An integrated theory of causal stories and evidential
arguments
Floris Bex
Department of Information and Computing Sciences
Utrecht University, The Netherlands

f.j.bex@uu.nl
ABSTRACTIn the process of proof alternative stories that explain ‘what
happened’ in a case are tested using arguments based on evidence. Building on the author’s earlier hybrid theory, thispaper presents a formal theory that combines causal stories
and evidential arguments, further integrating the different
types of reasoning in a framework for structured argumentation. This then allows for correct reasoning with causal
and evidential rules, and further integrates arguments andstories by grounding them both in well-known dialecticalargumentation semantics.

1.

INTRODUCTION

The process of legal proof is one of inference to the best
explanation (IBE), in which alternative stories or scenariosthat explain ‘what happened’ in a case are tested and compared using arguments based on evidence. This combinationof explanatory, story-driven and argumentative, evidencedriven reasoning is captured in the hybrid theory [4], which
combines theories of causal-abductive diagnosis [15, 10] witha framework for structured argumentation [3].
In the hybrid theory, causally coherent stories that explain the evidence can be abductively inferred: if we observe
some effect e (e.g. a body with gunshot wounds) we are allowed to hypothesise a possible cause c (e.g. the person wasshot). Stories can then be supported or attacked by arguments based on evidence. For example, a witness who saw
that the victim was shot provides evidence to support our
hypothesis that the victim died because he was shot, whilea coroner’s report stating that the victim was already deadwhen he sustained the gunshot wounds attacks the hypothesis. Such counterarguments can point to alternative explanations: maybe someone wanted it to look like the victim
was shot to cover up some other cause of death. Ultimately,
the alternative stories have to be compared according to howmuch of the evidence supports and attacks them.
The hybrid theory is significantly more expressive thanmost logical frameworks for IBE [15, 10], which were origPermission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others thanACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission
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DOI: http://dx.doi.org/10.1145/2746090.2746094 .

inally intended for automated diagnosis within relatively
small and pre-defined domains. It allows for reasoning withboth causal rules (c causes e) and evidential rules (e is evidence for c) and reasoning about the elements of stories,
such as the events and the causal links that comprise them.
Furthermore, where classical theories of IBE compare stories
on basic criteria such as the minimum number of assumptions, the hybrid theory also includes other more fine-grained
story comparison criteria based on the number of supportingor attacking arguments.
The hybrid theory stops short of truly integrating causal,
story-based reasoning and evidential argumentation. While
there is a clear link between arguments and stories — arguments can have elements of stories as their conclusion —arguments and stories remain completely separate entities,
each with their own definitions related to conflict and comparison. Thus, two contradictory arguments attack eachother but two contradictory stories do not, and two alternative stories are in competition, whilst two arguments that
present alternative explanations are not. Furthermore, thestatus of arguments is determined using the standard semantics of Dung [8], whereas the status of stories is determinedwith more absolute measures (i.e., by counting the numberof supporting and attacking arguments based on evidence[4]). The acceptability of a certain claim thus depends onwhether it is modelled as part of a story or an argument.
The outlines of the integrated theory were discussed ina previous position paper [2], in which both both causalabductive and evidential argumentation were captured ina structured argumentation framework similar to ASP IC +[11]. In this paper, this integrated framework is further extended and formalised as an instance of ASP IC + . This new
integrated theory allows for causal and evidential arguments
as well as mixed causal-evidential arguments, while still adhering to the constraints imposed by Pearl’s C-E system[12]. Furthermore, the integrated theory allows for a natural rendering of the different modes of reasoning employedin the process of proof, as will be demonstrated with a small
case-study that is used as a running example.
The case that is used concerns that of Danny Rijkbloem[23, 4], who has a considerable list of sentences (theft, robbery) starting when he was 15 years old. Nicole Lammers isa baker’s daughter who had a relationship with Rijkbloem
and lived together with him. At some point Nicole decided
under pressure of her parents to break up with Rijkbloem
and leaves him. A few days after the breakup, Nicole andher parents went to Rijkbloem’s house to pick up some of
Nicole’s stuff and got into an argument with Rijkbloem. Atsome point, a scuffle developed, which ended in the father
getting shot in the head. When the police, who had been informed by Rijkbloem, arrived, the father was already dead.
From this point onwards, the two women (Nicole and hermother) and Rijkbloem tell a different story. According to
the two women, Rijkbloem pulled out a gun with which heshot father Lammers. Rijkbloem, however, gives a different
account of the events. He said that during the fight, Mrs.
Lammers pulled a gun out of her purse and threatened toshoot Rijkbloem with it. Rijkbloem pushed the hand holding the gun away and in the struggle the gun went off andthe bullet hit father Lammers.

2.

A HYBRID THEORY OF STORIES AND
ARGUMENTS

Two influential methods for the analysis of evidence are
the argument-based approach [3], which focuses on arguments based on evidence, and the story-based approach [23,14], which uses hypothetical stories or scenarios to explain
the evidence. In the argument-based approach the reasoning goes from evidence to conclusions by the application ofevidential rules (e is evidence for c), and arguments for andagainst the conclusion are considered and compared. In thestory-based approach multiple scenarios consisting of causal
rules (c is a cause for e) are constructed to explain the evidence, and the scenarios are considered and compared.
Both the evidential-argument approach and the causalstory approach have advantages. An inference from a pieceof evidence (e.g. a witness testimony) to a conclusion isbest captured using evidential rules [4, 7], and arguments
based on evidence can be compared using Dung’s solid andintuitive framework [8]. However, certain parts of a case(such as the cause of death, or the behaviour of the suspect)
are more easily captured using causal rules [7]. Furthermore,scenarios help in providing an overview of the case as they
are understandable stories that explain the evidence in anatural way [14]. What is missing from purely causal, storybased approaches, however, is the possibility to talk about
elements (events, causal links) in a scenario using evidence.
The combination of evidential arguments and causal scenarios then seems to be an intuitive and analytically usefulperspective for looking at cases and evidence. Hence, we
proposed the hybrid theory [4], a combination of formal,
causal story-based reasoning (see e.g. [10, 15]) and formal

