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{{Paper
|id=Vol-2669/paper3
|storemode=property
|title=On Validating Theories of Abstract Argumentation Frameworks: The Case of Bipolar Argumentation Frameworks
|pdfUrl=https://ceur-ws.org/Vol-2669/paper3.pdf
|volume=Vol-2669
|authors=Henry Prakken
|dblpUrl=https://dblp.org/rec/conf/comma/Prakken20
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==On Validating Theories of Abstract Argumentation Frameworks: The Case of Bipolar Argumentation Frameworks==
https://ceur-ws.org/Vol-2669/paper3.pdf
On Validating Theories of Abstract
Argumentation Frameworks: the Case of Bipolar
Argumentation Frameworks
Henry Prakken12
1
Department of Information and Computing Sciences, Utrecht University, The
Netherlands h.prakken@uu.nl
2
Faculty of Law, University of Groningen, The Netherlands
Abstract. This paper discusses the validation of abstract formal or
computational theories of argumentation as normative theories of ar-
gumentation. Three validation approaches are discussed: instantiation
with a more concrete theory of argumentation (theory-based validation),
validating with intuitions about concrete examples (intuition-based vali-
dation) and comparing the theory with how humans actually argue (em-
pirical validation). It is argued that intuition-based validation can be
useful for validating structured but not for validating abstract accounts
of argumentation, that empirical validation can be used at all levels of
abstraction but at best as a partial validation method, and that a full
validation of abstract accounts of argumentation should include theory-
based validation. A case study of the ‘standard’ theory of bipolar frame-
works reveals that it is to a large extent still awaiting validation as a
good theory of rational argumentation.
Keywords: Validating abstract argumentation theories · Bipolar argu-
mentation frameworks
1 Introduction
When a formal or computational theory of argumentation is proposed, an im-
portant question is why it is a good theory of argumentation. In this paper I
discuss several ways of answering this question for abstract theories of argumen-
tation. By ‘abstract’ I mean theories that leave the nature of arguments and
their relations unspecified. With ‘goodness’ I do not mean whether the theory is
empirically adequate as a theory of how humans actually argue. Rather I mean
whether itis a good theory of rational reasoning and argumentation, that is,
of how people (or machines) should argue. I will in particular focus on bipolar
argumentation frameworks (BAF s, frameworks with both attack and support
relations between arguments), since these have recently been the subject of two
detailed validation studies [4, 11]. I will critically analyse these studies with two
aims in mind: a specific aim to see what these studies imply about whether
BAF s are a good normative account of argumentation, and a general aim to
develop guidelines for evaluating abstract theories of argumentation.
CMNA’20 - F. Grasso, N. Green, J. Schneider, S. Wells (Eds) - 8th September 2020
Copyright c 2020 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
22 Henry Prakken
The first attempt to validate an abstract theory of argumentation was made
by Dung in his seminal paper [5] on the theory of abstract argumentation frame-
works (AF s, with just attack relations between arguments). It is worth quoting
Dung in full [5, pp. 324-5].
Our main goal is to give an analysis of the nature of human argumen-
tation in its full generality. This is done in two steps. In the first step,
a formal, abstract but simple theory of argumentation is developed to
capture the notion of acceptability of arguments. In the next step, we
demonstrate the “correctness” (or “appropriateness”) of our theory. It is
clear that the “correctness” of our theory cannot be “proved” formally.
The only way to accomplish this task is to provide relevant and convinc-
ing examples. Two “examples” are provided. The first one shows how our
theory can be used to investigate the logical structure of many human
economic and social problems. The second one shows that many ma-
jor approaches to nonmonotonic reasoning in AI and logic programming
(. . . ) are in fact different forms of our theory of argumentation.
