Artif Intell Law
DOI 10.1007/s10506-010-9092-x

A hybrid formal theory of arguments, stories
and criminal evidence
Floris J. Bex • Peter J. van Koppen • Henry Prakken •Bart Verheij Springer Science+Business Media B.V. 2010

Abstract This paper presents a theory of reasoning with evidence in order todetermine the facts in a criminal case. The focus is on the process of proof, in which
the facts of the case are determined, rather than on related legal issues, such as theadmissibility of evidence. In the literature, two approaches to reasoning with evidence can be distinguished, one argument-based and one story-based. In an argument-based approach to reasoning with evidence, the reasons for and against theoccurrence of an event, e.g., based on witness testimony, are central. In a storybased approach, evidence is evaluated and interpreted from the perspective of thefactual stories as they may have occurred in a case, e.g., as they are defended by theprosecution. In this paper, we argue that both arguments and narratives are relevant
and useful in the reasoning with and interpretation of evidence. Therefore, a hybrid
F. J. Bex (&)
Argumentation Research Group, School of Computing, University of Dundee,
Dundee, UK
e-mail: florisbex@computing.dundee.ac.uk
P. J. van Koppen
Faculty of Law, Maastricht University, Maastricht, The Netherlands
e-mail: peter.vankoppen@maastrichtuniversity.nl
P. J. van Koppen
Faculty of Law, Free University Amsterdam, Amsterdam, The Netherlands
H. Prakken
Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands
e-mail: henry@cs.uu.nl
H. Prakken
Department of Law and ICT, Faculty of Law, University of Groningen, Groningen, The Netherlands
B. Verheij
Department of Artificial Intelligence, University of Groningen, Groningen, The Netherlands
e-mail: b.verheij@ai.rug.nl

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approach is proposed and formally developed, doing justice to both the argumentbased and the narrative-based perspective. By the formalization of the theory andthe associated graphical representations, our proposal is the basis for the design ofsoftware developed as a tool to make sense of the evidence in complex cases.
Keywords

Argumentation  Stories  Legal evidence

1 Introduction
In this paper, we propose and formalise a general theory for reasoning with evidencein criminal cases. This theory, which is adapted from Bex’ thesis (Bex 2009),
models the process of proof, which concerns the facts of the case rather than legal
issues such as, for example, the admissibility of evidence. In this process of proof,
hypothetical stories or scenarios about ‘‘what happened’’ in a case are constructed
and arguments based on evidence or commonsense knowledge are used to support
or attack these stories. Thus, the stories can be compared according to the extent
they conform to the evidence and our generally accepted commonsense knowledge.
The distinction between the study of evidence law and the study of the process of
proof was made in the beginning of the twentieth century by Wigmore (1931), whoargued for the formulation of rational principles for reasoning with evidence andproof independent of the rules of law. Various theories on reasoning with evidence,both descriptive and normative, have since been proposed in legal theory and legal
psychology. Here, two trends can be distinguished: the argument-based approachand the story-based approach. Most of the research from legal theory (Anderson
et al. 2005; Schum 1994; Tillers 2005) focuses on the use of Wigmorean argument
charts to structure and analyse a mass of evidence and to expose sources of doubt in
the reasoning.1 In contrast, theories from a more psychological perspective(Wagenaar et al. 1993; Pennington and Hastie 1993) stress the use of stories toorganize and analyse available evidence. In AI and Law, formal theories influenced
by either the argument-based or the story-based approach have been proposed. For
example, Bex et al. (2003) have modelled Wigmorean argument charts in a formal
argumentation logic and Bayesian interpretations of Wigmore charts have beenstudied by, amongst others, Hepler et al. (2007). With regards to the story-based
approach, Verheij (2000) compares the Anchored Narratives Theory to formal
logics for argumentation, Keppens and colleagues (e.g. Keppens and Schäfer 2006)
and Josephson (2002) provide logical, model-based approaches to reasoning withcrime scenarios and Thagard (2004, 2005) model stories and evidence inconnectionist ‘‘coherence networks’’.
A central theme in both the argument-based and the story-based theories is the
structuring and analysis of one’s reasoning so as to make sense of the case. In alarge case it is important that the reasoning, evidence, hypotheses and backgroundknowledge are all made explicit, so that sources of doubt in the reasoning can be
1

A notable exception here are Pardo and Allen (2007), who advocate using stories to explain theevidence.

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identified and reasoned about (Anderson et al. 2005). Furthermore, explicitly
identifying and analysing the hypothetical stories lessens the danger of so-calledtunnel vision, where the most likely story is taken as the leading hypothesis andalternatives are insufficiently considered (Wagenaar et al. 1993; Heuer 1999).
A relatively new development in this respect is the emergence of computer-based
support tools for investigators and decision makers,2 which allow for the electronic
management of, for example, evidence and timelines. The aim of such tools is totake some of the cognitive load off the users and to provide a basic tool for
structuring a case. The interest in sense-making and visualization tools has also
grown in AI (and Law) (Reed and Rowe 2004; Verheij 2005; Gordon 2007). Basedon ideas from critical thinking and argumentation theory, these tools allow the user
to structure and visualize their reasoning, often according to some underlying theoryof reasoning. Through this underlying theory, a tool essentially enforces a basic
standard of rationality by requiring that the user’s reasoning stays within theboundaries set by the theory.
A complex process such as the process of proof ideally has a specialised sensemaking tool, in which not only the evidence and stories can be structured in a simple
way but in which it is also possible to reason about the evidence and stories using asound underlying theory. Such a specialised tool for sense-making should not bebased on a general theory of reasoning but rather on a specific model of reasoning in
the process of proof. This model should be formally specified (in order to facilitate
implementation) and at the same time it should be natural so that it can be used byan everyday reasoner such as a crime analyst, who cannot be expected to have indepth knowledge of mathematical or formal models. In his dissertation, Bex (2009)shows that when dealing with the complex reasoning prevalent in the process of
proof, a story-based approach works best for some points of a case, while in otherinstances an argumentative approach is the most natural. Arguments and stories
therefore need to be combined into one hybrid theory, where stories are used tocausally explain the explananda (facts to be explained) and arguments based onevidence are used to support and attack these stories. Stories can thus be used forconstructing intelligible hypotheses about what happened in an intuitive way andarguments can be used to connect the evidence to these stories and to reason about
the stories and the evidence in greater detail.
This paper aims to expound mainly the formal logical version of the hybrid
theory developed by Bex (2009), previous versions of which were discussed in (Bexet al. 2007a, b). This formal theory, which has served as the basis of the sensemaking and visualization tool AVERS,3 models reasoning with arguments as
defeasible argumentation and reasoning with stories as abductive (model-based)inference to the best explanation. The paper is organized as follows. In Sect. 2, wegive a brief overview of the process of proof, that is, the typical aspects and2

Examples are CaseMap (http://www.casemap.com; accessed on February 5th, 2010) and Analyst’s Notebook (http://www.i2group.com/products/analysis-product-line/analysts-notebook; accessed on February 5th,2010).
3

AVERS stands for Argument Visualization for Evidential Reasoning based on Stories and was
specifically developed for crime analysis by van den Braak and Vreeswijk, see van den Braak et al.
(2008), van den Braak (2010) and Bex et al. (2007a).

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elements of reasoning with evidence. In light of these aspects, Sect. 3 discusses
the advantages and disadvantages of the argument-based and the story-based
approaches and argues why a combined approach best captures the process of proof.
Section 4 presents the hybrid theory. The paper concludes with a discussion andsome ideas for future research.

