Abductive argumentation with stories
Floris Bex
Humanities Computing Department
University of Groningen
Groningen, The Netherlands
florisbex@gmail.com

ABSTRACTThe process of proof is one of inference to the best explanation, in
which alternative stories are supported and attacked by arguments.
This combination of stories and arguments was previously
presented as a formal hybrid theory. In this paper, the aim is tofurther integrate stories and arguments by treating stories as
abductive arguments and by showing how arguments can be
explanations, just like stories. Thus, stories and arguments can beincorporated in the same dialectical framework.

Keywords
Argumentation, stories, evidence, abductive reasoning.

1. INTRODUCTION
The process of proof involves can be viewed as inference to thebest explanation (IBE): there are alternative hypothetical stories
about “what happened?” in a case, these stories are tested using
the evidence and ultimately the best story is chosen. In IBE,
hypotheses that explain initial observations are inferred using
abductive reasoning: if we observe some effect (e.g. a body withgunshot wounds), we are allowed to hypothesise its possible
causes (e.g. the person was shot). This cause can be a single event,but it can also be a sequence of events, a story.
Most of the examples of IBE in the AI literature (see e.g. [8] foran overview) do not incorporate a way to effectively reason about
the alternative hypothetical stories using the evidence. This has
been remedied in Bex’ recent work on the hybrid theory [1],which combines classic theories of model-based causal reasoning(stories) with the ASPIC+ framework [11] (arguments). Thus,
evidential arguments based on, for example, testimonies orforensic reports can be used to support and attack the alternative
stories in a case.
One shortcoming of the hybrid theory is that the combination of
the two reasoning frameworks leads to two different ways ofdetermining the “status” of stories and arguments, that is, whether
we are justified in accepting them as true. For arguments, thisstatus is relative to the available counterarguments and the statusof these counterarguments. For stories, this status is determinedwith a more absolute measure: the more observations a story
explains, the better. What is still missing from the hybrid theory is
a way to properly compare alternative stories relative to eachother and the available arguments.
The present paper aims to further integrate arguments and stories
by considering alternative stories as counterarguments, thus
combining abductive and evidential argumentation in one
framework. Thus, the status of both stories as well as arguments
can be determined using well-known argumentation-theoretic
semantics [6].
The rest of this paper is structured as follows. Section 2 discusses
the existing hybrid theory as proposed in [2]. Section 3 thendiscusses how stories and arguments can be used interchangeably
in reasoning with evidence. Section 4 presents an integrated
theory of stories and arguments which expands the hybrid theory.
Section 5 concludes the paper.

2. A HYBRID THEORY OF STORIES AND
ARGUMENTS
It is possible to distinguish at least two1 separate methods for the
analysis of evidence and legal cases. The first method [1] has its
roots in Wigmore’s evidence charts and focuses on argumentsfrom evidence to an ultimate claim that is to be proven. The
second analytical method [9][18] is inspired by experiments on
how people structure the evidence in a case [11], and hence uses
stories to structure and analyse the available evidence.
In the argument-based approach, arguments based on evidence areoffered for a conclusion, which is usually a main claim in thecase. The arguments evidential rules of the form “e is evidence forp”, and this argumentative reasoning is therefore characterized as
evidential reasoning. Reasoning with arguments is dialectical, inthat not only arguments for a conclusion but also arguments
against the conclusion and other kinds of counterarguments are
considered.
The story-based or narrative approach involves constructing
stories about what (might have) happened in a case that explain
the evidence. The relations between the events and the evidence
are expressed as causal rules of the form “c is a cause for e”, andthis reasoning is therefore characterized as causal reasoning. The
narrative approach also has a clear dialectical component, as in acase multiple incompatible accounts of the events are considered
and compared.
Both the argumentative and the narrative approach can beseparately applied to any one case, and they each have their own
advantages. Arguments provide a natural way of reasoning about,for example, witness statements and the conclusions that can be
drawn from such statements. The possibility of constructing
counterarguments allows one to ask critical questions and thus
expose sources of doubt in the reasoning [19]. The mathematicalproperties of argument and counterargument are well-developedin the literature on computational argumentation [6].
In a purely argument-based approach the overview of the casetends to be lost. Stories can help here as they are understandable
scenarios that explain the evidence in a natural way [11].
However, one aspect that is relatively underdeveloped in AI isexactly how the quality of the different stories should bedetermined. In model-based diagnosis [8] often the simplest (interms of subset minimality) and most complete (in terms of
1

