# Ranking-based Argumentation Semantics Applied to Logical Argumentation (full
version)

Jesse Heyninck  Open Universiteit, the Netherlands  Badran Raddaoui  SAMOVAR,
Télécom SudParis, Institut Polytechnique de Paris, France  Institute for
Philosophy II, Ruhr-Universität Bochum, Germany  Christian Straßer  Institute
for Philosophy II, Ruhr-Universität Bochum, Germany

###### Abstract

In formal argumentation, a distinction can be made between extension-based
semantics, where sets of arguments are either (jointly) accepted or not, and
ranking-based semantics, where grades of acceptability are assigned to
arguments. Another important distinction is that between abstract approaches,
that abstract away from the content of arguments, and structured approaches,
that specify a method of constructing argument graphs on the basis of a
knowledge base. While ranking-based semantics have been extensively applied to
abstract argumentation, few work has been done on ranking-based semantics for
structured argumentation. In this paper, we make a systematic investigation
into the behaviour of ranking-based semantics applied to existing formalisms
for structured argumentation. We show that a wide class of ranking-based
semantics gives rise to so-called culpability measures, and are relatively
robust to specific choices in argument construction methods.

##  1 Introduction

A central concept in formal argumentation [25] is that of an _argumentation
graph_ , that consists of a set of nodes interpreted as arguments, and edges
interpreted as argumentative attacks. Within the area of formal argumentation,
one can distinguish between _abstract_ and _structured_ approaches. Abstract
argumentation abstracts away from the content of arguments and thus can be
seen as taking argumentation graphs as primitive objects of study. Structured
approaches, on the other hand, specify construction methods defining how an
argumentation framework is generated on the basis of a knowledge base
consisting of formulas specified in a logical language. _Plausible reasoning_
[17, 6, 7] is a sub-class of structured argumentation formalising reasoning on
the basis of defeasible premises and non-defeasible rules or logical systems
such as classical logic.

_Argumentation semantics_ specify the acceptability of arguments given an
argumentation graph. One can distinguish between so-called _extension-based_
semantics, that specify which sets of arguments are jointly acceptable, and
_gradual_ or _ranking-based_ semantics, that express acceptability in terms of
a ranking over the arguments. Recent years have seen an intensive study of
both kind of semantics in both breath and depth (see [20, 14] for some
overviews) from the abstract perspective. For extension-based semantics, the
structured perspective has played a major role. For ranking-based semantics,
on the other hand, virtually no work has been carried out from the structured
perspective (the only exception being [2]). Such a gap is rather surprising,
as structured argumentation still provides one of the gold standards of the
construction or generation of abstract argumentation graphs. Indeed, it is not
clear whether and how the ranking-based semantics provide intuitive meaning
from the logical perspective of structured argumentation. Furthermore, the
study of ranking arguments in a structured setting could be helpful for
numerous real-world applications of structured argumentation [23, 39, 44]. For
instance, in medicine and legal reasoning, it might be more appropriate to
assign arguments a gradual score instead of distinguishing merely between
acceptance and rejection.

In this paper, we make the first general study of ranking-based semantics
applied to structured argumentation frameworks. We focus on logic-based
argumentation, an important and well-studied sub-class of structured
argumentation. In particular, we answer the following questions:

  1. 1.

What is the logical meaning of ranking-based semantics applied to
argumentation graphs generated on the basis of strict and defeasible
information?

  2. 2.

What is the influence of choices underlying the construction of the argument
graph (e.g. the form of argumentative attacks) on the outcome of ranking-based
semantics?

The main results are the following: (1) a large family of ranking-based
semantics induce so-called _culpability measures_ , and behave better than
many existing culpability measures, (2) ranking-based semantics applied over
argumentation graphs constructed using approaches from structured
argumentation are robust under many choices considered in the literature such
as different argumentative attack forms or argument construction methods,
apart from the filtering out of inconsistent arguments.

Outline of the paper. In Section 2, the relevant preliminaries on simple
contrapositive assumption-based frameworks (Subsection 2.1), ranking-based
argumentation semantics (Subsection 2.2) and culpability measures (Subsection
2.3) are recalled. We review postulates for ranking-based semantics in Section
3. In Section 4, we show that ranking-based semantics applied to argumentation
frameworks obtained on the basis of simple contrapositive assumption-based
frameworks induce culpability measures. In Section 5, we look at other
structured argumentation formalisms. We review related work in Section 6 and
conclude in Section 7.

##  2 Preliminaries

This section contains background material about simple contrapositive
assumption-based argumentation frameworks (more details can be found in [10,
31]), ranking-based semantics, and culpability measures.

###  2.1 Simple Contrapositive Assumption-based Frameworks

We shall denote by ${\cal L}$ an arbitrary propositional language. Atomic
formulas in ${\cal L}$ are denoted by $p,q,r,$ etc., arbitrary ${\cal
L}$-formulas (atomic or compound) are denoted by $\psi,\phi,\delta,$ etc., and
sets of formulas in ${\cal L}$ are denoted by $\Gamma$, $\Delta$, $\Theta,$
etc. (possibly primed or indexed). $\wp(\mathcal{L})$ denotes the power-set of
${\cal L}$.

Instead of restricting ourselves to one specific logic, we merely assume some
common properties of logics in order to keep our results as general as
possible.

######  Definition 1 (Logic).

A logic for a language ${\cal L}$ is a pair ${\mathfrak{L}}={\langle{\cal
L},\vdash\rangle}$, where $\vdash$ is a (Tarskian) consequence relation for
${\cal L}$, that is, a binary relation among sets of formulas and individual
formulas in ${\cal L}$, satisfying the following conditions:

  * Reflexivity:

if $\psi\in\Gamma$, then $\Gamma\vdash\psi$.

  * Monotonicity:

if $\Gamma\vdash\psi$ and $\Gamma\subseteq\Gamma^{\prime}$, then
$\Gamma^{\prime}\vdash\psi$.

  * Transitivity:

if $\Gamma\vdash\psi$ and $\Gamma^{\prime},\psi\vdash\phi$ then
$\Gamma\cup\Gamma^{\prime}\vdash\phi$.

In what follows, we shall assume that the language ${\cal L}$ contains at
least the following connectives and constant:

a $\vdash$-negation $\neg$, satisfying: $p\not\vdash\neg p$ and $\neg
p\not\vdash p$ (for every atomic $p$).

a $\vdash$-conjunction $\wedge$, satisfying: $\Gamma\vdash\psi\wedge\phi$ iff
$\Gamma\vdash\psi$ and $\Gamma\vdash\phi$.

a $\vdash$-falsity ${\sf F}$, satisfying: ${\sf F}\vdash\psi$ for every
formula $\psi$.111Particularly, ${\sf F}$ is not a standard atomic formula,
since ${\sf F}\vdash\neg{\sf F}$.

Our results only assume the above connectives with the respective properties.
This means that these results apply to any language with additional
connectives or additional properties assumed on the connectives. We abbreviate
$\\{\neg\psi\mid\psi\in\Gamma\\}$ by $\neg\Gamma$, and when $\Gamma$ is finite
we denote by $\bigwedge\\!\Gamma$ the conjunction of all the formulas in
$\Gamma$.

Given a logic ${\mathfrak{L}}={\langle{\cal L},\vdash\rangle}$, the
$\vdash$-closure of a set $\Gamma$ of ${\cal L}$-formulas is defined by
$Cn_{\vdash}(\Gamma)=\\{\psi\mid\Gamma\vdash\psi\\}$. When $\vdash$ is clear
from the context or arbitrary, we will sometimes simply write $Cn(\Gamma)$.
Now,

  * •

We shall say that $\psi$ is a $\vdash$-theorem if $\psi\in
Cn_{\vdash}(\emptyset)$, and that $\Gamma$ is $\vdash$-consistent if
$Cn_{\vdash}(\Gamma)\neq{\cal L}$ (thus, $\Gamma\not\vdash{\sf F}$, otherwise,
we say that $\Gamma$ is $\vdash$-inconsistent).

  * •

${\mathfrak{L}}$ is called explosive, if for every ${\cal L}$-formula $\psi$
the set $\\{\psi,\neg\psi\\}$ is $\vdash$-inconsistent.