2 Reasoning with criminal evidence
In this section we will give an overview of the typical aspects and elements of theprocess of proof in a criminal case. We will clarify this overview by means of an
example, which will be used throughout the paper. The example is a simplified
version of the Rijkbloem case, which was first presented by Wagenaar et al. (1993)
and later adapted by Bex et al. (2007b). The case concerns Danny Rijkbloem, a 23year old Surinamese man living in the Netherlands. He has a considerable list ofsentences (theft, robbery) starting when he was 15 years old. Nicole Lammers is a20-year old baker’s daughter who had a relationship with Rijkbloem and lived
together with him. At some point Nicole decided, under pressure of her parents that
it is best to break up with Rijkbloem and she leaves him. A few days after the breakup, Nicole and her 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 hit by a gunshot to the head. When the police, who
had been informed by Rijkbloem, arrived, the father was already dead.
2.1 The process of proof
In the process of proof, evidence and general commonsense knowledge of the world
around us is used to establish the facts of the case. The process usually starts whensome initial observations are made, that is, when some initial evidence is found.
These clues or observations then become explananda, facts that need to be
explained. In the Rijkbloem case, for example, the main explanandum would be thedead body of the father in Rijkbloem’s house. In order to explain the explananda,
hypothetical stories about what caused them need to be constructed. In the
Rijkbloem case, for example, there were two main stories. The first story has
Rijkbloem pulling out a gun and shooting the father during the fight. In the otherstory Mrs. Lammers pulled a gun out of her purse and threatened to shoot Rijkbloem
with it. Rijkbloem pushed the hand holding the gun away and in the struggle the gun
accidentally went off, killing father Lammers. Some of the initial hypotheses may
perhaps be immediately discarded as implausible (i.e. extraterrestrial aliens killed
Mr. Lammers), while others have to be tested by searching for further evidence andthen determining which of them are compatible with the new evidence. For
example, the police could check physical evidence such as the angle of impact of
the bullet and the type of casings discarded by the gun to see whether a smallconcealable gun accidentally went off (as would be the case had the mother got out
her gun) or a larger gun was deliberately fired (as would be the case had Rijkbloem
shot the father). When a particular hypothesis has been chosen as the most likely,123

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this choice should be justified by explicitly showing that it is coherent and conforms
to the evidence. Thus a proof should be constructed, consisting of a story and ajustification of why the particular story was chosen. In this respect, Nijboer andSennef (1999) speak of explanatory justification. According to this purpose ofjustification, a proper justification should not only meet some standard of rationality
but it should also provide a clear explication of one’s reasons for the choice of
hypothesis which makes sense to not just the reasoner but also to third parties.
2.2 Evidence
The term evidence stands for the available body of information indicating whether abelief in some proposition is justified. When people talk about ‘the evidence in acase’ they usually mean the evidential data (Anderson et al. 2005, p. 382), that is,
the primary sources of evidence. This evidential data, also called items of evidence,pieces of evidence or sources of evidence, is perceived by the reasoner with his own
senses (Wigmore 1931 calls this ‘‘autoptic proference’’). Notwithstanding sensory
defects on the part of the reasoner, the existence of the evidential data itself cannot
be sensibly denied. In other words, if we hear a testimony of a witness who says thatshe saw someone who looked like the suspect jump in a red car, the proposition‘there is a testimony by a witness who saw someone who looked like the suspectjumped into a red car’ can be accepted as justified.
In addition to evidential data, the term evidence can also point to otherpropositions which have been accepted as true and which might have been inferredfrom evidential data themselves. For example, the proposition that ‘someone wholooked like the suspect jumped into a red car’, which has been inferred from theabove testimony, is itself evidence for the proposition that ‘the suspect jumped into
a red car’. It is important that the evidential data and the propositions that are
inferred from it are not confused. Anderson et al. (2005, p. 60) denote this asfollows: E* stands for evidential data about event E; in the above example, E* is thetestimony itself and E is the event that ‘someone who looked like the suspectjumped into a red car’. As was already noted, this event E can then be evidence foranother event F: ‘the suspect jumped into a red car’. The separation between an
event and the evidential data from which the event is inferred is important because
the existence of the evidential data does not mean that the event actually happened.
In the above example, the witness may be lying or he may misremember or the
person who looked like the suspect may not be the suspect at all. In sum, the term
evidence stands for the information that (positively or negatively) influences ourbelief about a particular proposition. This information can be a piece of evidential
data but also a proposition which has itself been inferred from data.
2.3 Generalizations and commonsense knowledge
That a fact at issue in the case follows from the evidence is by no means self-evident
and often involves constructing complex chains of inferences using commonsense
knowledge. Furthermore, the coherence of the hypothetical scenarios in the processof proof also depends on the degree to which they conform to our commonsense123

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knowledge and expectations. So in addition to knowledge gained from (casespecific) evidence, reasoning with evidence also involves reasoning with commonsense knowledge which is not somehow based on evidential data. This can begeneral knowledge, which is widely accepted in a certain community (e.g. the dateof Christmas is the 25th of December) or personal knowledge gained throughfirsthand experience (e.g. the University of Groningen is usually closed between
Christmas and New Year’s Eve).
Commonsense knowledge can take the form of simple facts (e.g. Amsterdam is
the capital of the Netherlands) or be expressed as generalizations. Generalizations
are statements about how we think the world around us works, about human actions
and intentions, about the environment and about the interaction between humans
and their environment (Cohen 1977, pp. 274–276). Examples of generalizations are‘being shot in the head can cause a person to die’, ‘witnesses usually speak thetruth’, ‘mothers usually don’t carry guns in their purses’ and ‘people from Suriname
are more prone to becoming involved in crime than native Dutch people’.
Generalizations are almost never universally true and there are often exceptions to
the generalization. For example, witnesses under oath do not always make true
statements, they can lie or misremember. Because they are not universally true,
generalizations are often qualified with terms such as usually, often and sometimes.
Also, generalizations such as the ones given above can be rewritten as conditionalrules of the form ‘if…then…’; for example: ‘if there is a forceful impact then thiscan cause a person’s skull to break’ but also ‘if person w is a witness under oath andw says that event e happened then usually e will have happened’.

3 Argumentative and story-based analysis
In this section we first briefly and informally discuss two dominant approaches toreasoning with legal evidence in AI and Law, namely argumentation (Bex et al.
2003; Verheij 2000; Walton 2002) and inference to the best explanation or
abductive-causal reasoning (Josephson 2002; Thagard 2004, 2005; Keppens andSchäfer 2006). After this we will discuss some of the advantages and disadvantagesof the two approaches and argue that a combined approach best captures the aspectsof reasoning with criminal evidence as discussed in Sect. 2.
3.1 Evidential arguments
In logics for argumentation (Prakken and Vreeswijk 2002), the rules of classicallogic are augmented with rules for defeasible inference. Associated with adefeasible inference is an underlying evidential generalization. Toulmin (1958/
2003) and Verheij (2005) speak of this generalization as a warrant, Hage (1996),
Prakken and Sartor (1997) and Prakken (2010) call it a rule, Pollock (1995) uses theterm rule of inference and Walton (2002) interprets them as schemes. Arguments
can be constructed by chaining applications of inferences and such lines ofargument can be combined into an argument graph or inference graph (Pollock1995) with as its premises the pieces of evidence and as its ultimate conclusion one

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P: It was Rijkbloem who killed Mr. Lammers by shooting at him from close range with a gun
1: Rijkbloem fired at Mr. Lammers from close range
NT1: Nicole’s testimony to 1
2: Mrs. Lammers heard a shot when Rijkbloem charged at Mr. Lammers
MT1: Mother’s testimony to 2
3: Rijkbloem had a gun
NT2: Nicole’s testimony to 3
MT2: Mother’s testimony to 3
4: The bullet which killed Lammers was of the same type as the bullets found in
Rijkbloem’s house
5: Bullet 1 taken out of Mr. Lammers’ head
6: Bullet 2, taken from .22 rounds which were found in Rijkbloem’s house
7: The chemical composition taken from bullet 1 and bullet 2 is significantly similar
FR: Forensics report
8: Mr. Lammers died because of Bullet 1 in his brain
CR: Coroner’s report

Fig. 1 An evidential argument graphof the explananda. Such a graph is very similar to a Wigmore chart (Wigmore 1931;
Anderson et al. 2005). In Fig. 1, an example of an argument graph is given. In thisfigure, some of the evidence for the main hypothesis, that it was Rijkbloem who shot
Mr. Lammers (P), is given. The inferences in the argument are all of an evidential
nature: a piece of evidence e and the evidential generalisation ‘e is evidence for p’
allows us to infer p. For example, the inference from NT1 to 1 is justified by thegeneralization ‘a witness testimony that a certain event e happened is (usually)
evidence for the occurrence of e’.
Arguments can be attacked. They can be rebutted with an argument for the
opposite conclusion and they can be undercut (Pollock 1995) with an argument for
why an inference is not allowed (usually because a generalization does not apply in
the given circumstances). For example, an argument for ‘Rijkbloem did not shootthe Mr. Lammers’ rebuts ‘Rijkbloem shot Mr. Lammers’ and an argument for‘Nicole is lying (for example, because she wants to frame Rijkbloem)’ undercuts theinference from NT1 to 1, because the situation where the witness lies is an exception
to the general rule that ‘a witness testimony that a certain event e happened is(usually) evidence for the occurrence of e’.
In the argument-based approach, arguments like the one shown in Fig. 1 arebuilt, supporting the propositions that have to be proven, and other arguments
are also built to attack these arguments. In the example, the prosecution has to123

