=Paper=
{{Paper
|id=Vol-3086/paper7
|storemode=property
|title=A Dialectical Characterisation of Argument Game Proof Theories for Classical Logic Argumentation
|pdfUrl=https://ceur-ws.org/Vol-3086/paper7.pdf
|volume=Vol-3086
|authors=Federico Castagna
|dblpUrl=https://dblp.org/rec/conf/aiia/Castagna21
}}
==A Dialectical Characterisation of Argument Game Proof Theories for Classical Logic Argumentation==
https://ceur-ws.org/Vol-3086/paper7.pdf
A Dialectical Characterisation of Argument Game
Proof Theories for Classical Logic Argumentation
Federico Castagna1
1
King’s College London, Bush House, 30 Aldwych, London, WC2B 4BG, United Kingdom
Abstract
Argument game-based proof theories provide procedural structures capable of determining the status
of an argument. Given an argumentation framework, argument games identify the membership of an
argument to a specific extension simulating a dialogue between two opposing contenders. The semantics
meant to be captured dictates the rules of the played game, which serve to describe how the players
can achieve victory. Dialectical Classical logic Argumentation (Dialectical Cl-Arg) is a recent approach
that provides real-world dialectical characterisations of Cl-Arg arguments by resource-bounded agents
while preserving the rational criteria established by the rationality postulates. This paper combines both
subjects and introduces argument games for Dialectical Cl-Arg, highlighting the properties and strengths
enjoyed by these games in comparison with the standard ones. The result will be a proof theory capable
of approximating real-world non-monotonic single-agent reasoning processes.
Keywords
Argumentation, Argument game, Classical logic, Dialectic, Bounded reasoning
1. Introduction
Human reasoning evolved to produce and evaluate arguments [1]. Trying to consolidate
possessed information by formulating reasons (arguments) that challenge or defend it is an
ordinary procedure in which humans engage. This process is not only common but even
necessary: how could it be possible, otherwise, to decide what to believe or trust without
being misled by a non-reliable source of information? This ‘scaffolding’ (as defined in [2])
role of dialogues and arguments can be seen in social and lone thinking practices where
the reasoner(s) evaluates the possessed information by constructing counter-arguments that
assesses its acceptability. That is to say, every reasoning process entails a dialogue, and every
dialogue entails arguments irrespective of the type of interacting agents: be they humans and
AIs, among themselves and with humans. Thanks to its important role, argumentation has been
developed as a theory able to characterize the essence of non-monotonic reasoning through the
dialectical interplay of arguments [3]. Intuitively, in order to determine if a piece of information
is acceptable, it will suffice to prove that the argument (in which the specific information is
embedded) is justified under one of Dung’s semantics. A way of doing this is to show the
membership of the argument to a winning strategy of an argument game (as described, for
example, in [4], [5] and [6]).
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Although a plethora of works has successfully shown various instantiations of Dung’s abstract
argumentation framework (AF) and reached different goals, none of these approaches managed
to provide a full rational account of real-world resource-bounded agents. Undoubtedly, the
introduction of the rationality postulates [7][8] has allowed eschewing the arising of counter-
intuitive results in AFs instantiations. However, the rationality postulates require a consumption
of resources that far exceed the availability of real-world agents.
1.1. Contribution
The main contribution of this research is the development of argument games for Dialectical
Classical Logic Argumentation (Dialectical Cl-Arg [9]), a recent approach that provides real-
world dialectical characterisations of AFs by resource-bounded agents. This approach satisfies
the rationality postulates (under minimal requirements) and revolves around the core notion
of dialectical defeats. Such defeats enable argumentative interactions more aligned with the
dialectical reasoning of real-world resource-bounded agents. Thus, their presence requires the
implementation of dialectical argument game proof theories capable of conveying the same idea
as single-agent reasoning processes.
2. Background
Argumentation has been developed as a theory of reasoning able to characterize the essence of
non-monotonic logic via the dialectical interplay of arguments. According to Dung’s seminal
paper [3], an Argumentation Framework (AF) is composed of a set of arguments ‘AR’ and a
binary ‘attacks’ relation, which denotes the conflicts existing among the arguments in AR, i.e.,
AF = ⟨AR, attacks⟩. Various semantics s have also been presented and each of them specifies
the status of (sceptically or credulously) justified (i.e., acceptable) arguments. Several works
stemmed from [3], some of which introduced different ways of structuring arguments and
instantiating Dung’s abstract AF [10][11][12]. For example, Classical Logic Argumentation
(Cl-arg) [13][14] is one of such instantiation that builds AFs using classical logic as its underlying
language.
2.1. Dialectical Cl-Arg
Unlike standard formalisation of Cl-Arg, real-world agents behave pragmatically and do not
need to: (i) always construct every argument defined by a base of wff and (ii) enforce consistency
and subset minimality checks on their arguments (nor they have enough computational power
to do these checks, given their limited resources). Dialectical Cl-Arg provides a formalisation of
real-world modes of dialectical reasoning from resource-bounded agents whilst satisfying the
rationality postulates [7][8].
Definition 1. [Dialectical Arguments] X = (Δ, Γ, 𝛼) is a dialectical argument defined by a
base ℬ of classical wff, if (Δ ∪ Γ) ⊆ ℬ, Δ ∩ Γ = ∅, and Δ ∪ Γ ⊢𝑐 𝛼 1 . If 𝛼 = ⋏, then 𝑋 is said to be a
1
The symbol ‘⊢𝑐 ’ denotes the consequence relation of classical logic.
falsum argument. If Γ = ∅, then 𝑋 is said to be unconditional; else 𝑋 is conditional. Finally, if Δ =
∅, then 𝑋 is said to be unassailable.