Women
testimony1

Rijkbloem
and father
arguedRijkbloem
testimony1

evidential argumentation [3].
The hybrid theory consists of a set of evidence, a set of hypotheses and a set of defeasible inference rules, both causal(c causes e) and evidential (e is evidence for c). Scenarios
are built by taking some observations and abductively inferring hypotheses. So given an observation o and a causal
rule ‘c causes o’, we can abductively infer c. Arguments canbe built by taking evidence and consecutively applying evidential inference rules to infer conclusions. So given a pieceof evidence e and an evidential rule ‘e is evidence for c’ we
can conclude c. The connection between arguments and stories lies in the conclusions of arguments. These conclusions
can be elements of a story (i.e., the evidence on which theargument is based supports the story) or the negation of elements of a story (i.e., the evidence on which the argument
is based contradicts the story).
Figure 1 shows the Rijkbloem case as a combination ofarguments and stories, where arrows with open arrowheads
stand for evidential rule applications, and arrows with closed
arrowheads stand for causal rule applications. The main observation that needs to be explained in this case is Father
died. Now, if we assume the hypothesis Rijkbloem and father argued, we can build two stories Sr and Sm that explain this observation. Story Sr says that Rijkbloem shot
father is the cause of Father was hit, while story Sm argues that Father was hit was caused by Mother shot father, which itself was caused by Rijkbloem pushed mother.
Stories Sr and Sm are alternative explanations for thesame facts. What this means is that, in absence of any explicit links between them, we consider the chances of both
explanations being true at the same time being very low(i.e., either Rijkbloem shot father or mother accidentally
shot father, but not both). In Pearl’s terms [12], two causesof a common consequence compete with each other. Thus,
accepting one explanation will severely diminish our belief
in the other. In the literature on Bayesian networks [13,20], this phenomenon is called ‘explaining away’. In logical accounts of abductive reasoning like the hybrid theory,
this can be enforced by only accepting the simplest explanations, that is, those that are (subset) minimal in terms of
the hypotheses and rules they use [15, 16, 10].
The evidence is rendered as grey boxes in figure 1. Fromthe evidence, elements of the stories can be inferred using
evidential reasoning. For example, from Women testimony3 ,a statement by Nicole and mother that it was Rijkbloem

Womentestimony2

Womentestimony3

Rijkbloem
pulled gun

Rijkbloem
shot father

Motherpulled gun

Rijkbloem
pushed mother

Mothershot father

Rijkbloem
testimony2

Rijkbloem
testimony3

Rijkbloem
testimony4

Coronerreport2

Arguments

Fatherhit

Fatherdied

Coronerreport1

Coronerreport3

Figure 1: The two Rijkbloem scenarios as a hybrid case

StoriesArguments

who shot father, we can infer the conclusion that Rijkbloem
shot father because the testimony is evidence for the factthat the event happened. It is also possible to support or
attack (applications of) causal inference rules in the stories.
For example, the evidence Coroner Report2 , an autopsy report by the coroner that father was killed by a gunshot tothe head, allows us to conclude that there is a causal link
between Father hit and Father died.
Given alternative scenarios such as Sr and Sm , the question is how to compare them. In the hybrid theory, an important role is played by evidential support, the evidence
that supports a scenario because it is part of the scenario
or because it is a premise of an argument that has as its
conclusion an element in the scenario. Similarly, anotherimportant concept is evidential contradiction, evidence that
contradicts a scenario because it is a premise of an argument that contradicts the scenario. These two concepts canbe used to compare scenarios; for example, we can look how
much of the total evidence in a case supports or attacks astory, or we can see whether a story is supported or attacked
by an important piece of evidence, thus getting an indication of the relevance and coverage of a scenario [14]. Notethat the status of arguments, that is, whether they are partof an extension [8] may influence these comparison criteria[4]. For example, we could say that only evidence whichis part of an argument in an extension counts towards theevidential support or contradiction.

2.1

Stories and arguments: two sides of thesame coin

The two separate approaches to reasoning with evidence,arguments and stories, often blend into each other, both in
the literature and in real cases. For example, the AnchoredNarratives Theory [23] has clear argumentative aspects, as
it is possible to ‘argue’ for a particular position in a caseusing causal stories instead of evidential arguments. Thisis because ultimately, it is not the structure of the reasoning (i.e. evidential, causal) but rather the intention of the
speaker that determines whether a story is used to, for example, explain or argue [5]. This is already evident in thewording chosen earlier in this paper: story S2 argues thatfather was shot accidentally.
Further overlap between the two parts of the hybrid theory
concerns the fact that causal and evidential reasoning areclosely entwined: if we have a causal rule c causes e then we
will usually also accept that, in some way, e is evidence for c[12]. For example, father getting shot can cause him to die,so the observation that father died can be seen as evidence
for the fact that he was shot. This can also be true the otherway: if we have an evidential rule, we will often also accept
some sort of causal relation, precisely because the evidential
rule is based on a causal rule. However, this depends on ourinterpretation of the evidential rule e is evidence for c. If weinterpret the rule broadly, as e being any sign that increases
our belief in c, it need not be based on a causal relation.
For example [24]: dark clouds are evidence for the fact that
it will rain later, but we would not say that the fact that it
will rain later causes dark clouds. However, if we interpretthe meaning of e is evidence for c as e being evidence forsome event c that happened earlier, then we are quite safein assuming that e was caused (at least partly) by c.
The close relation between causal and evidential reasoningis also important for formal models: Poole [15, 16] argues

that information in causal models can equally well be represented by evidential rules; instead of abductive reasoning
we then perform modus-ponens style reasoning (‘deduction’)to infer the hypotheses based on our observations. So, forexample, instead of abductively inferring Rijkbloem shot
father from the observation Father was hit and a causal
rule ‘Rijkbloem shot father causes Father was hit’, we
use the observation Father was hit and the evidential rule
‘Father was hit is evidence for Rijkbloem shot father’to infer the conclusion Rijkbloem shot father.
The relation between causal and evidential reasoning is
also evident in the different ways in which, for example, theinferences and attacks based on witness testimony can be
captured. One way to do this is to use an evidential rule,
like in the hybrid case in figure 1. An argument based on
this rule can be undercut (attacked) by claiming that, forexample, the women had a good reason to give false statements. This way of modelling inference based on witness
testimony is captured in figure 2a, where the arrow withthe open arrowhead is again an evidential inference and thearrow with the circle at the end is an attack relation. A second way to model testimonies is by regarding a testimony as
being caused either by the event to which it testifies, or byother causes, such as a reason to lie. There are then two alternative abductive explanations for Women testimony3 (figure 2): Rijkbloem shot father and Women had reason tolie, and accepting one will deny the other. This link between explaining away and undercutting has recently beenformalised in [20], who show that a Bayesian network withtwo competing causal explanations can be translated into
two attacking arguments.