Interestingly, as regards reasoning and argumentation, Dung did not instantiate
his abstract theory with concrete natural-language examples but with logical
systems, namely, default logic [19], Pollock’s [12] argumentation system and two
logic-programming semantics. In this paper I argue that this is the best way of
validating an abstract theory of argumentation, i.e., instantiating it with more
concrete theories that have been independently validated as “correct” norma-
tive accounts of argumentation. I will call this theory-based validation. As a
case study I will investigate whether there are such theory-based “proofs of
correctness” for the ‘standard’ semantics of BAF s, and argue that these only
partially exist. Then I will review two alternative ways in which BAF s have been
validated, validating with intuitions in concrete examples (intuition-based vali-
dation) and Polberg & Hunter’s [11] recent experimental comparison with how
humans argue (empirical validation). For BAF s I will argue that these valida-
tion attempts are inconclusive. More generally I will argue that intuition-based
validation can at best be a partial validation method at more concrete levels
of abstraction, that empirical validation can be used at all levels of abstraction
but at best as a partial validation method, and that a full validation of abstract
accounts of argumentation should include theory-based validation.
The style of my paper will be largely informal, paraphrasing rather than
stating formal results and observations, and assuming that the reader is familiar
with the discussed works. Nevertheless, for ease of reference I will start with a
brief recapitulation of the main definitions of the theories of AF s and BAF s.
2 Formal preliminaries
In this section the basics of Dung’s theory of abstract argumentation frameworks
and the main theories of bipolar frameworks are reviewed.
On Validating Theories of Abstract Argumentation Frameworks 23
An abstract argumentation framework (AF ) is a pair hA, Ci, where A is
a set of arguments and C ⊆ A × A is a relation of attack. Dung’s theory of AF s
identifies sets of arguments (called extensions) which are internally coherent and
defend themselves against attackers. An argument A ∈ A is defended by a set
by S ⊆ A if for all B ∈ A: if B attacks A, then some C ∈ S attacks B. Then
relative to a given AF , E ⊆ A is admissible if E is conflict-free and defends all
its members; E is a complete extension if E is admissible and A ∈ E iff A is
defended by E; E is a preferred extension if E is a ⊆-maximal admissible set;
E is a stable extension if E is admissible and attacks all arguments outside it;
and E ⊆ A is the grounded extension if E is the least fixpoint of operator F ,
where F (S) returns all arguments defended by S. It holds that any preferred,
stable or grounded extension is a complete extension. Finally, for T ∈ {complete,
preferred, grounded, stable}, X is sceptically or credulously justified under the
T semantics if X belongs to all, respectively at least one, T extension.
Bipolar frameworks add a binary support relation S to AF s. Thus BAF s
are a triple (A, C, S). In [2] a sequence of supports for argument B by argument
A is a sequence ASB1 , . . . SBn SB (it is said that A supports B). Often the
semantics of BAF s is defined in terms of constraints on the attack relation given
sets of attack and support relations between arguments, specifying which attack
relations should also hold given other attack and support relations. Arguments
are then evaluated with a given semantics for AF s applied to BAF s that are
closed under these constraints. The following constraints are the most important
ones that have been considered in the literature (the formulations below are
adapted from [11]). Accordingly, I will call them the ‘standard semantics’ for
BAF s. A semantics of BAF s can use any subset of these constraints.
– There is a supported attack from A to B iff there exists an argument C such
that there is a sequence of supports from A to C and C attacks B.
– There is a secondary attack from A to B iff there exists an argument C such
that there is a sequence of supports from C to B and A attacks C.
– There is an extended attack from A to B iff there exists an argument C such
that there is a sequence of supports from C to A and C attacks B.
– There is a mediated attack from A to B iff there exists an argument C such
that there is a sequence of supports from B to C and A attacks C.
3 Dung-style validation attempts of theories of bipolar
argumentation frameworks
To the best of my knowledge, only two works following Dung’s validation strategy
for bipolar argumentation frameworks exist: my own preliminary research in [15]
and the recent excellent comprehensive study of Cohen et al. in [4]. Below I will
mainly discuss the latter paper, since it includes the main positive result of [15]
as a special case.