Δ, Γ and 𝛼 are respectively referred to as the premises (Prem(𝑋 )), suppositions (Supp(𝑋 )) and
conclusion (Con(𝑋 )) of 𝑋 = (Δ, Γ, 𝛼).
Attacks and defeats for Dialectical Cl-Arg work differently than their respective counterparts
for Cl-Arg. The reason is the presence of suppositions embedded in the internal structure of the
arguments. This epistemic distinction between information considered true (i.e., the premises of
an argument) and information supposed true for the sake of the argument (i.e., the supposition
of the argument) proves useful also in solving the so-called ‘foreign commitment problem’ 2 .
Definition 2. [Attacks and Defeats][9] Let AR be a set of dialectical arguments defined by
a base ℬ. The attack relation ‘attacks’ ⊆ AR × AR is defined as follows. For any 𝑋 = (Δ, Γ, 𝛼),
𝑌 = (Π, Σ, 𝛽) ∈ AR: attacks(𝑋 , 𝑌 ) iff:
• if 𝛼 ≠ ⋏ then 𝛼 ∈ Π (𝑋 attacks 𝑌 on 𝛼, equivalently on 𝑌 ′ = ({𝛼}, ∅, 𝛼));
• if 𝛼 = ⋏ (𝑋 attacks 𝑌 on any 𝜙 ∈ Γ ∩ Π, equivalently on any 𝑌 ′ = ({𝜙}, ∅, 𝜙)).
Let ≺ be a strict partial ordering over AR. Then, for every 𝑋, 𝑌 such that attacks(𝑋 , 𝑌 ), defeats(𝑋 , 𝑌 )
iff exactly one of the following holds:
• either 𝑋 is an argument of the form (∅, Γ, ⋏);
• else, ∃𝜓 ∈ Π such that attacks(𝑋 , 𝑌 ) on 𝜓, and 𝑋 ⊀ ({𝜓 }, ∅, 𝜓 ).
𝑋 ⇒ 𝑌 will stand for “defeats(𝑋 , 𝑌 )”, and 𝑋 ⇏ 𝑌 will stand for “¬defeats(𝑋 , 𝑌 )”.
Cl-Arg assumes instantiation of an AF by all arguments defined by a base ℬ of classical wff, a
task that proves to be unfeasible for agents with limited resources. As such, dialectical argu-
ments (Definition 1) along with the described defeat relation (Definition 2) allow us to introduce
a dialectical AF as an argumentation framework ⟨AR, defeats⟩ where AR is any subset of the
dialectical arguments defined by a base ℬ.
The difference between a defeat and a dialectical defeat for dialectical AFs is the refer-
ence to a set 𝒮 of arguments. The general idea is that, when challenging the acceptability of
an argument with respect to a set 𝒮, the defeating argument can suppose premises from all
the arguments in 𝒮. Whereas, the argument that defends 𝒮 can only suppose the premises of
the defeating argument. This new kind of defeat forces us to adjust the standard semantics
accordingly.
Definition 3. [Dialectical defeats and semantics for dialectical AFs][9]
Let ⟨AR, defeats⟩ be a dialectical AF, 𝒮 ⊆ AR and 𝑋, 𝑌 ∈ AR. Then:
1) 𝑋 dialectically defeats 𝑌 with respect to 𝒮, denoted 𝑋 ⇒𝒮 𝑌, if defeats(𝑋 , 𝑌 ) and Supp(𝑋 )
⊆ Prem(𝒮 ∪{𝑌 }).
2
As extensively explained in [15], the foreign commitment problem is the issue that arises in dialogical ap-
plications when agents are forced to commit to the premises of their interlocutors in order to challenge their
arguments.
2) 𝒮 is conflict free if ∀𝑍, 𝑌 ∈ 𝒮, 𝑍 ⇏𝒮 𝑌 .
3) 𝑌 is acceptable with respect to 𝒮 if ∀𝑋 such that 𝑋 ⇒𝒮 𝑌, ∃𝑍 ∈ 𝒮 such that 𝑍 ⇒{𝑋 } 𝑋.
4) Let 𝒮 be conflict free. Then 𝒮 is: an admissible extension iff 𝑋 ∈ 𝒮 implies 𝑋 is acceptable
with respect to 𝒮; a complete extension iff 𝒮 is admissible and if 𝑋 is acceptable with respect
to 𝒮 then 𝑋 ∈ 𝒮; a preferred extension iff it is a set inclusion maximal complete extension;
the grounded extension iff it is the set inclusion minimal complete extension.
Dialectical AFs may enjoy some specific properties, as explained in [9]. Here we are going to
outline four of them (P1, P2, P3, P4), which will prove useful in the next sections.
Remark 1. Given a dialectical AF = ⟨AR, defeats⟩:
(𝑃1) ∀𝑋 ∈ 𝐴𝑅: 𝛼 ∈ Prem(𝑋 ) implies that ({𝛼}, ∅, 𝛼) ∈ 𝐴𝑅;
(𝑃2) ∀𝑋 ∈ 𝐴𝑅: if 𝑋 ′ is the logically equivalent argument of 𝑋 (i.e., the only difference between 𝑋
and 𝑋 ′ is the different distribution of premises and supposition), then 𝑋 ′ ∈ 𝐴𝑅;
(𝑃3) If (Δ, ∅, 𝛼) ∈ 𝐴𝑅 and (Γ, ∅, 𝛼) ∈ 𝐴𝑅, then either (Δ, ∅, ⋏) ∈ 𝐴𝑅 or (Γ, ∅, ⋏) ∈ 𝐴𝑅 or
(Δ ∪ Γ, ∅, ⋏) ∈ 𝐴𝑅;
(𝑃4) If (Γ, ∅, 𝛼) ∈ 𝐴𝑅, Δ ⊆ Γ, Δ ≠ ∅ and Δ is such that it does not share any propositional atomic
formulae with (Γ ⧵ Δ) ∪ {𝛼} 3 , then either (Δ, ∅, ⋏) ∈ 𝐴𝑅 or (Γ ⧵ Δ, ∅, 𝛼) ∈ 𝐴𝑅.