An influential third method is the Bayesian approach [13]. See
recent work by [17] on the connection between Bayesian
reasoning, stories and arguments.

explaining the most evidence) scenario is chosen, but this need
not necessarily be the best scenario [15].
The combination of evidential arguments and causal stories thenseems to be an intuitive and analytically useful perspective forlooking at cases and evidence, as the weaknesses of one approach
are alleviated by the strengths of the other and vice versa. Hence,[2] proposes a hybrid theory for reasoning with arguments andstories. The hybrid theory is a theory of Inference to the Best
Explanation (IBE), where causal stories are hypothesised to
explain the evidence, after which these stories can be supported
and attacked using evidential arguments. For example, arguments
can be used to further support a story with evidence or to reason
about the plausibility of a causal link in a story.

2.1 A formal hybrid theory
The formal hybrid theory is a combination of a formal model of
causal-abductive reasoning CT, which is based on model-based
approaches to causal-abductive inference [8] and an evidential
argumentation theory ET, which takes its inspiration fromstandard accounts of structured and abstract argumentation[6][12]). The logic L of this theory is a combination of theinference rules of classical logic and a modus ponens inference
rule for the connective  (defeasible implication). Object-level
rules in CT and ET are formalised using this connective: ri: p1
... pn  q. Here ri is the name of the rule, p1,…,pn and q are
literals. The type of rule is indicated with a subscript: E denotes
an evidential rule used in arguments and C denotes a causal ruleused in stories. As usual, a rule with variables is a scheme
standing for all its ground instances.

2.1.1 Causal reasoning with stories
A criminal case usually starts when some initial evidence is foundthat points to the possibility that a crime has been committed. On
the basis of this preliminary evidence, we can formulate one ormore explananda, facts that can be explained by constructing
stories – coherent sequences of states and events – about whatmight have happened. A coherent story is causally connected inthat the individual events are connected by (sometimes implicit)
causal links. These causal links allow us to perform abductive
reasoning: if we have a general rule s1 C s2 and we observe s2,
we are allowed to infer s1 as a possible hypothetical explanationof s2. The cause s1 can be a single state or event, but it can also bea causally connected chain of events, a story.
The causal theory CT consists of a set of causal rules RC, a set of
hypothetical events H (ground literals) and a set of explananda E(ground literals). A story is then defined as follows.
Definition [Story] A story S based on a causal theory CT is a
finite sequence [φ1,…,φn], where n > 0, such that for all φi (1  i n): φi  H  RC; or φi follows from ψ1, …, ψn  {φ1, …,φi-1} by application of
defeasible modus ponens.
The ordering of the events in the sequence suggests theirchronological succession. Stories(CT) denotes the set of all stories
that can be constructed from a theory CT.
As an example of a story, take the following story S1, which saysthat rain makes the grass wet, which causes one’s shoes to get
wet.







rain
rc1: rain c grass_wet,
grass_wet,
rc2: grass_wet c shoes_wet,
shoes_wet

A story serves as an explanation for the explananda in F if these
explananda causally follow from the story. That is, a story S
explains an explanandum f iff S ⊢C f, where ⊢C stands for logicalconsequence according to the set of common deductive inference
rules extended with modus ponens for C. For example, theabove story explains why my shoes are wet.