  * •

${\mathfrak{L}}$ is called contrapositive, if (i) $\vdash\neg\mathsf{F}$, and
(ii) for every non-empty set $\Gamma$ and $\psi$ it holds that
$\Gamma\vdash\neg\psi$ iff either $\psi={\sf F}$ or for every $\phi\in\Gamma$
we have that $(\Gamma\setminus\\{\phi\\})\cup\\{\psi\\}\vdash\neg\phi$.

  * •

${\mathfrak{L}}$ is called uniform if for any
$\Gamma\cup\Gamma^{\prime}\cup\\{\phi\\}\subseteq{\cal L}$, if
$\Gamma^{\prime}\not\vdash{\sf F}$ and $\Gamma\cup\\{\phi\\}$ and
$\Gamma^{\prime}$ have no atoms in common, it holds that: $\Gamma\vdash\phi$
iff $\Gamma\cup\Gamma^{\prime}\vdash\phi$.

#### Remark.

Perhaps the most notable example of a logic which is explosive, contrapositive
and uniform, is classical logic (in short, ${\sf CL}$)222In fact, by the Łos-
Suszko theorem [36] any structural logic with a characteristic matrix is
uniform.. Intuitionistic logic and standard modal logics are other examples of
well-known formalisms having these properties.

######  Note 1.

A useful property of an explosive logic ${\mathfrak{L}}={\langle{\cal
L},\vdash\rangle}$ is that for every set $\Gamma$ of ${\cal L}$-formulas and
any ${\cal L}$-formulas $\psi$ and $\phi$, if $\Gamma\vdash\psi$ and
$\Gamma\vdash\neg\psi$, then $\Gamma\vdash\phi$.

The following family of assumption-based argumentation frameworks [19] has
been shown to be a useful setting for argumentative reasoning [31]. In more
detail, a simple contrapositive assumption-based framework is a knowledge base
consisting of a set of strict premises $\Gamma$, a set of defeasible premises
$Ab$, a logic $\mathfrak{L}$, and a contrariness operator specifying which
conflicts are taken into account.

######  Definition 2 (Simple contrapositive ABFs).

An _assumption-based framework_ (ABF, for short) is a tuple ${\sf
ABF}={\langle\mathfrak{L},\Gamma,Ab,\sim\rangle}$ where:

  * •

${\mathfrak{L}}={\langle{\cal L},\vdash\rangle}$ is a propositional Tarskian
logic.

  * •

$\Gamma$ (the _strict assumptions_) and $Ab$ (the _candidate/defeasible
assumptions_) are distinct (countable) sets of $\mathcal{L}$-formulas, where
the former is assumed to be $\vdash$-consistent and the latter is assumed to
be nonempty.

  * •

$\sim\>:Ab\rightarrow\wp(\mathcal{L})$ is a _contrariness operator_ ,
assigning a finite set of ${\cal L}$-formulas to every defeasible assumption
in $Ab$, such that for every consistent and non-theorem formula $\psi\in Ab$
it holds that $\psi\not\vdash\bigvee\\!\sim\\!\psi$ and
$\bigvee\\!\sim\\!\psi\not\vdash\psi$.

A simple contrapositive ${\sf ABF}$ is an assumption-based framework ${\sf
ABF}={\langle\mathfrak{L},\Gamma,Ab,\sim\rangle}$, where $\mathfrak{L}$ is an
explosive, contrapositive and uniform logic, and
$\sim\\!\psi=\\{\lnot\psi\\}$.333Notice that uniformity is not required
explicitly in [30]. To avoid clutter, we choose to adapt the definition here
instead of using “simple contrapositive uniform assumption-based frameworks.”

######  Definition 3.

Let ${\sf ABF}={\langle\mathfrak{L},\Gamma,Ab,\sim\rangle}$ be an assumption-
based framework, $\Delta,\Theta\subseteq Ab$, and $\psi\in Ab$. Then, $\Delta$
_attacks_ $\psi$ iff $\Gamma\cup\Delta\vdash\phi$ for some
$\phi\in\>\sim\\!\psi$. Then, $\Delta$ attacks $\Theta$ if $\Delta$ attacks
some $\psi\in\Theta$.

The following useful notions are used in the paper:

######  Definition 4.

Let ${\sf ABF}={\langle\mathfrak{L},\Gamma,Ab,\sim\rangle}$. A set
$\Delta\subseteq Ab$ is maximal consistent in ${\sf ABF}$, if (i)
$\Gamma\cup\Delta\not\vdash{\sf F}$ and (ii)
$\Gamma\cup\Delta^{\prime}\vdash{\sf F}$ for every
$\Delta\subsetneq\Delta^{\prime}\subseteq Ab$. The set of all the maximal
consistent sets in ${\sf ABF}$ is denoted ${\sf MCS}({\sf ABF})$. A set
$\Delta\subseteq Ab$ is minimal inconsistent in ${\sf ABF}$, if (i)
$\Gamma\cup\Delta\vdash{\sf F}$ and (ii)
$\Gamma\cup\Delta^{\prime}\not\vdash{\sf F}$ for every
$\Delta\supsetneq\Delta^{\prime}\subseteq Ab$. The set of all the minimal
inconsistent sets in ${\sf ABF}$ is denoted ${\sf MIC}({\sf ABF})$.
We define the set of free formulas in ${\sf ABF}$ as ${\sf FREE}({\sf
ABF})=Ab\setminus\bigcup{\sf MIC}({\sf ABF})$.

For a set of formulas $Ab\subseteq{\cal L}$, we will denote the set of all
classically maximal consistent subsets of $Ab$, defined as ${\sf
MCS}({\langle\mathsf{CL},\emptyset,Ab,\sim\rangle})$, by ${\sf MCS}(Ab)$.
Similarly for the minimal inconsistent subsets (${\sf MIC}(Ab)$) and the free
formulas (${\sf FREE}(Ab)$) of $Ab$.

######  Example 1.

Let $\mathfrak{L}={\sf CL}$, $\Gamma=\emptyset$, $Ab=\\{p,\lnot p,q\\}$, and
$\sim\\!\psi=\\{\neg\psi\\}$. An attack diagram for this ABF is shown in
Figure 1. In fact, an attack diagram consists of all subsets of assumptions,
and the attacks according to Definition 3 between these assumptions. Note that
since in ${\sf CL}$ inconsistent sets of premises imply any conclusion, the
classically inconsistent set $\\{p,\neg p,q\\}$ attacks all the other sets in
the diagram (for instance, $\\{p,\neg p,q\\}$ attacks $\\{q\\}$, since
$\\{p,\neg p,q\\}\vdash\neg q$).

$\\{q\\}$$\\{p,\lnot p\\}$$\\{p,\lnot p,q\\}$$\\{p\\}$$\\{\lnot
p\\}$$\\{p,q\\}$$\\{\neg p,q\\}$$\emptyset$ Figure 1: An attack diagram for
Example 1

###  2.2 Ranking-based Argumentation Semantics

Ranking-based semantics have been mainly studied from the abstract
perspective, that focuses on abstract argumentation frameworks:

######  Definition 5.

An _argumentation framework_ ${\sf AF}=({\cal A},\rightarrow)$ consists of a
set of arguments ${\cal A}$ and an attack relation
${\rightarrow}\subseteq{\cal A}\times{\cal A}$. Where $a\in{\cal A}$,
$a^{-}=\\{b\in{\cal A}\mid b\rightarrow a\\}$.

Notice that for any ${\sf ABF}=\langle\mathfrak{L},\Gamma,Ab,\sim\rangle$,
$(\wp(Ab),\rightarrow)$ is an argumentation framework, where $\rightarrow$ is
defined as in Definition 3. In other words, an assumption-based argumentation
gives a method of how to construct an argumentation framework. We now discuss
ranking-based semantics, that allow to determine the _acceptability_ of
arguments for a given ${\sf AF}$.