2.2. Standard Argument Games
Finally, let us also review the fundamental notions of the standard argument games as described
in [4]. Notice that these definitions have been modified to accommodate dialectical AFs (which
is a fair straightforward adaptation). However, recall that the main contributions of this paper
concern the development of argument games for Dialectical Cl-Arg that involves dialectical
defeats (Definition 3): this entails a non-trivial modification of the standard games.
In a nutshell, an argument game is played by two players, PRO (for proponent) and OPP (for
opponent), each of which is referred to as the other’s ‘counterpart’. PRO starts the game by
moving an initial argument X that it wants to test. After that, both players take turns in moving
arguments against their counterpart’s moves. This generates disputes:
Definition 4. [Dispute] A sequence of moves in which each player moves against its counter-
part’s argument is referred to as a dispute. Formally, d = 𝑋—𝑌—𝑍— ⋯ is a dispute, and 𝑋—𝑌 denotes
a player moving argument 𝑌 against an argument 𝑋 played by its counterpart (similarly, 𝑌—𝑍 ). A
sub-dispute 𝑑 ′ of a dispute 𝑑 is any sub-sequence of 𝑑 that starts with the same initial argument.
For example, if 𝑑 = 𝑋—𝑌—𝑍, then 𝑑 ′ = 𝑋—𝑌 would be a sub-dispute of 𝑑.
Notice that, to avoid ambiguity, each argument of a dispute will be labelled with either P or
O (that stands for either one of the two players, PRO or OPP). Hence, 𝑑 = (P)𝑋—(O)𝑌—(P)𝑍 is
a dispute where PRO moves the argument 𝑋, followed by 𝑌 played by OPP and countered by
another move from PRO, 𝑍.
3
Alternatively, if we are dealing with first order logic, Δ is such that it does not share functions and predicates
with (Γ ⧵ Δ) ∪ {𝛼}.
We can now introduce the notion of the (unique) dispute tree, which represents the ‘playing
field’ of the standard argument games. In other words, the dispute tree is the data structure
that contains all the potential moves (and sequences of moves) available to the players.
Definition 5. [Dispute Tree] Let AF = ⟨AR, defeats⟩ be a finite dialectical argumentation frame-
work, and let A ∈ AR. The dispute tree induced by A in the AF is the tree 𝒯 of arguments, such
that 𝒯’s root node is A, every branch of the tree is a different dispute, and ∀𝑋, 𝑌 ∈ AR: 𝑋 is a child
of 𝑌 in 𝒯 iff defeats(𝑋 , 𝑌 ).
From here on, we are going to write PRO(∗) and OPP(∗) to denote the sets of all PRO and OPP
arguments in ∗, where ∗ can be replaced with 𝑑, 𝒯 or any other tree that will be introduced in
the remainder of the paper. Also, LAST(𝑑) will identify the last argument played in a dispute 𝑑.
An argument game is said to be won by the proponent only if it has a winning strategy. That
is to say, only if it can successfully defend the argument it wants to test (i.e., the root of 𝒯)
against any possible arguments moved by the opponent. PRO loses otherwise.
Definition 6. [Winning Strategy] Let 𝒯 be the dispute tree induced by A in a finite dialectical
AF = ⟨AR, defeats⟩. Let also 𝑑 be a dispute in 𝒯. Then, a winning strategy 𝒯 ′ for A is the dispute
tree 𝒯 pruned in a way such that:
(6.1) The set 𝒯D′ of disputes in 𝒯 ′ is a non-empty finite set such that each dispute 𝑑 ∈ 𝒯D′ is
finite and is won by PRO (i.e., LAST(𝑑) ∈ PRO(𝒯 ));
(6.2) ∀𝑑 ∈ 𝒯D′ , ∀𝑑 ′ such that 𝑑 ′ is some sub-dispute of 𝑑, LAST(𝑑 ′ ) = 𝑋 and 𝑋 ∈ PRO(𝒯 ), then
∀𝑌 ∈ OPP(𝒯 ) such that 𝑌 ⇒ 𝑋, there is a 𝑑 ″ ∈ 𝒯D′ such that 𝑑 ′ — 𝑌 is a sub-dispute of 𝑑 ″ .
Informally, the previous definition states that a winning strategy is the dispute tree 𝒯 pruned
in a way such that (6.1) 𝒯D′ is a non-empty finite set, its disputes are finite, end with a PRO
argument and (6.2) are such that OPP has moved exhaustively (i.e., all the moves that OPP could
have played, had been played) and also PRO has countered every defeating argument moved by
OPP.
3. Developing Dialectical Argument Games
In the following sections, we are going to develop argument games for Dialectical Cl-Arg that
accommodate the dialectical defeats and semantics introduced in Definition 3. The resulting
proof theory will present some specific features that will distinguish it from the standard argu-
ment games, although the general structure remains similar. Intuitively, winning a dialectical
game for an argument 𝐴 means having a specific ‘dialectical way’ (depending on the semantics
that the proof theory is meant to capture) for defending the information contained in 𝐴, hence
showing the admissibility of the encoded data.