Rijkbloem
shot father

Rijkbloem
shot fatherc

Women hadreason to lie
(a)

e

Women
testimony3cWomen hadreason to lie

Womentestimony3(b)

Figure 2: Witness testimony as arguments (a) or
stories (b)The above examples show that causal stories and evidential arguments are essentially two sides of the same coin: intheory any case can be modelled using either causal stories
or evidential arguments. With respect to causal and evidential, it is in some ways arbitrary how we should capturea relation. Pearl [12] argued that people have a hard time
expressing their knowledge just as either causal or evidential rules, and van den Braak et al. [7] found that, while
there are situations in which people consistently choose either causal or evidential modelling techniques, there are also
many examples where people interpret a relation evidentiallyor causally in equal measure.
Similarly, some aspects of a case will be more conveniently
modelled in terms of arguments and others in terms of stories, For example, if the decision whether the suspect is
guilty or not hinges on a single fact (e.g. whether the suspect
was at a particular location at a particular time), alternative
stories are less important and the main analytical technique
is to carefully consider all the arguments for and against
this fact. In other cases, such as cases where there is no di-rect evidence for the main event, it is important to consider
the alleged events as a coherent story in order to be able todecide on the basis of more circumstantial evidence.
In sum, it is difficult to say that there is one ‘correct’
way of reasoning in the process of proof. The exact choice
whether to use causal rules, evidential rules, stories or arguments is often determined by personal preference as much asfeatures of the case. Hence, we need a theory that capturesand integrates both types of reasoning. While the hybrid
theory is a good first step, there are some problems, as explained in the next section.

2.2

Rijkbloem
shot father

Women
testimony3e

Rijkbloem
shot fathercCasings at
crime scene

Mother shot
fathere

rules, we get two arguments Ar and Am that do not attack
each other (figure 5b), even though the stories Sr0 and Sm
are in competition.

Rijkbloem
shot father

Rijkbloem
shot father

S’rc

Father was
hit in head
Mother shot
father

Ar e

c

S’m

Mother shot
fathere

Am

Father was
hit in head

(a)

(b)

Figure 5: Alternative stories (a) or arguments (b)Another problem is that the comparison of alternative scenarios and the comparison of attacking arguments is defineddifferently, and that thus the acceptability of a claim maythus depend on whether it is modelled as part of a story or
an argument. For example, an argument B with the conclusion ¬(Rijkbloem shot father) attacks story Sr0 (as well as
the main story Sr ), adding to its evidential contradiction.
B will also attack argument Ar , but since determining the‘winning’ argument is done differently than determining the‘winning’ story, it is not guaranteed that a case with Sr0 andB has the same outcome as a case with Ar and B.

Rijkbloem
shot father

Figure 3: Mixed evidential and causal reasoning
Whilst mixed C-E reasoning is thus a desirable feature ofa theory for reasoning in the process of proof, care must be
taken that only correct or valid lines of reasoning are allowed[12]. Take, for example, the reasoning in figure 4. Here, wefirst causally infer Father hit from Rijkbloem shot father, after which we infer the explanation Mother shot father from Father hit using an evidential reasoning. By
itself, the evidential step from Father hit to Mother shot
father is fine, as the latter is a possible cause of the former.
However, the problem here is that we have already assumed
that Rijkbloem shot father, and we should not be able toinfer the alternative, competing cause for father hit (i.e.
Mother shot father) from this. In the hybrid theory, thisis not a problem: mixed arguments are not possible, and if
there are two different causes they will be part of alternative explanations due to the subset minimality requirementof explanations. However, an integrated theory that does
allow for mixed C-E reasoning will have to explicitly takethis into consideration.
An integrated theory should also allow us to emphasize
argument-oriented elements or story-oriented elements ina case analysis without changing the essential conclusions.
However, in the current hybrid theory this is not the case.
Take, for example, the two alternative causes for F was hit
0in the head as two (simple) causal stories Sr0 and Sm(figure 5a). If we change the causal rules used into evidential

Fatherhit

Figure 4: Invalid mixed evidential and causal reasoning

Problems for the hybrid theory

In an integrated theory of causal stories and evidential
arguments, it should be possible to combine reasoning withboth evidential and causal rules in a correct way. For example, in the Rijkbloem case the two women testified that
Rijkbloem used a ‘black pistol’. From this we can evidentially infer that Rijkbloem probably shot father with atype of pistol that usually ejects its cartridges when fired.
We can then predict a so-called ‘story consequence’: if thestory about the type of pistol is really true, then we shouldfind empty cartridges at the scene of the crime. Figure 3
shows this line of reasoning: from Women testimony3 we evidentially infer Rijkbloem shot father using an evidential
rule, and we then subsequently causally infer casings at
crime scene from Rijkbloem shot father using a causal
rule. While these types of mixed causal-evidential (C-E)
arguments are very common in the process of proof, theycannot be constructed in the hybrid theory.

cMother shot
father

c

S’’
r
c

S’’
m

Casings at
crime scene
No casings at
crime scene

Figure 6: Two contradictory stories
Related to this is the problem that contradictory stories
do not attack each other. For example, Rijkbloem argued
that mother used a ‘small revolver’ of the type that does noteject empty cartridges when shot. Thus, we can build two
small causal stories that explain contradictory observations(figure 6): Sr00 predicts that there should be shell casings at00
the crime scene whilst Sm
predicts that there should not beshell casings at the crime scene. According to the definitionsof the hybrid theory, these two stories do not contradict eachother, even though they are clearly incompatible. Thus,
it is not possible to capture, for example, the idea of analibi story in the hybrid theory, which is not an alternative
explanation (it does not explain the main questions in acase, e.g. why the victim died) but it does contradict theprosecution’s story (that the suspect killed the victim).