There has been a wide range of ranking-based semantics proposed in the
literature to rank order arguments from the most acceptable to the weakest
arguments [20]. Following Baroni, Rago and Toni [14], we consider ranking-
based semantics as mappings $\sigma$ of arguments to a set $\mathbb{I}$
preordered by $\leq$ relative to an argumentation framework ${\sf AF}=({\cal
A},\rightarrow)$. Given $a,b\in{\cal A}$, $\sigma(a)\geq\sigma(b)$ means that
the argument $a$ is at least as acceptable as the argument $b$ according to
$\sigma$. For simplicity, we assume the image of $\sigma$ is always
$\mathbb{R}_{\geq 0}$. In the remainder of the paper, we will denote
$\sigma_{{\sf AF}}$ as the ranking-based semantics relative to ${\sf AF}$. A
high ranking $\sigma_{{\sf AF}}(a)$ represents a high quality of an argument
$a$ w.r.t. ${\sf AF}$ according to $\sigma$.

As an illustration of a ranking-based semantics, we look at the _categorizer
semantics_ , first presented in [15] for acyclic frameworks but generalized to
any argumentation framework in [3]. This semantics formalizes the idea that
the number of attackers decreases the quality of an argument. Formally, it
works by taking the inverse of the sum of the score of the attacking arguments
iteratively.

######  Definition 6.

Given an ${\sf AF}=({\cal A},\rightarrow)$ and $a\in{\cal A}$:

$\displaystyle{\tt cat}_{0}(a)$ $\displaystyle=$ $\displaystyle 1$ (1)
$\displaystyle{\tt cat}_{i+1}(a)$ $\displaystyle=$
$\displaystyle\frac{1}{1+\Sigma_{b\in a^{-}}{\tt cat}_{i}(b)}$ (2)

It has been shown in [3] that ${\tt cat}_{i}$ converges as $i$ approaches
infinity. We therefore define $\sigma^{\tt cat}_{{\sf
AF}}(a)=\lim_{i\to\infty}{\tt cat}_{i}(a)$.

######  Example 2.

Consider the argumentation framework from Figure 1. It can be verified (for
example using the Tweety-library [41]) that: $\sigma_{{\sf AF}}^{\tt
cat}(\emptyset)=1$, $\sigma_{{\sf AF}}^{\tt cat}(\\{q\\})=0.71$, $\sigma_{{\sf
AF}}^{\tt cat}(\\{p\\})=\sigma_{{\sf AF}}^{\tt cat}(\\{\lnot p\\})=0.52$, and
$\sigma_{{\sf AF}}^{\tt cat}(\\{p,\lnot p,q\\})=0.41$.

###  2.3 Culpability Measures

A fundamental research topic in reasoning with conflicting information is the
measure of inconsistencies. A lot of work focuses on _inconsistency measures_
that allow to quantify _how inconsistent_ [34, 42] a _set_ of formulas
(typically propositional) is.

On the other hand, _culpability measures_ quantify the degree of
responsibility of a formula in making a set of formulas inconsistent. In more
detail, a culpability measure for a knowledge base $Ab\subseteq{\cal L}$ is a
function ${\cal C}_{Ab}:Ab\rightarrow\mathbb{R}_{\geq 0}$ that satisfies the
following postulates:

Consistency.

If $Ab$ is consistent, then ${\cal C}_{Ab}(\phi)=0$ for every $\phi\in Ab$.

Blame.

If $\phi\in\bigcup{\sf MIC}(Ab)$, then ${\cal C}_{Ab}(\phi)>0$.

Thus, _consistency_ states that the culpability measure of premises in a
consistent set are minimal (i.e. in consistent sets, no formula is culpable
for inconsistency), whereas _blame_ states that any formula involved in an
inconsistency will receive a non-zero culpability degree.

In [33], culpability measures are defined as _inconsistency values_ 444To
comply with the notation in this paper, higher inconsistency values
intuitively represent a higher degree of inconsistency, in contradistinction
to the work of [33].. The following properties for inconsistency values are
defined in [33] (we adapt them to our notation).555In [33], other properties
of culpability measures are discussed, but they describe the relation between
inconsistency and culpability measures, and are therefore not relevant to our
study.

Free Formula.

If $\phi\in{\sf Free}(Ab)$666Recall that ${\sf FREE}(Ab)={\sf
Free}({\langle\mathsf{CL},\emptyset,Ab,\sim\rangle})=Ab\setminus\bigcup{\sf
MIC}(Ab)$., then ${\cal C}_{Ab}(\phi)\leq{\cal C}_{Ab}(\psi)$ for any $\psi\in
Ab$.777In [33], this property is called _minimality_ , and it is required that
a free formula evaluates to 0. We use a weaker condition that allows to
differentiate between the score of $p$ in $\\{p,q,\lnot q\\}$ and in
$\\{p,q\\}$.

Dominance.

Where $\phi,\psi\in Ab$, if $\phi\vdash\psi$ and $\phi\not\vdash{\sf F}$, then
${\cal C}_{Ab}(\phi)\geq{\cal C}_{Ab}(\psi)$.

Thus, _free formula_ states that formulas not involved in any inconsistency
are at most as culpable as any other formula, and _dominance_ states that
logically stronger formulas are at least as culpable as their weaker
counterparts.

By ways of illustration, we consider the following culpability measures ${\cal
C}^{d}_{Ab}$, ${\cal C}^{\ast}_{Ab}$ and ${\cal C}^{c}_{Ab}$ from [33]:888In
our presentation, unlike in [33], the measures are normalized to ensure a
mapping to $[0,1]$. We follow the convention that $\sum\emptyset=0$ which
means that ${\cal C}^{c}_{Ab}(\phi)=0$ for any free formula $\phi$. For ${\cal
C}^{\ast}_{Ab}$ and ${\cal C}^{c}_{Ab}$, the knowledge base is supposed to be
finite.

$\displaystyle{\cal C}^{d}_{Ab}(\phi)$ $\displaystyle=$
$\displaystyle\begin{cases}1&\mbox{if }\phi\in\bigcup{\sf MIC}(Ab)\\\
0&\mbox{otherwise}\end{cases}$ $\displaystyle{\cal C}^{\ast}_{Ab}(\phi)$
$\displaystyle=$ $\displaystyle\frac{|\\{\Delta\in{\sf
MIC}(Ab)\mid\phi\in\Delta\\}|}{|{\sf MIC}(Ab)|}$ $\displaystyle{\cal
C}^{c}_{Ab}(\phi)$ $\displaystyle=$
$\displaystyle\displaystyle\left(\sum_{\Delta\in{\sf MIC}(Ab)\mbox{ and
}\phi\in\Delta}\frac{1}{|\Delta|}\right)\frac{1}{\sum_{\Delta\in{\sf
MIC}(Ab)}|\Delta|}$

######  Example 3.

Consider the knowledge base $\Gamma=\\{p\land\lnot p,q,r,\lnot q\lor\lnot
r,s\\}$. The minimal inconsistent sets of $\Gamma$ are ${\sf
MIC}(\Gamma)=\\{\\{p\land\lnot p\\},\\{q,r,\lnot q\lor\lnot r\\}\\}$. This
means that ${\cal C}^{d}_{\Gamma}(s)={\cal C}^{\ast}_{\Gamma}(s)=0$, and
${\cal C}^{d}_{\Gamma}(\phi)=1$ and ${\cal C}^{\ast}_{\Gamma}(\phi)=0.5$ for
any $\phi\in\Gamma\setminus\\{s\\}$. Also, we have ${\cal
C}^{c}_{\Gamma}(s)=0$, ${\cal C}^{c}_{\Gamma}(p\land\lnot p)=\frac{1}{4}$ and
${\cal C}^{c}_{\Gamma}(\phi)=\frac{1}{12}$ for any
$\phi\in\Gamma\setminus\\{p\land\lnot p,s\\}$.

##  3 Postulates for Ranking-based Semantics

In this section, as a first illustration of the difference between abstract
and structured argumentation in the context of ranking-based semantics, we
look at some common postulates proposed for ranking-based semantics and
evaluate their usefulness in a structured setting. We will see that, even
though these postulates are of limited meaning in the structured
argumentation, they can be easily adapted to a variant that is more meaningful
in the structured setting. We restrict attention to postulates that will be
relevant in the rest of the paper, and leave an exhaustive study of the whole
range of postulates for ranking-based semantics (as found in e.g. [14]) for
future work.

An important postulate in the literature on ranking-based semantics is the
principle of _void precedence_ [1]. It ensures that unattacked arguments are
maximally preferred.