F. J. Bex et al.
Fig. 2 An abstract
argumentation framework

A3

A2

A1

prove that Rijkbloem killed the father. In order to do this, arguments like theone in Fig. 1 will have to be constructed. The defence will try to attack thesearguments by saying, for example, that it was not Rijkbloem but the mother who
shot Mr. Lammers, or by arguing that the testimonies of the two women cannot
be trusted.
After it has been determined which arguments attack which other arguments, thedefeat relations and thus the dialectical status of arguments can be assessed (seePrakken and Vreeswijk 2002). In this respect, arguments can be classified into threekinds: the justified arguments (those that survive the competition with theircounterarguments), the overruled arguments (those that lose the competition with
their counterarguments) and the defensible arguments (those that are involved in atie). For example, assume that an argument for the conclusion ‘Rijkbloem did notshoot Mr. Lammers’ (A1) is rebutted by an argument for the conclusion that
‘Rijkbloem shot Mr. Lammers’ (A2) and vice versa. Furthermore, assume that in A2
the conclusion has been derived from Nicole’s testimony and that this argument isundercut by an argument that Nicole is a liar (A3). This situation can be rendered in
an abstract argumentation framework (Dung 1995) as follows:
In Fig. 2, each individual argument is rendered as a single node; the exact
internal structure of each argument is for current purposes less important. The
arrows do not denote a support relation, as in Fig. 1, but rather an attack relation
between the arguments. Because A3 is itself not attacked, it is justified and(assuming it is strong enough) defeats A2, which is in turn overruled. A1 is now also
justified because its only attacker is overruled. If A3 had not attacked A2, A1 andA2 would both have been defensible.
3.2 Causal stories
In the story-based approach hypothetical stories or scenarios, coherent sequences ofstates and events, are used to explain the observations, the evidence that has been
observed in the case. This explaining is done through abductive reasoning. The
basic idea of abductive inference is that if we have a general causal rulecause ? effect and we observe effect, we are allowed to infer cause as a possible
explanation of the effect. This cause which is used to explain the effect can be asingle state or event, but it can also be a sequence of events, a story.
Stories are hence modelled (Josephson 2002; Thagard 2004, 2005) as causal
networks, where the events in the story are connected by causal (or explanatory)
links. These links may denote physical or psychological causation, but sometimes
stand for not much more than some kind of temporal precedence between events.
These causal relations can be expressed as causal generalizations; for example,‘Rijkbloem shooting the father causes the father to die’. Figure 3 depicts a smallpart of the prosecution’s story in the Rijkbloem case.

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Lammers andRijkbloem
were arguing

Rijkbloem
pulled out
his gunRijkbloem
shotLammers at
close range

Lammers
was hit in
the head

Lammers
died

Rijkbloem
had a gunFig. 3 A causal story network

Note the different type of arrow than in Fig. 1, which distinguishes the relationsin Fig. 3 as causal instead of evidential (as in Fig. 1). The causal network shown in
Fig. 3 explains the observations in the case. For example, Mr. Lammers’ dead bodyis explained by him being shot in the head.
Taken by itself, the abductive scheme is nothing but the fallacy of affirming theconsequent (i.e. if p ? q and q then p). However, in a setting where alternative
explanations are generated and compared, it can still be rational to accept one ofthese explanations. In the story-based approach, the idea is that the explananda(which means ‘‘facts to be explained’’), which are the important observations in thecase, are explained by different explanations, the best of which has to be chosen.
This type of reasoning is therefore often called inference to the best explanation(IBE). IBE is a form of defeasible reasoning, as additional observations might giverise to a new (best) explanation. In the Rijkbloem case, the explanandum would be
the dead body of Mr. Lammers. There are two different stories that each explain
this: the prosecution’s story that it was Rijkbloem who shot Mr. Lammers andRijkbloem’s story that it was Mrs. Lammers who (accidentally) shot her husband.
This is visualized in Fig. 4.
An important question in IBE is how it can be determined what the ‘‘best’’explanation is. Naturally, a good story should explain as much of the observed
evidence as possible. The simplest way of determining the best explanation is thuschoosing a subset-minimal explanation that explains all the observations in the case.
However, this is often not possible in real-world cases, as conflicting evidence leads
to conflicting observations. For example, if there are two expert reports, one whichsays that Mr. Lammers was shot at close range and one which says that he was shotfrom afar, it is impossible to explain both with the same story, as then the story
would be inconsistent. Prakken and Renooij (2001) discuss a more refined method
where there are certain observations that must be explained (i.e. the explananda) andLammers
died

Fig. 4 Two explanations

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other additional observations which are ideally explained; the more additional
observations explained, the better. Thagard (2004) has a wholly different way ofdetermining the best explanation. He computes the activation of the individual statesand events, where acceptance or rejection of an explanation is represented by the
degree of activation of the individual events in the explanation.
Pennington and Hastie (1993) argued that, in addition to explaining the observed
evidence, a story should also be coherent. This coherence depends on three subcriteria. First, a story should be internally consistent in that it does not containinternal contradictions; a story which says that Rijkbloem at the same time shot
father from up close and from afar is not consistent. Second, a story is plausible if itconforms to our general knowledge of the world. Wagenaar et al. (1993) argued thata story is plausible if it is anchored in commonsense knowledge which can be
expressed as generalizations. For example, one might argue that the defence’s storyis less plausible because it appeals to the generalization ‘women (often, sometimes)
have guns in their handbags’. The prosecution’s story, on the other hand, seems
much more believable as it is anchored in generalizations such as ‘petty criminals
often solve problems with violence’. Finally, Pennington and Hastie’s coherence
also depends on a story’s completeness. A story is complete if it mentions some
initiating events that cause the main actor to have goals, which give rise to actions
that have consequences. If one of these elements is missing, a story is less coherent.
The prosecution’s story, for example, mentions the Lammers getting into anargument with Rijkbloem (initiating events), Rijkbloem subsequently pulling out a
gun and shooting Mr. Lammers (actions) and Mr. Lammers’ death (consequence).
The story could be made more complete by explicitly incorporating that it was
Rijkbloem’s goal to kill Mr. Lammers.
3.3 Advantages and disadvantages of the two approaches
Both the argumentative and the narrative approach can be separately applied to any
one case. In real cases argumentative and narrative aspects will often blend into
each other in a natural way (as will be shown in Sect. 4). In Sect. 3.2, however, wehave strived to maximally distinguish the two approaches so that their respective
advantages and disadvantages become more clear.
Argumentative reasoning provides us with a transparent, natural and rationallysound way of analysing and assessing reasoning with evidence. The effect of theevidence on a conclusion can be shown in an intelligible way and the evidential
generalizations warranting the inferences play an important role in explicating why
one thinks the conclusion can be inferred from the particular evidence. Case studiesof legal judgements (Bex 2009; Bex et al. 2009; Bex and Verheij 2010) andempirical research (van den Braak 2010) has shown that evidential reasoning is anatural way of reasoning about, for example, witness statements and the conclusions
that can be drawn from such statements. The possibility of constructing
counterarguments to attack a conclusion or an inference allow one to expose
sources of doubt in the reasoning, and the dialectical process of argument andcounterargument, if conducted properly, provides a rationally justified conclusion.

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The ideas on argumentation presented here are almost all logically andconceptually well-developed in the literature. For example, standard patterns of(evidential) reasoning and their typical sources of doubt have been studied
extensively in the work on argumentation schemes (Bex et al. 2003; Walton 2002;
Walton et al. 2008). In the field of computational argumentation, formal
argumentation-theoretic semantics have been developed which allow for thecomputation of the acceptability of arguments in light of counterarguments and thus
provide a mathematically sound way to determine the status of arguments (i.e.
justified, defensible, overruled).
The atomistic nature of arguments makes them very useful for carefully
analysing each piece of evidence, the conclusions and the general knowledge usedin reasoning from this evidence to the conclusions. However, in a purely argumentbased approach the overview of the case tends to be lost. In a real case, the various
hypotheses about what (might have) happened are usually not simple propositions
but rather hypothetical scenarios, complex sets of propositions, the elements ofwhich are related in various ways. The conclusion of an evidential argument isusually only a single element of such a scenario, an individual state or event, while
the scenarios about ‘what happened’ in the case are cut into pieces. This can be seenin Fig. 1, where it is not clarified how the propositions 1–4, which are offered as
evidence for P, relate to each other or can be combined into a coherent whole. For
instance, that Rijkbloem had a gun (3) is clearly a prerequisite for him firing it at
Mr. Lammers (1), but this (causal) relation is not made explicit. Furthermore,certain events which have no direct bearing on the conclusion but which might help
in understanding the case (e.g. ‘Lammers and Rijkbloem were arguing’) are notincluded in the argument. In a sense, the individual arguments from evidence only
provide us with the pieces of the puzzle; without an idea of what the eventual image
should look like, it can be very hard to put together these pieces.
Abductive, story-based reasoning can be used to construct understandable
scenarios in a natural way. An important feature of explanatory abduction is that itprovides us with new hypotheses. These new hypothetical scenarios may then point
to new avenues of investigation by predicting observations that could be made. In
this sense, abductive reasoning can be classified as being imaginative, creative or‘ignorance preserving’ (Gabbay and Woods 2006). This is in contrast to evidential
reasoning with arguments, which is more ‘likelihood enhancing’ (each piece ofevidence enhances the likelihood that the conclusion is or was the case). Stories also
provide an overview of the case and clearly show how the course of events
surrounding the crime might have unfolded. They help organize the evidence andhelp people to fill gaps in a case. For example, the story in the Rijkbloem case(Fig. 3) makes it clear what happened and allows us to consider the situation ‘‘as awhole’’.
An important advantage of the narrative approach is that it is perhaps closest to
how investigators and decision-makers actually think about a case. Experiments by
Bennett and Feldman (1981) and Pennington and Hastie (1993) suggest that when
making a decision, jurors construct and compare stories which explain mostevidence and then choose the most coherent and plausible story that explains themost evidence. Studies in story recall and understanding have also shown that our123