2.1.2 Evidential reasoning with arguments
Given a piece of evidence, we can treat it as an explanandum andtry to find a causal story that explains it. However, we can alsoinfer conclusions from the evidence using evidential rules, thus
creating arguments, which can then be used to support or attack
stories.
The evidential theory ET consists of a set of evidential rules RE
and a consistent set of evidence K (ground literals). Arguments
are then defined as follows.
Definition [Argument] An argument based on an evidential
theory ET is a finite sequence [φ1, …, φn], where n > 0, such thatfor all φi (1  i  n): φi  K  RC; or φi follows from ψ1, …, ψn  {φ1, …,φi-1} by application of
defeasible modus ponens.
We say that Args(ET) denotes the set of all arguments that can beconstructed from a theory ET.
As an example of an argument, take A1, where my observationthat my shoes look wet is used as evidence for that fact that myshoes are wet:

shoes_look_wet,
re1: shoes_look_wet  shoes_wet,
shoes_wet

Arguments are a tool to deal with the contradictory evidence thatis often available in a case. Conflict between arguments based onevidence leads to one argument attacking the other. An argument
A1 can attack another argument A2 in two ways. A1 and A2 rebut
each other if they have an opposite (intermediate) conclusion; forexample, an argument with the conclusion that my shoes are dry(based on, for example, me feeling whether my shoes are wet) will
rebut the above argument. A1 undercuts A2 if there is a conclusion
ri in A1 and an application of defeasible modus ponens to ri in A2(where ri is the name of a rule in RE). Here, ri generalises the twocases where a rule is invalid (inapplicable in all cases) and where
there is an exception to the rule (inapplicable in some cases). As
an example, consider the knowledge that my shoes always gleam
in the light, making them look wet. In this case, the rule re1 will be
undercut.
An advantage of formal models of argumentation is that they canbe used to evaluate a particular argument given a set of otherevidential arguments. In other words, given a collection ofarguments and their attack relations, the dialectical status of thearguments can be determined. Following [12], we assume that ourtheory for arguments instantiates one of [6]’s semantics. For
present purposes, the exact type of semantics is not important andwe simply assume that arguments can be justified, which means

that they are not defeated by other justified arguments, overruled,which means that they are defeated by other justified arguments,
or defensible, which means that they are neither justified nor
overruled.

2.1.3 Combining stories and arguments in a hybrid
theory of IBE
An important part of the hybrid theory is the consideration ofalternative stories. In most cases there are alternative stories, foreach of which there is some support (but not conclusively
justifying) and for each of which there is some attack (but notconclusively defeating). Moreover, a search and consideration ofalternative stories is a good method to lessen the danger of socalled confirmation bias. Take our example where our shoes arewet. One alternative explanation S2, is that the sprinkler was on:






sprinkler_on
rc1: sprinkler_on c grass_wet,
grass_wet,
rc2: grass_wet c shoes_wet,
shoes_wet

In addition to the search for alternatives, the stories themselvesalso have to be thoroughly analysed and compared. There arevarious criteria with which stories can be compared. The mostimportant of these criteria are about to what extent a story
conforms to the evidence in a case. In [2], there are two criteriathat pertain to the relations between a story and the evidence in acase. The first is evidential support, the number of pieces ofevidence that support a story, and the second is evidential
contradiction, the number of pieces of evidence that contradict astory. Note that these numbers are not objective measures for thequality of stories, but rather aids in the comparison of stories.
Furthermore, by contrasting the evidential support orcontradiction to the total number of pieces of evidence in a case
we get an indication of the relevance and coverage of a story [11].
The above criteria can be defined in the hybrid theory as follows.
For a piece of evidence to support a story there should be a nonoverruled argument A  Args(ET) such that e  K is a premise of
A and some s  S is a conclusion of A. Evidence e  K contradicts
a story S if there is a non-overruled argument A  Args(ET) suchthat e is a premise of A and s is a conclusion of A, where s  S.
Note that only arguments which are not overruled support orcontradict an explanation: if an argument based on evidence isitself defeated, the evidence does not support the explanation.
The criteria of evidential support and contradiction can be used
for a direct comparison of stories. For example, all other things
being equal, a story S can be considered better than a story S’ if
the number of sources of evidence that support S is higher than S’
and the number of sources of evidence that contradict S are equal
or lower to S’.