######  Definition 7.

A semantics $\sigma$ satisfies _void precedence_ iff $a^{-}\neq\emptyset$ and
$b^{-}=\emptyset$ implies $\sigma_{{\sf AF}}(a)<\sigma_{{\sf AF}}(b)$.

When moving to logic-based argumentation, void precedence is often trivially
satisfied as there are virtually no unattacked arguments (except for
$\emptyset$) due to inconsistent arguments “contaminating” [21] innocent
bystanders:

######  Example 4.

Consider again Example 1. We see that $\\{p,\lnot p\\}$ attacks $q$ due to
logical explosion.

It is reasonable to expect an innocent bystander to be ranked higher than
other arguments. We thus obtain the following postulate. We first introduce
the notation $\sigma_{\sf ABF}$ to denote the ranking-based semantics
$\sigma_{\sf AF}$ for ${\sf AF}=(\wp(Ab),\rightarrow)$, where $\rightarrow$ is
defined as in Definition 3.

######  Definition 8.

A semantics $\sigma$ satisfies _logical void precedence_ iff for any ABF ${\sf
ABF}={\langle\mathfrak{L},\Gamma,Ab,\sim\rangle}$, if $\Theta\subseteq{\sf
Free}({\sf ABF})$, then $\sigma_{\sf ABF}(\Theta)\geq\sigma_{\sf ABF}(\Delta)$
for any $\emptyset\neq\Delta\subseteq Ab$.

Notice that logical void precedence immediately implies that every free
formula is ranked at least as high as any other formulas.

Logical void precedence is satisfied by any ranking-based semantics that
satisfies the following property of monotony:

######  Definition 9.

A semantics $\sigma$ satisfies _monotony_ iff $a^{-}\subseteq b^{-}$ implies
$\sigma_{{\sf AF}}(a)\geq\sigma_{{\sf AF}}(b)$.

In Remark 1, we give some examples of ranking-based semantics that satisfy
monotony. Monotony of a ranking-based semantics implies logical void
precedence.

######  Proposition 1.

For simple contrapositive ABFs, any ranking-based semantics that satisfies
monotony satisfies logical void precedence.

###### Proof.

Suppose $\Theta\subseteq{\sf Free}({\sf ABF})$. Let
$\emptyset\neq\Delta\subset Ab$ be arbitrary. We show that every attacker
$\Phi\subseteq Ab$ of $\Theta$ also attacks $\Delta$ and so, with monotony,
$\sigma_{{\sf AF}}(\Theta)\geq\sigma_{{\sf AF}}(\Delta)$. Note that
$\Gamma\cup\Phi\vdash{\sim}\psi$ for some $\psi\in\Theta$. Since $\psi\in{\sf
Free}({\sf ABF})$, $\Gamma\cup\Phi\vdash{\sf F}$ and so
$\Gamma\cup\Phi\vdash{\sim}\delta$ for any $\delta\in\Delta$. So, $\Theta$
also attacks $\Delta$. ∎

Another property that will be useful below is _counter-transitivity_ [1],
which makes use of the following comparison of sets of arguments:

######  Definition 10.

Let a ranking-based semantics $\sigma$ and an argumentation framework ${\sf
AF}=({\cal A},\rightarrow)$ be given. For any $S_{1},S_{2}\subseteq{\cal A}$,
$S_{1}\succeq_{\sigma}S_{2}$ is a _group comparison_ if there exists an
injective mapping $f:S_{2}\to S_{1}$ s.t. for every $a\in S_{2}$, $\sigma_{\sf
AF}(f(a))\geq\sigma_{\sf AF}(a)$.

######  Example 5.

Consider again Example 1. As $\\{q\\}^{-}\subset\\{p,\lnot p\\}^{-}$, the
identity function ${\sf id}$ is an injective mapping s.t. (for any semantics
$\sigma$), $\sigma_{\sf AF}(f(\Delta))\geq\sigma_{\sf AF}(\Delta)$ for every
$\Delta\in\\{q\\}^{-}$ and thus $\\{p,\lnot
p\\}^{-}\succeq_{\sigma}\\{q\\}^{-}$.

Thus, a group comparison allows expressing that one set $S_{1}$ is better than
another one $S_{2}$ in the sense that for every element of $S_{2}$ we can find
an element of $S_{1}$ that is ranked better or equal according to $\sigma$. We
now recall the property of _counter-transitivity_ [1] that expresses that an
argument with numerous and better attackers is ranked lower:

######  Definition 11.

A semantics $\sigma$ satisfies _counter-transitivity_ if for any ${\sf
AF}=({\cal A},\rightarrow)$ and any $a,b\in{\cal A}$,
$b^{-}\succeq_{\sigma}a^{-}$ implies $\sigma_{\sf AF}(a)\geq\sigma_{\sf
AF}(b)$.

It has been shown in [20] that the categoriser, discussion-based [1] and the
burden-based semantics satisfy the postulate of counter-transitivity. Notice
that counter-transitivity implies monotony.

######  Remark 1.

The postulate of monotony is not generally used in the literature on
postulates for ranking-based semantics999The property of (strict) monotony is
studied in [5] (see principle 2, page 5).. However, several postulates that
imply monotony have been studied. We already noted the postulate of counter-
transitivity. Another example is the postulate of _cardinality precedence_
[20], satisfied among others by the categoriser, discussion-based [1] and the
burden-based semantics. A second example is the _general principle 7_ [14]
which is satisfied by e.g. the categoriser semantics, inverse equational
systems [26], LocSum [4], and social models [35].

##  4 Ranking-based Semantics as Culpability Measures

In this section, we look at the logical meaning of ranking-based semantics.
More precisely, we ask the following question: when applying ranking-based
semantics to an argumentation framework generated using a structured
argumentation methodology, what is the logical meaning of the ranking over
arguments? It turns out that a ranking-based semantics has a clear logical
meaning: it induces an inversed culpability measure. Thus, the only difference
between a culpability measure and a ranking-based semantics is the
“direction”: high ranking scores equal low culpability values and vice versa.

Now, in more formal detail, as is common in structured argumentation, we
consider an ${\sf ABF}$ as a knowledge base that consists of strict and
defeasible premises, equipped with a logic. We note that the fact that we
allow for strict premises and a wide choice of logics is a generalization
w.r.t. the assumptions usually made for culpability measures (i.e., existing
measures are always defined for a specific logic, in most cases classical
logic.). Then, ranking-based semantics are applied to the argumentation
framework generated on the basis of the ABF. Afterwards, we look at the
arguments $\Delta\subseteq Ab$ (i.e. sets of defeasible premises) and will see
that the rankings over these arguments induce a culpability measure. As we
allow for strict premises and a wide variety of logics, the postulates from
Subsection 2.3 have to be generalized, as we do in Proposition 2. We also
generalize the postulates to sets of defeasible assumptions instead of single
assumptions.

In culpability measures, the value $0$ is seen as a “best rank”, i.e., the
rank assigned to the maximally plausible formulas. For the work done below, it
is therefore often useful to assume a best rank w.r.t. a ranking-based
semantics.

######  Definition 12.

A semantics $\sigma$ satisfies _void best rank_ if there is an
$r\in\mathbb{R}_{\geq 0}$ s.t. for every ${\sf AF}=({\cal A},\to)$ and
$a\in{\cal A}$ with $a^{-}=\emptyset$, $\sigma_{{\sf AF}}(a)=r$. In that case,
we say $r=\mathsf{{\mathsf{BestScore}}}(\sigma)$.

It can be easily observed that the void best rank property is satisfied by the
categoriser semantics, as unattacked arguments are assigned the rank $1$ at
any step in the computation. Likewise, this property holds for any
instantiation of the social models from [35], the burden-based semantics and
the discussion-based semantics [1].

The next result shows that a ranking-based semantics is a culpability measure
satisfying dominance and free formula postulates. Notice that all postulates
of culpability measures are generalized to the setting of simple
contrapositive ABFs, which have as a special case the culpability measures as
defined in Section 2.3 (with ${\sf ABF}=({\sf CL},\emptyset,Ab,\sim)$).