3.

AN INTEGRATED THEORY

The above discussion indicates that we need to reconsider
the relation between arguments and scenarios: we need tobe able to reason with alternative explanations even if they
are modelled as evidential arguments and we need to be able
to consider contradictory causal scenarios. Furthermore, alternative and contradictory claims should be defined as incompatible irrespective of whether they are part of an argument or a story, and their evaluation should always follow
the same rules. Finally, both evidential and causal inference
should be allowed in an argument, so that we have a fully integrated theory instead of a hybrid of causal, scenario-basedand evidential argumentative reasoning.
In this section, the integrated theory for causal stories andevidential arguments is presented. This integrated theory is
explicitly based on the ASPIC + framework [11]. As discussed in section 2.1 (see also [2]), the formal structure ofarguments and stories is very similar: both are derivations
in some defeasible logic, the difference being that scenarios
use causal rules and are based on hypotheses and arguments
use evidential rules and are based on evidence. This meansthat we need at least a logical language, causal and evidential inference rules and a knowledge base of hypotheses andevidence, all of which are captured by the ASPIC + notionof an argumentation system.
Definition 1. [Argumentation systems] An argumentation system is a triple AS = (L, R, n) where:• L is a logical language closed under negation (¬).
• R = Rs ∪Rd is a set of strict (Rs ) and defeasible (Rd )inference rules, where Rd = Rc ∪ Re is a set of causal(Rc ) and evidential (Re ) defeasible inference rules andRs ∩ Rd = ∅.
• n : Rd −→ L is a naming convention for defeasible
rules.
We write ψ = −ϕ just in case ψ = ¬ϕ or ϕ = ¬ψ.

3.1

Causal and evidential inference rules

Definition 1 is similar to the original ASPIC + definition
save for the addition of sets Rc and Re for causal and evidential rules. The rules can be defined as follows.
Definition 2. [Inference Rules] Let ϕi , ψ be elementsof L.
• A strict rule is of the form ϕ1 ,. . . , ϕn → ψ. Informally,
if ϕ1 ,. . . , ϕn holds then without exception ψ holds.
• A causal defeasible rule is of the form ϕ ⇒c ψ. Informally, if ϕ holds then this has (presumably) causedψ.
• An evidential defeasible rule is of the form ϕ ⇒e ψ.
Informally, if ϕ holds then (presumably) this is evidencefor ψ.
For the strict rules it is assumed that the common inference rules of deductive logic are in Rs . Note that whereas
strict rules can have more than premise, causal and evidential rules have only one wff from L as a premise. This isa change from ASPIC + which was made to simplify several definitions in the integrated theory, most notably definition 3, which compares premises and conclusions of causal
and evidential rules. This does not diminish the expressivity of the integrated framework: multiple premises can be
combined into one conjunction using a strict inference rule,and this conjunction can then serve as the single premise ofa causal or evidential rule.

In the Rijkbloem case, we have the following causal inference rules (where R is Rijkbloem, F is father, M is motherand g is gun).
rc1
rc2
rc3
rc4
rc5
rc6
rc7
rc8
rc9
rc10

R and F argued ⇒c R pulled g
R pulled g ⇒ R shot F
R shot F ⇒c F hit
R and F argued ⇒c M pulled g
M pulled g ⇒c R pushed M
R pushed M ⇒c M shot F
M shot F ⇒c F hit
F hit ⇒c F dies
R shot F ⇒c casings
M shot F ⇒c ¬(casings)

There are also multiple evidential rules in the case, someexamples of which are as follows.
re1
re2
re3
re4
re5
re6
re7

Rijkbloem testimony4 ⇒e M shot F
Women testimony3 ⇒e R shot F
Coroner report1 ⇒e F hit
Coroner report2 ⇒e rc3
Women lied ⇒e ¬re2
F hit ⇒e R shot F
F hit ⇒e M shot F

Note that causal as well as evidential rules can be supported- rule re4 , for example, allows us to infer rc3 from Coroner
report2 - as well as attacked - rule re5 , for example, denies
rule re2 (see figure 2.1 .
It is important to note that there is an implicit causal theory underlying the causal and evidential rules, that is, thedomain can sensibly be thought of in terms of causes and effects (cf. [16]). Hence, we assume that a causal rule c causes
e is based on a causal relation this underlying theory andthat if we say that e is evidence for c there is a causal relation between c and e in the underlying theory. The causal
theory has only one constraint, namely that it does not allow
for causal cycles: if a is a cause of b, then b cannot also bea cause of a. Causality is a contentious topic and it is very
easy to disagree about whether one event causes another,or whether one event is evidence for another. The causal
theory does not enforce exactly which causal and evidential
rules are ‘correct’ and which are not - disagreement about
rules can be expressed by counterarguments to causal rules.

3.2

Capturing alternative causes

When modelling the causal and evidential relations basedon an underlying causal theory, it is important to consider
the definition of alternative causes. Like in the hybrid theory, two different causes with the same consequences are
considered to be alternative explanations. As was discussed
in section 2.2, it is also possible to have two alternativecauses in evidential arguments, because an evidential ruleis essentially nothing more than an explicit expression of an
abductive reasoning step applied to a causal rule. Thus,
evidential alternatives are exactly the opposite of causal alternatives: two different conclusions that are evidentially inferred from the same premises are considered as alternatives(see figure 5). Finally, it is also possible to have alternative
causal-evidential arguments: if some conclusion e is causally1

For simplicity, no distinction is made between a rule being
inapplicable in all cases and a rule being inapplicable in only
some casesinferred from a cause c1 and another cause c2 is then evidentially concluded from e, causes c1 and c2 are alternatives.
Definition 3. [Alternatives] Causes ϕ and χ are alternatives if for any pair of rules ri , rj ∈ Rc ∪ Re .
1. ri = ϕ ⇒c ψ and rj = χ ⇒c ψ and ϕ 6= χ; or
2. ri = ψ ⇒e ϕ and rj = ψ ⇒e χ and ϕ 6= χ; or
3. ri = ϕ ⇒c ψ and rj = ψ ⇒e χ and ϕ 6= χ.
We can see that R shot F and M shot F are alternatives
because for any pair of the following rules either definition 3(1), (2) or (3) holds.
rc3
rc7
re6
re7

R shot F ⇒c F hit
M shot F ⇒c F hit
F hit ⇒e R shot F
F hit ⇒e M shot F

Note that in the hybrid theory, alternatives are not explicitly defined. Rather, the idea of definition 3(1) is implicitly
enforced by comparing only subset-minimal causal explanations of a consequence. Furthermore, evidential or C-E
alternatives as defined in (2) and (3) are not distinguishedin the hybrid theory.