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memory is organized through episodes or stories (Schank and Abelson 1977). Other
psychological research (Simon 2001) has shown that a holistic coherence approach(which is part of the narrative approach, see also Thagard 2004, 2005) is the natural
way of modelling decision making in legal trials.
Furthermore, there seems to be a consensus in the literature on evidence analysis
that fact investigators work with causal story structures and timelines (Heuer 1999;
de Poot et al. 2004). Finally, Pardo and Allen (2007) have argued that a theory of
IBE is a natural way to analyse legal trials; although their focus is on civil trials,
they show how a general theory of IBE can be used to, for example, model theshifting of the burden of proof.
The holistic approach of stories, where the elements of the case are notconsidered separately but rather as a whole, also has some inherent disadvantages
and dangers. Often, the evidential data has no clear and separate place in the modelof the case (cf. Pennington and Hastie 1993). Purely story-based theories requirethat the observations are explained by the story, but it is not made clear whether
these observations are the actual evidential data itself or the events that follow fromthe evidence. This an implicit connection to the evidence makes stories dangerous,
as a coherent story can be more believable than a story which is incoherent but
supported by evidence. In other words, there is a danger of ‘‘good stories’’ pushingout ‘‘true stories’’ (Bennett and Feldman 1981). Work in AI by Josephson (2002)
and Thagard (2004) has elaborated on the narrative approach and given theevidential data a clear place, namely as the causal effects of (events in) the story.
This allows for reasoning about the reliability of individual pieces of evidence.
However, there are indications that this does not adequately capture reasoning with
implicit default knowledge (see Bex 2009, p. 78). By contrast, in an argumentationbased approach, sources of evidence are used to infer the occurrence of events, andthe defeasible nature of these inferences gives a natural place to examining thesource’s reliability, namely, as the search for counterarguments based on criticalquestions of argument schemes.
An aspect of the story-based approach which is underdeveloped is exactly how
stories should be rationally compared. As Thagard and Shelley (1997) noted, thesimplest (in terms of subset minimality) and most complete (in terms of explaining
the most observations) explanation is not always the best. For example, a complex
explanation that explains only a few important pieces of evidence can be better thana simple explanation that explains many less important pieces of evidence.
Furthermore, the link between a story and one piece of evidence may be very strong
whereas another piece of evidence may only have a weak connection to the story.
Pennington and Hastie’s coherence principles provide some pointers for determining the inherent plausibility of an explanation, but they do not precisely define
criteria. Here, a particular disadvantage of a purely story-based approach is that it is
impossible to reason about the (elements of a) story. For example, one might want
to argue that the story in Fig. 3 is not plausible as people normally do not shootother people, even when they are arguing.
Above, the argument- and story-based approach were presented as two different
approaches. On a conceptual level, this makes sense: the sequence of events in
Fig. 3 is clearly not an argument and the argument in Fig. 1 has, considered on its

123A hybrid formal theory

own, nothing to do with a story. In practice the distinction is similarly noticeable:
for example, a judge considering a single event and the evidential data for andagainst this event is engaged in atomistic and more argumentative reasoning whilst
an investigator trying to determine what course of events led to the current situationis comparing various stories in a more holistic way.
However, it can be argued that the distinction between the two approaches is an
artificial one. For example, it is perfectly possible to reason about causality or
intentions, goals and actions using just arguments (see e.g. Bex et al. 2009).
Alternatively, reasoning about a single witnesses’ credibility can be done in an
abductive approach: a testimony will then be caused by the event to which it testifies
and attackers of this witness’ credibility are then modeled as alternative
explanations (Thagard 2005).4 The exact way in which the two approaches aremodelled further amplifies or downplays their advantages and disadvantages. For
example, an argument-based approach which uses only more abstract argument
frameworks (Fig. 2) can be said to be more holistic, as it provides an overview of
how all the evidence in the case interacts while sacrificing the details on how
exactly the conclusions follow from the evidence. Similarly, a story-based approachin which all the causal links between the events and the observations are rendered indetail can be said to be more atomistic than holistic.
Because of this overlap between the purely argumentative and narrative
approaches, Bex and Verheij (2010) have argued that stories and arguments are‘‘communicating vessels’’; in some instances a causal, holistic and more story-based
approach works best and in other instances an evidential, atomistic and argumentative approach is the most natural. Here the context and the aims of the analyst canplay an important role. For example, when not much evidence is available to theanalyst, it makes sense to hypothesize one or more stories to steer the investigation.
On the other hand, when the main goal is to organize the reasons for and against asingle (important) witness’ credibility, arguments may be more suitable.

4 A hybrid theory
In the previous section we argued that both the argumentative and the narrative
approach have their own advantages. The argumentative approach, which builds ona significant academic tradition of research on informal and formal argumentation,is well suited for a thorough analysis of the individual pieces of evidence, whilst theempirically tested narrative approach is appreciated for its natural account of crime
scenarios and causal reasoning. In some cases, a causal, holistic and more storybased approach works best and in other instances an evidential, atomistic andargumentative approach is the most natural.
Hence, arguments and stories need to be combined into one hybrid theory which
fully combines the two separate approaches. In this hybrid theory, stories in theform of causal networks are used to explain the explananda. Arguments based on
4Simon (2001) has argued that ‘‘… [people] defy the syntactic rules of unidirectional inference…’’, i.e.
that causal and evidential reasoning are both used in conjunction.

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F. J. Bex et al.

NT1

MT1
gunshot noise

Lammers andRijkbloem
were arguing

Rijkbloem
pulled out
his gunRijkbloem
shotLammers at
close range

Rijkbloem
had a gunNT2

Lammers
was hit in
the head

Lammers
died

CR

MT2
FR

Fig. 5 Combining arguments and explanations

evidence are used to support and attack these stories. Thus, stories can be used as anintelligible overview of what might have happened in the case and arguments can beused to support or attack these stories and to reason about the inherent plausibility
and coherence of the stories in greater detail. In this combined approach all thefeatures and advantages of the individual argumentative and narrative approaches
remain intact; combining the two modes of reasoning only extends the possibilities.
Figure 5, which combines Figs. 1 and 3, illustrates these ideas.
In the rest of this section, the formal version of the hybrid theory is presented.
The basic idea of the formalized hybrid approach is as follows. A logical model ofabductive inference to the best explanation (IBE) takes as input a causal theory(a set of causal rules or generalizations) and a set of observations that has to be
explained, the explananda, and produces as output a set of hypotheses that explain
the explananda in terms of the causal theory. The combination of hypotheses andcausal theory can be seen as a story about what might have happened. Arguments
based on evidence can be used to support and attack stories. These arguments canthemselves be attacked and defeated, thus making it possible to reason in detailabout the extent to which a story conforms to the evidence. The coherence of a storyis also a subject of argumentation: arguments which are not based on evidence but
rather on general commonsense knowledge can be given to support or attack a story.
In this way, the plausibility of a story (i.e. its conformance with our commonsenseworld knowledge) can also be discussed in detail. Finally, the stories can becompared according to their coherence and the extent to which they conform to theevidence in a case.
This section is structured as follows. First, the logic underlying the hybrid theory
will be briefly presented. Section 4.2 discusses the argumentative part of the hybrid
theory and shows how arguments based on evidence can be constructed andattacked. Section 4.3 shows how the basics of the causal (narrative) part of thehybrid theory can be modelled. Section 4.4 discusses the hybrid theory, and inparticular the criteria for determining a story’s quality given the evidence andarguments.