######  Proposition 2.

Any ranking-based semantics that satisfies monotony, void precedence and void
best rank satisfies the following properties (for a simple contrapositive ABF
${\sf ABF}=({\mathfrak{L}},\Gamma,Ab,\sim)$):

Freeness.

If $\Psi\subseteq{\sf Free}({\sf ABF})$, then $\sigma_{\sf
ABF}(\Psi)\geq\sigma_{\sf ABF}(\Delta)$ for any $\emptyset\neq\Delta\subseteq
Ab$.

Dominance.

If $\Gamma\cup\\{\phi\\}\vdash\psi$ and $\Gamma\cup\\{\phi\\}\not\vdash{\sf
F}$, then $\sigma_{\sf ABF}(\\{\psi\\})\geq\sigma_{\sf ABF}(\\{\phi\\})$.

Blame.

If $\phi\in\bigcup{\sf MIC}({\sf ABF})$, then $\sigma_{\sf
ABF}(\\{\phi\\})<{\mathsf{BestScore}}(\sigma)$.

Consistency.

If $\Gamma\cup Ab\not\vdash{\sf F}$, then $\sigma_{\sf
ABF}(\Phi)={\mathsf{BestScore}}(\sigma)$ for every $\Phi\subseteq Ab$.

###### Proof.

Freeness: Follows immediately from Proposition 1. Dominance: Consider
$\phi,\psi\in Ab$ s.t. $\Gamma\cup\\{\phi\\}\vdash\psi$ and
$\Gamma\cup\\{\phi\\}\not\vdash{\sf F}$. Suppose some $\Delta\subseteq Ab$
attacks $\\{\psi\\}$, i.e., $\Gamma\cup\Delta\vdash{\sim}\psi$. So with
contraposition, $\Gamma\cup\\{\psi\\}\vdash{\sim}\bigwedge\Delta$, and thus,
$\Gamma\cup\\{\phi\\}\vdash{\sim}\bigwedge\Delta$ and thus,
$\Gamma\cup\Delta\vdash{\sim}\phi$. This means
$\\{\psi\\}^{-}\subseteq\\{\phi\\}^{-}$ and thus with monotony, $\sigma_{\sf
ABF}(\\{\phi\\})\leq\sigma_{\sf ABF}(\\{\psi\\})$. Blame: Suppose
$\phi\in\bigcup{\sf MIC}({\sf ABF})$. Then, there is some $\Delta\in{\sf
MIC}({\sf ABF})$ s.t. $\phi\in\Delta$. As $\Gamma\cup\Delta\vdash{\sim}\phi$,
$\\{\phi\\}^{-}\neq\emptyset$ and thus with void precedence, void best rank
and since $\emptyset^{-}=\emptyset$, $\sigma_{\sf ABF}(\phi)<\sigma_{\sf
ABF}(\emptyset)={\mathsf{BestScore}}(\sigma)$. Consistency: Suppose
$\Gamma\cup Ab\not\vdash{\sf F}$. Then, there are no attacks between sets in
$(\wp(Ab),\rightarrow)$, and thus with void best rank, we have that
$\sigma_{\sf ABF}(\Phi)={\mathsf{BestScore}}(\sigma)$ for every $\Phi\subseteq
Ab$. ∎

Besides establishing that ranking-based semantics induce culpability measures,
we show that these induced culpability measures perform better than existing
culpability measures based on minimal inconsistent subsets. This is
demonstrated using the following example:

######  Example 6.

Let $Ab=\\{p\land\lnot p,q,\lnot q\land r,\lnot q\land s\\}$. Then, ${\cal
C}^{\star}_{Ab}(p\land\lnot p)=\frac{1}{3}$ whereas ${\cal
C}^{\star}_{Ab}(q)=\frac{2}{3}$. This is counter-intuitive as $p\land\lnot p$
is inconsistent, whereas $q$ is consistent. For the culpability measures
${\cal C}^{d}$ and ${\cal C}^{c}$, the situation is comparable, as ${\cal
C}^{d}_{Ab}(\phi)=1$ and ${\cal C}^{c}_{Ab}(\phi)=\frac{1}{5}$ for any
$\phi\in\\{p\land\lnot p,q\\}$, i.e., these measures do not distinguish
between the self contradictory formula $p\land\lnot p$ and the consistent
formula $q$.

We will see below that many ranking-based semantics, including the categoriser
semantics, avoid this behaviour (Proposition 3).

We formalize this kind of behaviour, or rather, the avoidance of it, by the
following postulate of _falsity_ :

######  Definition 13.

A semantics $\sigma$ satisfies _falsity_ if for any ${\sf
ABF}={\langle\mathfrak{L},\Gamma,Ab,\sim\rangle}$, if
$\Gamma\cup\\{\phi\\}\vdash{\sf F}$, then for any $\psi\in Ab$, $\sigma_{{\sf
ABF}}(\phi)\leq\sigma_{{\sf ABF}}(\psi)$.

This postulate is satisfied by a large family of ranking-based semantics
(recall Remark 1).

######  Proposition 3.

A ranking-based semantics $\sigma$ satisfies falsity if it satisfies monotony.

###### Proof.

Consider a $\phi\in Ab$ s.t. $\Gamma\cup\\{\phi\\}\vdash{\sf F}$. Then,
$\Gamma\vdash{\sim}\phi$. Thus, for any $\Delta\subseteq Ab$, $\Delta$ attacks
$\\{\phi\\}$. So, $\\{\phi\\}^{-}\supseteq\Delta^{-}$ for any $\Delta\subseteq
Ab$. With monotony, for any such $\Delta$ (and thus for any
$\\{\psi\\}\subseteq Ab$ in particular), $\sigma_{{\sf
ABF}}(\phi)\leq\sigma_{{\sf ABF}}(\Delta)$. ∎

Another property related to the postulate of falsity is that of _self-
contradiction_ [37], requiring that self-attacking arguments are ranked lower
than any other arguments. It is clear that self-contradiction ensures falsity.
The self-contradiction postulate has been shown to hold for the nh-categoriser
semantics [43], and the ranking-based semantics from [37].

In this section, we have shown that ranking-based semantics applied to
argumentation frameworks generated on the basis of simple contrapositive ABFs
induce culpability measures, and thus have a clear logical meaning.
Furthermore, we show that, in contradistinction to many other culpability
measures, ranking-based semantics satisfy the postulate of falsity. Finally,
our method is applicable to a wider variety of settings than existing
culpability measures, since we allow for strict premises and any
contrapositive logic.

##  5 Logical Argumentation Based on Conclusion-Support Pairs

So far, we have considered the assumption-based argumentation framework, which
is one among many formalisms for structured argumentation. In this section, we
show that our results generalize to other settings based on arguments
conceived of support-conclusion pairs and using several different attack
forms, such as presented in [11, 17].

###  5.1 A Glimpse at Logical Argumentation

We now consider argumentation frameworks constructed on the basis of arguments
conceived of as assumption-conclusion pairs (see, e.g, [12, 18])101010See,
e.g., [7] for a more detailed comparison of the formalisms. While in such
systems typically only defeasible assumptions are considered, recently strict
assumptions have been included in [8]., which is another popular approach to
logical argumentation.

######  Definition 14 (Argument).

Given a logic ${\mathfrak{L}}={\langle{\cal L},\vdash\rangle}$ and a set of
strict premises $\Gamma\subseteq{\cal L}$, an argument is a pair
$a={\langle\Delta,\psi\rangle}$, where $\Delta$ is a finite set of ${\cal
L}$-formulas and $\psi$ is an ${\cal L}$-formula, such that
$\Gamma\cup\Delta\vdash\psi$. We denote the set of all arguments by
$\mathsf{Arg}(\Gamma)$.

In the following, we shall denote arguments by $a,b,c$, etc., possibly primed
or indexed.

  * •

Given an argument $a={\langle\Delta,\psi\rangle}$, we call $\Delta$ the
support set of $a$, and $\psi$ the conclusion (or the claim) of $a$ (denoted
by ${\sf Conc}(A)$).

  * •

The set of arguments whose supports are subsets of $Ab$ is denoted by ${\sf
Arg}(\Gamma,Ab)$. That is: ${\sf Arg}(\Gamma,Ab)=\\{a\in{\sf
Arg}(\Gamma)\mid\mathsf{Sup}(a)\subseteq Ab\\}$.