3. STORIES AND ARGUMENTS AS
COMMUNICATING VESSELS
The two separate approaches to reasoning with evidence,arguments and stories are abstract accounts of two dominant
trends in research about evidence and proof, which have beenvery clearly distinguished here to make a point. In real cases,
argumentative and narrative aspects will often blend into eachother. Furthermore, the literature mentioned for each of theseapproaches does usually not completely disregard the“competing” approach. For example, the Anchored Narratives

Theory [18] has clear argumentative aspects (cf. [16]) and the“New Wigmoreans” also discuss the use of stories and scenario’s
in evidential reasoning [1].
To distinguish arguments and stories, in the hybrid theory fairly“narrow” definitions of argument and story are used: an argument
is a tree-like structure in which one or more premises support asingle conclusion through one or more consecutive inference
steps, whilst a story is a coherent sequence of events inchronological order. However, if more broad definitions areadhered to, this distinction fades. For example, in [3] we showthat a story as a whole can be used as a premise to a legal
argument, and it is argued in [7] that stories play an important rolein arguments from analogy. As argued in [5], it is not the structureof the reasoning (i.e. evidential, causal) but rather the intention of
the speaker that determines whether something is an explanationor an argument. Thus, it is possible to use stories to explain, toentertain and to argue.
The overlap between story-based and argument-based reasoning is
also in the relationship between causal and evidential reasoning,which are closely related: if we have a causal rule ‘c causes e’then we will usually also accept that ‘e is evidence for c’. For
example, fire can cause visible smoke so the observation of smoke
can be seen as evidence for the fact that there is a fire. On thissubject, Note that both forms of reasoning are defeasible: fire does
not necessarily cause (visible) smoke as some types of fire (e.g. a
gas flame) do not cause smoke; similarly, observing smoke does
not necessarily imply that there is a fire as there are other causesof smoke (e.g. a smoke machine).
The close relation between causal and evidential reasoning is also
evident in the explains relation between a story and the
explananda in the hybrid theory. [14] argues that some
proposition c explains some other proposition e if the states ofaffairs described by c are part of the “causal process” that
produces the states of affairs described by e. For example, by
accepting a testimony E* as evidence for an event E we implicitly
make a causal judgment that the witness tells us E* because thatwitness really believes E happened.
This interweaving of causal and evidential reasoning is also
evident in formal models. For example, [4] show that the
information in causal models can equally well be represented byevidential rules; instead of abductive reasoning we then performargumentation to infer hypothetical scenarios for our explananda.
Alternative explanations then become counterarguments. As an
example, take Figure 1 where the same piece of evidential data isused in two different ways. On the left side, the fact that my shoes
look wet is used to evidentially infer that my shoes are wet (closed
arrowhead). This argument is undercut by an argument that I havespecial gleaming shoes which always look wet (dottedarrowhead). On the right side, the fact that my shoes are wet isused to explain the fact that my shoes look wet (open arrowhead).
The counterargument is here modelled as an alternative
explanation: that I have gleaming shoes also explains why theylook wet.
The situation in Figure 1 shows that, in a purely argumentative
approach, alternative interpretations of the evidence lead tocounterarguments and in a purely narrative approach, suchalternative interpretation lead to different explanatory stories. So
it would seem that in the hybrid approach, an alternative story canalso be expressed as a counterargument