3.3

Constructing arguments

Arguments and stories in the integrated theory are basedon a knowledge base. Like in the hybrid theory, this knowledge base contains both the evidence and the possible hypotheses.
Definition 4. [Knowledge bases] A knowledge base in
an AS = (L, R, n) is a set K ⊆ L consisting of two disjoint
subsets Ke (the evidence) and Kh (the hypotheses).
The evidence in Ke is similar to the axiom premises in ASPIC + in that they cannot be sensibly denied or attacked.
What this means is not that, for example, the veracity of awitness testimony cannot be questioned, but rather that theexistence of the testimony cannnot sensibly be denied. The
hypotheses in Kh are just that, and stories or arguments
based on hypotheses can be undermined by other stories orarguments (see definition 6).
The combination of a knowledge base and inference rules
allows us to build arguments and stories, derivations in theargumentation system. The integrated theory treats stories
as causal arguments, which will allow us to attack and compare them as if they were arguments (definition 6 and 8). In
definition 5, for a given argument the function Prem returns
all its premises, Conc returns its conclusion, Sub returns all
its sub-arguments, TopRule returns the last inference rule
applied in the argument, ERule returns all its evidential arguments and CRule returns all its causal arguments.
Definition 5. [Arguments] An argument A on the basis of a knowledge base KB in an argumentation system AS
is:
1. ϕ if ϕ ∈ K.
Prem(A) = {ϕ}; Conc(A) = ϕ; Sub(A) = {ϕ}; TopRule(A)= undefined; ERule(A) = ∅ CRule(A) = ∅2. A1 , . . . An → ψ if A1 , . . . , An are arguments s.t. there
exists a strict rule Conc(A1 ), . . . , Conc(An ) → ψ in Rs .

Prem(A) = Prem(A1 ) ∪ . . . ∪ Prem(An ); Conc(A) = ψ;Sub(A) = Sub(A1 ) ∪ . . . ∪ Sub(An ) ∪ {A};TopRule(A) = Conc(A1 ), . . . , Conc(An ) → ψ;ERule(A) = ERule(A1 ) ∪ . . . ∪ ERule(An );CRule(A) = CRule(A1 ) ∪ . . . ∪ CRule(An )3. A1 , . . . An ⇒C ψ if A1 , . . . , An are arguments s.t. there
exists a causal rule Conc(A1 ), . . . , Conc(An ) ⇒C ψ in Rc .
Prem(A) = Prem(A1 ) ∪ . . . ∪ Prem(An ); Conc(A) = ψ;Sub(A) = Sub(A1 ) ∪ . . . ∪ Sub(An ) ∪ {A};TopRule(A) = Conc(A1 ), . . . , Conc(An ) ⇒C ψ;
ERule(A) = ERule(A1 ) ∪ . . . ∪ ERule(An );CRule(A) = CRule(A1 ) ∪ . . . ∪ CRule(An )∪{Conc(A1 ), . . . , Conc(An ) ⇒C ψ}
4. A1 , . . . An ⇒E ψ if A1 , . . . , An are arguments s.t. there
exists an evidential rule Conc(A1 ), . . . , Conc(An ) ⇒E ψ
in Re .
Prem(A) = Prem(A1 ) ∪ . . . ∪ Prem(An ); Conc(A) = ψ;Sub(A) = Sub(A1 ) ∪ . . . ∪ Sub(An ) ∪ {A};TopRule(A) = Conc(A1 ), . . . , Conc(An ) ⇒E ψ;
ERule(A) = ERule(A1 ) ∪ . . . ∪ ERule(An )∪{Conc(A1 ), . . . , Conc(An ) ⇒E ψ};
CRule(A) = CRule(A1 ) ∪ . . . ∪ CRule(An )An argument A is causal if CRule(A) 6= ∅ and ERule(A) = ∅,evidential if ERule(A) 6= ∅ and CRule(A) = ∅, and mixed if
CRule(A) 6= ∅ and ERule(A) 6= ∅.
The type of argument is indicated as follows: causal arguments are named Ci , evidential arguments are named Ei
and mixed arguments are named Mi .
Like in the hybrid theory, we can build causal defeasible
arguments (i.e. stories). If we take Rc = {rc1 , . . . , rc10 }and assume that (R and F argued) ∈ Kh we can construct
the following causal arguments.
Cr1 :
Cr2 :
Cr3 :
Cr4 :
Cr :
Cr5 :

R and F argued
Cr1 ⇒c R pulled g
Cr2 ⇒c R shot F
Cr3 ⇒c F hit
Cr4 ⇒c F diedCr3 ⇒c casings

Cm1 :
Cm2 :
Cm3 :
Cm4 :
Cm5 :
Cm :
Cm6 :

R and F argued
Cm1 ⇒c M pulled g
Cm2 ⇒c R pushed M
Cm3 ⇒c M shot F
Cm4 ⇒c F hit
Cm5 ⇒c F diedCm4 ⇒c ¬casings

Arguments Cr and Cm are the two stories S1 and S2 thatwere also rendered in figure 1, and causal arguments Cr5
and Cm6 are expanded versions of the two stories S100 and
S200 presented in figure 6.
Because it is assumed that there are no causal cycles, there
is no causal argument C where Prem(C) = Conc(C). One difference between stories in the integrated theory and stories
in the hybrid theory is that in the latter, a story can havemore than one story consequence so, for example, both F
died and casings could both be part of one story. Because
the integrated theory treats stories as arguments, it is notpossible to have more than one conclusion for a causal argument. However, overlapping arguments with different conclusions are considered as-a-whole in the integrated theory,
namely through the idea of positions, which will be further
discussed below (definition 9).
Evidential arguments based can also be constructed in the
integrated theory. For example, if we take Re = {re1 , . . . , re7 }and assume Coroner report1 ∈ Ke we can construct the following evidential arguments.