The main difference between a dispute tree 𝒯 and a dialectical dispute tree 𝒟 can be identified
with the additional reference to a subset 𝒮 ⊆ PRO(𝒯 ). That is to say, 𝒮 represents a candidate
admissible set of PRO arguments such that PRO commits to their premises. Recall once again
that, when challenging the acceptability of an argument with respect to a set 𝒮, the defeating
argument can suppose premises from all the arguments in 𝒮. Whereas, the argument that
defends 𝒮 can only suppose the premises of the defeating argument. Another important
difference between standard and dialectical games is that the latter handles partially instantiated
dialectical AFs (pdAFs), i.e. dialectical AFs that satisfy properties P1, P2, P3 and P4 4 .
As a consequence, each dialectical game enjoys specific properties that encapsulate the di-
alectical uses of arguments by real-world resource-bounded agents, thus succeeding in better
approximating a process capable of bridging formal (proof-theoretical) and informal (real-world
exchange of arguments) single-agent reasoning.
We can now formally introduce the (unique) dialectical dispute tree induced by A wrt a
set 𝒮 5 :
Definition 7. [Dialectical Dispute Tree] Let 𝒯 be the dispute tree induced by A in a finite
pdAF = ⟨AR, defeats⟩. Let also 𝒮 ⊆ PRO(𝒯 ). Then, the dialectical dispute tree 𝒟 induced by A
with respect to 𝒮 is the dispute tree 𝒯 pruned in a way such that ∀𝑋, 𝑌 ∈ AR: 𝑋 is a child of 𝑌 in 𝒟
iff defeats(𝑋 , 𝑌 ) and:
1. If 𝑋 ∈ PRO(𝒟 ) and 𝑌 ∈ OPP(𝒟 ), then 𝑋 ⇒{𝑌 } 𝑌, i.e. 𝑋 defeats 𝑌 and Supp(𝑋 ) ⊆ Prem(𝑌 );
2. If 𝑋 ∈ OPP(𝒟 ) and 𝑌 ∈ PRO(𝒟 ), then 𝑋 ⇒𝒮 𝑌, i.e. 𝑋 defeats 𝑌 with respect to a set 𝒮 and
Supp(𝑋 ) ⊆ Prem(𝒮 ∪ {𝑌 }).
The ‘playing field’ of the dialectical argument games (i.e., the data structure on the basis of
which the games are played) is still depicted by the dispute tree 𝒯. Indeed, the relationship
existing between the dispute tree 𝒯 induced by 𝐴 in a finite pdAF and the dialectical dispute
tree 𝒟 induced by 𝐴 wrt 𝒮 is such that 𝒟 is ‘contained’ in 𝒯 (since 𝒟 is a pruned version of 𝒯),
as shown in the following example.
Example 1. Figure 1 presents the (incomplete) dispute tree 𝒯 induced by 𝐴1 in a finite pdAF =
⟨AR, defeats⟩ and the corresponding (incomplete) dialectical dispute tree 𝒟 induced by 𝐴1 wrt
𝒮 = {𝐴1 , 𝐺2 , 𝑂1 } in the same pdAF. Both trees are incomplete since the purpose of the example is
just to show the relationship existing between them. For the same reason, we also avoid listing all
the arguments of the pdAF.
Observe that, unlike 𝒯, where no set is taken into consideration, the defeats in 𝒟 are parametrized
to the set 𝒮. This implies that, when defeating PRO’s arguments, the arguments moved by OPP can
only suppose the premises of the arguments in the set 𝒮. No such restrictions exist for 𝒯. Notice
that, even if we keep extending both trees, dispute 𝑑 = (P)𝐴1 —(O)𝐹2 —(P)𝐺2 will never be part of 𝒟.
This is because, according to Definition 7 (which also emphasizes how dialectical defeats work),
PRO can move 𝐺2 only if Supp(𝐺2 ) ⊆ Prem(𝐹2 ). However, this is never going to be the case since the
formula ¬𝑎 ∨ ¬𝑏 ∉ Prem(𝐹2 ). Therefore, even if the two trees were identical in every other branch,
the absence of dispute 𝑑 will still make 𝒟 ‘contained’ in 𝒯.
4
Refer to Remark 1.
5
Notice, however, that there can be different dialectical dispute trees induced by A wrt to different sets 𝒮. That
is to say, a dialectical dispute tree 𝒟1 induced by A wrt 𝒮1 constitutes a different data structure than a dialectical
dispute tree 𝒟2 induced by A wrt 𝒮2 .
Figure 1: The (incomplete) dispute tree 𝒯 (on the left) induced by 𝐴1 in a finite pdAF = ⟨AR, defeats⟩ and the
corresponding (incomplete) dialectical dispute tree 𝒟 (on the right) induced by 𝐴1 wrt 𝒮 = {𝐴1 , 𝐺2 , 𝑂1 } in the same
pdAF = ⟨AR, defeats⟩.
Dialectical argument games share with the standard argument games the notion of winning
strategy: in order to win the game for an argument A, PRO must have a winning strategy for it.
It will lose otherwise. However, the two definitions slightly differ since a dialectical winning
strategy has to take into account the set 𝒮 targeted by the dialectical defeats:
Definition 8. [Dialectical Winning strategy] Let 𝒟 be the dialectical dispute tree induced by
A wrt 𝒮 in a finite pdAF = ⟨AR, defeats⟩ and let 𝑑 be a dispute in 𝒟. Then, a dialectical winning
strategy 𝒲 for A corresponds to the dialectical dispute tree 𝒟 pruned in a way such that:
(8.1) The set 𝒲D of disputes in 𝒟 is a non-empty finite set such that each dispute 𝑑 ∈ 𝒲D is finite
and is won by PRO (i.e., LAST(𝑑) ∈ PRO(𝒟 ));
(8.2) ∀𝑑 ∈ 𝒲D , ∀𝑑 ′ such that 𝑑 ′ is some sub-dispute of 𝑑, LAST(𝑑 ′ ) = 𝑋 and 𝑋 ∈ PRO(𝒟 ), then
∀𝑌 ∈ OPP(𝒟 ) such that 𝑌 ⇒𝒮 𝑋, there is a 𝑑 ″ ∈ 𝒲D such that 𝑑 ′ — 𝑌 is a sub-dispute of 𝑑 ″ .