Cohen et al. take ASPIC + [9] as the basis for their investigations. ASPIC +
instantiates AF s by giving definitions of the structure of arguments and the
24 Henry Prakken
nature of attack. Informally, in ASPIC + arguments are constructed from logical
knowledge bases by chaining applications of two kinds of inference rules, and can
be displayed as acyclic inference graphs (which are trees if no premise is used
more than once). Arguments have subarguments. Informally, every argument B
corresponding to subgraph of an argument A (viewed as a graph) that is also
an argument (so takes all its premises from the premises of A) is a subargument
of A. Moreover, if B does not equal A, then B is a proper subargument of A.
Arguments can be attacked on their premises, except on those that are declared
unattackable (undermining attack ), on the conclusions of their defeasible infer-
ences ([12]’s rebutting attacks) and on their defeasible inferences themselves by
arguing that it has an exception ([12]’s undercutting attacks). These attack re-
lations can be indirect in that if an argument is attacked, also all arguments
that have this argument as a proper subargument are attacked. Thus ASPIC + ’s
attack relation satisfies closure under secondary attacks. Attack relations can
be resolved into so-called defeat relations by applying a binary preference re-
lation on arguments to attacks. A set of ASPIC + arguments and their defeat
relations then induces an AF by letting the AF attack relations correspond to
the ASPIC + defeat relations. In this paper I will assume for convenience that
all attacks succeed as defeats, which is equivalent to assuming that there are
no strict preference relations. This allows me to speak of attack relations only.
However, my observations will generalise to any defeat relation.
Cohen et al. first define four ways in which ASPIC + arguments can support
each other. The first is the ASPIC + proper subargument relation between
arguments. The second notion of support is a notion of argument accrual: two
different arguments A and B conclusion-support each other if they have the
same final conclusion. A third notion is premise support. Argument A premise-
supports another argument B if A’s final conclusion is a premise of B. Fourth,
a variant of conclusion support is intermediate support: if A conclusion-supports
a proper subargument of another argument B that does not equal a premise of
B, then A intermediate-supports B.
Cohen et al. then consider three semantics for BAF s in terms of the AF s
generated by sets of attack constraints. A BAF , is for general support iff its
attack relation is closed under supported and secondary attack. As in [3] a BAF
is for deductive support iff its attack relation is closed under supported and
mediated attacks. And as in [10] a BAF is said to be for necessary support if
its attack relation is closed under secondary and extended attacks and, moreover,
its support relation is irreflexive and transitive.
Cohen et al. then investigate whether their four notions of support in AS-
PIC + can be related to these three BAF semantics. They do this for each of
the three ASPIC + attack relations separately. Like in the present paper they
assume that all ASPIC + attacks succeed as defeats. For each AF = (A, C)
induced by an ASPIC + instantiation and a particular ASPIC + attack rela-
tion (undermining, rebutting or undercutting attack), they first consider the
BAF = (A, C, Ss ) where Ss is the support relation on A according to ASPIC +
support type s (proper-subargument, conclusion, premise or intermediate sup-
On Validating Theories of Abstract Argumentation Frameworks 25
port). Then for each of the three BAF semantics x (general, deductive or neces-
sary) they consider BAFx = (A, Cx , Ss ), where Cx is the closure of C under the
attack constraints of semantics x. They then compare for each ASPIC + -induced
BAF = (A, C, Ss ) and each corresponding BAFx = (A, Cx , Ss ) the sets C and
Cx . The semantics x is an abstraction of support type s just in case C = Cx .
Cohen et al.’s findings on this question are largely negative. They identify
only one full correspondence, between ASPIC + proper-subargument support
and BAF s for necessary support, regardless of the type of ASPIC + attack. A
similar result was earlier proven by me in [15], except that I only assumed closure
of attack under secondary attack relations. Cohen et al.’s result implies that also
assuming closure under extended attacks does not affect this result.