shoes_wet
shoes_wet
shoes_look_wet

special_shoe
shoes_look_wet

special_shoe

Figure 1: Stories and arguments as communicating vessels
The exact choice whether to use causal or evidential rules in one’s
reasoning seems somewhat arbitrary and it has also been argued
that both causal and evidential reasoning is needed in complex
situations [10].
Since it is possible to emphasize either evidential, argumentoriented elements or causal, story-oriented elements in a caseanalysis, one could say that arguments and stories behave as"communicating vessels": a change in an argument-oriented
version of a case analysis requires a matching change in a storyoriented analysis. As a consequence of this communicating
vessels property it is possible to choose what works best when
performing an analysis: some cases or aspects of a case will moreconveniently be modelled in terms of evidential arguments, others
in terms of causal stories. For example, if the decision whether thesuspect is guilty or not hinges on a single fact (e.g. whether thesuspect was at a particular location at a particular time),
alternative stories are less important and the main analyticaltechnique is to carefully consider all the arguments for and against
this fact. In other cases, certain facts might not necessarily bejustified because they are supported by strong evidence but rather
because they are part of the most coherent story. Take, forinstance, a case with multiple contradicting claims based on
different testimonies. Simply considering these claims as counterarguments to each other does not tell us which one, if any, is true.
Rather, we should look which claim is part of the most coherent
story that is supported by the most additional evidence, that is,which claim is compatible with the totality of the evidence.

4. EXPANDING THE HYBRID THEORY
The fact that arguments and stories to a certain extent behave as
communicating vessels indicates that there is room for a deeper
understanding of the relation between arguments and stories inreasoning about evidence, in such a way that they are treated in a
genuinely integrated way, further building on the hybrid theory.
The first step towards an integrated theory of arguments andstories is to determine the dialectical component of story-based
reasoning, that is, exactly how different stories interact in adialectical way with each other and with the arguments in a case.
The current method for this is to consider alternative stories ascounterarguments, thus combining causal and evidential
argumentation in one framework.
One way to integrate causal and evidential reasoning into one
framework is to consider stories as causal arguments. This would
allow us to integrate stories into the ASPIC+ argumentation
frameworks [12], which allows us to determine their statusaccording to well-known argumentation-theoretic semantics [6].
When comparing the formal definitions of stories and arguments
given above, it is clear that they are similar in structure: both arederivations in the defeasible logic L, the difference being thatstories use causal rules and arguments use evidential rules.
Furthermore, as was already argued earlier, it is the intention of

the speaker that determines whether a story should be considered
as an explanation or an argument. So is it not possible to simply
have two types of argument: causal arguments and evidential
arguments?
When considering causal argumentation, it is instructive to look at
the argumentation schemes for causal reasoning. [19] consider
two schemes which are important for current purposes: a schemefrom cause to effect and a scheme from effect to cause. The
scheme from cause to effect is as follows ([19], p. 168).
Generally, if A occurs then B will (might) occur.
In this case, A occurs (might occur).
Therefore, in this case B will (might) occur.
This scheme can be captured by the causal rules in CT. Note,however, that this type of causal reasoning is essentiallyprediction, where one makes an observation and tries to predict
what will be the case as a consequence of this observation.
In contrast, in the hybrid theory we used the causal rules for
explanatory (abductive) reasoning, where one makes anobservation and tries to explain how this observation could havefollowed from some hypothesized states of affairs. Hence, in the
hybrid theory we mainly use stories and the causal rules containedin them to reason from effect to cause, viz. ([19], p. 170).
Generally, if A occurs then B will (might) occur.
In this case, B did in fact occur.
Therefore, in this case, A also presumably occurred.
Such abductive reasoning can be valid if multiple alternativecauses are considered. This requirement is contained in thescheme’s critical questions (adapted from [19]).
CQ1: Are there alternative explanations of B?
CQ2: How much better an explanation is A when compared to thealternatives?
Note that explanatory (abductive) reasoning and predictive
reasoning with causal rules are close intertwined: a hypothesis
explains some observations if the hypothesis correctly predicts
these observations. This means that abductive reasoning is similar
to a special subspecies of predictive cause-to-effect reasoning,where the premise “A occurs” is a hypothesis instead of anobservation. This clearly distinguishes this causal reasoning fromevidential reasoning, where the arguments are based on direct
evidence.
So in our integrated framework there must be a way of
distinguishing arguments based on evidence and arguments basedon hypotheses. This is made possible by the different types of
elements of the knowledge base in ASPIC+. We will return to this
distinction in section 4.1.
In addition, it is also necessary to revisit the attack relations
between arguments. Two evidential arguments attack each other ifthey have an opposite conclusion. Abductive causal arguments,
however, also attack each other if they are alternative explanationsof the same “conclusion” (cf. CQ1 of the “cause to effect” scheme
above). This will be discussed in sections 4.1.1 and 4.1.2.