Similarly to Definition 6, the previous definition states that a dialectical winning strategy cor-
responds to the dialectical dispute tree 𝒟 pruned in a way such that (8.1) 𝒲D is a non-empty
finite set, its disputes are finite, end with a PRO argument and are such that (8.2) OPP has
moved exhaustively and also PRO has countered each defeating argument moved by OPP. The
difference is in the dialectical defeats: the nodes are no more connected by means of the defeats
relations among arguments, but through dialectical defeats among arguments that target the
set 𝒮.
We now have all the elements needed to formally introduce the protocol of the dialecti-
cal admissible/preferred game. Similar to a list of instructions, this protocol determines the legal
moves that can be performed by the players. The game unfolds as a result of the legal arguments
played and terminates when there are no more valid moves available. When this happens,
the status of the root of the tree is evaluated. The presence of a winning strategy for such an
argument assigns the victory to PRO. Strictly speaking, OPP never wins: its purpose is to counter
each argument moved by the proponent in order to assist it in testing the admissibility of the
root argument (indeed, argument games are formalisations of single-agent reasoning processes).
Nevertheless, OPP can still prevent PRO’s victory by invalidating its winning strategy.
4. Dialectical Admissible/Preferred Games
When we play a Φ-dialectical game we are increasingly building, starting from the root A and
following the legal moves licensed by the protocol Φ, a dialectical dispute tree denoted as Φ-𝒟 n .
Each node of such tree corresponds to an argument progressively played by either PRO or
OPP that is labelled with a positive integer 𝑖 (with 1 ≤ 𝑖 ≤ 𝑛). These additional labels allow
identifying the order in which the arguments have been played, hence, also determining the
current stage (i.e., the 𝑛th-stage) of the Φ-dialectical game. Recall that the dispute tree 𝒯 induced
by A represents the playing field of the games and every Φ-dialectical game for A is contained
within its data structure (i.e., Φ-𝒟 n is a ‘pruned-version’ of 𝒯). Being a dialectical dispute tree,
even Φ-𝒟 n is constructed wrt a set 𝒮 ⊆ PRO(𝒯 ), however, such 𝒮 can gradually increase with
each new move made by PRO during the game. Indeed, 𝒮 is composed of the same arguments
moved by PRO in Φ-𝒲 n (i.e., a dialectical winning strategy for A of Φ-𝒟 n ), which can be
extended while the game proceeds 6 . As it will be shown, observe also that 𝒮 is still a different
set than PRO(Φ-𝒲 n ), meaning that it will modify its members according to the changes in
PRO(Φ-𝒲 n ), but it will never be empty even if there is no winning strategy Φ-𝒲 n . Formally:
Definition 9. [Φ-𝒟 n and Φ-𝒲 n ] A dialectical dispute tree Φ-𝒟 n induced by A wrt 𝒮 ⊆
PRO(𝒯 ) (with 𝒮 ≠ ∅) in a finite pdAF = ⟨AR, defeats⟩ is the tree that starts from the argument A
and it is progressively built up to the 𝑛th-move of one of the two players, such that each node of the
tree is labelled with a positive integer 𝑖 (for 1 ≤ 𝑖 ≤ 𝑛). Moreover, ∀𝑋, 𝑌 ∈ AR: 𝑋 is a child of 𝑌 in
Φ-𝒟 n iff 𝑋 is a legal move identified by Φ, defeats(𝑋 , 𝑌 ) and:
1. If 𝑋 ∈ PRO(Φ-𝒟 n ) and 𝑌 ∈ OPP(Φ-𝒟 n ) 7 , then 𝑋 ⇒ 𝑌, i.e. 𝑋 defeats 𝑌 and Supp(𝑋 ) ⊆
{𝑌 }
Prem(𝑌 );
2. If 𝑋 ∈ OPP(Φ-𝒟 n ) and 𝑌 ∈ PRO(Φ-𝒟 n ), then 𝑋 ⇒𝒮 𝑌, i.e. 𝑋 defeats 𝑌 with respect to a set
𝒮 and Supp(𝑋 ) ⊆ Prem(𝒮 ∪ {𝑌 }).
Finally, Φ-𝒲 n will denote a dialectical winning strategy for A of Φ-𝒟 n .
6
Although the set 𝒮 can increase the number of its members while the game goes on, it can never exceed the
size of PRO(𝒯 ). Keep in mind that every Φ-dialectical game for A is contained in the dispute tree 𝒯 induced by A
(since 𝒯 corresponds to the playing field of the game).
7
Similarly to PRO(𝒟 ) and OPP(𝒟 ), writing PRO(Φ-𝒟 n ) and OPP(Φ-𝒟 n ) denote the sets of all PRO, respectively
OPP, arguments in Φ-𝒟 n .
Every stage of a Φ-dialectical game can then be identified with a specific dialectical dispute
tree Φ-𝒟 n . As we are going to see, this notion is essential for a proper account of the dialectical
defeats in the game protocol 8 .