Ec0 :Ec1 :E1 :Coroner report1
Ec0 ⇒e F hit
Ec1 ⇒e R shot FE2 :Ec1 ⇒e M shot F

Because there are no causal cycles and the evidential rules
are all based on causal relations, there is no evidential argument E where Prem(E) = Conc(E).
Notice that evidential and causal arguments can be basedon both evidence or hypotheses (i.e. any element of K canserve as the premise of an any argument), whereas in the
hybrid theory evidential arguments can only be based onevidence and causal stories can only be based on hypotheses.
This is one example of how stories and arguments are further
integrated. Another example is that the integrated theory
allows for mixed causal-evidential (C-E) arguments. Assume
that Rc and Re are as above, and that Women testimony3∈ Ke .
Ew0 :
Ew1 :

Women testimony3
Ew0 ⇒e R shot F

M1 :
M2 :
M3 :

Ew1 ⇒c casings
Ew1 ⇒c F hit
M2 ⇒e M shot F

Definition 5 allows for all types of C-E reasoning: M1 (figure 3) is a correct way of reasoning with causal and evidential information. The reasoning captured by the mixed
argument M3 (figure 4), however, is a classic example of CE reasoning that is typically incorrect: as Pearl notes [12],an evidential rule e ⇒e c2 should only be invoked if noother explanation c1 of e is believed. In M3 , however, we already have explanation R shot F for F hit, so we should not
be able to use re7 to subsequently infer M shot F. In otherwords, the application of rule re7 should be suppressed by
the earlier application of causal rule rc3 in M2 .
There are essentially two ways to capture the suppressionof evidential rules. One way is to specifically disallow anevidential inference rule to be applied to a conclusion thathas been causally inferred; this is the way Pearl proposesin his C-E system [12]. However, in the integrated theory acause that has been evidentially inferred (in the example: M
shot F) is considered as a competing alternative to any othercause (e.g., R shot F). One reason to opt for this second
way of modelling is that, as Pearl also notes, not all causal
rules may present a clear separate cause that completely
suppresses the inference of alternatives. That is, we may
still want to infer the possible alternatives and then decide
whether or not the original explanation suppresses any otheralternatives.

3.4

AttackThe definition of alternatives is important for an integrated theory of stories and arguments, because in explanatory, abductive-causal reasoning this is where the dialecticalcomponent lies: two explanations (whether causal stories orevidential arguments) are in competition of they represent
alternative causes for the same consequence. Hence, it canbe said that alternative arguments attack each other.
In ASPIC + arguments can be attacked in three ways:
on their premises (undermining attack), on their conclusion(rebutting attack) or on an inference step (undercutting attack). The latter two are only possible on applications ofdefeasible inference rules. The integrated theory now adds
another type of attack, namely alternative attack.
Definition 6. [Attack] A attacks B iff A undercuts, rebuts, undermines or alternative-attacks B, where:• A undercuts argument B (on B 0 ) iff Conc(A) = −n(r)and B 0 ∈ Sub(B) such that B 0 ’s top rule r is defeasible.
• A rebuts argument B (on B 0 ) iff Conc(A) = −ϕ forsome B 0 ∈ Sub(B) of the form B 00 ⇒C/E ϕ.
• Argument A undermines B (on B 0 ) iff Conc(A) = −ϕfor some B 0 = ϕ, ϕ 6∈ Ke .
• Argument A alternative-attacks B (on B 0 ) iff Conc(A) =ϕ for some B 0 ∈ Sub(B) with Conc(B 0 ) = ψ and ϕ andψ are alternatives according to definition 3.
Because definition 6 does not specifically distinguish causal
and evidential arguments, we can have any combination ofarguments or stories attacking each other in the different
ways specified in definition 6. So, for example, the arguments Cr5 and Cm6 attack each other because they have
opposite conclusions casings and −casings. Alternatives
also attack each other, and because according to definition 3
they can be defined between any combination of evidential
and causal rules, it is also possible to have,for example, twoalternative evidential arguments that attack each other such
as E1 and E2 (cf. figure 5). Further alternatives are, forexample, the argument Cr3 that R shot F attacks all thecausal, evidential and mixed arguments in which M shot F(e.g. Cm , E2 , M3 ). Because the alternatives relation issymmetrical, we also have that, for example, the argument
Ec1 that M shot F attacks all arguments in which R shot
F. Interestingly, argument M3 attacks its own subarguments
Ew1 , M2 and is thus a self-attacking argument.
The above definition of attack puts evidential arguments
and causal arguments (stories) on the same level: in the
hybrid theory it was only possible for an evidential arguments to attack either another argument or a causal story,
but stories could not attack arguments or each other. Furthermore, evidential arguments that propose some causal
relation now compete with causal stories that propose analternative cause. Note that the important difference between arguments based on evidence and arguments basedon hypotheses remains: the former cannot be underminedbecause the evidence itself cannot be attacked.

3.5

Argumentation semantics

Given an argumentation system (definition 1) plus a knowledge base (definition 4), we can now form an integrated theory, which induces an integrated argumentation framework.
Definition 7. [Integrated Argumentation Frameworks] Let IT be an integrated theory (AS, KB). An integrated argumentation framework ( IAF) defined by AT , isa tuple hA, C i where A is the set of all finite arguments
constructed from KB in AS and (X, Y ) ∈ C iff X attacks
Y.
An integrated argumentation framework thus contains all
the arguments and stories in a case and, most importantly,
correctly indicates the attack relations between the contradictory and alternative arguments. Given such an IAF, we
can define admissible sets and extensions as in [8] and determine whether an argument is justified [11].
Definition 8. [Argumentation Semantics] Let hA, Ci
be an IAF.
• A set of arguments S ⊆ A is conflict free if there are
no X, Y in S such that (X, Y ) ∈ C.