######  Remark 2.

An alternative notation for an argument ${\langle\Delta,\psi\rangle}$ is
$\Delta\Rightarrow\psi$ (where $\Rightarrow$ is a new symbol, not appearing in
the language of $\Delta$ and $\psi$). The latter resembles the way sequents
are denoted in the context of proof theory [27]. This notation is frequently
used in sequent-based argumentation (see, e.g., [12, 13]) to emphasize the
fact that the only requirement on $\Delta$ and $\psi$ to form an argument is
that the latter follows, according to the base logic, from the former.

Various attack forms have been studied in the literature. In logical
argumentation, it is often assumed that attacks among arguments are usually
expressed in terms of attack rules (w.r.t. the underlying logic). Table 1
gives some of these attack relations. Notice that we restrict attention to
support-based attacks. Other attack rules between arguments are described,
e.g. in [28, 18, 12].

Rule Name | Acronym | Attacking Argument | Attacked Argument | Attack Conditions
---|---|---|---|---
Defeat | Def | ${\langle\Delta_{1},\psi_{1}\rangle}$ | ${\langle\Delta_{2},\psi_{2}\rangle}$ | $\Gamma\cup\psi_{1}\vdash\sim\\!\bigwedge\\!\Delta_{2}$
Direct Defeat | DirDef | ${\langle\Delta_{1},\psi_{1}\rangle}$ | ${\langle\\{\delta_{2}\\}\cup\Delta^{\prime}_{2},\psi_{2}\rangle}$ | $\Gamma,\psi_{1}\vdash\sim\\!\delta_{2}$
Undercut | Ucut | ${\langle\Delta_{1},\psi_{1}\rangle}$ | ${\langle\Delta^{\prime}_{2}\cup\Delta^{\prime\prime}_{2},\psi_{2}\rangle}$ | $\psi_{1}\equiv_{\Gamma}\sim\\!\bigwedge\\!\Delta^{\prime}_{2}$
Canonical Undercut | CanUcut | ${\langle\Delta_{1},\psi_{1}\rangle}$ | ${\langle\Delta_{2},\psi_{2}\rangle}$ | $\psi_{1}\equiv_{\Gamma}\sim\\!\bigwedge\\!\Delta_{2}$
Direct Undercut | DirUcut | ${\langle\Delta_{1},\psi_{1}\rangle}$ | ${\langle\\{\delta_{2}\\}\cup\Delta^{\prime}_{2},\psi_{2}\rangle}$ | $\psi_{1}\equiv_{\Gamma}\sim\\!\delta_{2}$
Table 1: Some attack rules in the context of the strict premises $\Gamma$. We
let $\phi\equiv_{\Gamma}\psi$ iff $\Gamma\cup\\{\phi\\}\vdash\psi$ and
$\Gamma\cup\\{\psi\\}\vdash\phi$. We assume that the support sets of the
attacked arguments are nonempty (to avoid attacks on theorems).

Rules like those specified in Table 1 form attack schemes that are applied to
particular arguments according to the underlying logic. For instance, when
classical logic is the underlying formalism, the attacks of
${\langle\\{p\\},p\rangle}$ on ${\langle\\{\neg p\\},\neg p\rangle}$ and of
${\langle\\{\neg p\\},\neg p\rangle}$ on ${\langle\\{p\wedge q\\},p\rangle}$
are obtained by applications of the Defeat rule (or other rules in Table 1).
When an attack rule ${\cal R}$ is applied, we say that its attacking argument
${\cal R}$-attacks the attacked argument. Notice that assumption-based
argumentation makes use of the attack form of direct defeat.

###  5.2 Considerations on Ranking-based Semantics

One of the benefits of assumption-based argumentation is the fact that the
resulting argumentation frameworks are finite when the set of assumptions $Ab$
is finite. To the best of our knowledge, all ranking-based semantics are
defined only for finite argumentation frameworks, and therefore this property
of assumption-based argumentation ensures that ranking-based semantics are
applicable to argumentation frameworks constructed in assumption-based
argumentation. This stands in contrast with the systems based on support-
conclusion pairs, which even for finite sets of assumptions give rise to
infinite sets of arguments (as e.g. the argument
$\langle\emptyset,p\lor\ldots\lor p\rangle$ will be part of any argumentation
framework with $p\lor\ldots\lor p$ having length $n$ for any
$n\in\mathbb{N}$). One way to circumvent this is found in [2]. In that paper,
Amgoud and Ben-Naim assume that there exists a well-order $w$ on ${\cal L}$
and restrict attention to the following set of arguments:111111Amgoud & Ben-
Naim [2] require that arguments are additionally consistent and minimal, but
we do not generally require this.

${\cal A}^{\sf fn}(\Gamma,Ab)=\\{\langle\Delta,\phi\rangle\in{\cal
A}(\Gamma,Ab)\mid\phi=\min_{w}\\{\phi^{\prime}\mid\phi^{\prime}\equiv\phi\\}\\}$

Thus, the well-ordering $w$ allows selecting one among the infinitely many
equivalent conclusions $\phi$.

The first observation we make is that excluding inconsistent arguments, as is
done in e.g. [1, 16], makes the resulting frameworks useless for obtaining
culpability measures, for the simple reason that arguments with inconsistent
premises are not allowed, and thus the argumentation framework will not
contain arguments that conclude an inconsistent premise. We will therefore not
filter out inconsistent arguments.

######  Example 7.

Consider $Ab=\\{p\land\lnot p,q,\lnot q\land r\\}$. Then, any argument using
$p\land\lnot p$ in the support set will have an inconsistent support and thus
not be assigned any ranking.

We now show that a wide variety of attack rules ensure satisfaction of
postulates for culpability measures.

Given a set of strict premises $\Gamma$ and a set of defeasible premises $Ab$,
we apply a ranking-based semantics over the argumentation framework ${\sf
AF}=({\cal A}^{\sf fn}(\Gamma,Ab),{\cal R})$, in which the arguments ${\cal
A}^{\sf fn}(\Gamma,Ab)$ are obtained as defined above and the attacks ${\cal
R}$ are obtained by applying a set of attack rules ${\sf
R}\subseteq\\{\mathrm{Def},\mathrm{DirDef},\mathrm{Ucut},\mathrm{CanUcut},\mathrm{DirUcut}\\}$.
Recall that we denote the resulting ranking by $\sigma_{\sf AF}$.

######  Proposition 4.

Let some $\Gamma,Ab\subseteq{\cal L}$, ${\sf
R}\subseteq\\{\mathrm{Def},\mathrm{DirDef},\mathrm{Ucut},\mathrm{CanUcut},\mathrm{DirUcut}\\}$,
${\cal A}={\cal A}^{\sf fn}(\Gamma,Ab)$, and ${\sf AF}=({\cal A},{\sf R})$ be
given. Any ranking-based semantics $\sigma$ that satisfies counter-
transitivity, void precedence and void best rank satisfies the following
properties:

Freeness.

If $\Psi\subseteq{\sf Free}(\Gamma,Ab)$121212Recall that ${\sf
FREE}(\Gamma,Ab)={\sf Free}({\langle\mathfrak{L},\Gamma,Ab,\sim\rangle})$.,
then $\sigma_{\sf AF}(\langle\Psi,\psi\rangle)\geq\sigma_{\sf
AF}(\langle\Delta,\delta\rangle)$ for any
$\langle\Delta,\delta\rangle,\langle\Psi,\psi\rangle\in{\cal A}$ with
$\Delta\neq\emptyset$.

Dominance.

Where $\psi,\phi\in Ab$, if $\Gamma,\psi\vdash\phi$ and
$\Gamma\cup\psi\not\vdash{\sf F}$, then $\sigma_{\sf
AF}(\langle\\{\psi\\},\psi^{\prime}\rangle)\geq\sigma_{\sf
AF}((\langle\\{\phi\\},\phi^{\prime}\rangle)$ for all
$\langle\\{\phi\\},\phi^{\prime}\rangle,\langle\\{\psi\\},\psi^{\prime}\rangle\in\mathcal{A}$.

Blame.