It is interesting to notice that, during a Φ-dialectical game, a dialectical defeat that occurred in
an early stage of the game might not take place in a more advanced phase of the same game.
This can be caused by an update of the current 𝒮, the set parametrized by OPP for performing
dialectical defeats. We denote this anomaly as ‘disqualified defeats’.
Definition 10. [Disqualified dialectical defeats] Let Φ-𝒟 n be the dialectical dispute tree of
a Φ-dialectical game where 𝑋 and 𝑌 denote arguments played respectively by OPP and PRO in
Φ-𝒟 n . Let also 𝑋 ⇒𝒮 𝑌 by supposing 𝛼 ∈ Prem(𝒮 ). If, after the game goes on, we will reach
a stage Φ-𝒟 n+k (for 𝑘 > 0) where 𝛼 ∉ Prem(𝒮 ), then the defeat moved by 𝑋 against 𝑌 will be
invalidated and will be denoted as ‘disqualified’. As such, 𝑋 and all the arguments following it in
the same dispute will be (temporary) pruned from the tree.
Consider indeed that the status of disqualified defeats might be temporary and be updated again
in a further stage of the game (when these defeats will become valid once more).
Definition 10 entails the following proposition:
Proposition 1. Let Φ-𝒟 n be the dialectical dispute tree of a Φ-dialectical game:
(𝐼 ) If the nth-move is an argument 𝑋 played by OPP, then moving 𝑋 cannot disqualify the
dialectical defeat that 𝑋 performs against a PRO argument.
(𝐼 𝐼 ) The presence of OPP arguments whose defeats have been disqualified will not affect the
dialectical winning strategy.
Proof 1.
(𝐼 ) Since 𝑋 is the last argument (legally) played in Φ-𝒟 n , it trivially does not comply with
Definition 10.
(𝐼 𝐼 ) Even if some of the dialectical defeats moved by OPP arguments have been disqualified
(hence are no more a threat for PRO), the defeat of the last argument played by OPP has not
been disqualified due to (𝐼 ). Notice also that every argument that follows X, whose defeat
has been invalidated, will not be part of the dialectical winning strategy Φ-𝒲 n (including
X). Otherwise, we would violate Definition 10.
Notice that every dialectical game protocol Φ takes into account disqualified defeats, which are
then also contemplated by the dialectical dispute tree Φ-𝒟 n (and dialectical winning strategy
Φ-𝒲 n ) of Definition 9.
8
Observe that it is possible for one (or more, depending on the protocol) dialectical winning strategy Φ-𝒲 n for
A of Φ-𝒟 n to exist, although there is no dialectical winning strategy 𝒲 for A of 𝒟. This can happen, for example,
when 𝒟 is composed only by infinite disputes (recall that we need finite disputes to have winning strategies, as
stated by Definition 8.1), whilst Φ-𝒟 n is composed by finite disputes, due to the restrictions imposed by the protocol
Φ. In this situation, it is possible to identify in Φ-𝒟 n a winning strategy Φ-𝒲 n . Such an example is illustrated in
Figure 2(b).
4.1. Protocol
We can now formally introduce the protocol for the dialectical admissible/preferred game. As
already stated, during the dialectical argument game, the players have to comply with a protocol
Φ that identifies the legal moves allowed.
Definition 11. [Dialectical Admissible Game legal moves] Let Φ-𝒟 n and Φ-𝒲 n be defined
as in Definition 9, let 𝑑 be a dispute of Φ-𝒟 n and 𝑑 ′ be a sub-dispute of 𝑑. Let also (PL𝑛 )𝑋 (for
𝑛 > 0) denotes the argument 𝑋 played by either one of the two players (P or O) as the (last) 𝑛th-move.
Then Φ𝑃 identifies legal moves in the following way:
(11.0) PRO moves the first argument.
(11.1) If (PL𝑛 )𝑋 and 𝑛 = 2𝑘 (for 𝑘 > 0), then the next move 𝑛+1, say Y, is by PRO and it is such
that:
(𝑎) 𝑌 ⇒{𝑍 } 𝑍, where 𝑍 ∈ OPP(Φ-𝒟 n );
(𝑏) There exists a Φ-𝒲 n+1 for 𝐴 of Φ-𝒟 n+1 .
(11.2) If (PL𝑛 )𝑋 and 𝑛 = 2𝑘 + 1 (for 𝑘 ≥ 0), then the next move 𝑛+1, say Y, is by OPP and it is such
that:
(𝑎) 𝑌 ⇒𝒮 𝑍, where 𝑍 ∈ 𝒮 and 𝒮 ∶= PRO(Φ-𝒲 n ) 9 ;
(𝑏) If 𝑑 = 𝑑 ′ —𝑍, then 𝑌 ∉ OPP(𝑑 ′ ).
A Φ𝑃 -dialectical game is said to be terminated when, during its turn, the corresponding player
runs out of the legal moves identified by (11.1) or (11.2) of the protocol Φ𝑃 . PRO wins only if it
has a winning strategy once the game terminates. It loses otherwise.
The previous protocol can be informally summarised as follows. PRO starts the game by
playing the first argument [(11.0)] and, after that, OPP will make its move. Then, the two
players alternate in playing only one argument at a time to reply to one of their counterpart’s
arguments. Observe that when 𝒮 is initialized in the game and, subsequently, every time its
arguments are updated by the changes in PRO(Φ-𝒲 n ) [(11.2(a))], it is always the beginning of
OPP’s turn. This means that the condition for which 𝒮 ≠ ∅ is continuously respected 10 .