• An argument X ∈ A, X is acceptable w.r.t. someset of arguments S ⊆ A iff for all arguments Y s.t.
(Y, X) ∈ C there is an argument Z ∈ S s.t. (Z, Y ) ∈ C.
• A conflict-free set of arguments S is admissible iff eachargument X ∈ S is acceptable w.r.t. S.
• An admissible set of arguments S is a complete extension iff X ∈ S whenever X is acceptable w.r.t. S; S isa preferred extension iff it is a set inclusion maximal
complete extension; S is the grounded extension iff itis the set inclusion minimal complete extension; andS is a stable extension iff it is preferred and ∀Y ∈/ S,
∃X ∈ S s.t. (X, Y ) ∈ C.
For T ∈ {complete, preferred, grounded, stable}, an argument X is sceptically or credulously justified under the
T -semantics if X belongs to all, respectively at least one,
T -extension.
With the argumentation semantics, we can compare causal
and evidential arguments in a case. As an example, let’s
go back to the main Rijkbloem case in figure 1. Assume
that Kh = {R and F argued}, Ke contains the women’s three
testimonies (Women testimony1,...,3 ), Rijkbloem’s four testimonies (Rijkbloem testimony1,...,4 ) and the coroner’s threereports (Coroner report1,...,3 ). Furthermore, assume that
Rc is as on page and that Re contains all the rules on
page plus all the evidential inference rules for inferring conclusions from the testimonies and the reports (re1 ,. . . , re4
are four examples of such rules). From this, we can construct the two main stories in the case Cr and Cm and theirsubarguments, as well as Cr5 and Cm6 (and subarguments)
representing the possible predictions about the bullet casings at the scene of the crime (cf. figure 6). Furthermore,
we have the evidential and mixed arguments based on thewitness testimonies. There are then two preferred extensions, one that claims that father died because Rijkbloem
shot him (i.e. containing all arguments except the ones using rule rc3 and the ones with −casings as a conclusion)and one that claims that father died because mother shot
father (i.e. containing all arguments except the ones using
rc7 and the ones with casings as a conclusion).
Whilst the alternative stories Cr and Cm and the contradictory stories Cr5 and Cm6 are (correctly) not in thesame extension, the various sub-stories and the evidential
arguments supporting the different stories are in both extensions because they are, considered outside of the context
of their overarching alternative stories, not explicitly contradictory. So, for example, an extension which includes thestory Cr3 , that Rijkbloem shot Father, will not contain any
stories or arguments arguing that it was mother who shot father. However, any sub-stories - for example, the story thatmother pulled a gun - or evidential arguments - for example,
the argument consisting of just women testimony3 - will bein the same extension as Cr3 . Thus, individual extensionsof an IAF allow for fairly broad views on a case, capturing
every element of a case that is not explicitly contradicted by
the current assumptions.

3.6

Causal-Evidential constraints

Given these argumentation semantics, it is possible toshow that the integrated theory conform to Pearl’s C-E constraint [12] that it should in general not be possible to evidentially infer a conclusion from something that has itself
been inferred using causal reasoning. More precisely:

Proposition 1. [C-E constraint] Given an integrated
argumentation framework hA, C i, an argument A ∈ A will
not be (sceptically or credoulously) justified under any semantics if there is an argument B ∈ Sub(A)) such that
B : B 0 ⇒e χ and B 0 : B 00 ⇒c ψ and Conc(B 00 ) = ϕ where
ϕ 6= χ.
According to the definition of alternatives (definition 3), ϕand χ will be alternatives, which means that B 00 and B will
attack each other (definition 6). Because B 00 is a subargument of B, this means that that argument B (and therefore
also A) will attack itself. Clearly, self-attacking arguments
are not justified, since any set containing such arguments is
not conflict-free.
In our example, M3 was an example of incorrect C-E reasoning. Given the above definitions of an IAF and the corresponding semantics, M3 self-attacks since it alternativeattacks its subargument Ew1 . If we now consider an IAF
where A = {Ew0 , Ew1 , M2 , M3 } (page ), we have the complete extension {Ew0 }, which is also the grounded extension,and complete extension {Ew0 , Ew1 , M2 }, which is also the
only preferred extension and the stable extension.

3.7

Positions

In the hybrid theory, the view on a case is often morenarrow: the idea that an explanatory causal story needs
to be subset-minimal only allows us to accept the directly
relevant stories and arguments. While in some ways thisdoes not provide the broad view that the extensions of theintegrated theory do, it does provide more focus around theseparate main stories. Hence, it would be informative todefine positions in the integrated theory, admissible sets that
present a meaningful and coherent view on the evidence andthe facts of the case. For example, it makes sense that not
all the arguments based on the women’s testimonies and thearguments based on Rijkbloem’s testimonies are in the same
position, as, even though they do not directly contradict
each other, they support different (alternative) explanations.
The way to look at positions is to consider a particularpreferred extension in a case and determine what makes it
different from other preferred extensions. In other words,
what is the main focus of this view on the case?
Definition 9. [Positions] Given an integrated argumentation framework IAT with one preferred extension P E,
the position P is equal to P E. For an IAT with multiple preferred extensions P E1 , . . . , P En , each position is a
set Pi ⊆ P Ei such that for each argument A ∈ Pi .
• ∃B s.t. A ∈ Sub(B) and B ∈ P Ei and B 6∈ P Ej
(i 6= j); or• ¬∃B s.t. A 6∈ Sub(B 0 ) and B 0 6∈ P Ei and B 0 ∈ P Ej
(i 6= j).
Definition 9 effectively ensures that all the arguments ina position are either unique to the associated preferred extension or that they are not part of an argument that is
unique to another extension. A position does not include
the subarguments of arguments that are contradictory to
the current position and thus lays the focus the specific arguments that make the associated extension unique. So, forexample, any position which contains the main story that
Rijkbloem shot father (Cr ) will contain the arguments that
support this view but not the arguments that are part of theopposite view, that mother shot father (e.g. the argument
Cm2 that mother pulled a gun).

3.8

Comparing positions

Because alternative-attack is symmetrical, we will often
end up with different explanations of the evidence that areincompatible, different positions. The question is then how
to compare these positions. Here it might be interesting
to recall the criteria and concepts used to compare the stories in the original hybrid theory, like evidential support orevidential contradiction: which evidence from Ke directly
supports or contradicts an element in a position? The evidential support for a position would then be, for example,
the set of evidence that are the premise of an argument inthat position. Like in the hybrid theory, we can then use
these concepts to compare positions and thus indicate preferences between positions; for example, the position with thelarger (set-inclusive) evidential support is preferred. In thisway, we can define preferences not only between arguments
but also between particular sets of arguments [1].