If $\phi\in\bigcup{\sf MIC}(\Gamma,Ab)$, then $\sigma_{\sf
AF}(\langle\\{\phi\\},\phi^{\prime}\rangle)\><\>{\mathsf{BestScore}}(\sigma)$
for all $\langle\\{\phi\\},\phi^{\prime}\rangle\in\mathcal{A}$.

Furthermore, if $\sigma$ satisfies void best rank, it satisfies:

Consistency.

If $\Gamma\cup Ab\not\vdash{\sf F}$, then
$\sigma_{\mathsf{AF}}(\langle\Phi,\phi\rangle)={\mathsf{BestScore}}(\sigma)$
for every $\langle\Phi,\phi\rangle\in\mathcal{A}$.

###### Proof.

The following lemma will be useful.

######  Lemma 1.

Where $\langle\Psi,\psi\rangle,\langle\Phi,\phi\rangle\in\mathcal{A}$, if for
every $\langle\Delta,\delta\rangle\in\mathcal{A}$ that attacks
$\langle\Psi,\psi\rangle$ there is an
$\langle\Delta,\delta^{\prime}\rangle\in\mathcal{A}$ that attacks
$\langle\Phi,\phi\rangle$, then $\sigma_{\sf
AF}(\langle\Phi,\phi\rangle)\leq\sigma_{\sf AF}(\langle\Psi,\psi\rangle)$.

Note that for each such $\langle\Delta,\delta\rangle$ and
$\langle\Delta,\delta^{\prime}\rangle$ we have
$\langle\Delta,\delta\rangle^{-}=\langle\Delta,\delta^{\prime}\rangle^{-}$.
So, by monotony (which is implied by counter-transitivity), $\sigma_{\sf
AF}(\langle\Delta,\delta\rangle)=\sigma_{\sf
AF}(\langle\Delta,\delta^{\prime}\rangle)$. Thus, by counter-transitivity,
$\sigma_{\sf AF}(\langle\Phi,\phi\rangle)\leq\sigma_{\sf
AF}(\langle\Psi,\psi\rangle)$, which proves our lemma.

We now prove the properties of Proposition 4.

For Freeness, suppose $\Psi\subseteq\mathsf{Free}(\Gamma,Ab)$ and
$\langle\Psi,\psi\rangle\in\mathcal{A}$. Consider some
$\langle\Delta,\delta\rangle\in\mathcal{A}$ with $\emptyset\neq\Delta$.
Suppose $\langle\Phi,\phi\rangle$ attacks $\langle\Psi,\psi\rangle$ with
respect to any of the attack rules in $\mathsf{R}$. Since
$\Psi\subseteq\mathsf{Free}(\Gamma,Ab)$, $\Gamma\cup\Phi\vdash\mathsf{F}$. So,
$\Gamma\cup\Phi\vdash\sim\\!\epsilon$ for any $\epsilon\in\Delta$. Thus, there
is a $\epsilon^{\prime}$ for which $\epsilon^{\prime}\equiv_{\Gamma}\epsilon$
and $\langle\Phi,\sim\\!\epsilon^{\prime}\rangle\in\mathcal{A}$. We note that
$\langle\Phi,\sim\\!\epsilon^{\prime}\rangle$ attacks
$\langle\Delta,\delta\rangle$ with respect to any of the attack rules in
$\mathsf{R}$. Thus, by Lemma 1,
$\sigma(\langle\Psi,\psi\rangle)\geq\sigma(\langle\Delta,\delta\rangle)$.

For Dominance, suppose $\langle\Delta,\delta\rangle$ attacks
$\langle\\{\phi\\},\phi^{\prime}\rangle$. So, (1)
$\Gamma\cup\\{\delta\\}\vdash{\sim}\phi$ (in case of Def and DirDef) or (2)
$\delta\equiv_{\Gamma}{\sim}\phi$ (in case of Ucut, ConUcut and DirUcut).
Since $\Gamma\cup\\{\psi\\}\vdash\phi$, by contraposition
$\Gamma\cup\\{{\sim}\phi\\}\vdash{\sim}\psi$. So,
$\Gamma\cup\\{\delta\\}\vdash{\sim}\psi$ by the transitivity of $\vdash$. So
$\langle\Delta,\delta\rangle$ also attacks
$\langle\\{\psi\\},\psi^{\prime}\rangle$ in case (1). Also, there is a
$\epsilon$ for which $\epsilon\equiv_{\Gamma}\sim\\!\psi$ and
$\langle\Delta,\epsilon\rangle\in\mathcal{A}$. In case (2),
$\langle\Delta,\epsilon\rangle$ attacks
$\langle\\{\psi\\},\psi^{\prime}\rangle$. By Lemma 1, $\sigma_{\sf
AF}(\langle\\{\phi\\},\phi^{\prime}\rangle)\geq\sigma_{\sf
AF}(\langle\\{\psi\\},\psi^{\prime}\rangle)$.

For blame, suppose $\phi\in\bigcup{\sf MIC}(\Gamma,Ab)$. Notice that
$\langle\emptyset,\psi\rangle\in{\cal A}$ for any $\psi\in{\cal L}$ s.t.
$\Gamma\vdash\psi$. Furthermore, inspection of all the attack rules confirms
that $(\langle\emptyset,\psi\rangle)^{-}=\emptyset$. Since $\phi\in\bigcup{\sf
MIC}(\Gamma,Ab)$, there is a $\Delta\in{\sf MIC}(\Gamma,Ab)$ s.t.
$\Gamma\cup\Delta\vdash{\sf F}$ and $\phi\in\Delta$. Thus,
$\Gamma\cup\Delta\vdash{\sim}\phi$ (with contraposition), which means
$\langle\Delta,\lnot\phi^{\star}\rangle\in{\cal A}$ (for
$\phi^{\star}\equiv_{\Gamma}\phi$) attacks
$\langle\\{\phi\\},\phi^{\prime}\rangle\in{\cal A}$ and thus
$(\langle\\{\phi\\},\phi\rangle)^{-}\neq\emptyset$. With void precedence and
void best rank, this means $\sigma_{\sf
AF}(\langle\\{\phi\\},\phi\rangle)<\sigma_{\sf
AF}(\langle\emptyset,\psi\rangle)={\mathsf{BestScore}}(\sigma)$.

For Consistency, suppose $\Gamma\cup Ab\not\vdash{\sf F}$ and
$\langle\Phi,\phi\rangle\in\mathcal{A}$. Then,
$\langle\Phi,\phi\rangle^{-}=\emptyset$ and by void best rank,
$\sigma_{\mathsf{AF}}(\langle\Phi,\phi\rangle)={\mathsf{BestScore}}(\sigma)$.
∎

##  6 Related Work

Interestingly enough, one of the earliest ranking-based semantics, namely the
categoriser semantics [15] has been proposed in the context of logical
argumentation. Since then, ranking-based semantics have been studied in breath
and depth, but from an abstract level. [15] does not use Dungean argumentation
frameworks, but rather is based on argumentation trees for a given conclusion.
To the best of our knowledge, ranking-based semantics have been applied to
structured argumentation in only one other work, namely [2]. One difference is
that we do not assume that arguments have a consistent or minimal set of
supports. In our paper, we do not filter out inconsistent arguments, as we
need to be able to assign culpability measures to (potentially inconsistent)
premises. In the work of Amgoud and Ben-Naim, this is not possible, as an
argument like $\langle\\{p\land\lnot p\\},p\land\lnot p\rangle$ is not allowed
in their framework. Another benefit is that we allow for strict premises. A
final difference with [2] is that Amgoud and Ben-Naim adopt only the direct
undercut attack rule between argument, whereas we consider various attack
forms.

Culpability measures have been investigated in several proposals. To the best
of our knowledge, we have studied all postulates proposed for culpability
measures in the literature. In future work, we want to compare ranking-based
semantics with more involved culpability measures, such as the _tacit
culpability measures_ from [40].

##  7 Conclusion and Future Work

This paper contains a principled investigation of ranking-based semantics
applied to structured argumentation. Rather than having the last word on this
topic, we hope to have established that ranking-based semantics for structured
argumentation are meaningful and useful, and thus open new ground for
research. We believe this work can be extended in various directions, e.g. to
ranking-semantics over sets of attacking logical arguments, other frameworks
for structured argumentation [19, 22, 11, 38, 29, 10, 24, 32, 9], which give a
natural account of the notion of so-called intrinsic strength of arguments
often assumed in ranking-based logics [14].