Cohen et al. remain neutral on the methodological implications of their find-
ings but in my opinion they provide strong support for the claim that the ‘stan-
dard’ semantics for BAF s has not yet been validated as a good theory of rational
argumentation. Of course, this conclusion presupposes that ASPIC + is itself a
good structured account of rational argumentation, although it does not presup-
pose that ASPIC + is the only such good account; nor does it presuppose that
the only theory-based way to validate a theory of BAF s is by instantiating it
with ASPIC + . In Section 5 I will return to these issues.
4 Alternative validation attempts of theories of bipolar
argumentation frameworks
In the present section I discuss two main alternative validation approaches that
have been employed in the literature: appeals to intuition in concrete examples
and comparisons with how humans actually argue.
4.1 Validation with intuitions in concrete examples
An often applied strategy in validating the normative adequacy of theories of
reasoning is applying them to concrete examples and checking whether they
validate one’s intuitions about these examples. In the literature on computational
argumentation this is occasionally used but not in a very systematic way. For
structured accounts of argumentation (which specify the structure of arguments
and the nature of their relations) it can be a useful aspect of validation but it
should be used with care. One problem is that it is not obvious whose intuitions
should count [21, Ch. I.1]. Those of logicians or argumentation researchers are
seriously biased by overexposure to formalism. So should we ask the ‘average
language user’, hoping that they are not infected by theoretical bias? Then
the problem often arises that their answers reveal a lacking understanding of
the reasoning patterns; however, teaching them about these reasoning patterns
infects them with the theoretical bias we were hoping to avoid. Also, whether
we ask researchers or lay people, intuitions of different persons often conflict.
In validating theories of defeasible reasoning another problem is that it is
often hard to distinguish between intuitions that a particular reasoning pattern is
26 Henry Prakken
invalid and implicitly assumed additional information that invalidates inferences
that are defeasibly valid in the absence of that information [13].
When validating abstract theories argumentation there is the problem that
if the theory is validated with natural-language examples of argumentation that
are directly translated into AF s instead of through a theory of the nature of
arguments and attacks, then the resulting modellings may be ad-hoc, so that
the observations made about these modellings have no general validity. Prakken
& Winter [18] illustrate this problem with several examples from the literature
on probabilistic abstract argumentation frameworks and bipolar frameworks.
In conclusion, intuition-based validation can, if used with care, be useful for
validating structured accounts of argumentation but they should not be used for
validating abstract accounts of argumentation, since directly encoding natural-
language examples in abstract frameworks, without guidance by theories of the
nature of arguments and their relations, risks in yielding ad-hoc modellings.
4.2 Empirical validation
Polberg & Hunter [11] empirically evaluate both AF s and BAF s in experiments
with human test subjects, who were instructed to identify support and attack
relations between the arguments in two natural-language persuasion dialogues.
They call their experiments “explorative” and warn that their results are merely
“indicative” and should serve as “a basis for further studies, rather than as an
“indisputable proof for or against a given argumentation approach”. I will not
discuss the ‘strength’ of their findings in this sense but rather whether their
experiments can in principle support conclusions of the kind they draw.
Their overall conclusion is that the data from their experiments support the
need for bipolar approaches. The question arises: the need for bipolar approaches
for what purpose? Are they needed to specify how people should argue (the
topic of the present paper) or are they needed to model how people actually
argue? These things are quite different. It is not entirely clear from their paper
which of these questions Polberg & Hunter are interested in. Yet it is important
to disambiguate, since if their claims are interpreted as being about normative
adequacy, then the question arises how empirical findings on how people actually
argue can be relevant at all for a normative theory on how they should argue.