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A hybrid formal theory

4.1 A logic for argumentative-narrative reasoning
Logic basically consists of a combination of a formal object language and a notionof valid consequence expressed in a metalanguage. In this paper, the object
language, from here on referred to as the logical language L, is largely left
unspecified. As in Prakken (2010), the assumption that L is closed under classicalnegation is generalized in two ways. Firstly, non-symmetric conflict relations
between formulas will be allowed (to capture, for instance, negation as failure).
Secondly, in addition to classical negation, other symmetric conflict relations will beallowed, so that, for example, formulas like ‘bachelor’ and ‘married’ can, if desired,be declared contradictory.
Definition 1 [Contrariness] Let L, a set, be a logical language and - a contrariness
function from L to 2L . If u [ w; then if w 62 u; then u is called a contrary of w,
otherwise u and w are called contradictory. The latter case is denoted by u = -w
(i.e., u [ w then if w [ u).
We will not explicitly mention that u [ w if this is obvious from the text (forexample, Rijkbloem shot Mr. Lammers and Rijkbloem did not shoot Mr. Lammers).
Now that the notion of negation has been generalised, the same must be done with
the notion of consistency.
Definition 2 [Consistent set] Let P ( L. P is consistent iff there is no u, w [ Psuch that w [ u, otherwise it is inconsistent.
Since reasoning about evidence is defeasible, our logic must be nonmonotonic.
Therefore, we augment the inference rules of classical logic with defeasible rules. In
the hybrid theory, it should be possible to reason with generalizations: evidential
generalizations are used to warrant inferences in an argument and causal
generalizations are used to express causal relations in a story. As was discussed
in Sect. 2.3, generalizations can be expressed as conditionals of the form ‘if p thenq’. These conditional generalizations can be modelled as object-level conditionals
or as (metalinguistic) rules of inference. Bex et al. (2007b) model them as objectlevel conditionals which are premises for a defeasible modus ponens inference rule.
Bex (2009) uses a mixed approach, were oft-used generalizations such as the one forwitness testimony (see Sect. 3.1) are modelled as inference rules and more casespecific ones are modelled as premises at the object level. Following Prakken (2010)and van den Braak (2010) in this paper generalizations are modelled as defeasible
rules.
Definition 3 [Defeasible rules]••

Let u1, …, un, w be elements of L.

An evidential defeasible rule is of the form u1, …, un e w, informallymeaning that if u1, …, un hold, then (presumably) this is evidence for w.
A causal defeasible rule is of the form u1, …, un c w, informally meaning that
if u1, …, un hold, then this has (presumably) caused w.
u1, …, un are called the antecedents of the rule and w its consequent.

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F. J. Bex et al.

As usual in logic, inference rules will often be specified by schemes in which a
rule’s antecedents and consequent are metavariables ranging over L. Furthermore, it
is assumed that applications of inference rules can be expressed in the object
language. In this paper we assume that this can be done in terms of a subset LR of
L containing formulas of the form ri. For convenience we will also use elements of
LR at the metalevel as names for inference rules, letting the context disambiguate.
4.2 A theory for evidential argumentation
Using the logic as defined in Sect. 4.1, arguments based on some input information
can be built. In accordance with the standard reasoning in the process of proof (Sect.
2), this input consists of evidential data and commonsense knowledge.
Definition 4 [Evidential theory]
where
•
•

An evidential theory is a tuple ET = (Re , K)

Re is a set of evidential defeasible rules.
K = Ke [ Ka is a knowledge base, where:– Ke , the evidence, is a consistent set of literals from L:– Ka is the set of commonsense assumptions.

In an evidential theory, commonsense knowledge is represented in both Re andKa ; conditional generalizations are in Re and other assumptions are in Ka . Given
this commonsense knowledge and the evidence in Ke , arguments can be constructed
as in the following definition.
Definition 5 [Argument] An argument based on an evidential theory ET is a finite
sequence [u1, …, un], where n [ 0, such that for all ui (1 B i B n):••ui [ K; orThere exists a rule in Re such that w1, …, wn e ui and w1, …, wn [ {u1, …,
ui-1}.

The elements of the sequence are also called lines of argument. Alternatively, an
argument could be defined as a tree, as in e.g. Prakken (2010). As is well-known, thesequence- and tree-form of an argument are equivalent in that every sequence
captures one way to construct a tree.
According to the above definition, a line of argument is a proposition from the
input information in ET or is derived from preceding lines of argument by theapplication of some evidential inference rule. In what follows, for a given argument A,
Prem(A) denotes the set of all the elements in A which are also in K (i.e. A’s premises),
Conc(A) denotes the set of A’s conclusions, lines of argument which are not in K andRules(A) is the set of inference rules from Re used in the argument. Furthermore,Args(ET) denotes the set of all arguments that can be constructed from a theory ET.
For an example of an evidential argument, assume that ET is as follows:
Ke = {Nicole testified that ‘I remember Rijkbloem shooting my father’}
Re = {rwitness: w testifies that ‘p’ e p is true,
rmemory: w remembers that p happened e p happened}123

A hybrid formal theory

These two inference rules for witness testimony and memory are well-known
from the literature (Pollock 1995; Bex et al. 2003; Walton et al. 2008). Now, thefollowing argument Ashot for the conclusion that Rijkbloem shot Mr. Lammers canbe constructed:
1.
2.
3.

Nicole testified that ‘I remember Rijkbloem shooting my father’ (Ke )Nicole remembers that Rijkbloem shot Mr. Lammers (1, rwitness)
Rijkbloem shot Mr. Lammers (2, rmemory)

Note that the lines of argument have been numbered and that at the end of eachline of argument it is noted how the line was inferred.
Now the notion of attack between arguments must be formally defined.
Normally, attacking an argument does not mean that the argument is automaticallydefeated. For argument A to defeat argument B, A has to successfully attack B andfor this, A has to be somehow stronger than B or preferred to B. However, because
currently no notion of strength or preference of arguments is defined, the notions ofattack and defeat will simply be equated. Extending the framework to include
preferences between arguments (which can themselves be based on, for example,
preferences between evidence or inference rules) can be done along the lines of
Prakken (2010).
Definition 6 [Defeat]Given two arguments A and B:•

A rebuts B iff Au [ A and Aw [ B such that u [ w and w follows from theapplication of a rule ri [ Re .
• A undercuts B iff Au [ A and Ari [ Rules(B) such that u [ ri .
• A undermines B iff Au [ A and Aw [ B such that u [ w and w [ Ka .
A defeats B iff A either rebuts, undercuts or undermines B.
With the above definition of defeat, the current argumentation theory combines
two ways to capture the defeasibility of reasoning, called plausible and defeasible
reasoning by Vreeswijk (cf. Prakken and Vreeswijk 2002). In plausible reasoning,arguments can only be attacked on their (defeasible) premises (e.g. Bondarenko
et al. 1997). In defeasible reasoning, arguments can only be attacked on theapplications of defeasible rules (e.g. Pollock 1995; Vreeswijk 1997). In Verheij(2003), the distinction between plausible and defeasible reasoning is downplayed as
the applications of rules are expressed as conditional sentences in the object
language. The current approach follows Prakken (2010) and allows arguments to be
attacked on the applications of inference rules (rebutting and undercutting) as wellas on defeasible premises (undermining).
As an example of a rebuttal, consider the following argument Anot_shot.
1.
2.

Rijkbloem testified that ‘I never shot Mr. Lammers’ (Ke )Rijkbloem never shot Mr. Lammers (1, rwitness)

Line 2 of this argument rebuts line 3 of the argument Ashot. Now assume that thefollowing inference rule is added to Re :rsuspect: w is a suspect in the case e w is not trustworthy

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F. J. Bex et al.

Furthermore, assume that ‘w is not trustworthy’ [ rwitness , i.e. if someone isuntrustworthy then this contradicts any application of the witness inference rule.
The following argument Asuspect, which undercuts Anot_shot can now be constructed:
1.
2.

Rijkbloem is a suspect (Ke )Rijkbloem is not trustworthy (2, rsuspect)

However, this argument possibly is a fallacy: if we were to believe that suspects
are always untrustworthy, then they would never be able to defend themselves incourt. So what we want to do is give an argument for the fact that rsuspect expresses
an invalid generalization. First, we add the following rules to Re :rexpert: e is an expert in criminal psychology, e says p e p is true
rinvalid_suspect: suspects are no less trustworthy than any other witness e
the rule rsuspect expresses an invalid generalization
Furthermore, assume that ‘the rule ri expresses an invalid generalization’ [ ri , i.e.
if a particular rule is deemed invalid it is contradicted. Now, the following argument
Ainvalid_suspect undercuts Asuspect.
1.
2.
3.
4.