4.1 An integrated formal framework
In this section I propose an integrated formal framework for
causal and evidential reasoning, based on the ASPIC+ framework.
The logic and object-level language of this integrated theory are
the same as for the hybrid theory: there are connectives E andC for evidential and causal rules, and a modus ponens inferencerule for these connectives. Like in the hybrid theory, there are setsof causal and evidential rules.
Definition [Rules] The set of rules is R = RE  RC, where
RE is the set of evidential rules.

RC is the set of causal rules.
Following the ASPIC+ framework, in the integrated theory there
is one knowledge base (with appropriate subsets).
Definition [Knowledge base] A knowledge base is a set K = KE 
KH, whereKn is the set of necessary axioms, which cannot be
attacked. This set contains the evidence, of which theexistence cannot be sensibly denied.
Kp is a set of normal premises, which can be attacked as
explained in section 4.1.2.
Using this knowledge base and the rulesets, we can build
evidential and causal rules.
Definition [Arguments] A causal/evidential argument based on
is a finite sequence [φ1, …, φn], where n > 0, such that for all φi (1 i  n): φi  Kp  RC (causal argument), φi  Kn  RE (evidential
argument); or φi follows from ψ1, …, ψn  {φ1, …,φi-1} by application of
defeasible modus ponens.
Basically, this definition is the same as the definitions ofarguments and stories in section 2.1. Note that causal arguments
are based on normal premises that can be attacked; after all, causal
arguments are hypotheses that explain their conclusion and weshould be able to attack these hypotheses. We also define some
auxiliary notions related to arguments, which are needed for later
definitions.
Definition [Elements of arguments] For any argument A and A’

1.
2.
3.
4.

The set of premises of A is Prem(A) = {φ  A | φ  K};
The conclusion of A is Conc(A) = {φ  A | φ is the lastelement in the sequence};
The set of evidence in A is Evidence(A) = {ei  A | ei  Kn};An argument A’ is a (proper) subargument of A iff A’ is a
(proper) subsequence of A.

but S1 has a subargument with conclusion rain and S2 has asubargument with conclusion sprinkler_on (Figure 2).
rain

S1

sprinkler_on

grass_wet

S2

shoes_wet

Figure 2: alternative causal arguments
Note that evidential arguments can also be alternatives. An
evidential rule is essentially nothing more than an explicit
expression of an abductive reasoning step applied to a causal rule.
For example, in Figure 1 it was shown that the evidential A1 canbe represented as a causal argument. Similarly, we can rewrite thecausal arguments S1 and S2 as evidential arguments A2:[shoes_wet, shoes_wet e grass_wet, grass_wet, grass_wet erain, rain] and A3: [shoes_wet, shoes_wet e grass_wet,
grass_wet, grass_wet e sprinkler_on, sprinkler_on] respectively.
Here, A2 and A3 are alternatives: A2 says that the fact that thegrass is wet is evidence for rain, whilst A3 argues that grass_wet
is evidence for the fact that the sprinkler was on.
Definition [Alternative Evidential Arguments] An evidential
argument E1 is an alternative to an evidential argument E2 if
Conc(E1) ≠ Conc(E2) and Conc(E1’) = Conc(E2’), where E1’
and E2’ are (proper) subarguments of E1 and E2, respectively.
A2 and A3 have different conclusions, rain and sprinkler_on,
which have been derived from the same proposition, grass_wet(Figure 3).
rain

