because its attacks could be defended acceptably,
as before in the basic logic of 𝒜ℒ, or its attacks are non-acceptable in 𝒜ℒ.
The following result shows that the extended argumentation logic, 𝒜ℒ+ , is a “proper” extension of
𝒜ℒ when the given premises 𝑇 are classically consistent.
Theorem 1. Let 𝑇 be a classically consistent theory and 𝐴𝐹 𝑇 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ its corresponding
argumentation logic framework. Let also 𝜑 be a propositional formula such that ({𝜑}, {}) ̸∈ 𝒜ℒ holds.
Then 𝒜ℒ+ ({𝜑}, {}) does not hold.
Proof. This is a technical result whose proof is in the Appendix.
This means that the extension of the logic does not trivialize the original logic and specifically
classical Propositional Logic for consistent premises.
Corollary 1. Let 𝑇 be a classically consistent theory and 𝐴𝐹 𝑇 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ its corresponding
argumentation logic framework. Let also 𝜑 be a propositional formula such that 𝑇 ̸|= 𝜑. Then 𝑇 ̸|=𝐴𝐿+ 𝜑2 .
Proof. Follows directly from the theorem and the equivalence of propositional logic with
argumentation logic, i.e. that 𝑇 ̸|= 𝜑 iff 𝑇 ̸|=𝐴𝐿 𝜑.
We also know from proposition 3 that 𝒜ℒ+ contains the original logic of 𝒜ℒ. Hence 𝒜ℒ+ , is a
conservative extension of Propositional Logic and of 𝒜ℒ when the given premises 𝑇 are classically
consistent.
The following example taken from [5] clarifies the link between the extended 𝐴𝐿+ and the original
𝐴𝐿 and how it gives genuinely new cases of acceptable formulae.
Example 7.
Consider the following two theories of propositional logic:
• 𝑇1 = {¬(𝛽 ∧ 𝛼), ¬𝛼} 𝑇2 = {¬(𝛽 ∧ 𝛼), ¬(𝛼 ∧ 𝛾), ¬(𝛼 ∧ ¬𝛾)}
It is easy to see that the argument {𝛽} is acceptable in 𝒜ℒ relative to theory 𝑇1 . Its minimal attack {𝛼} is
directly self-inconsistent and hence self-attacking (i.e. 𝑇1 ∪ {𝛼} ⊢𝑀 𝑅𝐴 ⊥) and so it can be defended by {}.
The argument {𝛽} is also acceptable in 𝒜ℒ relative to theory 𝑇2 even though its attack 𝛼 is not directly
inconsistent. The defense against the attack of {𝛼}, namely {¬𝛼}, is such that ({¬𝛼}, {𝛽}) ∈ 𝒜ℒ. Notice,
however, that this attack of {𝛼}, is itself a non-acceptable self-defeating argument, as it cannot defend
acceptably against its attack by {𝛾}: the only possible defense of {¬𝛾} in non-acceptable in the context
of {𝛼} because {𝛼} attacks {¬𝛾}. Therefore recognizing the non-acceptability of the attack {𝛼} is an
alternative way to enforce the acceptability of {𝛽}. The extended acceptability semantics of 𝒜ℒ+ uses this
alternative way. Importantly, it does so in the same way for both theories 𝑇1 , 𝑇2 .
The extended acceptability semantics becomes relevant when the theory of premises is inconsistent,
and attacks like {𝛼} above cannot be defended acceptably by {¬𝛼}.
Example 8. [Example 7 cnt.]
Consider the following theory, obtained from 𝑇2 by making also ¬𝛼 non acceptable: 𝑇3 = 𝑇2 ∪ {¬(¬𝛼 ∧
𝛿), ¬(¬𝛼 ∧ ¬𝛿)}. The attack {𝛼} cannot be acceptably defended because the possible defense of {¬𝛼} is
non-acceptable in a way similar to the non-acceptability of {𝛼} shown above (replacing {𝛾} with {𝛿}).
Nevertheless, as {𝛼} is by itself non-acceptable it is reasonable to accept {𝛽} as acceptable, as 𝒜ℒ+ does.
Finally, we point out that in the extended Argumentation Logic, 𝐴𝐿+ , we can drop the second
element of the defense relation on the formulation of AL in definition 4, namely that now the defence
relation does not need to contain that the empty argument defends against any self-attacking argument.