Observe that the established protocol allows backtracking to other arguments. That is to say,
when PRO moves it can either target the last argument played by OPP or another argument
moved by OPP in the dialectical dispute tree generated thus far (i.e., an argument member
of the set OPP(Φ-𝒟 n )) [(11.1(𝑎))]. Similarly, when OPP moves it can either target the last
argument played by PRO or another argument moved by PRO in the current dialectical win-
ning strategy (i.e., an argument member of the set PRO(Φ-𝒲 n )) [(11.2(𝑎))]. The relevance
conditions [(11.1(𝑏)) for PRO; (11.2(𝑎)) for OPP] ensure that: after PRO has made its move,
there will be a winning strategy Φ-𝒲 n+1 , hence providing the victory to PRO; after OPP has
moved, instead, the previous winning strategy will cease to exist, thus preventing PRO from
9
The symbol ‘∶=’ denotes a variable initialization rather than an equivalence relation. That is to say, at the
beginning of each OPP’s turn, the content of 𝒮 is initialized to the current PRO(Φ-𝒲 n ), i.e, the arguments member
of 𝒮 are the same as PRO(Φ-𝒲 n ). This operation overwrites the previous contents of 𝒮.
10
That is because a situation in which 𝒮 = PRO(Φ-𝒲 n ) = ∅ never occurs at the beginning of OPP’s turn.
winning. That is to say, PRO will be forced to generate a dialectical winning strategy during each
of its turns, while OPP will have to invalidate such winning strategy during every one of its turns.
Finally, the restriction (11.2(𝑏)) on the moves played by OPP is necessary (as also shown in
the standard games of [4], [5] and [6]). Indeed, allowing OPP to repeat its arguments, since
OPP is required to move exhaustively, could imply the generation of infinite disputes. To see
why let us suppose that (PL𝑛 )𝑋 (for 𝑛 > 0) identifies an argument 𝑋 played by either one of the
two players (denoted as P or O) as its 𝑛th move in a Φ-dialectical game. Then, there could be an
infinite dispute 𝑑 like the following:
𝑑 = (𝑃1 )𝐴 − ⋯ − (𝑂𝑛 )𝑌 − (𝑃𝑛+1 )𝑍 − (𝑂𝑛+2 )𝑌 − (𝑃𝑛+3 )𝑍 − (𝑂𝑛+4 )𝑌 − ⋯
Intuitively, since 𝑍 is capable of defending itself by defeating 𝑌, there is no need to further
extend the dispute by repeating the same arguments: this is because 𝑍 has already shown its
acceptability wrt PRO(Φ-𝒲 n+1 ). Therefore, the only way for avoiding infinite disputes (and
infinite dialectical admissible/preferred games) is to prevent OPP from repeating its arguments
in the same disputes.
Remark 2. Similarly to the standard argument games presented in [4], the protocol of the di-
alectical admissible games is identical to the protocol of the dialectical credulous preferred games.
Indeed, it suffices to show the membership of an argument A in an admissible extension to show
also that A is credulously justified under the preferred semantics as well. That is because every
admissible extension of a dialectical AF is a subset of a preferred extension. This is a consequence
of the Fundamental Lemma (FL) introduced by Dung [3] and adapted for Dialectical Cl-Arg in [9].
As it has been defined, the admissible/preferred game satisfies the properties of soundness
and completeness. This proves the equivalence existing between the victory of the Φ𝑃 -dialectical
game for an argument A and the membership of the same A to the admissible/preferred exten-
sions of the corresponding finite pdAF.
Theorem 1. Let Φ𝑃 -𝒟 n identifies a terminated Φ𝑃 -dialectical game. Then, there exists a dialecti-
cal winning strategy Φ𝑃 -𝒲 n for A, such that the set PRO(Φ𝑃 -𝒲 n ) of arguments moved by PRO
in Φ𝑃 -𝒲 n is conflict-free, iff A is included in an admissible extension Adm of the pdAF.
Proof 2. [Soundness] It follows directly from 8.1 and 8.2
Proof 3. [Completeness](Sketched). Assume that A ∈ Adm. Since the pdAF is finite, then it
is also finitary. We can build a dialectical winning strategy Φ𝑃 -𝒲 n for A if, for each argument
Y dialectically defeating A and moved by OPP, PRO chooses one argument X from Adm (even
A itself) such that X⇒{𝑌 } Y (Definition 11.2(𝑏) prevents the generation of infinite disputes). This
procedure can be repeated for every defeating argument moved by OPP.
Figure 2: Figure a) illustrates a pdAF with a list of its arguments and the set 𝒮 that is parametrized by the
dialectical defeats. Consider also that 𝑋2 is defeated by all the arguments of the pdAF, except 𝐴1 , 𝐵1 , 𝐶1 (the arrows
that should have highlighted such defeats have been omitted to avoid unnecessary graphical confusions). Figure b)
displays the dialectical dispute tree 𝒟 induced by 𝐴1 wrt 𝒮 in the pdAF of Figure a). Notice that 𝒟 is composed of
infinite disputes (the vertical dots represent the endless continuation of the disputes). A dialectical dispute tree
Φ-𝒟 n , with 𝑛 = 4, is depicted in Figure c) and corresponds to a Φ-dialectical game played up to the 𝑛th-move.
Observe that the number of each move (next to the label P or O) represents the order in which the arguments have
been played in the game. In this example, we are assuming a protocol Φ that licenses legal moves where PRO can
play more than one argument per turn, therefore, Φ-𝒟 n has two winning strategies (both of which are encircled in
the figure).
4.2. Main Features of Dialectical Argument Games
Dialectical admissible/preferred argument games hold specific features that differentiate them
from the standard argument games of [4][6][5] and depend upon their protocol and the proper-
ties possessed by each pdAF (especially P1, P2 and P3). In the following, we are going to list all
of such features. Notice that the proofs of the results in this section are left out due to space
constraints.