Peter is an expert in criminal psychology (Ke )Peter says suspects are no less trustworthy than any other witness (Ke )Suspects are no less trustworthy than any other witness (2, rexpert)
The rule rsuspect expresses an invalid generalization (3, rinvalid_suspect)Thus, we can talk about the invalidity of defeasible rules. It is also possible togive an argument for the validity of an inference rule. Assume that the following
rules are added to Re :Re = {rgen_knowl: It is general knowledge that p e p is true,
rvalid_suspect: suspects are not trustworty ethe rule rsuspect expresses a valid generalization}
The inference rule for general knowledge was first proposed by Bex et al. (2003).
The inference rule rvalid_suspect shows that it is possible to talk about the validity ofgeneralizations using arguments. Essentially, this allows one to provide an inference
rule with what Toulmin calls a backing, a reason for the validity of a rule, viz.
argument Avalid_suspect:
1.
2.
3.

It is general knowledge that ‘suspects are not trustworthy’ (Ka )Suspects are not trustworthy (1, rgen_knowl)
The rule rsuspect expresses a valid generalization (2, rvalid_suspect)

This argument ‘‘defends’’ the argument Asuspect that uses rmemory, because it
defeats any argument that argues that the rule does not express a valid generalization:line 3 in Avalid_suspect rebuts line 4 in Ainvalid_suspect and vice versa. In this case,however, we assume that Avalid_suspect is undermined, which is possible because thepremise at line 1 is a commonsense assumption. Undermining Avalid_suspect would
involve arguing for the contrary of this assumption (e.g. by giving an argument thatsays that general knowledge may not be used as a basis for reasoning); assume for theexample’s sake that we have such an argument Anot_valid_suspect.

123A hybrid formal theory

Ashot

Anot_
shot

AsuspectAinvalid_

Avalid_Anot_valid

suspectsuspectsuspect

Fig. 6 An abstract argumentation framework

The arguments and defeat relations specified above provide a set of arguments,
rendered in Fig. 6 as an abstract argumentation framework.
Here, Ashot rebuts Anot_shot and vice versa, Asuspect undercuts Anot_shot, Ainvalid_suspect
undercuts Asuspect, Avalid_suspect and Ainvalid_suspect rebut each other and Anot_valid_suspect
undermines Avalid_suspect. Based on these defeat relations, constraints on the
dialectical status of arguments can be defined.
Definition 7 [Dialectical status assignment] A dialectical status assignment is alabelling of arguments with labels either in, out or undecided, such that thefollowing hold:•••An argument is in iff all arguments defeating it (if any) are out.
An argument is out iff it is defeated by an arguments that is in.
An argument is undecided in all other cases.

Using such labelling-based status assignments, any semantics S proposed byDung (1995) can be used to determine the arguments’ status (cf. Verheij 1996;
Caminada 2006). Here we take it that an argument is justified if it is ‘in’ in all
S status assignments, it is overruled if it is ‘out’ in all S status assignments and it isdefensible if it is ‘in’ in some but not all S status assignments. For example,
Ainvalid_suspect and Anot_valid_suspect are justified and the rest of the arguments areoverruled according to grounded semantics. For preferred semantics, these twoarguments are also justified but Ashot and Anot_shot are defensible; the rest areoverruled.
4.3 A theory for explanatory stories
As was indicated at the beginning of this section, hypothesized stories should
causally explain the explananda, the facts to be explained in the case. As in
traditional models of abductive model-based reasoning (Console and Torasso 1991),
this explains relation between a story and the explananda can be defined through a
notion of logical consequence: the explananda should follow from a combination of
hypothesized events and causal rules (expressing causal relations between events).
A causal theory that is to be used as the basis for stories should therefore contain the
explananda as well as the events and causal rules from which the explananda follow:
Definition 8 [Causal Theory]•••

A causal theory is a tuple CT = (Rc , H, F ) where

Rc is a set of causal rules;
H, the hypotheticals or hypothetical events, is a set of ground literals;
F , the explananda which have to be explained, is a consistent set of ground firstorder literals.

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F. J. Bex et al.

The causal rules in Rc denote causal generalizations, which in turn express somekind of (assumed) causal relation in the world and the hypotheticals H denoteassumed events. The explananda F is a set that has to be explained by the
combination of hypotheticals and rules (in standard definitions of causal-abductive
reasoning these are usually called observations). The causal theory CT is in a sense
analogous to the evidential theory ET: an ET supplies the input on the basis of whicharguments can be constructed and a CT supplies the input on the basis of which
hypothetical stories can be constructed, namely assumed events (hypotheticals) andcausal rules.
Standard accounts of abductive model-based reasoning simulate the abductive
inference with classical-logical derivation: some hypothesis S explains an observation o if o is a logical consequence of S. In order to define how stories explain
explananda in the current logic, first the structure of a story and the way in which
propositions can be derived from a story should be defined.
Definition 9 [Story] A story S based on a causal theory CT is a finite sequence[u1,…,un], where n [ 0, such that for all ui (1 B i B n):••ui [ H; orThere exists a rule in Rc such that w1, …, wn c ui and w1, …, wn [ {u1, …,
ui-1}.

Here, a story is a set of events put in (chronological) sequence. These events may
simply be assumed (i.e. in H) or they can be causally inferred from a previous eventin the sequence. The elements of the sequence are called events; the function
Events(S) returns all events for a given story S and, similar to arguments, Rules(S) is
the set of inference rules from Rc used in the story and Stories(CT) denotes all
stories which can be constructed from some CT. Notice the similarity of a causal
story to an evidential argument: both are derivations in the logic as defined in Sect.
4.1. This structural similarity does, however, not mean that stories and arguments
are used in the same way: where arguments are used to evidentially argue for aparticular conclusion, stories are used to causally explain explananda. As an
example of a story SP, take the following sequence:
1.
2.
3.
4.
5.
6.

Lammers and Rijkbloem were arguing (H)Rijkbloem had a gun (H)Rijkbloem pulled out his gun (1, 2, causal inference)
Rijkbloem shot Lammers at close range (3, causal inference)
Lammers was hit in the head (4, causal inference)
Lammers died (5, causal inference)

Note that we did not first specify the causal theory. The elements which are from
H are indicated in the sequence and it is assumed that Rc contains the inference ruleneeded to infer the derived events which are not in H (indicated by causal
inference). For example, the inference from 5 to 6 is justified by the causal rule
Lammers was hit in the head c Lammers died. This means that the inference rulesin the above story are case-specific (i.e. they are not expressed as schemes and do
not contain variables) unless explicitly mentioned otherwise.

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A hybrid formal theory

Some of the causal generalizations in the above explanation stand for not much
more than some kind of temporal precedence5 between events and the actual causal
relations between events may be debated. For example, one could argue that asimple argument is not a good motivating cause for Rijkbloem to decide to shoot
Mr. Lammers. In the current model, this naı̈ve view on causal reasoning is
purposeful, as it allows for the quick and easy construction of stories and thus
retains the holistic flavour of narrative reasoning. The ad-hoc interpretation of
causality is not a problem, as any objection to doubtful causal links can explicitly be
expressed using arguments (see Sect. 4.4).
Now, a story explains some explanandum if the explanandum is an (explicitly)
mentioned event in the story. This is slightly different than in traditional models of
explanation, where hypotheticals together with rules explain the explananda whenthe explananda are simply derivable from the hypotheticals and the generalizations(i.e. hypotheticals [ generalizations ‘ explananda).
Definition 10 [Explanation]
for a set of literals E iff•••

Given a causal theory CT, a story S is an explanation

for all e, if e [ E then e [ Events(S); andS is consistent; andS contains no two generalizations with the same consequent.

The second condition in the above definition, which is standard in models of
abductive causal reasoning, ensures that the explanation does not lead toinconsistencies. Notice that this condition effectively models one of the criteria
for the quality of a story, namely the criterion that a story should be internallyconsistent. The third condition ensures that two different explanations for F arereally seen as two separate explanations: two stories with the same consequences are
considered as alternative stories.
In the Rijkbloem case, the story SP explains that Mr. Lammers died. Recall that
Rijkbloem himself recounted another story about what happened that day, namely
one in which Mrs. Lammers pulled out a gun and accidentally shot Mr. Lammers.
A simplified version of his story SD is as follows.
1.
2.
3.
4.
5.
6.
7.

Lammers and Rijkbloem were arguing (H)Mrs. Lammers had a gun (H)Mrs. Lammers pulled out her gun (1, 2, causal inference)
Rijkbloem hit Mrs. Lammers hand in which she was holding her gun (3, causal
inference)
The gun Mrs. Lammers was holding went off (4, causal inference)
Lammers was hit in the head (5, causal inference)
Lammers died (6, causal inference)

This story now also explains the explanandum, Lammers died. The two stories SP
and SD now have to be compared. Because arguments play a major role in this5

In the current framework, time is not explicitly represented. Rather, it is implicitly assumed that thefurther in the sequence an event is, the later it takes place.