Indeed, this is now covered by the extended semantics, as any directly self-inconsistent arguments are
non-acceptable because they are self-attacking and hence such attacks will be taken into account as
harmless by the semantics of 𝐴𝐿+ .
2
Here |= denotes the classical entailment of Propositional Logic. The entailment relation |=𝐴𝐿+ in 𝒜ℒ+ , is defined (as it is
for |=𝐴𝐿 in 𝒜ℒ) by 𝑇 |=𝐴𝐿+ 𝜑 iff 𝒜ℒ+ ({𝜑}, {}) holds and 𝒜ℒ+ ({¬𝜑}, {}) does not hold.
5. Conclusions
We have shown how to extend Argumentation Logic to capture the intuitive idea that for attacks
which are by themselves self-defeating it is not necessary to defend against. The proposed extended
Argumentation Logic clarifies how reasoning to a conclusion can be achieved either by showing
explicitly how the conclusion is supported or by showing that opposing conclusions are by themselves
invalid. This resonates with classical logical reasoning where conclusions are either directly derived
from the premises or their oppositions are shown, via Reductio ad Absurdum, to be inconsistent with
the given premises.
The extended argumentation semantics is based on definition 6. We can then consider applying this
definition iteratively to give possible further extensions of acceptability and study the properties of such
extensions. We can also study how we can further extend the framework to allow directly inconsistent
premises. For example, we can examine how we can accommodate this by simply extending the defense
relation so that any two subsets of the premises which are directly inconsistent with each other are
able to defend against each other.
References
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2 (1992) 719–770.
[2] A. C. Kakas, P. Mancarella, P. M. Dung, The acceptability semantics for logic programs, in: Proc.
of 11th Int. Conf. on Logic Programming, 1994, pp. 504–519.
[3] A. C. Kakas, F. Toni, Computing argumentation in logic programming, J. Log. Comput. 9 (1999)
515–562.
[4] A. Kakas, P. Mancarella, On the semantics of abstract argumentation., Journal of Logic and
Computation [electronic only] 23 (2013) 991–1015.
[5] A. Kakas, P. Mancarella, F. Toni, On Argumentation Logic and Propositional Logic, Studia Logica
106 (2018) 237–279.
[6] P. M. Dung, On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic
Reasoning, Logic Programming and n-Person Games, Artificial Intelligence 77 (1995) 321 – 357.
[7] R. Baumann, G. Brewka, M. Ulbricht, Shedding new light on the foundations of abstract argumen-
tation: Modularization and weak admissibility, Artif. Intell. 310 (2022) 103742.
[8] P. Baroni, M. Giacomin, G. Guida, Scc-recursiveness: a general schema for argumentation seman-
tics, Artif. Intell. 168 (2005) 162–210.
[9] G. A. Bodanza, F. A. Tohmé, Two approaches to the problems of self-attacking arguments and
general odd-length cycles of attack, J. Appl. Log. 7 (2009) 403–420.
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(2009) 109–145.
6. Appendix: Proof of Theorem 1
We will use the following Lemma.
Lemma 1. Let 𝑇 be a directly consistent theory and 𝐴𝐹 𝑇 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ its corresponding argu-
mentation logic framework. Let also
𝜑1 , 𝜑2 , . . . , 𝜑𝑘 , 𝑘≥1
be a sequence of formulae such that:
(i) 𝜑𝑖 ̸= 𝜑𝑗 , for each 𝑖 ̸= 𝑗
(ii) {𝜑𝑖 } attacks {𝜑𝑖−1 }, for each 𝑖 = 2, . . . , 𝑘
(iii) ({𝜑𝑘 }, {𝜑1 , . . . , 𝜑𝑘−1 }) ̸∈ 𝒜ℒ
Then
(𝜑1 , {}) ̸∈ 𝒜ℒ
Proof. If 𝑘 = 1 the required result is given by condition (iii). Hence let 𝑘 ≥ 2. We show
({𝜑𝑘−1 }, {𝜑1 , . . . , 𝜑𝑘−2 }) ̸∈ 𝒜ℒ (*)
Let ∆0 = {𝜑1 , . . . , 𝜑𝑘−2 } and ∆ = {𝜑𝑘−1 }. By condition (ii), {𝜑𝑘 } attacks {𝜑𝑘−1 }. The only defence
against this attack is {𝜑𝑘 } which is non-acceptable with respect to ∆ ∪ ∆0 by (iii). Hence (*) holds. By
iterating this process 𝑘 − 1 times, we obtain the sequence of facts
({𝜑𝑘−2 }, {𝜑1 , . . . , 𝜑𝑘−3 }) ̸∈ 𝒜ℒ
...