Polberg & Hunter write “Without empirical evidence, we can accidentally
increase the gap between applying argumentation and successfully applying ar-
gumentation in real life situations” (their emphasis, HP). In my opinion, what
this hints at is an empirical constraint on normative theories of reasoning that
they should refer to the concepts that people use when reasoning or arguing,
on the penalty of not being applicable by people. The principles formulated by
the theory may still deviate from how people actually reason, as long as they
are stated in terms that are natural to people. Otherwise it may happen that a
theory, even if it is normatively adequate, is so foreign to people that they are
not able to apply the theory, so that they are left without any useful guidance
for their reasoning. It may be that Polberg & Hunter have this kind of empirical
naturalness constraint on the usefulness of normative theories of argumentation
On Validating Theories of Abstract Argumentation Frameworks 27
in mind. When interpreted thus, their results do not inform us about whether
a semantics of BAF s is normatively correct but about whether it is stated in
terms that are natural to people. Henceforth I will call this kind of validation
quasi-empirical validation of (the human naturalness of) normative theories.
Let us see whether Polberg & Hunter’s results can be seen as quasi-empirical
validation of BAF s in this sense. In my opinion, this is not the case, since Polberg
& Hunter state their claims in absolute terms while their data at best support a
relative conclusion, namely, that using BAF s is more quasi-empirically adequate
than using AF s, since their experiments are designed to compare AF s and
BAF s. However, there are other alternatives, namely, using AF s in combination
with a theory of the structure of arguments and the nature of attack. There are
quite a few such theories, dating back to the seminal work of Pollock in [12]. For a
historical overview see my [16]. All these theories allow for support relations that
are not between but inside arguments, namely as inferential relations between
(sets of) statements in some logical language. Examples are assumption-based
argumentation, Defeasible Logic Programming [7] and ASPIC + . The arguments
of [11] for their claims are thus instances of the fallacy of the excluded middle.
What is quasi-empirically more adequate: modelling support relations pri-
marily as relations between arguments or primarily as relations between state-
ments? Polberg & Hunter’s experiments are inherently unsuitable for answering
this question, since they did not instruct their test subjects to identify support
relations between statements in the sense of a structured account of argumenta-
tion, so between premises and conclusions inside arguments. In fact, Polberg &
Hunter’s use of the terms ‘statement’ and ‘argument’ is sometimes ambiguous.
They refer to the elements of their persuasion dialogues as “statements”. How-
ever, most of these “statements” are what argumentation theorists would call
arguments, with statements as premises, another statement as a conclusion and
a claimed relation of support between premises and conclusion. For example:
Hospital staff members are exposed to the flu virus a lot. Therefore, it
would be good for them to receive flu shots in order to stay healthy (my
emphasis, HP).
On the other hand, some of their ‘statement’ are just claims. For example:
Hospital members do not need to receive flu shots.
Since Polberg & Hunter do not systematically distinguish between statements
as full arguments or as elements of arguments, they also do not systematically
distinguish between support relations between and inside arguments. In any
case, no conclusion can be drawn from their experiments on the relative quasi-
empirical adequacy of BAF -style approaches and structured approaches to the
modelling of support relations in argumentation.
5 Validation of the instantiations
As noted above at the end of Section 3, and as also acknowledged by [11, p.
488], the success of theory-based validation depends on whether the more con-
28 Henry Prakken
crete theories with which abstract theories are instantiated are themselves valid
as normative theories of reasoning or argumentation (or at least accepted as
valid by a substantial part of the research community). A full treatment of this
issue is beyond the scope of this paper but some observations can be made. Dung
instantiated his theory with one of the best known nonmonotonic logics, default
logic, with two semantics for logic programming, which is a well-established
computational theory, and with a system for defeasible argumentation (Pol-
lock’s system) with foundations in epistemology which has been very influential
in the study of formal argumentation. Others have instantiated Dung’s theory
with, for instance, assumption-based argumentation [6] and the ASPIC + frame-
work [14]. Both of these frameworks have in turn been validated to some ex-
tent. Assumption-based argumentation has been used for reconstructing several
well-known nonmonotonic logics and it has been successfully used in many ap-
plications [20], which contributes to its quasi-empirical validation. ASPIC + has
been used for reconstructing assumption-based argumentation [14] and several
variants of classical-logic argumentation [8], which is a Dung-style validation of
ASPIC + . Moreover, it has received quasi-empirical validation in several appli-
cations or case studies, as summarised in Section 6.3 of [9]. Of course, opinions
may differ on the extent to which validation studies have been successful but in
my opinion most of this other work has been sufficiently validated so that Dung’s
theory of AF s has received theory-based validation to a very high degree.