123F. J. Bex et al.

comparison, first the combination of explanatory stories and arguments based onevidence will have to be defined.
4.4 A hybrid theory of argumentation and explanation
So far, two separate parts of the hybrid theory have been defined. The argumentative
part consists of input data ET from which arguments can be constructed. Thesearguments can defeat each other and they can be assigned a status overruled,
defensible or justified accordingly. The causal part CT consists of input data CT
from which hypothetical causal stories that explain the explananda F can beconstructed. The hybrid theory is a combination of an evidential theory and a causal
theory.
Definition 11 [Hybrid theory]
HT = (ET, CT), where••

A hybrid argumentative-narrative theory is a tuple

ET is an evidential theory
CT is a causal theory such that every f [ F is the conclusion of a justified
argument A in Args(ET) for which there is a u [ Prem(A) which is in Ke .

In the hybrid theory the argumentative part directly influences the composition of
the narrative part: the explananda should follow from evidential data through ajustified argument. The reason for this is that we are only interested in states or
events which have actually happened, so events which do not follow from evidence(i.e. we are not sure they happened) do not have to be explained.
Given sets of arguments and stories, essentially three criteria for determining theextent to which a story conforms to the evidence can be given. The first is evidential
support, the extent to which the events or the causal relations in a story aresupported by evidence from Ke through an argument. The second is evidential
contradiction, the extent to which the events or the causal relations in a story arecontradicted by an argument based on evidence from Ke . The third is evidential
gaps, the number of the events in a story about which there is no evidence, that is,
the events which are neither supported nor contradicted by evidence.
In order to determine evidential support and contradiction, it first needs to be
defined how arguments based on evidence can support or contradict a story.
Definition 12 [Support] Given a story S, u (justifiably or defensibly) supports S iff
there is a (justified or defensible) argument A [ Args such that u [ Prem(A) and w [Conc(A), where w [ S or w is some ri [ Rules(S).
Definition 13 [Contradiction] Given a story S, u (justifiably or defensibly)
contradicts w [ S iff u (justifiably or defensibly) supports w.
So a premise of a non-overruled argument supports its conclusions andcontradicts anything that is contradictory to its conclusions. Notice that it is also
possible to support or contradict an inference rule and that there are two degrees of
support and contradiction (justified and defensible) depending on whether theargument itself is justified or defensible.

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Evidential support and contradiction can now be defined as the set of all pieces ofevidence that support or contradict some event or causal relation in a story,
respectively.
Definition 14 [Evidential support]
E?(S) = {u [ Ke | u supports S}.

The evidential support of a story S is the set

Definition 15 [Evidential contradiction] The evidential contradiction of a story S
is the set E- (S) = {u [ Ke | u contradicts S}.
These two definitions allow us to record the extent to which the evidence
supports or contradicts a story. Take, as an example, the story SP. This story is
supported by the evidence Nicole testified that ‘I remember Rijkbloem shot myfather’ because argument Ashot has as its conclusion the event in line 4 from thestory. In a similar way, an argument Acod from expert opinion can be constructedthat supports the causal rule about the cause of Lammers’ death rcod: Lammers was
hit in the head c Lammers died.
1.
2.
3.
4.
5.

Bob is a coroner (Ke )Bob says the cause of Lammers’ death was the bullet lodged in his brain (Ke )The cause of Lammers’ death was the bullet lodged in his brain (2, rexpert)
The rule rcod is valid (3, evidential inference)rcod (4, evidential inference)In this argument, the two inference rules justifying the inferences from 3 to 4 and
4 to 5 have been implicitly assumed. By arguing for the validity of the causal rule,
we have thus increased the evidential support of both SP and SD.
As an example of evidential contradiction, take argument Anot_shot. In addition todefeating argument Ashot, this argument also makes that Rijkbloem’s testimony(about him not shooting) contradicts the story SP. Like with support, evidence could
also be used to contradict causal rules in the story SP, further increasing its
evidential contradiction.
The third criterion that pertains to a story’s conformance to the evidence is that ofevidential gaps, those events for which there is no evidence from which we may
infer either that the event happened or did not happen.
Definition 16 [Evidential gaps] The evidential gaps of a story S is the set
EG(S) = {w [ S | Au [ Ke such that u supports or contradicts w}.
Evidential gaps are hypothesized events for which there is no direct evidence andwhich therefore have to be inferred from plausible circumstances, that is, the story
as a whole. Thus the circumstances detailed in the story make it plausible that theevents in question happened. This is the ‘gap-filling’ function of stories: gaps in theevidence are filled with events that fit the total picture painted by the story.
Recall from Sect. 3.2 that a good story not just conforms to the evidence but isalso coherent. Here, two criteria for coherence will be given, which are similar to
Pennington and Hastie’s (1993) criteria but defined in more detail: internalconsistency and plausibility. The criterion of consistency is already incorporated
into Definition 10 of an explanation. The criterion of plausibility is less obvious.

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Pennington and Hastie argue that a story is plausible if it conforms to our generalcommonsense knowledge of the world. That is, the states, events and causal
relations that are not based on evidence should be plausible in that they follow fromour stock of knowledge. Such events or relations can be simply assumed: inprinciple, the causal theory CT places no restrictions on the hypothesized events in
H or rules in Rc . However, the plausibility of the story can also be explicitly
reasoned about, thus drawing out sources of doubt or arguing why a certain story isinherently plausible. In this reasoning, an event which is explicitly supported by anargument from general knowledge is more plausible than an event which is
implicitly assumed. The rationale behind this is that if an explicit argument is given,a consensus about the assumed event is more easily reached because explicit
arguments can be tested in the dialectical process.
Thus the plausibility of a story can be defined as the events and thegeneralizations expressing causal relations are supported by explicit arguments
based on the commonsense assumptions Ka . In a similar way, the implausibility of astory is the extent to which the events and the generalizations expressing causal
relations are attacked by explicit arguments based on Ka . Whilst, for example, animplicitly assumed rule expressing a causal relation can be implausible, this is notevident unless an argument contradicting the relation has been given.
Definition 17 [Plausibility]
Kg | u supports S}.

The plausibility of a story S is the set P?(S) = {u [Definition 18 [Implausibility]{u [ Kg | u contradicts S}.

The implausibility of a story S is the set P-(S) =Note that arguments based on evidence do not directly increase or decrease astory’s plausibility, as they are already directly taken into account with evidential
support and contradiction. Thus, plausibility is a notion that can be establishedindependently of the evidential data. The reason for this is that when there isevidence for a particular part of a story, its coherence is of secondary importance: if,for example, an event is supported by evidential data, we do not need to reason withour commonsense knowledge in order to make this particular event or generalization more plausible.
As an example of an argument relevant for the plausibility of SP, consider thefollowing argument for the validity of rthreat: Lammers and Rijkbloem were arguing,
Rijkbloem had a gun c Rijkbloem pulled out his gun.
1.
2.
3.
4.

It is general knowledge that people like Rijkbloem threaten with violence at theslightest provocation (Ka )Rijkbloem can be expected to threaten at the slightest provocation (2, evidential
inference)
The rule rthreat is valid (3, evidential inference)
rthreat (4, evidential inference)
A similar argument relevant for the implausibility of SD can be constructed.

1.
2.

It is general knowledge that most women do not have guns (Ka )Most women do not have guns (2, evidential inference)123

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3.

Mrs. Lammers did not have a gun (3, evidential inference)

This argument contradicts an element in SD, Mrs. Lammers had a gun, with acommonsense assumption, thus adding to the implausibility of the story.
An important question is when one story is better than another story. One way tocompare stories is to determine the extent to which the stories conform to theevidence with the criteria of evidential support, contradiction and gaps and theextent to which a story is coherent with the criteria of consistency, plausibility andimplausibility and then provide an ordering on stories (for example, the moreevidential data that supports the story, the better the story or the less evidential gaps
the better the story). This comparison could be done by simply counting, forexample, the total number of evidential data that support story S1 and S2; if S1 is
supported by more data it is better than S2 and vice versa.
Definition 19 [Comparing stories] Given two stories Si and Sj, a total preorderingfunction B ht can be defined as follows:•••

If |E?(Si)| \ |E?(Sj)| and |E-(Si)| C |E-(Sj)| then Si \ ht Sj
If |E?(Si) B |E?(Sj)| and |E-(Si)| [ |E-(Sj)| then Si \ ht Sj
If |E?(Si)| = |E?(Sj)| and |E-(Si)| = |E-(Sj)| then
– If |P?(Si)| \ |P?(Sj)| and |P-(Si)| C |P-(Sj)| then Si \ a Sj
– If |P?(Si)| B |P?(Sj)| and |P-(Si)| [ |P-(Sj)| then Si \ a Sj
– If |P?(Si)| = |P?(Sj)| and |P-(Si)| = |P-(Sj)| then
– If |EG(Si)| [ |EG(Sj)| then Si \ a Sj
– If |EG(Si)| = |EG(Sj)| then Si = a Sj
where |V| stands for the cardinality of a set V.