(𝜑1 , {}) ̸∈ 𝒜ℒ
Theorem 1 Let 𝑇 be a classically consistent theory and 𝐴𝐹 𝑇 = ⟨𝐴𝑟𝑔, 𝐴𝑡𝑡, 𝒟𝑒𝑓 ⟩ its corresponding
argumentation logic framework. Let also 𝜑 be a propositional formula such that ({𝜑}, {}) ̸∈ 𝒜ℒ holds.
Then ({𝜑}, {}) ̸∈ 𝒜ℒ+ also holds.
Proof. Theorem 2 of [5] allows us without loss of generality to restrict consideration to singleton sets
of arguments both for attacks as well as defenses when examining the acceptability or non-acceptability
of argument sets. Let us denote by ∆ the set {𝜑}. We need to show that when (∆, {}) ̸∈ 𝒜ℒ then
(∆, {}) ̸∈ 𝒜ℒ+ . We will show this by contradiction. Assume both (∆, {}) ̸∈ 𝒜ℒ and (∆, {}) ∈ 𝒜ℒ+
hold. By the definition of 𝒜ℒ, (∆, {}) ̸∈ 𝒜ℒ implies that there exists a sequence of tuples of formulae
𝑎𝑖 = ⟨𝜑𝑖 , 𝜑𝑖 ⟩ 𝑖 = 1, ..., 𝑘 for some natural number 𝑘 ̸= 0 (†)
such that {𝜑1 } attacks ∆ and for each 𝑖 = 2, ..., 𝑘:
• {𝜑𝑖 } attacks {𝜑𝑖−1 } (a)
• {𝜑𝑖 } defends against {𝜑𝑖 } (b)
• ({𝜑𝑖 }, ∆ ∪ {𝜑𝑗 | 𝑗 = 1, . . . , 𝑖 − 1}) ̸∈ 𝒜ℒ (c)
• ∆ ∪ {𝜑𝑗 | 𝑗 = 1, ..., 𝑘 − 1} attacks {𝜑𝑘 } (d)
This is a sequence of consecutive defenses and attacks, starting from ∆ as the first defense, such that
all defenses in the sequence are non-acceptable with respect to the union of defenses that are prior to
them. By the definition of non-acceptability, the third condition above also means that all these defences
are not equal to each other, i.e. 𝜑𝑛 ̸= 𝜑𝑚 for any 𝑛 ̸= 𝑚, and hence also 𝜑𝑛 ̸= 𝜑𝑚 for any 𝑛 ̸= 𝑚.
Let us also assume that 𝑘 is the smallest number for which such a sequence exists for the non-
acceptability of ∆ = {𝜑}. If 𝑘 = 1, then the non-acceptability of ∆ comes from the fact that ∆
is self-attacking (i.e., {𝜑1 } = ∆). By the definition of 𝒜ℒ+ , this is in direct contradiction with
(∆, {}) ∈ 𝒜ℒ+ .
Hence 𝑘 > 1. Notice also that, for any 𝑛 = 2, ..., 𝑘 − 1, 𝜑𝑛 ̸= 𝜑, as otherwise 𝑘 would not be the
least value for such a sequence.
we have two cases:
• Case 1: 𝜑𝑗 attacks 𝜑𝑘 for some 𝑗 ≤ 𝑘
i.e. some previous defense 𝜑𝑗 or itself attacks 𝜑𝑘 , or
• Case 2: ∆ attacks 𝜑𝑘 .
Case (1): We can show that ({𝜑1 }, {}) ̸∈ 𝒜ℒ holds as follows. First notice that 𝜑𝑗 attacks 𝜑𝑘 implies
({𝜑𝑘 }, {𝜑𝑗 }) ̸∈ 𝒜ℒ, by definition of 𝒜ℒ. Since all 𝜑𝑖 are different between them, we have that
({𝜑𝑘 }, {𝜑1 , 𝜑2 , ..., 𝜑𝑗−1 , 𝜑𝑗 , 𝜑𝑗+1 , ..., 𝜑𝑘−1 }) ̸∈ 𝒜ℒ
This, along with (a) above and, again, the fact that all 𝜑𝑖 are different between them, allow us to
apply Lemma 1 and to conclude
({𝜑1 }, {}) ̸∈ 𝒜ℒ.