## References

  * [1] Leila Amgoud and Jonathan Ben-Naim.  Ranking-based semantics for argumentation frameworks.  In SUM, pages 134–147. Springer, 2013.
  * [2] Leila Amgoud and Jonathan Ben-Naim.  Argumentation-based ranking logics.  In AAMAS, pages 1511–1519, 2015.
  * [3] Leila Amgoud, Jonathan Ben-Naim, Dragan Doder, and Srdjan Vesic.  Acceptability semantics for weighted argumentation frameworks.  In IJCAI, 2017.
  * [4] Leila Amgoud, Claudette Cayrol, Marie-Christine Lagasquie-Schiex, and Pierre Livet.  On bipolarity in argumentation frameworks.  International Journal of Intelligent Systems, 23(10):1062–1093, 2008.
  * [5] Leila Amgoud and Victor David.  A general setting for gradual semantics dealing with similarity.  In AAAI, pages 6185–6192, 2021.
  * [6] Ofer Arieli, AnneMarie Borg, and Jesse Heyninck.  A review of the relations between logical argumentation and reasoning with maximal consistency.  Annals of Mathematics and Artificial Intelligence, 87(3):187–226, 2019.
  * [7] Ofer Arieli, AnneMarie Borg, Jesse Heyninck, and Christian Straßer.  Logic-based approaches to formal argumentation.  FLAP, 8(6):1793–1898, 2021.
  * [8] Ofer Arieli, AnneMarie Borg, and Christian Straßer.  Characterizations and classifications of argumentative entailments.  In Proceedings of KR, pages 52–62, 2021.
  * [9] Ofer Arieli and Jesse Heyninck.  Simple contrapositive assumption-based argumentation part II: reasoning with preferences.  Int. J. Approx. Reason., 139:28–53, 2021.
  * [10] Ofer Arieli and Jesse Heyninck.  Simple contrapositive assumption-based frameworks, part II: Reasoning with preferences.  Journal of Approximate Reasoning, 139:28–53, 2021.
  * [11] Ofer Arieli and Christian Straßer.  Sequent-based logical argumentation.  Journal of Argument and Computation, 6(1):73–99, 2015.
  * [12] Ofer Arieli and Christian Straßer.  Sequent-based logical argumentation.  Argument & Computation, 6(1):73–99, 2015.
  * [13] Ofer Arieli and Christian Straßer.  Logical argumentation by dynamic proof systems.  Theoretical Computer Science, 781:63–91, 2019.
  * [14] Pietro Baroni, Antonio Rago, and Francesca Toni.  From fine-grained properties to broad principles for gradual argumentation: A principled spectrum.  International Journal of Approximate Reasoning, 105:252–286, 2019\.
  * [15] Philippe Besnard and Anthony Hunter.  A logic-based theory of deductive arguments.  Artificial Intelligence, 128(1):203–235, 2001.
  * [16] Philippe Besnard and Anthony Hunter.  Argumentation based on classical logic, pages 133–152.  Springer, 2009.
  * [17] Philippe Besnard and Anthony Hunter.  A review of argumentation based on deductive arguments.  Handbook of Formal Argumentation, 1:435–482, 2018.
  * [18] Philippe Besnard and Anthony Hunter.  A review of argumentation based on deductive arguments.  In Handbook of Formal Argumentation, pages 437–484. College Publications, 2018.
  * [19] Andrei Bondarenko, Phan Minh Dung, Robert Kowalski, and Francesca Toni.  An abstract, argumentation-theoretic approach to default reasoning.  Artificial Intelligence, 93(1):63–101, 1997.
  * [20] Elise Bonzon, Jérôme Delobelle, Sébastien Konieczny, and Nicolas Maudet.  A comparative study of ranking-based semantics for abstract argumentation.  In AAAI, 2016.
  * [21] Martin Caminada.  Contamination in formal argumentation systems.  In BNAIC, pages 59–65, 2005.
  * [22] Kristijonas Čyras, Quentin Heinrich, and Francesca Toni.  Computational complexity of flat and generic assumption-based argumentation, with and without probabilities.  Artificial Intelligence, 293:103449, 2021.
  * [23] Kristijonas Cyras, Tiago Oliveira, Amin Karamlou, and Francesca Toni.  Assumption-based argumentation with preferences and goals for patient-centric reasoning with interacting clinical guidelines.  Argument Comput., 12(2):149–189, 2021.
  * [24] Kristijonas Čyras and Francesca Toni.  ABA+: assumption-based argumentation with preferences.  arXiv preprint arXiv:1610.03024, 2016.
  * [25] Phan Minh Dung.  On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games.  Artificial Intelligence, 77:321–358, 1995.
  * [26] Dov M Gabbay.  Equational approach to argumentation networks.  Argument & Computation, 3(2-3):87–142, 2012.
  * [27] Gerhard Gentzen.  Untersuchungen über das logische Schließen I, II.  Mathematische Zeitschrift, 39:176–210, 405–431, 1934.
  * [28] Nikos Gorogiannis and Anthony Hunter.  Instantiating abstract argumentation with classical logic arguments: Postulates and properties.  Artificial Intelligence, 175(9–10):1479–1497, 2011.
  * [29] J. Heyninck and C. Straßer.  Revisiting unrestricted rebut and preferences in structured argumentation.  In Proc. IJCAI’17, pages 1088–1092. AAAI Press, 2017.
  * [30] Jesse Heyninck and Ofer Arieli.  On the semantics of simple contrapositive assumption-based argumentation frameworks.  In Proc. COMMA, pages 9–20, 2018.
  * [31] Jesse Heyninck and Ofer Arieli.  Simple contrapositive assumption-based argumentation frameworks.  Journal of Approximate Reasoning, 121:103–124, 2020.
  * [32] Jesse Heyninck and Christian Straßer.  A comparative study of assumption-based argumentative approaches to reasoning with priorities.  FLAP, 8(3):737–808, 2021.
  * [33] Anthony Hunter and Sébastien Konieczny.  Measuring inconsistency through minimal inconsistent sets.  In KR, pages 358–366, 2008.
  * [34] Saïd Jabbour, Yue Ma, Badran Raddaoui, Lakhdar Sais, and Yakoub Salhi.  A MIS partition based framework for measuring inconsistency.  In KR, pages 84–93, 2016.
  * [35] João Leite and João G. Martins.  Social abstract argumentation.  In IJCAI, pages 2287–2292, 2011.
  * [36] J. Los and R. Suszko.  Remarks on sentential logics.  Indagationes Mathematicae (Proceedings), 61:177–183, 1958.
  * [37] Paul-Amaury Matt and Francesca Toni.  A game-theoretic measure of argument strength for abstract argumentation.  In JELIA, volume 5293, pages 285–297, 2008.
  * [38] Henry Prakken.  An abstract framework for argumentation with structured arguments.  Argument and Computation, 1(2):93–124, 2010.
  * [39] Henry Prakken.  A new use case for argumentation support tools: supporting discussions of bayesian analyses of complex criminal cases.  Artif. Intell. Law, 28(1):27–49, 2020.
  * [40] Jandson S Ribeiro and Matthias Thimm.  Consolidation via tacit culpability measures: Between explicit and implicit degrees of culpability.  In KR, pages 529–538, 2021.
  * [41] Matthias Thimm.  Tweety: A comprehensive collection of java libraries for logical aspects of artificial intelligence and knowledge representation.  In KR, 2014.
  * [42] Matthias Thimm.  Inconsistency measurement.  In SUM, volume 11940, pages 9–23, 2019.
  * [43] Bruno Yun, Srdjan Vesic, and Madalina Croitoru.  Ranking-based semantics for sets of attacking arguments.  In AAAI, pages 3033–3040, 2020.
  * [44] Zhiwei Zeng, Zhiqi Shen, Benny Toh Hsiang Tan, Jing Jih Chin, Cyril Leung, Yu Wang, Ying Chi, and Chunyan Miao.  Explainable and argumentation-based decision making with qualitative preferences for diagnostics and prognostics of alzheimer’s disease.  In KR, pages 816–826, 2020.