6 Conclusion
In this paper I have argued that the best way of validating the ‘correctness’ of
an abstract normative theory of argumentation is theory-based validation, that
is, instantiation with more concrete theories of reasoning and argumentation
that have themselves been sufficiently validated in other ways. As a case study I
have investigated how current bipolar approaches, which extend Dung’s abstract
argumentation approaches with abstract support relations, have been validated
in this way. It turned out that the only two papers that, to my knowledge,
apply theory-based validation to bipolar approaches, [15] and [4], yield negative
results for BAF semantics for general and deductive support but positive results
for necessary support, which has been shown to be an abstraction of ASPIC + ’s
subargument relation.
The largely negative results of [15] and [4] are telling. They should warn
us that developing a theory of argumentation without taking the structure of
arguments and the nature of attack into account is dangerous: it increases the
risk of developing a piece of mathematics that is elegant and seductive but has
no clear relations to real argumentation phenomena. Moreover, a ‘social’ risk
is that it sends the wrong message to young researchers that this is the way
research in argumentation should be done, namely, by ignoring the structure of
arguments. It may make that not enough effort is devoted to the study of genuine
argumentation phenomena. I therefore repeat my proposal from [16] that any
newly proposed abstract formalism for argumentation should be accompanied
On Validating Theories of Abstract Argumentation Frameworks 29
with at least one meaningful instantiation. In other words, we should follow the
example that Dung set in his classic 1995 paper.
After applying Dung’s validation strategy to bipolar approaches, I investi-
gated two common alternative validation strategies of abstract approaches. As
regards validation of BAF semantics by intuitions in concrete examples, I re-
peated and summarised observations in the literature that this validation strat-
egy is ad-hoc and unreliable and that modellings of concrete examples should be
guided by accounts of the structure of arguments and the nature of their rela-
tions. As regards validation of BAF s by empirical comparison with how people
actually argue I have made the following contributions. I first distinguished sev-
eral senses of validation and argued that Polberg & Hunter’s experiments in [11]
can best be interpreted not as validation of these theories as normative theo-
ries of rational argumentation but as validation of their naturalness to human
arguers. The benefit of this kind of quasi-empirical validation is that it guides
researchers in developing normative theories of argumentation that are not only
rationally well-founded but also applicable by humans. Such theories can still de-
viate from the way humans actually argue but since they are stated in terms that
are natural to humans, they have a higher chance of being applied by humans
than possibly better but less natural normative theories.
Next, I argued that Polberg & Hunter’s results do not contribute to this kind
of quasi-empirical validation, since their paper is silent about the vast body of
work in AI on the structure of arguments and the nature of attack and support.
I agree with them that support relations are important in argumentation but
this has been known for a long time. All informal definitions of argumentation in
the argumentation-theoretic literature and all formal definitions of the structure
of arguments in the AI literature are variations on the theme that argumenta-
tion involves supporting claims with grounds. The question is how such support
relations can best be modelled. One main issue here is whether support can best
be modelled primarily as a relation between arguments or primarily as a relation
between grounds and claims inside arguments. Polberg & Hunter do not answer
to this question, since they do not consider the latter alternative.
Finally, an interesting question is what is the role of postulate-based ap-
proaches (e.g. [1]) in validating abstract theories of argumentation. In my opin-
ion, this depends on knowing what abstract theories abstract from, since other-
wise it is hard to say whether a postulate makes sense. So in this sense postulates
for abstract accounts also require theory-based validation. In [17] I made this
point in more detail for abstract accounts of probabilistic argumentation.
To conclude, although the the present paper by no means offers a compre-
hensive account of validating abstract accounts of argumentation, it provides
some arguably useful distinctions between various kinds of validation and some
guidelines for how to conduct validation attempts.
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