We also know from (∆, {}) ∈ 𝒜ℒ+ that the attack of 𝜑1 against ∆ must also be non-acceptable,
i.e., we also have ({𝜑1 }, {}) ̸∈ 𝒜ℒ holds. These two facts lead (via Theorem 2 in [5]) to 𝑇 |= 𝜑1 and
𝑇 |= 𝜑1 contradicting the classical consistency of 𝑇 .
Case 2: ∆ attacks 𝜑𝑘
By the symmetry of the attack we also have that 𝜑𝑘 attacks ∆. This is another attack against ∆. We
will show that its defense 𝜑𝑘 is non-acceptable both in the empty context, i.e., ({𝜑𝑘 }, {}) ̸∈ 𝒜ℒ holds,
and is also non-acceptable with respect to ∆, i.e., also ({𝜑𝑘 }, ∆) ̸∈ 𝒜ℒ holds. The latter means that
the attack of 𝜑𝑘 cannot be acceptably defended against and so by (∆, {}) ∈ 𝒜ℒ+ this attack must be
non-acceptable, i.e., ({𝜑𝑘 }, {}) ̸∈ 𝒜ℒ. These two results, as above in case 1, using Theorem 2 in [5])
give 𝑇 |= 𝜑𝑘 and 𝑇 |= 𝜑𝑘 leading to a contradiction of the classical consistency of the premises 𝑇 .
To show these two non-acceptability results we consider the sequence of formulae, 𝜓1 , ..., 𝜓𝑘 where
𝜓𝑖 = 𝜑𝑘+1−𝑖 for 𝑖 = 1, ..., 𝑘. The sequence starts with 𝜓1 = 𝜑𝑘 that defends against the attack of 𝜑𝑘
on ∆. From the way that the formulae 𝜑𝑖 have been constructed above and by the symmetry of attacks
and the definition of defense in argumentation the formulae 𝜓𝑖 are different and the following hold:
(i) 𝜓𝑖 ̸= 𝜓𝑗 , for each 𝑖 ̸= 𝑗
(ii) {𝜓𝑖 } attacks {𝜓𝑖−1 }, for each 𝑖 = 2, . . . , 𝑘
(iii) ({𝜓𝑘 }, {}) ̸∈ 𝒜ℒ
We also have the following two conditions:
(iii) ({𝜓𝑘 }, {}) ̸∈ 𝒜ℒ
(iv) ∆ attacks 𝜓𝑘
These two conditions hold by the construction of the sequence of 𝜑𝑖 and 𝜓𝑘 = 𝜑1 (𝜑1 attacks ∆ so
the reverse also holds and 𝜑1 cannot be acceptably defended by ∆ so this attack must be non-acceptable
for (∆, {}) ∈ 𝒜ℒ+ hold to hold).
From the forth condition it follows trivially that ({𝜓𝑘 }, ∆ ∪ {𝜓1 , ..., 𝜓𝑘−1 }) ̸∈ 𝒜ℒ. Then as in
Lemma 1 it follows that ({𝜓1 }, ∆) ̸∈ 𝒜ℒ (i.e., ({𝜑𝑘 }, ∆) ̸∈ 𝒜ℒ as required) since also the formula 𝜑 in
∆ is different from all the 𝜑𝑖 and hence the 𝜓𝑖 .
Finally, we consider the third condition (iii) above and extend the sequence of 𝜓𝑖 with the sequence of
attacks and defenses that the non-acceptability of 𝜓𝑘 entails. This is a sequence of formulae 𝜎1 , ..., 𝜎𝑚
such that 𝜎1 attacks 𝜓𝑘 and 𝜎𝑗 attacks 𝜎𝑗−1 for 𝑗 = 2, . . . , 𝑚. All the formulae 𝜎𝑗 are different between
them but some may be equal to some of the 𝜓𝑖 formulae. If this is the case we can iterative replace such
formulae 𝜓𝑖 , starting with the one that appears deepest in 𝜎𝑗 sequence, with the subsequence of the
sequence of 𝜎𝑗 formulae until we have a sequence of formuale which are all different. We can then
apply again Lemma 1 to arrive at the result of ({𝜓1 }, {}) ̸∈ 𝒜ℒ, i.e., that ({𝜑𝑘 }, {}) ̸∈ 𝒜ℒ, as required.