Notice that if the criteria are used to provide an ordering on stories, they should be
ranked according to their importance. Here, the extent to which a story conforms to
the evidence is ranked as being more important than the coherence of the story. The
danger of a ‘‘good’’ story pushing out a ‘‘true’’ story is real. If a more evidentially
supported story is, according to the ordering, always better than another story nomatter the coherence of both stories, this danger is obviated.
Take the two stories SP and SD. Both stories are supported by Bob’s expert
testimony and both are supported by one witness testimony (through Ashot andAnot_shot). No arguments based on evidence have been given that contradict either ofthe stories. In the end, SP is the better story because it has one explicit argument for
its plausibility while SD has an explicit argument for its implausibility.
Interpreting the various criteria in this discrete, almost mathematical way as
discussed here is not without problems. It is, for example, not always the case that ifstory A is supported by more evidential data than story B, story A is better thanstory B. If story A is supported by only one piece of evidence that is deemed highly
credible and relevant and story B is supported by multiple pieces of evidence whose
relevance to the main explananda is slight, we would not say that story B is
automatically better because it has a higher evidential support.
A different way of using the criteria is by not interpreting them as providing
hard-and-fast rules for the comparison of stories, but rather to regard them as123

F. J. Bex et al.

providing guidelines for reaching a rational and well-thought-out decision about thefacts of the case. Bex (2009) uses orderings like the one defined above to guide theanalytic process with evidence stories.

5 Conclusion
The current hybrid theory’s contribution lies in particular in its full integration ofarguments and stories into one theory for reasoning with criminal evidence. Whilst
Modified Wigmorean Analysis (exemplified by the book by Anderson et al. 2005)
and the Anchored Narratives Theory (Wagenaar et al. 1993) both allow the use ofstories as well as arguments (or at least some sort of argumentative inference),neither of these theories fully integrates the two. The hybrid theory bridges the gap
between the Modified Wigmorean Analysis and the Anchored Narratives Theory.
Proponents of the first approach have argued that reasoning with (argument) charts
is the best rational way of reasoning with evidence. They argue that stories aremainly useful for organizing and presenting the evidence and that they play alargely psychological (as opposed to a critically rational) role in evidence
evaluation. Here we argue that, when combined with arguments and open to
criticism, causal stories in inference to the best explanation are not only a natural
but also a rational way of reasoning. Proponents of the more story-centred
approaches have argued that the only natural way in which people reason withevidence is through stories and that all reasoning in a case takes the form of a story.
However, often reasoning from evidential data to some conclusion takes the form ofa syllogistic argument, and the hybrid theory shows how stories can be grounded inevidence through such arguments.
The discussion of the separate argument-based and story-based approaches has
also shown that stories and arguments are both necessary for a natural and rationallywell-founded theory of reasoning with criminal evidence. By retaining theadvantages and flexibility of the separate approaches whilst at the same timesolving some of their problems by integrating the two approaches, the hybrid theory
acknowledges this interaction between evidence, arguments and stories. Holistic
stories provide an overview of the various possible scenarios in a case and can beused to make sense of a complex mass of evidence. They can be used in a relatively
simple way to construct complex new hypotheses in the discovery phase of theprocess of proof. Furthermore, when combined with arguments, stories are usefulwhen justifying one’s decision in a complex case because they help make sense of
the evidence and the events that can be inferred from the evidence. Thus, a decision
can be more easily checked by third parties. Arguments provide a naturalconnection between the evidential data and the facts of the case. Notions such as
critical questions and attackers can be used to reason about the relevance andinferential force of an argument and the focused and atomistic way of reasoning
with arguments allows for a detailed analysis of the individual pieces of evidence,
the hypothetical stories and the commonsense world knowledge that is used in thereasoning. The direct interaction between arguments and stories, that is, the ways inwhich arguments can directly support and contradict stories and the ways in which123

A hybrid formal theory

the criteria for judging a story’s quality are dependent on evidential arguments, areunique to the hybrid theory and allow for a natural and rationally well-foundedtheory of inference to the best explanation.
In a sense, stories and arguments in the hybrid theory act as communicating
vessels. At some point in a case, only one individual state or event will be at issue
and in such cases it is most natural to reason with evidential arguments. For
example, we might want to know if a suspect was at a particular location at aparticular time and consider all the evidence for and against this fact. However, atother points in an investigation, we might use a story to fill an evidential gap and see
whether assuming the gap-filling event still allows for a plausible story. For
example, if no evidence for the suspects location at a particular important time isavailable we can assume that the suspect actually was at the location and, given
other evidence about the suspect’s whereabouts at other points in time, see if thecausal and temporal structure of the story is still plausible.
The theory as developed here has been expressly developed to underpin a sensemaking tool. Hence, it is one of the few logical theories which has conceptual,
cognitive as well as computational aims. In the current project, the logic wasdeveloped concurrently with the prototype system AVERS (van den Braak 2010).
The logical model of causal stories combined with evidential arguments is directly
implemented in the system; the results of the tests performed with the system andthe informal contact with various teams of the Dutch police force and Dutch police
academy have influenced the theory and strengthened the claim that the hybrid
theory is close to how actual reasoning with evidence in an investigation context is
performed.
The design of the hybrid theory provides new insights which are interesting forformal defeasible reasoning in general and logical inference to the best explanationin particular. In real domains such as criminal investigation, the logical ormathematical modelling of inference to the best explanation is a hard enterprise.
One reason for this is that a logical or mathematical theory which is used to model
the explanations in the domain is itself complex. Hence, a model of the domain in a
logical theory is too complex to be constructed or understood by (logical ormathematical) laymen. For example, a proper Bayesian Network cannot be
constructed or fully appreciated without in depth knowledge of the probabilities anddependencies expressed in the mathematical model underlying the network. In sucha case, the user of a model or system based on a logical model does not know if hisor her beliefs about the case are correctly expressed.
The hybrid theory is based on a logical model that models explanations as stories,
causal networks in which events are connected with simple, case-specificgeneralizations. The theory is based on research on how we understand the world
around us and how the human memory is organized; stories and story schemes arenatural tools we use in our everyday reasoning about a complex world. Therefore,
no formal mathematical training is needed and models of relatively complex cases(such as the Anjum murder case, which is analyzed using the present theory by Bex
2009) can be constructed and understood by laymen. A formal model ofargumentation, based on the intuitive concepts of argument and counterargument,
is used to reason about the formal causal model (i.e. a hypothetical story). This123

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combination of stories and critical argumentation makes the hybrid theory ideal for
sense-making: the knowledge represented in the system (i.e. the stock of knowledge
and the hypothetical stories) can be constructed incrementally and is defeasible inthat knowledge which is accepted at one point may be rejected at a later time when
new evidence that contradicts this knowledge becomes available. Thus the users of
the system can try to reach a cognitive consensus about the model of a case that iscompatible with the evidence in a natural and rational way.
When used for factual proof instead of investigation, a limitation of our approachis that it does not tell the trier of fact when the relevant standard of proof has been
met. To model reasoning with standards of proof, our model should be extended but
we believe that this is rather straightforward. Firstly that a story is the best does not
imply that it is sufficiently good to be held true, so the IBE part of our model must
be extended with standards of proof for the ‘absolute’ quality of a story. It should be
noted that setting such a standard of proof is a substantial legal issue and not matterof inference, so it lies outside any formal model of reasoning.
Second, in the argumentation part of our model we have not modelled how achoice can be made between rebutting arguments (for example, between arguments
based on conflicting evidential sources). Such choices can be expressed if our modelis extended with a preference relation on arguments along the lines of e.g. Prakken(2010).6 Here too it holds that the decision whether one of two rebutting arguments
is preferred over the other and so defeats it is a substantial one and not a matter ofinference. However, once such a decision has been expressed, our model captures its
impact on other issues.
Acknowledgments The research reported in this paper has been performed in the project ‘Making senseof evidence. Software support for crime investigations’, supported by the Netherlands Organisation for
Scientific Research (NWO) (ToKeN2000, project number 634.000.429).

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