A2

sprinkler_on

grass_wet

A3

4.1.1 Alternatives
Before we determine when causal and evidential arguments attack
each other, it has to be determined when two explanations arealternatives. The reason for this is that in explanatory, abductive
reasoning this is where the dialectical component lies: two
explanations (whether stories or arguments) are in competition of
they are alternative causes for the same consequence. For
example, the two example stories S1 and S2 are not compatible,
nor are the two explanations in Figure 1.
For causal arguments, alternatives can be defined as follows.
Definition [Alternative Causal Arguments] A causal argument
C1 is an alternative to a causal argument C2 if Conc(C1) =Conc(C2) and Conc(C1’) ≠ Conc(C2’), where C1’ and C2’ are(proper) subarguments of C1 and C2, respectively.
Two causal arguments are alternatives if their conclusion (the lastelement in the sequence) is the same but they have a subargumentof which the conclusion is not the same. In our example, S1 andS2 are thus alternatives: they both have the conclusion shoes_wet,

shoes_wet

Figure 3 alternative evidential arguments
Note that it is even possible for an evidential argument and acausal argument to represent alternatives. For example,
Definition [Alternative Evidential/Causal Arguments] An
evidential argument E is an alternative to a causal argument C if
Conc(C’) = Conc(E’) and Conc(C’’) ≠ Conc(E’’), where C’ andE’ are subarguments of C and E, C’’ is a subargument of C’ andE’ is a subargument of E’’.
So, for example, arguments S1 and A3 are alternatives. Consider
the different subarguments from the definition:
S1: [rain, rain c grass_wet, grass_wet, grass_wet c shoes_wet,
shoes_wet]
A3:[shoes_wet, shoes_wet e grass_wet, grass_wet,
grass_wet e sprinkler_on, sprinkler_on]

S1’: [rain, rain c grass_wet, grass_wet]

A4 shoes_look_dusty

A3’: [shoes_wet, shoes_wet e grass_wet, grass_wet]
S1’’: [rain](grass_wet)

A3’’: [shoes_wet, shoes_wet  grass_wet, grass_wet,
grass_wet  sprinkler_on, sprinkler_on]

S3

So Conc(S1’) = Conc(A3’) and Conc(S1’’) ≠ Conc(A3’’), where
where S1’ and A3’ are subarguments of S1 and A3, S1’’ is a
subargument of S1’ and A3’ is a subargument of A3’’. In Figure
4, the situation where evidential (closed arrowhead) and causal(open arrowhead) arguments are combined is shown.

forecast_yesterday A3

rain

sprinkler_on

no_rainshoes_dusty

rain

S1

shoes_wet

grass_wet

S2 sprinkler_on
special_shoe

A2

S1

grass_wet

A1 shoes_look_wet

A3

Figure 5: stories and arguments

shoes_wet

Figure 4: alternative causal and evidential arguments

4.1.2 Attack and defeat
In ASPIC+, arguments attack each other if they, or theirsubarguments, have opposing conclusions. Furthermore, anargument further attacks another if it undercuts it, that is, if it
denies an inference. In the hybrid theory these types of attacks areallowed between evidential arguments. In the integrated theory,causal arguments (stories) can also attack each other.
Definition [Attack] An argument A1 attacks an argument A2 iff

There are two subarguments A1’ and A2’ that have an
opposite conclusion, and the conclusion of A2’ is not in
Kn (rebut); or
There is a subargument A1’ which has conclusion ri inand there is an application of defeasible modus ponens
to ri in A2 (where ri is the name of a rule in R)(undercut); or
A1 and A2 are alternatives (alternative attack).
Note that attack relations can be between two causal arguments,
two evidential arguments or a causal and an evidential argument.
Also note that it is not possible to attack a piece of evidence.
As an example of these new attack relations between arguments
and stories, consider Figure 5. The different labels S1, S2, S3 forstories indicate the maximal causal sequence (open arrowhead)starting at that particular proposition, and the argument labels
indicate the maximal evidential sequence (closed arrowhead)starting at the labelled proposition. There are three stories: one(S1) that the grass is wet because it rained, one (S2) that the grass
is wet because the sprinkler was on and one (S3) that the grass is
not wet. The stories are supported and attacked (indicated by thearrow with the dot for its head) by arguments. One of thesearguments (A1) is undercut.