4.2.1. Features F1-F2
(F1) The set of all the arguments moved by PRO in a dialectical winning strategy of a Φ𝑃 -dialectical
game (i.e., PRO(Φ𝑃 -𝒲 n )) is always conflict-free;
(F2) The relevance conditions, i.e., the conditions of the admissible/preferred protocol that compel
both players to change the outcome of the game at the end of every turn, are essential to the
unfolding of the Φ𝑃 -dialectical games. This also justifies why the set 𝒮 cannot be initialized
with any set other than PRO(Φ𝑃 -𝒲 n ).
4.2.2. Feature F3
Before the introduction of the third feature (F3) enjoyed by the dialectical admissible/preferred
argument games, we need to formally define what is meant by the uniqueness of the dialectical
winning strategy Φ-𝒲 n .
Definition 12. [Uniqueness of the dialectical winning strategy Φ-𝒲 n ] Let Φ-𝒟 n and let
Φ-𝒲 n be defined as in Definition 9. Then Φ-𝒲 n is said to enjoy the uniqueness property if there
is no other dialectical winning strategy for A wrt 𝒮 simultaneously present in Φ-𝒟 n .
Let us consider the dialectical dispute tree Φ-𝒟 n of Figure 2(c). This tree has two winning
strategies, say Φ-𝒲1n and Φ-𝒲2n , each of which is composed of a single dispute. That is to say:
𝑑1 = (𝑃1 )𝐴1 —(𝑂2 )𝐹1 —(𝑃3 )𝐺1 and 𝑑2 = (𝑃1 )𝐴1 —(𝑂2 )𝐹1 —(𝑃4 )𝐺2 , such that Φ-𝒲1n is composed of
𝑑1 , while Φ-𝒲2n is composed of 𝑑2 . Obviously, Φ-𝒟 n does not enjoy the uniqueness property.
Indeed, both 𝐺1 and 𝐺2 defeat the same argument 𝐹1 , whereas only one of such defeats is actually
needed. This implies that it suffices that either Φ-𝒲1n or Φ-𝒲2n is present for PRO to win
the game. For the final outcome of the game, it is pointless to have both winning strategies
simultaneously. It is also resources consuming, meaning that it does not comply well with the
Dialectical Cl-Arg purpose of capturing resource-bounded real-world agents reasoning.
(F3) Any dialectical winning strategy Φ -𝒲 n enjoys the uniqueness property.
𝑃
4.2.3. Advantages of F1-F3
Each of these features proves to be useful in some specific way. F1 allow eschewing consistency
checks on the premises of arguments moved by PRO in any dialectical winning strategy Φ𝑃 -𝒲 n .
In addition, it is of particular importance in providing the games with a wide range of efficiency
improvements. Without the need to perform any extra check or to enforce further restrictions in
the protocol (unlike in [4]), this property allows every dialectical game to prevent the proponent
from: playing self-defeating arguments, playing arguments already moved by the opponent
and playing arguments that defeat (or are defeated by) other arguments already moved by the
proponent. F2 revolves around the relevance conditions (11.1(b) and 11.2(a) of Definition 11)
that can be summarised as the conditions that force the two players to change the outcome of
the game at the end of every turn. These requirements are fundamental for real-world agents
that reason with limited availability of resources. Indeed, it would be nonsensical to allow such
agents to play arguments irrelevant for the outcome of the game: this would mean wasting
valuable resources. The last feature, F3, concerns the uniqueness property. Uniqueness, enforced
on a dialectical winning strategy Φ𝑃 -𝒲 n by the protocol of the dialectical admissible/preferred
argument game, is certainly a desirable feature since allows for shorter and simpler games. This
means that it is possible to evaluate the status of the dialectical dispute tree root faster than the
standard games.
5. Conclusions and Future Work
The main aspects of the real-world uses of argumentation by resource-bounded agents include:
(i) showing arguments inconsistencies by supposing the premises of the opponent’s arguments;
(ii) handling only finite subsets of the arguments of the AFs; (iii) reducing resources consumption
by employing dialectical means (while still satisfying the rationality postulates) [9]. These
features would constitute the hallmarks of an argument game based on Dialectical Cl-Arg, thus
capable of better approximating non-monotonic single-agent real-world reasoning processes
than the standard argument games. In this paper, we have achieved some important results.
We have developed argument game proof theories (denoted as dialectical argument games) for
the admissible and preferred semantics of Dialectical Cl-Arg. The limited space prevented us
from entirely appreciating the extent of the formalism that would have also included a specific
protocol for Dung’s grounded semantics and a fully-fledged list of efficiency improvements
for the introduced games. Incorporating dialectical defeats in the standard structure of the
argument games proved to be a non-trivial process which yielded the discovery of interesting
properties that differentiate dialectical games from the standard ones. That is to say, dialectical
games enjoy (a) specific relevance conditions that identify their protocols and yield (b) the
uniqueness of their winning strategies, whilst property F1 ensures (c) the conflict-freeness of
the set of arguments moved by the proponent in the winning strategy. The latter is of particular
importance since it provides the games with a various range of efficiency improvements.
Extending further the study commenced in [16], we plan to increase the range of dialectical
argument games protocols investigating the stable [6], semi-stable [17] and ideal semantics
[18] [19]. Another research direction that will be pursued involves generalising the developed
dialectical argument games to dialogues, following the guidelines of the already existing liter-
ature in the field (mainly [20]). This would have the interesting consequence of allowing to
move from non-monotonic single-agent inference to distributed non-monotonic reasoning.
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