4.

DISCUSSION AND RELATED WORK

Aside from the hybrid theory [4], there have been otherattempts to combine causal and evidential reasoning, mostnotably by Poole [15, 16]. The hybrid and integrated theories are based on these early ideas, but they expand on themsignificantly. For example, because these frameworks wereoriginally intended for automatic diagnosis within relatively
small and pre-defined domains, they do not incorporate aspects like reasoning about the rules that comprise the scenarios and more fine-grained scenario comparison criteria,
and hence they are less suited to modelling and supporting
more complex tasks such as analysing and making sense of
the evidence in large criminal cases [17].
Interestingly, Poole’s abductive-causal approach has also
been modelled as an argumentation framework in [6]. Theinterplay between stories and arguments and causal and evidential rules is, however, not directly captured by theseframeworks. Another relevant argumentation-theoretic approach - though one that stays exclusively at the level ofabstract argumentation - is proposed by [19], who combine
explanatory reasoning with argumentation. An interesting
avenue for future research would be to investigate how theirformal account of the explanatory power of arguments relates to the arguments and positions in the integrated theory.
One approach which does capture the different subtle influences in causal and evidential reasoning is the Bayesian
approach [13, 22, 20]. In this approach, which has steadily
gained influence since the mid 1990’s at the expense of logical models such as [16, 12], the stories and arguments in acase are modelled as a Bayesian Network, a graphical representation of a joint probability distribution, where the nodes
are variables (e.g., events, evidence) and the links represent
the (in)dependencies between the variables. The directed
links in a Bayesian network can be interpreted as causal
rules, though a less strict criterion of ‘correlation’ is often
also used.
Because the integrated theory is a logical interpretationof reasoning with causal information, it necessarily abstracts
from many subtleties of causal inference captured by Bayesian
reasoning. For example, the integrated framework does not
capture the idea of accruing causes, or ‘explaining in’ as it

is sometimes called, where two causes c1 and c2 individually explain a consequence e but taken together make theprobability of e even higher.
A valid question is then: why pursue the logical approachat all? Are we not merely building a, as Pearl [12] notes,‘coarse logical abstraction’ of the type of reasoning that is
easily captured by Bayesian inference? One answer is that
one of the goals of the integrated theory is to capture andfacilitate the way in which investigators and decision makers
reason in real cases. Much of our reasoning is qualitative,
and it therefore makes sense to capture it in a qualitative
framework. In fact, one major shortcomings of Bayesian
reasoning is that, in order to calculate the probability of theclaims represented in the network, many conditional andprior probabilities are needed, which are often simply notavailable. Furthermore, these probabilities and the relations
between them make Bayesian approaches difficult to understand for people that are not mathematically schooled, likepolice investigators and judges.
Bayesian approaches often struggle with the same issues
regarding knowledge representation as logical approaches do:
how do we represent stories and arguments, and how do we
capture the interplay between them? Interestingly, manyof the tentative proposed solutions to these questions originate from qualitative, (semi-)formal logical approaches. For
example, the structures for Bayesian arguments proposedby [9] draw from work in argumentation theory [24], andthe ‘scenario schemes’ for Bayesian networks in [22] were directly preceded by logical approaches to story schemes basedon early qualitative work in AI [18]. Hence, it seems thata formal, logical theory is a good intermediary between on
the on hand the informal, more natural conceptions of stories and arguments and on the other hand the quantitative
mathematical models. This cross-fertilization between probabilistic and logical approaches is also the subject of recent
work in argumentation [21, 20].

5.

CONCLUSIONS AND FUTURE RESEARCHThis paper has expanded and further formalised the integrated theory for reasoning with evidence that was first
proposed in a recent short paper [2]. In this integrated theory, it is possible to reason with both causal and evidential
information, and to employ stories (causal arguments) andevidential arguments in one’s reasoning whenever needed.
Thus, it offers the ‘best of both worlds’ with respect to storybased reasoning [23, 14] and argumentative reasoning [3] inthe process of proof.
The integrated theory offers a dialectical framework for
abductive argumentation which instantiates the ASPIC +framework for structured argumentation. In addition todefining a combination of abductive and non-abductive argumentation, the integrated theory also correctly captures
argumentative causal reasoning. Furthermore, the positions(definition 9) represent coherent views on a case which canbe compared using different criteria. Thus, this adds a new
way of defining preferences between sets of arguments in anargumentation framework.
The integrated theory is a natural continuation of thework on the hybrid theory [4], which was the first formal
theory that combined causal stories and evidential arguments. However, as the name suggests, the hybrid theory
stops short of truly integrating stories and arguments intoa single theory. The integrated theory proposed in the cur-rent paper therefore improves on the hybrid theory in a few
important ways.
• It is now possible to construct mixed causal-evidential
arguments as well as causal stories and evidential arguments. This mixed causal-evidential reasoning adheres
to Pearl’s [12] constraints for C-E reasoning.
• In addition to alternative causal stories, which werepart of the hybrid theory, the integrated theory also
defines alternative evidential and mixed arguments.
Thus, it is possible to emphasize causal or evidential
parts of a case as needed whilst keeping the case outcome the same.
• The integrated theory treats stories as causal arguments that can attack not only alternative explanations of the evidence but also contradictory stories (e.g.
alibi stories). Thus arguments and stories can both becompared using the common semantics for argumentation [8].
As for future research, one of the most urgent and interesting subjects is perhaps the exploration of how various otherphenomena that play an important role in causal reasoning
can be captured in a logical framework, such as accrual andthe distinction between ‘strong’ causal rules, which preclude
any subsequent evidential inference (as in proposition 1) and‘weak’ causal rules, which allow evidential inferences to bemade after they have been applied. Such an extended logical framework can then serve as a gateway between natural,
informal reasoning with causal stories and evidential arguments and more probabilistic and mathematically inclined
accounts of causal reasoning such as Bayesian networks.

6.

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