Not all the attacks between the different stories and arguments areindicated in Figure 5. Figure 6 shows the different arguments andstories as an argumentation framework in the style of [6]. The
arrows here stand for “attack” relations.
S1

A3
A2

S3
S2

A1

A4

Figure 6: Attack relations between arguments and stories
Stories S1 and S2 attack each other because they are alternatives.
Story S3, however, attacks S1 and S2 because it has acontradictory subconclusion, (grass_wet). Thus, it is possible toindicate incompatible stories even if they are not alternatives,
something which was not possible in the standard hybrid theory.
This is an important improvement, as now alibi stories can becorrectly represented. An alibi story usually does not explain anyof the evidence that the main prosecution’s story does because it
does not have to. If the suspect shows he was somewhere else, hedoes not have to explain why, for example, the victim died.

4.2 Positions
While it is interesting to see the different attack relations between
the stories, it is even more informative to consider positions,particular combinations of stories and arguments in a case.
Definition [Position] A position P in a case is a conflict-free setof arguments such that for each evidential argument E in P, there
is a subargument E’ which has as its conclusion an element incausal argument C, where C is part of P.
What this means is that a position in a case contains stories andthe arguments supporting them. Thus, a position is essentially aconsistent view on the case at hand, given the evidence,arguments and stories. In Figure 7, three positions that correspondto the different stories from our example are shown. Notice theattacks between the elements of the positions.

[2] Bex, F.J. (2011) Arguments, Stories and Criminal Evidence:
A Formal Hybrid Theory, Springer, Dordrecht.

P2
S2

A1

P1
S1

[4] Bex, F.J. & Verheij, B. (2011) Solving a Murder Case by
Asking Critical Questions: An Approach to Fact-Finding in
Terms of Argumentation and Story Schemes. Argumentation,
26:3, 325-353.

A3

P3

S3

A4

[3] Bex, F.J. & Verheij, B. (2012) Legal Stories and the Processof Proof. Artificial Intelligence and Law. Online first, doi:
10.1007/s10506-012-9137-4.

A2

Figure 7: Three cases
Because a position is a set in an argumentation framework, we cancheck whether, for example, a position corresponds to an
extension of the argumentation framework. Furthermore, it now
becomes possible for a position to defend itself by reinstating thestory when it is attacked by an argument from another position.
Thus, the dialectical characteristics of stories have been further
detailed compared to the original hybrid theory.

5. CONCLUSION
The hybrid theory [2] provides a way to reason with stories,
arguments and evidence. The ideas presented in this paper further
integrate the reasoning with stories and arguments. By treating
stories as causal-abductive arguments, they can be integrated into
the dialectical framework of [6]. This allows for the “calculus of
opposition” to be applied to the different viewpoints in a case.
Note that the original criteria like evidential support andevidential contradiction (section 2.1.3) are still valid. The
integrated theory simply adds new ways of comparing stories.
Most importantly, stories that provide, for example, an alibi cannow be incorporated into the dialectical process. In the hybrid
theory these stories were ignored as they do not explain the main
explananda, but in the integrated theory they attack other stories
because they claim that, for example, the suspect could not havecommitted the crime because he was somewhere else.
The integrated theory allows us to correctly reason with stories
and arguments as communicating vessels. As was argued above, achange in an argument-oriented version of a case analysis requires
a matching change in a story-oriented analysis. Furthermore, if
stories and arguments are truly communicating vessels thenchanging a case from story-oriented to argument-oriented should
not change its outcome. The hybrid theory has aspects where howa case is modelled influences the conclusions one can draw. For
example, two contradictory stories do not attack each other whilst
two contradictory arguments do. On the other hand, twoalternative stories are in competition, whilst two arguments that
present alternatives are not. By embedding stories in the
dialectical framework and allowing evidential arguments to attack
each other as “alternatives”, the integrated theory allows stories
and arguments to be treated as equals in the dialectical process.

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